Schramm-Loewner Evolution and long-range correlated systems

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Schramm-Loewner Evolution andlong-range correlated systems

Doctoral Thesis

Author(s):Posé, Nicolas

Publication date:2015

Permanent link:https://doi.org/10.3929/ethz-a-010552449

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Diss. ETH No. 22774

Schramm-Loewner Evolutionand long-range correlated

systems

A thesis submitted to attain the degree of

DOCTOR OF SCIENCES of ETH ZURICH

(Dr. sc. ETH Zurich)

presented by

Nicolas Pose

MSc ETH Physics,

Ingenieur diplome de l’Ecole Polytechnique

born on 24.10.1987

citizen of France

accepted on the recommendation of

Prof. Dr. Hans J. Herrmann, examiner

Prof. Dr. Alexander K. Hartmann, co-examiner

2015

Acknowledgment

I have to thank my supervisor Professor Hans J. Herrmann for allowing me

to work on a challenging topic in an inspiring environment.

I also would like to thank Professor Alexander K. Hartmann for having ac-

cepted to review my work.

Thanks are also due to my collaborators, with whom I worked on the different

projects presented in this thesis. First I would like to thank Professor Nuno

A. M. Araujo for the numerous discussions we had, the ideas we discussed,

his knowledge and experience he shared. I also would like to thank Dr. Ken

J. Schrenk for the work we started together and his help. I also want to

thank Dr. Miller Mendoza for the work we did together on graphene, the

inspiring discussions we had and his advices. I also thank my collaborators

Ilario Giordanelli, Laurens V. M. van Kessenich and Julian J. Kranz.

I am also grateful to my coworkers, Dominik and Trivik with whom I shared

the office in the past two years, Miller, Nuno, Julian, Vitor, Ilias, Farhang,

Roman, Fabrizio, Jan, Oliver, Ryuta, Sergio, Jens-Daniel, Ilario, Robin, Gau-

tam and the whole Comphys-Group for the nice time I spent with them.

I am also grateful to Professor Denis Bernard and Professor Wendelin Werner

for helpfull discussions and to all my teachers at the Ecole Polytechnique in

Paris and ETH Zurich for their enthousiasm in transmitting their knowledge.

Last but not least, I would like to thank my relatives, especially my parents,

for their unconditionnal help, Sara for her support during these years, and all

my friends not only in Zurich but wherever they might be now for spending

so many nice moments together.

3

4

Contents

1 Introduction 1

2 Schramm-Loewner Evolution theory 7

2.1 Conformal transformations and holomorphic functions . . . . . 9

2.2 The Riemann mapping theorem and its consequences . . . . . 10

2.3 Half-plane capacity parametrization . . . . . . . . . . . . . . . 12

2.4 The Loewner differential Equation and the driving function . . 13

2.5 Schramm-Loewner Evolution theory . . . . . . . . . . . . . . . 16

2.6 Mapping of curves generated in a rectangle into the upper-half

plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.7 Numerical tests of Schramm-Loewner Evolution . . . . . . . . 23

2.7.1 The fractal dimension . . . . . . . . . . . . . . . . . . 25

2.7.2 The winding angle . . . . . . . . . . . . . . . . . . . . 25

2.7.3 The left-passage probability . . . . . . . . . . . . . . . 26

2.7.4 The direct SLE algorithm . . . . . . . . . . . . . . . . 29

2.8 Known results in SLE . . . . . . . . . . . . . . . . . . . . . . 32

3 Shortest path in percolation and Schramm-Loewner Evolu-

tion 35

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 Winding angle method . . . . . . . . . . . . . . . . . . . . . . 39

3.4 Left passage probability method . . . . . . . . . . . . . . . . . 39

3.5 Direct SLE method . . . . . . . . . . . . . . . . . . . . . . . . 41

3.5.1 In chordal space . . . . . . . . . . . . . . . . . . . . . . 41

i

3.5.2 In dipolar space . . . . . . . . . . . . . . . . . . . . . . 43

3.6 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

4 Long-range correlated landscapes and correlated percolation 49

4.1 Correlated percolation . . . . . . . . . . . . . . . . . . . . . . 50

4.2 The Hurst exponent of long-range correlated landscapes . . . . 52

4.3 The Fourier Filtering Method for correlated landscapes . . . . 53

4.4 Correlated surfaces, Fourier Filtering Method and fractional

Gaussian fields . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5 Long-range correlated percolation 63

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2 Correlated percolation and the extended Harris criterion . . . 65

5.3 Percolation threshold . . . . . . . . . . . . . . . . . . . . . . . 67

5.4 Maximum cluster size and cluster size distribution . . . . . . . 69

5.5 Cluster perimeters . . . . . . . . . . . . . . . . . . . . . . . . 74

5.6 Transport properties: shortest path, backbone and cluster

conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.7 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6 Schramm-Loewner Evolution on long-range correlated land-

scapes 89

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.2 Isoheight lines on correlated landscapes with negative Hurst

exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.2.1 SLE and fractal dimension . . . . . . . . . . . . . . . . 92

6.2.2 Winding angle . . . . . . . . . . . . . . . . . . . . . . . 92

6.2.3 Left Passage Probability . . . . . . . . . . . . . . . . . 94

6.2.4 Direct SLE . . . . . . . . . . . . . . . . . . . . . . . . 96

6.3 Isoheight lines on correlated landscapes with positive Hurst

exponent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.1 Winding angle . . . . . . . . . . . . . . . . . . . . . . . 98

6.3.2 Left-Passage probability . . . . . . . . . . . . . . . . . 98

6.3.3 Direct SLE method . . . . . . . . . . . . . . . . . . . . 98

ii

6.4 Markovian properties of the driving functions . . . . . . . . . 103

6.5 Other critical curves . . . . . . . . . . . . . . . . . . . . . . . 103

6.5.1 Winding angle measurements for the shortest path and

the watersheds . . . . . . . . . . . . . . . . . . . . . . 105

6.5.2 Direct SLE test for the shortest path . . . . . . . . . . 108

6.5.3 Direct SLE and left-passage probability for watersheds 110

6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7 Conformal Invariance in Graphene 119

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

7.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

7.3 Scale invariance in graphene . . . . . . . . . . . . . . . . . . . 123

7.4 Conformal invariance and SLE properties in graphene . . . . . 126

7.5 Difference between graphene sheets and Gaussian random sur-

faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

7.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

8 Discussion and Outlook 135

References 139

iii

iv

List of Figures

1.1 Loop-Erased Random Walk Discreate Loop-Erased Ran-

dom Walk in the upper half-plane. It has been generated us-

ing the transition probabilities defined in Ref. [1] for a random

walk half-plane excursion. The discreate upper half-plane is

defined as Z+ iZ+ = {j + ik, j ∈ Z, k ∈ Z+}. . . . . . . . . . . 4

1.2 Percolation Percolation interface generated on a triangular

lattice in a rectangle with Dirichlet boundary conditions, i.e.

with zero value (red sites) on half of the border and one (blue

sites) on the other half. The interface, displayed as a black

solid line, is defined such that the interface path starting from

the bottom line has always a red site on its left and a blue site

on its right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 A growing curve (γt)t≥0 in the domainD, in this case the upper

half-plane. The hull Kt is here defined as the curve taken till

time t, and D \Kt is the domain D from which one substracts

Kt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Conformal mapping of the upper half plane H = {z ∈C, Im(z) > 0} into the unit disk {z ∈ C, |z| < 1}, using

f(z) = z−iz+i

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Mapping gt : H \ Kt → H. The hull is shown in blue and is

mapped to the real line by gt. . . . . . . . . . . . . . . . . . . 12

2.4 Mapping of the compact H-hull K ′ = K ∪ (K ′ \K) using gK′

or the composition of gK with gK , f = gK ◦ gK . . . . . . . . . 14

v

2.5 For 0 ≤ κ ≤ 4, the curves are simple. For 4 < κ < 8 they are

self touching and for κ ≥ 8 they are space filling . . . . . . . . 17

2.6 The two probability measures in the domains D and D \ γ[a,c].

If the Domain Markov Property is satisfied, these two proba-

bilities are equal. . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.7 Under the assumptions of conformal invariance and Domain

Markov Property, the curves γ(t, t + s] knowing γ[0, t] and

γ(0, s] are equal in law. From this one deduces that the driv-

ing function has stationary independant identically distributed

increments, i.e. (Wt+s −Wt)s≥0 has the same law as (Ws)s≥0. . 21

2.8 Mapping from a rectangle into the upper half plane

Rectangle mapped to the upper half-plane. (a) Rectangle of

aspect ratio 1 and (b) mapped points into the upper half-

plane. (c) Rectangle of aspect ratio 2 and (d) mapped points

into the upper half-plane. (e) Rectangle of aspect ratio 4 and

(f) mapped points into the upper half-plane. . . . . . . . . . . 24

2.9 (Winding angle) The winding angle is defined iteratively,

as the sum of the turning angles αi between two consecutive

segments (zi−1, zi) and (zi, zi+1). . . . . . . . . . . . . . . . . . 27

2.10 (Left passage) Curve separating the space into the points

belonging to the right side of the curve (space marked in grey)

and points belonging to the left side of the curve (space left

blank). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.11 (Left passage probability) Probability of a point belonging

the right side of the curve dependant on the angle φ. The left-

passage probability is independant on the value of the radius

R of the point z ∈ H. . . . . . . . . . . . . . . . . . . . . . . . 29

2.12 Iteration of the conformal maps gti : H \ gti−1◦ . . . ◦

g1(γ([ti−1, ti])) → H and extraction of the underlying driving

function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.13 Approximation made in the ’vertical’ slit map algorithm . . . 31

vi

3.1 A spanning cluster on the triangular lattice in a strip of ver-

tical size Ly = 512. The shortest path is in red and all the

other sites belonging to the spanning cluster are in blue. . . . 37

3.2 Variance of the winding angle against the lattice size Ly. The

analysis has been done for Ly ranging from 16 to 16384. The

statistics are computed over 104 samples. The error bars are

smaller than the symbol size. By fitting the results with

Eq. (2.25), one gets κwinding = 1.046± 0.004. In the inset, the

probability distribution of the winding angle along the curve

is compared to the predicted Gaussian distribution, drawn in

green, of variance κ4

ln(Ly) with κ = 1.046 and Ly = 16384. . . 40

3.3 Left passage probability test. (a) Weighted mean square de-

viation Q(κ) as a function of κ, for Ly = 16384. The vertical

blue line corresponds the minimum of Q(κ), and the green

vertical line is a guide to the eye at κ = κfract. The minimum

of the mean square deviation is at κLPP = 1.038± 0.019. The

light blue area corresponds to the error bar on the value of

κLPP . (b) Computed left passage probability as a function of

φ/π for R ∈ [0.70, 0.75] and κ = 1.038. The blue line is a

guide to the eye of Schramm’s formula (2.26) for κ = 1.038. . . 42

3.4 Driving function computed using the slit map algorithm. (a)

Mean square deviation of the driving function 〈ξ2t 〉 as a func-

tion of the Loewner time t. The diffusion coefficient κ is given

by the slope of the curve. In the inset we see the local slope

κdSLE(t). The thick green line is a guide to the eye corre-

sponding to κdSLE = 0.92. (b) Plot of the correlation c(t, τ)

given by Eq. (3.1), and averaged over 50 time steps. The

averaged value is denoted c(τ). In the inset are shown the

probability distributions of the driving function for three dif-

ferent Loewner times t1 = 1.2 × 10−3, t2 = 3.7 × 10−3 and

t3 = 9.95 × 10−3. The solid lines are guides to the eye of the

form P (ξt) = 1√2πκti

exp(− ξ2t

2κti

), for i = 1, 2, 3. . . . . . . . . . 44

vii

3.5 Driving function computed using the dipolar slit map given

by Eq. (3.3), for Ly = 16384. The mean square deviation of

the driving function 〈ξ2t 〉− 〈ξt〉2 is plotted as a function of the

Loewner time t. The diffusion coefficient κ is given by the

slope of the curve. . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.1 A random landscape and its percolation clusters. In blue are

the empty sites and in brown the occupied ones. The sites

in green belong to the percolating cluster. The paths in red

represent the left and right accessible perimeters and the paths

in yellow the one site wide fjords. . . . . . . . . . . . . . . . . 51

4.2 Random landscapes generated with the Fourier filtering

method for different values of the Hurst exponent: (a) H =

−1, (b) H = −0.5, (c) H = 0, and (d) H = 0.5. . . . . . . . . 55

4.3 Properites of the Fourier transform of one random landscape of

Hurst exponentH = −0.5 generated with the Fourier Filtering

Method. (a) Power spectrum Sq = |h(q)|2 as a function of |q|,where h is the Fourier transform of the surface h. The red line

is a guide to the eye of the form |q|−βc with βc = 2(H + 1).

(b) Phase φ = arg(h(q)) as a function of q = (qx, qy). . . . . . 56

4.4 Generating long range correlated surfaces with fixed

boundary conditions. Square surface of lattice size L = 32

and Hurst exponent H = −0.1. The boundary conditions are

L on one half of the boundary and −L on the other half. (a)

Random surface h(x) generated with free boundary coundi-

tions. (b) Correction field h satisfying the boundary value

problem Eq. (4.13). . . . . . . . . . . . . . . . . . . . . . . . . 61

viii

4.4 Generating long range correlated surfaces with fixed

boundary conditions. (continued) Square surface of lattice

size L = 32 and Hurst exponent H = −0.1. The boundary

conditions are L on half of the boundary and −L on the other

half. (c) Summing the two fields one obtains the random sur-

face hfinal(x) = h(x) + h(x) with the desired fixed boundary

conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.1 Convergence of the percolation threshold estimator pc,J . The

difference between the estimator and the threshold |pc,J − 1/2|is shown as function of the lattice size L for H = −1, H =

−0.85, −0.7, −0.4, and −0.1. The data is shifted vertically

to improve visibility. Results are averages over at least 104

samples. We keep track of the cluster properties with the

labeling method proposed by Newman and Ziff [2, 3], as in

Ref. [4]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Fraction of sites in the largest cluster smax/N as function of

the lattice size L for different values of H. The data is shifted

vertically to improve visibility. Solid black lines are guides to

the eye. Results are averages over at least 104 samples. . . . . 70

5.3 Fractal dimension of the largest cluster df as function of the

Hurst exponent H. For H > −1/3, the solid line show the

expressions of Eq. (5.7) and and for H < −1/3 the constant

value duncorrf = 91/48 is assumed. . . . . . . . . . . . . . . . . 71

5.4 Cluster size distribution ns as a function of the number of

occupied sites s, for different values of the Hurst exponent

H. The data is shifted vertically to improve visibility. Solid

black lines are guides to the eye. Results are averages over 104

samples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.5 Fisher exponent t as function of the Hurst exponent H. For

H > −1/3, the solid line show the expressions of Eq. (5.9)

and and for H < −1/3 the constant value τuncorr = 187/91 is

assumed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

ix

5.6 Complete and accessible perimeter. The blue (filled) sites of

the triangular lattice are part of the largest cluster, while the

white (empty) sites are unoccupied. Bonds of the dual lattice

are shown as dashed lines. Assume that the largest cluster

percolates in the vertical direction and does not touch the left

or right boundaries of the lattice. Consider a walker start-

ing on the left-bottom side of the lattice, which never visits

a bond twice and traces out the complete perimeter, turning

left or right depending on which of the two available bonds

separates an occupied from an empty site. The complete

perimeter is fully determined when the top side of the lattice

is reached. Performing the same walk, but with the additional

constraint that fjords with diameter ≤√

3/3 (in lattice units)

are not accessible, yields the accessible perimeter. The solid

green (thick) lines on the honeycomb lattice form the accessi-

ble perimeter, while dashed green (thick) lines indicate bonds

that are part of the complete perimeter but not of the accessi-

ble one. A similar walk yields the two perimeters on the right

hand side of the cluster. . . . . . . . . . . . . . . . . . . . . . 75

5.7 Snapshots of typical complete and accessible perimeters. The

accessible perimeter is shown in bold solid blue lines. In addi-

tion, the parts of the complete perimeter that do not belong

to the accessible perimeter are drawn with thin black lines.

The snapshots are taken for (a) H = −1, (b) −0.5, (c) −0.25,

and (d) 0, on a lattice of (vertical) length L = 128. . . . . . . 76

x

5.8 Main plot: Fractal dimension of the complete perimeter

dcp and of the accessible perimeter dap as function of the

Hurst exponent H. For H = −1 (uncorrelated), our results,

dcp = 1.75± 0.02 and dap = 1.34± 0.02, are in agreement with

values previously reported [5–11]. With increasing H, both

fractal dimensions seem to approach 3/2, compatible with the

data of Kalda et al. [12–15]. In the range −1/νuncorr2D ≤ H ≤ 0,

the solid lines show the expressions dcp = 3/2 − H/3 and

dap = (9− 4H)/(6− 4H). Insets: Length of the complete and

of the accessible perimeters as function of the lattice size L for

the values of H shown in the main plot. . . . . . . . . . . . . 77

5.9 Yardstick method to measure the fractal dimension of the com-

plete perimeter. The number of sticks needed to follow the

perimeter S is shown as function of the stick length m, for dif-

ferent lattice sizes L, and H = 0. The numerical value of the

complete perimeter fractal dimension dcp(H) obtained with

the yardstick method, dcp(0) = 1.49± 0.03, agrees, within er-

ror bars, with the results of the analysis of the local slopes of

the perimeter length, see Fig. 5.8, as well as with the litera-

ture [13–15]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.10 Left hand side of the duality relation for cluster perimeters,

(dap − 1)(dcp − 1) = 1/4 [11, 16], as function of the Hurst ex-

ponent H. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

xi

5.11 Fractal dimension of the shortest path dsp of the largest clus-

ter as function of the Hurst exponent H. The inset shows the

number of sites in the shortest path as function of the lattice

size L for the same value of H as in the main plot. For uncor-

related disorder, i.e. H = −1, we find dsp = 1.130± 0.005,

in agreement with the literature [17–20]. With increasing

Hurst exponent, dsp approaches unity [21]. The solid line is

the graph of the proposed behavior of the shortest path frac-

tal dimension: dsp(H) = 147/130− (3/4 +H)/(195/34 +H),

for −3/4 ≤ H ≤ 0, and dsp(−1 ≤ H ≤ −1/νuncorr2D ) =

dsp(−1/νuncorr2D ) = 147/130. . . . . . . . . . . . . . . . . . . . 81

5.12 Fractal dimension of the backbone dbb as function of the Hurst

exponent H. With increasing H, the backbone becomes more

compact and, consequently, dbb increases, while the fractal di-

mension of the shortest path, see Fig. 5.11, decreases [22]. For

uncorrelated disorder, H = −1, we measure dbb = 1.64± 0.02,

compatible with the results reported in Refs. [17, 19, 23–26].

The solid line is the graph of the following interpolation:

dbb(H) = 39/20(1 + H) − 166/101H. Inset: Backbone size

as function of the lattice size L for the same values of H as in

the main plot. . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.13 Reduced conductivity exponent tH/νH as function of the Hurst

exponent H. For uncorrelated disorder, we find tH/νH(−1) =

−0.992 ± 0.027 in agreement with Ref. [25]. The solid line

corresponds to the expression tH/νH = 16/41 −H − 7H2/25

in the range −1/νuncorr2D ≤ H ≤ 0 and tH/νH = t/νuncorr

2D for

−1 ≤ H ≤ −1/νuncorr2D . Inset: Conductivity C as function of

the lattice size L, for the same values of the Hurst exponent

H as in the main plot. . . . . . . . . . . . . . . . . . . . . . . 85

xii

6.1 Rescaled variance of the winding angle along

the curve for different Hurst exponents H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. The relationσ2θ−bκ/4

= lnLy is displayed by the black solid line. In the

inset, the rescaled probability distributions are plotted and

compared to a normal distribution for H = −1,−0.55, 0. . . . 93

6.2 Measured rescaled mean square deviation Q(κ)/Q(κmin) as

a function of κ/κmin with κmin the value of κ where the

minimum of Q(κ) is attained, for different Hurst exponents

H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. In the in-

set, the difference between the measured left passage prob-

abilities P (θ) and the left passage probabilities Pκ(θ) pre-

dicted by Schramm’s formula of Eq. (2.26) with κ = κmin,

for H = −0.85,−0.55,−0.25, 0. For convenience, we chose

502 points in the range [−0.025Lx, 0.025Lx]× [0.15Ly, 0.25Ly]

with Ly = 1024 and Lx = 8Ly, which are then mapped to

the upper half-plane through an inverse Schwarz-Christoffel

transformation, see Section 2.6. . . . . . . . . . . . . . . . . . 95

6.3 Rescaled variance of the driving functions for different values

of H = −1,−0.85,−0.7,−0.55,−0.25,−0.1, 0. In the upper

inset, we present the rescaled probability distributions of the

driving functions and compare them to a Gaussian distribution

for H = −1,−0.55, 0. In the lower one, one sees the auto-

correlation function c(τ) = 〈c(t, τ)〉τ of the increments for the

same values of H, averaged over 50 time steps. . . . . . . . . . 96

6.4 Estimated diffusion coefficients κ from the fractal dimension

(fract. dim.), the winding angle, the left-passage probability

(lpp), and the direct SLE (dSLE) methods for H ≤ 0. The

red crosses correpond to the analytically known results. . . . . 97

xiii

6.5 Rescaled variance of the winding angle along the curve for

different Hurst exponents H = 0, 0.1, 0.25, 0.55, 0.7, 0.95. The

relationσ2θ−bκ/4

= lnLy is displayed by the black solid line. In

the inset, the rescaled probability distributions are plotted and

compared to a normal distribution for the same values of H. . 99

6.6 Measured rescaled mean square deviation Q(κ)/Q(κmin) as

a function of κ, with κmin the value of κ where the min-

imum of Q(κ) is attained, for different Hurst exponents

H = 0, 0.1, 0.25, 0.55, 0.7, 0.95. For convenience, we chose 502

points in the range [−0.025Lx, 0.025Lx]×[0.15Ly, 0.25Ly] with

Ly = 1024 and Lx = 8Ly, which are then mapped to the upper

half-plane through an inverse Schwarz-Christoffel transforma-

tion, see Section 2.6. . . . . . . . . . . . . . . . . . . . . . . . 100

6.7 Rescaled variance of the driving functions for different val-

ues of H = 0, 0.1, 0.25, 0.55, 0.7, 0.95. In the upper inset, we

present the rescaled probability distributions of the driving

functions and compare them to a Gaussian distribution for

the same values of H. . . . . . . . . . . . . . . . . . . . . . . . 101



(lpp), and the direct SLE (dSLE) methods for H ≥ 0. The

red cross correponds to the analytically known result for H = 0.102

6.9 Auto-correlation function c(τ) = 〈c(t, τ)〉τ , averaged over 50

time steps, displayed in log-log scale for values of the Hurst

exponent H = −0.7,−0.55,−0.25, 0, 0.25, 0.55, 0.7. . . . . . . 104

6.10 Rescaled variance 4(σ2θ − b)/κ of the winding angle for the

shortest path in the percolating cluster for values of the Hurst

exponent H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0.

In the inset, the variance σ2θ is shown. . . . . . . . . . . . . . . 106

6.11 Rescaled variance 4(σ2θ − b)/κ of the winding angle for the

watershed lines in random surfaces of Hurst exponents H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. In the inset, the

variance σ2θ is shown. . . . . . . . . . . . . . . . . . . . . . . . 107

xiv

6.12 Fractal dimension computed using the variance of the wind-

ing angle (w.a.) compared with the fractal dimension

measured using the yardstick method (y.s.) for the wa-

tershed lines and the shortest path in the cases H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. The black lines

are the conjectured dependence of the fractal dimension of

the watersheds [28] and the shortest path Eq. (5.17) on the

value of the Hurst exponent H. . . . . . . . . . . . . . . . . . 108

6.13 Variance of the driving function for the shortest path in the

cases H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0 and

Ly = 1024. In the inset, one rescales the variance by dividing

the variance through the value of κ estimated by the winding

angle. The data do not collapse. . . . . . . . . . . . . . . . . . 109

6.14 Variance of the driving function for the watershed lines in the

cases H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0 with

Ly = 8192. In the inset, the probability distribution of the

driving function for a fixed Loewner time is displayed for

H = −1,−0.5, 0. . . . . . . . . . . . . . . . . . . . . . . . . . 110

6.15 Auto-correlation function of the driving function extracted

from the watershed lines c(τ) = 〈c(t, τ)〉τ , averaged over 50

time steps, displayed in log-log scale for values of the Hurst

exponent H = −1,−0.8,−0.6,−0.5,−0.4,−0.2, 0. . . . . . . . 111

6.16 Rescaled mean square deviation of the left-

passage probability of watersheds for H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0 and Ly = 4096. . . 112

6.17 Estimated values of κ obtained with the fractal dimen-

sion (fract. dim.), winding angle, left-passage probabil-

ity (lpp) and direct SLE (dSLE) measurements for H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. The black solid

line corresponds to the value of κ deduced from the conjec-

tured dependence of the fractal dimension df on H [28], using

κ(H) = 8(df (H)− 1). . . . . . . . . . . . . . . . . . . . . . . . 113

xv



(lpp), and the direct SLE (dSLE) methods for H ∈ [−1, 1].

The red crosses correpond to the analytically known result for

H = −1, i.e. percolation, and H = 0, i.e. the GFF. We

see that the results are compatible with SLE for H ≤ 0 but

incompatible for H > 0. . . . . . . . . . . . . . . . . . . . . . 116

7.1 Graphene membrane after thermalisation. Inset: The blue

points represent carbon atoms that are above the isoheight

plane. The red line shows the extracted path along the inter-

section between the membrane and the isoheight plane. . . . . 121

7.2 The function H(r), as defined in the main text, computed

for the inner carbon atoms of the graphene membrane. Main

panel: The function H(r) for different system sizes at T = 300

K. Inset: The function H(r) for different temperatures for a

system of 800A× 800A. . . . . . . . . . . . . . . . . . . . . . 124

7.3 Main panel: Fractal dimension of the isoheight contour lines

computed with the yardstick method for a fixed temperature

T = 300 K and different system sizes. We find a universal

fractal dimension of df = 1.28±0.05. Inset: Fractal dimension

df for different temperatures and fixed system size of 800A×800A. The solid line denotes the slope −df = −1.28. . . . . . 125

7.4 Fractal dimension of the area enclosed by the isoheight contour

lines of the spanning clusters, computed with the box counting

method. The number N of boxes used to cover the atoms

belonging to the spanning cluster is displayed as a function

of the lateral size r of the boxes. Main panel: Box counting

method for different system sizes at T = 300 K. Inset: Box

counting method for different temperatures for a system size

of 800A× 800A. . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xvi

7.5 Main panel: Mean 〈θ〉 and variance 〈θ2〉 of the winding angle θ

of the isoheight contour lines for three different temperatures

T = 100 K, T = 300 K, and T = 600 K, in a semi-log plot.

The mean is approximately zero and the variance linear in

lnL. The solid line denotes the best fitting. Inset: Rescaled

probability distribution of the winding angle for L = 13.37 A,

compared to a Gaussian distribution of variance one (solid line).128

7.6 Main panel: Variance of the driving function 〈ξ2t 〉 for three

different temperatures T = 100 K, 300 K, 600 K. The solid

line shows the linear dependence of the variance of a driving

function with diffusivity κ = 2.24. Upper-left inset: the prob-

ability distribution of the driving function at t = 29. The

solid line is the probability distribution of a Gaussian random

variable of zero mean and variance 2.24. Bottom-right inset:

The autocorrelation function of the increments of the driving

function, averaged over the range t = 29 to 49. . . . . . . . . . 129

7.7 Measured rescaled mean square deviation Q(κ)/Qmin as a

function of κ with Qmin the minimum value of Q, for temper-

atures T = 100K, T = 300K, and T = 600K. Inset: the mea-

sured left-passage probabilities are compared with Schramm’s

formula for κ = 2.24 (displayed as the solid line). . . . . . . . 130

7.8 Probability distribution of the modulus |uq| of the Fourier

coefficients of the Fourier transforms in the case of Gaussian

random surfaces and graphene sheets for a fixed q. . . . . . . 132

xvii

xviii

Zusammenfassung

In statistischer Physik interessiert man sich hauptsachlich dafur, Zufalls-

pfade und deren Prozesse durch ihre kritischen Eigenschaften zu klassi-

fizieren, um Ahnlichkeiten zwischen Prozessen herauszufinden, die bei er-

ster Betrachtung unterschiedliche Eigenschaften zu haben scheinen. In der

vorliegenden Arbeit studieren wir Zufallspfade im Rahmen der Schramm-

Loewner Evolution Theorie (SLE). SLE stellt einen allgemeinen Rahmen dar,

der uber die traditionelle Analyse hinausgeht, die auf kritischen Exponenten

beruht. Die Analyse mithilfe von SLE fuhrt dazu, Pfade durch eine eindimen-

sionale Brownsche Bewegung zu beschreiben. Die Statistiken dieser Pfade

spiegeln sich im Diffusionskoeffizienten der Brownschen Bewegung wieder.

Um die SLE-Eigenschaften der stochastischen Prozesse zu untersuchen, be-

nutzen wir vier verschiedene Tests, um festzustellen, ob die Statistiken der

Zufallspfade mit der SLE- Statistik ubereinstimmen. Wir studieren Zufallsp-

fade, die entweder von Standard-Modellen der statistischen Physik stammen

oder die an stochastische Flachen gekoppelt sind. Eine der wenigen noch

unbeantworteten Fragen zur unkorrelierten Perkolationstheorie ist, welche

fraktale Dimension der kurzeste Pfad hat. Es gab mehrere Versuche, sie

exakt zu berechnen, was zu mehreren Vermutungen gefuhrt hat, die sich

aber alle als unrichtig erwiesen haben. Deswegen sollte man dieses Prob-

lem mit einem neuen Ansatz angehen, wobei die SLE-Theorie ein solcher

Ansatz sein konnte. Wir haben getestet, ob der kurzeste Pfad durch die

SLE-Theorie beschrieben werden kann, und haben numerische Beweise fur

SLE-Statistik gefunden. Dies lasst es als moglich erscheinen, eine analytis-

che Rechnung der fraktalen Dimension des kurzesten Pfades zu entwickeln.

1

Im Gegensatz zur unkorrelierten Perkolationstheorie haben Systeme aus der

Natur weitreichende Korrelationen. Deswegen haben wir uns auch dafur

interessiert, weitreichende korrelierte Systeme zu studieren. Insbesondere

haben wir Systeme analysiert, die an stochastische Flachen gekoppelt sind,

die weitreichende Korrelationen haben, die durch den Hurst-Exponenten H

beschrieben sind. Zuerst haben wir die Eigenschaften der Perkolation auf

solchen Flachen analysiert, darunter die fraktale Dimension des perkolieren-

den Clusters und dessen Grenzpfade. Wir haben auch die geometrischen

Eigenschaften des großten Clusters studiert. Da es mathematisch bewiesen

wurde, dass die Grenzpfade des perkolierenden Clusters fur unkorrelierte

Perkolation der SLE-Theorie folgen, haben wir uns gefragt, ob solch ein Re-

sultat auch fur korrelierte Perkolation gultig ist. Wir haben herausgefunden,

dass die Statistiken des zuganglichen Perimeters fur H ∈ [−1, 0] mit der

SLE-Statistik kompatibel sind, aber nicht fur H ∈ (0, 1], und haben eine

Abhangigkeit zwischen dem Hurst-Exponenten H ≤ 0 der Landschaft und

dem Diffusionskoeffizienten der Brownschen Bewegung aufgezeigt. Dieses

Resultat erweitert zwei analytisch bewiesene Ergebnisse und konnte zu inter-

essanten Entwicklungen im Bereich derjenigen Pfade fuhren, die an Flachen

gekoppelt sind. Dieses Resultat hat jedoch auch Folgen fur Systeme, die man

als stochastische Flachen betrachten kann. Wir haben die SLE-Theorie auf

eine spezifische Flache, und zwar Graphen, angewendet, und haben herausge-

funden, dass die Isolinien auch Statistiken darstellen, die mit SLE kompatibel

sind.

2

Summary

In Statistical Physics, one is usually interested in classifying random curves

and their associated processes according to their critical properties, in order

to draw similarities between processes that seem at first to have different

properties. Here we are interested in the study of random curves in the

framework of Schramm-Loewner Evolution (SLE) theory. SLE provides a

general framework that goes beyond the traditional analysis based on critical

exponents. In fact, it provides a way to describe curves starting from a

generalized one-dimensional Brownian motion, where the statistics of the

curve is encoded in the diffusivity. Here, in order to get insights into the

SLE properties of random processes, we use four different numerical tests to

verify if the statistics of the random paths are compatible with SLE statistics.

In this thesis we study random curves, whether related to standard Statistical

Physics models or coupled to random surfaces.

One of the open question regarding random uncorrelated percolation is the

value of the fractal dimension of the shortest path. There has been many

attempts to compute it exactly, leading to many conjectures that have been

ruled out. Therefore, it seems that a new approach has to be found to

tackle this problem, and the SLE theory might be one. We tested if the

shortest path might be described by SLE, and found numerical evidence for

SLE statistics. This result opens the possibility to develop an analytical

framework to compute the fractal dimension of the shortest path, one of the

last critical exponent in percolation, whose exact value is unknown.

But as in nature, systems exhibit long-range correlations, we went beyond

the framework of usual uncorrelated percolation, and studied long-range cor-

3

related systems associated to random surfaces that display long-range cor-

relations characterized by their Hurst exponent H. First, we studied the

critical properties of percolation associated to these surfaces, studying the

fractal dimension of the percolating cluster and its boundaries, as well as the

geometrical and transport properties of the largest cluster. As the cluster

boundaries of the percolating clusters have been shown analytically to be SLE

for uncorrelated percolation, we wondered if this property applies also for the

boundaries of clusters in correlated percolation. We found that the accessible

perimeter displays statistics compatible with SLE in the range H ∈ [−1, 0],

but not for H > 0, and got a dependance of the diffusion exponent of the

underlying Brownian motion on the value of the Hurst exponent H ≤ 0 of

the surface. This result might lead to interesting developments concerning

the coupling between random surfaces and SLE, as it extends two exactly

known analytical results. But it also has consequences on the properties of

physical systems that can be seen as random surfaces. We applied the SLE

theory to one specific rough surface, suspended graphene sheet, and found

that isoheight lines present statistics compatible with SLE.

4

Resume

En Physique Statistique, on s’attache communement a la classification de

courbes aleatoires et des processus qui leur sont associes en fonction de

leurs proprietes critiques, afin d’etablir des similarites entre des processus

qui semblent a priori tres differents. Dans la presente these, nous nous

interessons a l’etude de courbes aleatoires dans le contexte des evolutions

de Schramm-Loewner (SLE). La theorie SLE constitue un cadre general qui

depasse l’approche traditionnelle basee sur l’etude des exposants critiques.

Elle permet de decrire des courbes aleatoires a partir d’un mouvement brown-

ien unidimensionnel dont le coefficient de diffusivite encode les proprietes

statistiques des courbes aleatoires. Afin d’etudier les proprietes SLE de pro-

cessus stochastiques, nous utilisons quatre tests numeriques differents pour

verifier si les statistiques des courbes aleatoires sont compatibles avec les

statistiques de processus SLE. Dans le present travail, nous etudions des

courbes aleatoires associees soit a des modeles usuels de la Physique Statis-

tique, soit a des surfaces aleatoires.

La valeur de la dimension fractale du chemin le plus court constitue l’une des

dernieres questions ouvertes dans la theorie de la percolation. Il y a eu de

multiples approches pour tenter de la calculer, conduisant a l’etablissement de

nombreuses conjectures qui ont toutes ete ecartees. C’est pourquoi il semble

necessaire de trouver une nouvelle approche pour aborder ce probleme, et

la theorie SLE pourrait en etre une. Nous avons par consequent teste si le

chemin le plus court dans le modele de percolation peut etre decrit par la

theorie SLE. Nous avons trouve un accord numerique avec les predictions

de la theorie SLE. Ce resultat ouvre la possibilite de developper un cadre

5

analytique permettant de calculer la dimension fractale du chemin le plus

court, l’un des derniers exposants critiques de la percolation dont on ignore

encore la valeur exacte.

Mais comme dans la nature les systemes presentent des correlations longue

portee, nous avons depasse le cadre de la percolation non correlee usuelle,

et avons etudie des systemes correles associes a des surfaces presentant des

correlations longue portee caracterisees par leur exposant de Hurst H. Dans

un premier temps nous avons etudie les proprietes critiques du processus

de percolation associe a ces surfaces, en s’attachant plus particulierement

a l’etude des proprietes fractales de l’amas percolant et de ses contours,

mais aussi aux proprietes geometriques et de transport de l’amas le plus

large. Comme il a ete demontre que le contour de l’amas percolant est decrit

par une evolution de Schramm-Loewner dans le cas de la percolation non

correlee, on peut se demander s’il est possible d’etendre cette propriete aux

contours des amas percolant dans un modele de percolation correlee. Nous

avons trouve que le perimetre accessible presente des proprietes statistiques

compatibles avec une evolution de Schramm-Loewner pour des valeurs de

l’exposant de Hurst comprises entre −1 et 0 mais pas pour des valeurs de H

strictement positives. Nous avons trouve que la valeur du coefficient de diffu-

sion du mouvement Brownien de l’evolution de Schramm-Loewner depend de

la valeur de l’exposant de Hurst H ≤ 0 de la surface. Ce resultat peut con-

duire a des developpements interessants dans le domaine des courbes SLE

associees a des surfaces aleatoires. Cependant cette approche a aussi des

consequences sur l’etude des proprietes de certains systemes physiques vus

comme des surfaces aleatoires. Nous avons par exemple applique la theorie

SLE a des courbes extraites d’une surface particuliere, a savoir une feuille de

graphene suspendue, et avons montre que ses lignes de niveau presentent des

statistiques compatibles avec la theorie SLE.

6

Chapter 1

Introduction

Statistical Physics has been very much attached to study the properties of

curves coming from different physical models, in order to determine if they

share similar properties. This has lead to the classification of models, through

their critical exponents, into universality classes. Especially, the study of Sta-

tistical Physics models reveals interesting fractal curves with fractal dimen-

sions that are the same for different models. But can we find more universal

properties about those curves? A new theory has emerged recently, that

allows to describe curves by a more general framework. It has been discov-

ered by Schramm in 1999 [29] and is called Schramm-Loewner, or stochastic

Loewner, Evolution (SLE) theory. Knowing the fractal dimension of a curve

does not give as much information about the system as being SLE. The SLE

theory gives insights into the statistical distribution of the curves, and more-

over it allows to describe curves belonging to different universality classes by

a same process: a one-dimensional Brownian motion. Loop-Erased Random

Walk (see Fig. 1.1), Self-Avoiding Walk, percolation hulls (see Fig. 1.2), hulls

in the Ising model for example belong to different universality classes but can

be described through the same random process, a Brownian motion, in the

framework of SLE. The free parameter in the SLE theory is the diffusion

coefficient of the one-dimensional Brownian motion that controls the statis-

tical properties of the random curves. Therefore the SLE theory reduces the

properties of the curves to a single parameter. It opens new opportunities for

analytical work and numerical simulation methods. Problems resulting from

1

2 CHAPTER 1. INTRODUCTION

some kinds of optimization processes like watersheds [30], or the shortest

path [31], or from solving complex partial differential equations like in tur-

bulence [32–34] are computationally expensive, and by means of SLE theory

one might develop methods to simulate statistically equivalent curves with

less computational time.

SLE is not only usefull to describe hulls or random processes arizing in Sta-

tistical Physics models but can be also found in random surfaces. Curves

like isoheight lines coupled to random surfaces exhibit SLE properties. This

has first been studied in relation to the Gaussian Free Field (GFF), where

isoheight lines have been shown to be SLE [35, 36]. But it has also been

seen experimentaly or numericaly in physical systems like WO3 grown sur-

faces [37], Kardar-Parisi-Zhang surfaces [38], or turbulences seen as random

surfaces [32–34]. Therefore a natural question is whether isoheight lines give

any insight into the statistics of the surface itself. This question has been

tackled in a very specific case by mathematicians: in the case of the GFF.

There one is able to reconstruct the surface from its isoheights. But if it is

possible to characterize the statistics of other surfaces from their isoheight

lines is still an open question. Very recently, the field of fractional Gaus-

sian Fields [39] has been developed and might give some insights into this

problem.

In this thesis we studied both the SLE theory applied to a classical Statistical

Physics problem, the shortest path in percolation, and in relation to random

surfaces through the study of their isoheight lines.

Chapter 2 gives an introduction to the SLE theory. We derive the Loewner

differential equation, and introduce its stochastic version that is used in SLE.

We also detail the implications of SLE and define the numerical methods we

use to test the compatibility of the studied processes to the SLE theory.

In Chapter 3 we apply SLE theory to a classical Statistical Physics problem,

the shortest path in percolation. The fractal dimension of the shortest path

is one of the last critical exponent of random uncorrelated percolation that

is not known exactly. One wonders if one can apply the SLE theory to study

this critical curve, as it would give some insights into the critical exponent

of the shortest path. Therefore we test numerically the compatibility of the

2

3

statistics of the shortest path with SLE.

In Chapter 4 we recall some important characteristics of rough surfaces and

random fields. We also describe the so-called Fourier Filtering Method that is

used to generate random fields, and its connection to correlated percolation.

We also give some insights in its relation to so-called fractional Gaussian

fields.

In Chapter 5 we study extensively the properties of long-range correlated

percolation, and its dependance on the so-called Hurst exponent H control-

ling the strength of the correlations. One sees that the properties of the

system are very much dependant on H, and conjecture the dependancy of

the critical exponents of the system on H.

In Chapter 6 we study isoheight lines in long-range correlated surfaces in re-

lation to SLE properties. Isoheight lines on uncorrelated random landscapes

and on the GFF have been analyticaly proven to be SLE. We investigate if

we can extrapolate these two results by tuning the strenght of the correlation

from the uncorrelated case to the GFF case and further to rough surfaces

with strictly positive Hurst exponent. We do the same study for watersheds

in correlated landscapes and the shortest path in correlated percolation and

compare the results of the SLE analysis that we have done for these three

different kinds of paths.

In Chapter 7 we study the properties of suspended graphene sheets in relation

to conformal invariance and SLE. We apply the SLE theory to the isoheight

lines of graphene sheets to show that they satisfy more than scale invariance,

indeed they are conformally invariant. This might lead to a field theoretical

approach of graphene sheets. We also compare the results we have found

for graphene sheets with the results we obtained for the theoretical rough

surfaces that we studied in the previous chapter and study their differences,

opening up the question of the best characterization of rough surfaces, not

only through their Hurst exponent.

3


O

Fig. 1.1: Loop-Erased Random Walk Discreate Loop-Erased Random

Walk in the upper half-plane. It has been generated using the transition

probabilities defined in Ref. [1] for a random walk half-plane excursion. The

discreate upper half-plane is defined as Z+ iZ+ = {j + ik, j ∈ Z, k ∈ Z+}.

4

5

Fig. 1.2: Percolation Percolation interface generated on a triangular lattice

in a rectangle with Dirichlet boundary conditions, i.e. with zero value (red

sites) on half of the border and one (blue sites) on the other half. The

interface, displayed as a black solid line, is defined such that the interface

path starting from the bottom line has always a red site on its left and a blue

site on its right.

5


6

Chapter 2

Schramm-Loewner Evolution

theory

In this chapter, we review some aspects of the SLE theory that might be

usefull to understand the methods that we will use later. Some proofs will

be skipped, but we refer the interested reader to the following references

[1, 40–45]. The SLE theory is well suited to study curves growing in two-

dimensional domains and in the following the domains under consideration

will always be two dimensional.

Suppose that we have a random curve γt growing in a given domain D. One is

interested in the statistical properties of this curve. This is usually a difficult

problem. However, the Loewner theory simplifies this problem, by reducing

the dimensionality of the problem. It maps any non-intersecting curve γt to

a real function. The idea of the Loewner theory is to define a one-to-one

mapping between a growing set, the curve, and a time serie. In the case of

the SLE theory, the curves one considers are generated by a random process

and the time serie to which they are mapped is a Brownian motion. The

aim of the SLE theory is to encode the statistical properties of the random

curves into a Brownian motion.

Suppose that we have a growing set Kt in a Domain D such that the domain

D \ Kt is simply connected, see Fig. 2.1. The Riemann mapping theorem

ensures that there exists a conformal transformation gt from D \Kt onto D.

7

8 CHAPTER 2. SCHRAMM-LOEWNER EVOLUTION THEORY

Kt=γ[0,t]

D

(γt)t≥0

D\Kt

Fig. 2.1: A growing curve (γt)t≥0 in the domain D, in this case the upper

half-plane. The hull Kt is here defined as the curve taken till time t, and

D \Kt is the domain D from which one substracts Kt.

The idea is to study the properties of this conformal transformation gt and

to relate it to a real-valued path. In the case of stochastic Loewner Evolu-

tion also called Schramm-Loewner Evolution (SLE), this idea is extended to

random families of curves and their distributions. The aim is to encode the

statistical properties of the random curves into a real valued random process,

that corresponds to a Brownian motion in the case of SLE.

This theory brings new insights in the common classification of Statistical

Physics models in different universality classes and their characterization

through critical exponents. It improves the characterization of the processes

as it gives insights into the distribution of the random curves, but at the

same time reduces the properties of the curves to a single parameter and

opens opportunities for analytical work and numerical simulation methods.

The most common SLE processes are chordal SLE, related to curves joing the

origin to the point at infinity in the upper half-plane, radial SLE, related to

curves joining usually 0 to 1 in the open unit disk U, and dipolar SLE, related

to curves joing the origin to the upper border in a slit. In the following, we will

8

2.1. CONFORMAL TRANSFORMATIONS AND HOLOMORPHICFUNCTIONS 9

focus on chordal SLE in the complex upper half-plane H = {z, Im(z) > 0}.

But by conformal invariance, one can study the curves in any simply con-

nected open domain. Indeed, by the Riemann mapping theorem, any non-

empty open simply connected proper1 subset of C admits a bijective confor-

mal map to the open unit disk U in C, see Fig. 2.2. Therefore, one can study

the properties of SLE in a choosen domain, and by conformal invariance,

the results will hold for any conformally equivalent domain, though formulas

might change.

2.1 Conformal transformations and holomor-

phic functions

A conformal transformation of the plane is defined as a mapping w = f(z)

that preserves local angles, i.e. a transformation such that for any two smooth

curves γ and η intersecting at z0, the angle formed between the curves γ and

η at z0 is equal to the angle formed between the curves f ◦ γ and f ◦ η at

f(z0), where f ◦ γ denotes the image of the curve γ by the map f . The

conformal property might be described in terms of the Jacobian matrix be-

ing everywhere a scalar times a rotation matrix. The link between angle

preserving transformations and holomorphic2 transformations with non van-

ishing derivative is done through the Cauchy-Riemann Equations

∂u

∂x=∂v

∂y,

∂u

∂y= −∂v

∂x.

(2.1)

which translates into the Jacobian matrix of f(x + iy) = u(x, y) + iv(x, y)

being of the form: Jf (z0) =

(a b

−b a

), with a, b ∈ R, which corresponds to

a rotation composed with a scaling.

1A proper subset S ⊂ S′ of S′ is a subset that is stricly included in S′, sometimes

denoted as S S′.2A holomorphic function is a complex-valued function that is complex differentiable in

a neighborhood of every point in its domain of definition.

9


O

O

f(z)= z-iz+i

Fig. 2.2: Conformal mapping of the upper half planeH = {z ∈ C, Im(z) > 0}into the unit disk {z ∈ C, |z| < 1}, using f(z) = z−i

z+i.

Therefore one uses the following definition of a conformal map. A conformal

map or biholomorphism is a bijective holomorphic function f : U → V with

U, V open sets in C. U and V are said to be conformally equivalent if there

exists such an f . f has the property that f ′(z) 6= 0 for all z ∈ U and the

inverse of f is also holomorphic.

An important property of holomorphic functions is that holomorphic func-

tions are analytical3 functions everywhere in the domain of definition.

2.2 The Riemann mapping theorem and its

consequences

A fundamental result in complex analysis is the Riemann mapping theorem.

Let D be a proper open simply connected domain, i.e. D is included in

C but cannot be the whole complex plane. There exists a conformal map

φ : D → D, where D = {z ∈ C, |z| < 1}. Actually there exists many of these

maps, but one can make them unique through the following argument. Let

w ∈ D. Then there exists a unique conformal map φ : D → D such that

φ(w) = 0 and arg(φ′(w)) = 0, or φ′(w) > 0.

3An analytic function is locally given by a converging power serie. Moreover analytic

functions are infinitely differentiable.

10

2.2. THE RIEMANN MAPPING THEOREM AND ITSCONSEQUENCES 11

A subset K of H is called a hull4 if it is bounded in H (included in a ball of

finite radius), H \K is simply connected and K = K ∩H.

In the following, we will be interested in conformal maps gK : H\K → H. By

the Riemann mapping theorem there exists many of them. The uniqueness of

the conformal map results from the condition that gK looks like the identity

at infinity, i.e. gK(z) ∼ z for |z| → ∞.

If K is a hull, there exists a unique conformal map gK : H \ K → H such

that:

lim|z|→∞

gK(z)− z = 0. (2.2)

This is called the hydrodynamic normalization.

Set D = {z : −z−1 ∈ H \ K}. D ⊆ H is a simply connect domain, and a

neighborhood of 0 in H. Therefore, by the Riemann mapping theorem, there

exists a conformal map φ : D → H which, by Schwarz reflection principle

imposing φ(z) = φ(z), can be extended to the lower half-plane and admits

a Taylor expansion around 0. As by the reflection principle, φ is real on a

neighborhood of 0 on the real line, one has that the coefficients must be real,

and gets that for z → 0

φ(z) = a0 + a1z + a2z2 + a3z

3 +O(|z|4), (2.3)

with a0, a1, a2, a3 ∈ R. By fixing φ(0) = 0 and φ′(0) > 0, one gets that

a0 = 0 and a1 > 0. We define gK by gK(z) = −a1φ(−z−1)−1 − a2/a1, which

is a conformal map from H \ K onto H. To prove the uniqueness, set g, h

two conformal maps satisfying the hydrodynamic normalization. Therefore

f = g ◦ h−1 is a conformal automorphism of H such that f(z)− z → 0 when

|z| → ∞, and f(∞) =∞. But the conformal automorphisms of H are of the

form

f(z) =az + b

cz + d, (2.4)

with a, b, c, d ∈ R and ad − bc = 1. In order to have f(∞) = ∞, one needs

to have c = 0 and d 6= 0. In order to have that f(z)− z = 0 when |z| → ∞,

4It is sometimes called a compact H-hull.

11


Kt

O

gt:ℍ \ Kt →ℍ

ℍ

gt(Kt)

O Ut=gt(γ(t))

ℍ

Fig. 2.3: Mapping gt : H \Kt → H. The hull is shown in blue and is mapped

to the real line by gt.

one needs to have that a/d = 1 and b/d = 0. Therefore f is the identity and

g = h, proving the uniqueness.

Actually we have shown that for |z| → ∞,

gK(z) = z +aKz

+O(|z|−2), (2.5)

with aK ∈ R, called the half-plane capacity.

2.3 Half-plane capacity parametrization

For the following, we will prove that the half-plane capacity aK is positive

and increasing in the sense that aK ≥ 0, aK > 0 if K 6= ∅, and aK ≤ aK′ if

K ⊂ K ′. This will allow us to make a reparametrization of the curve.

Let K be a hull, Bt a complex Brownian motion starting at z ∈ H \K, τ be

the hitting time of Bt for R ∪K, and gK be the conformal map from H \Konto H as defined before. Then for z = iy ∈ H\K, with y > 0, the half-plane

capacity aK is given by

aK = hcap(K) = limy→∞

yEiy(Im(Bτ )). (2.6)

Let us consider z 7→ gK(z)−z. It is a bounded analytic function ofH\K = H,

because it is bounded at ∞ and continuous. Therefore5 z 7→ Im(gK(z)− z)

5The real and imaginary parts of holomorphic functions are harmonic, see Eq. (2.1).

12

2.4. THE LOEWNER DIFFERENTIAL EQUATION AND THEDRIVING FUNCTION 13

is a bounded harmonic function. Let us consider a Brownian motion started

at z ∈ H and τ = inf{t > 0 : Bt /∈ H}. By the optional stopping theorem

Im(gK(z)− z) = Ez (Im(gK(Bτ )−Bτ )) . (2.7)

But by definition of gK , Im(gK(Bτ )) = 0. If one sets z = iy, one gets

Im(gK(iy)− iy) = −Eiy(Im(Bτ )). (2.8)

As y → ∞, gK(iy) = iy + aKiy

+ O(1/y2). Therefore yEiy(Im(Bτ )) = aK +

O(1/y) and one recovers Eq. (2.6). We have that aK ≥ 0 and aK > 0 if

K 6= ∅.Now, let us suppose that K ⊂ K ′ and denote K = K ′ \ K. One considers

the conformal maps defined as before gK : H \K → H and gK : H \ K → Hwith K = gK(K), see Fig. 2.4. One has gK(z) = z + aK

z+ o(1

z) and gK(z) =

z+aKz

+o(1z) for |z| → ∞. Now, one considers f = gK◦gK : H\K ′ → H. It has

the following limited development for |z| → ∞: f(z) = z+aK+aK

z+ o(1

z), i.e.

f satisfies the hydrodynamic normalization. One can also directly consider

gK′ : H\K ′ → H which is the unique conformal map from H\K ′ onto H with

the hydrodynamic normalization. Therefore, by uniqueness: aK′ = aK + aK ,

and aK′ > aK for K ⊂ K ′ and K 6= K ′.

This result is very usefull if one considers a family of increasing hulls (Kt)t≥0

such that Ks is strictly contained into Kt for s < t. Let us consider a simple

curve γ[0, t] growing in the upper half-plane. One defines gt : H \ γ[0, t]→ Hand for |z| → ∞, gt(z) = z +

aγ[0,t]z

+ o(1/z). Indeed it can be shown that

t 7→ aγ[0,t] is increasing in time and continuous. One reparameterize the curve

such that aγ[0,t] = 2t.

2.4 The Loewner differential Equation and

the driving function

The idea behind the Loewner differential Equation is to have a one-to-one

correspondence between a continuous real valued path Wt = gt(γt), where

13


ℍ

K K'\KK'

ℍ ℍ

K^

gKg

gK

gK^gKg

gK'

Fig. 2.4: Mapping of the compact H-hull K ′ = K ∪ (K ′ \K) using gK′ or the

composition of gK with gK , f = gK ◦ gK .

γt is the tip of the curve grown till time t, and increasing families of hulls

having a certain local growth property, the hulls Kt of the growing curve γt.

Here we show how a random curve satisfying some given conditions that we

will explain later can be mapped onto a time serie (Wt)t≥0. Also the time

serie (Wt)t≥0 characterizes the curve uniquely, i.e. one can go back from Wt

to the curve.

Let us consider a familiy of hulls (Kt)t≥0, increasing in time, i.e. Ks ( Kt

for s < t. Set Ks,t = gKs(Kt \ Ks). We say that (Kt)t≥0 has the local

growth property if diam(Kt,t+ε)→ 0 when ε→ 0+, where diam(Kt,t+ε) is the

diameter of the smallest disk encompassing the compact set Kt,t+ε.

Then one can show that there exists a unique Wt ∈ R such that:

Wt = gt(γt) :=⋂s>0

gt(γ(t, t+ s]). (2.9)

This defines a mapping between the curve and a real-valued path (Wt)t≥0

called the driving function of the curve.

Actually gt satisfies the Loewner differential Equation with the identity map

14

2.4. THE LOEWNER DIFFERENTIAL EQUATION AND THEDRIVING FUNCTION 15

as initial value. For a fixed z ∈ H,

∂tgt(z) =2

gt(z)−Wt

, with g0(z) = z. (2.10)

The function gt(z) is defined till time T (z) =: inf{t ≥ 0 : z ∈ Kt}.One can show that, see for example Prop. 3.46 in [1], there exists C ∈ Rsuch that for all r ∈ R+ and all ξ ∈ R, and for any hull K ⊂ D(ξ, r) and

z /∈ D(ξ, 2r),

|gK(z)− z − aKz − ξ

| ≤ CraK|z − ξ|2

. (2.11)

We consider the curve being parametrized with half-plane capacity, i.e. aKt =

2t, and applies Eq. (2.11) to gKt,t+ε(zt) = zt+ε = gt+ε(z) and zt = gt(z). One

has that Kt,t+ε ⊂ D(Wt, 2diam(Kt,t+ε)), and aKt,t+ε = 2ε. As z ∈ H \Kt, zt

is in H and for small enough ε, zt /∈ D(Wt, 4diam(Kt,t+ε)). Therefore one can

apply Eq. (2.11) to obtain:

|gt+ε(z)− gt(z)− 2ε

gt(z)−Wt

| ≤ 4Cdiam(Kt,t+ε)ε

|gt(z)−Wt|2. (2.12)

By the local growth property, one has that gt+ε(z)−gt(z)ε

= 2gt(z)−Wt

+ o(1),

and one finds the differential Eq. (2.10) satisfied by gt(z) by letting ε going

to zero. This actually only shows that the right side derivative satisfies

Loewner’s differential equation. For a full proof, see Ref. [1].

Eq. (2.10) is very usefull as if one starts from Wt one can solve the Loewner

differential equation and compute gt from which one deduces the curve. But

also, starting from the curve, one has gt and one can compute Wt = gt(γ(t)).

This leads to the one-to-one relation between the growing curve (γt)t≥0 and

a real valued path (Wt)t≥0.

In the cases we will study, simple or self-touching curves, the local growth

property will be fulfilled. In the case of a continuous simple path, Kt =

γ[0, t] ∈ H. But if one starts from the real-valued path (Wt)t≥0, it has to be

smooth enough in order to generate a simple curve γ(t) = g−1t (Wt) [46].

From (Wt)t≥0, one can deduce gt and from g−1t : H → H \ Kt construct

back Kt, i.e. γ(t) in the case of a simple curve. For all y ∈ H and t ≥ 0,

15


one constructs g−1t (H) =

⋃y∈H{g

−1t (y)}. Then reconstructs the curve as

γ(0, t] = H \ g−1t (H).

Eq. (2.11) has implications for numerical methods used to solve the Loewner

differential equation Eq. (2.10). From Eq. (2.11) one deduces that the solu-

tion of the Loewner differential equation has the following property:

gt(z) = z +2t

z+ o(t) (2.13)

for t small. But if one considers the following vertical slit

γ(0, t] = 2i√t for t > 0,

then gt : H\γ(0, t]→ H satisfying the Loewner differential Equation is given

by:

gt(z) =√z2 + 4t

as it has the following limited development gt(z) = z + 2t/z + O(1/|z|2) for

|z| → ∞. But it also has the same limited development gt(z) = z+2t/z+o(t)

as t → 0. Therefore Eq. (2.13) shows that as a first order approximation,

one can approximate the solution of the Loewner differential Equation by

a vertical slit map for short times, and that the error is of the order of

o(t). This gives an indication on the error of the vertical slit map method

described later. Let us suppose we are given the time serie (Wnδt)n≥0. Then

gnδt = gδt ◦ . . .◦gδt and at each time step if one approximates gδt by a vertical

slit one makes an error of the order of o(δt).

2.5 Schramm-Loewner Evolution theory

In 1999 Schramm considered a stochastic version of Eq. (2.10) and conjec-

tured that is was describing the scaling limit of the Loop-Erased Random

Walk and the Uniform Spanning Tree [29].

If we choose Wt =√κBt, with κ ≥ 0 and (Bt)t≥0 a one-dimensional standard

Brownian motion, then by Eq. (2.10) one constructs a random family of hulls

16

2.5. SCHRAMM-LOEWNER EVOLUTION THEORY 17

0<κ≤4 4<κ<8 8≤κ

Fig. 2.5: For 0 ≤ κ ≤ 4, the curves are simple. For 4 < κ < 8 they are self

touching and for κ ≥ 8 they are space filling

(Kt)t≥0 called SLEκ. Rohde and Schramm [47], and Lawler et al. [48] in the

case κ = 8, showed that SLEκ generates continuous curves.

Let κ ≥ 0 and gt : Ht → H be the solution of the stochastic Loewner

differential equation

∂tgt(z) =2

gt(z)−Wt

,

g0(z) = z.

(2.14)

defined for z ∈ Ht and up to time T (z) := sup{t > 0 : inf [0,t]|Wt−gt(z)| > 0}.There exists almost surely (a.s.) a continuous curve (γt)t≥0 in H such that for

all t ≥ 0, Ht is the unbounded connected component of H\γ[0, t]. Kt = H\Ht

is called the hull of the curve. In case the curve (γt)t≥0 is simple, Kt is the

curve itself. Indeed the curve (γt)t≥0 is defined as

γt = g−1t (Wt). (2.15)

Actually, in the following, SLEκ will refer whether to the increasing family

(Kt)t≥0, or to the conformal maps (gt)t≥0 or to the curve (γt)t≥0 depending

on the context.

Depending on the value of κ, the curve (γt)t≥0 exhibits different properties.

For κ ∈ [0, 4], the curve γ is a.s. simple. For κ ∈ (4, 8) the curve is a.s.

self-touching, whereas for κ ≥ 8 the curve is a.s. spacefilling, see Fig. 2.5.

Suppose the curve γ has a double point, such that γ(t1) = γ(t2) with t1 < t2.

Then if one maps γ(0, t1] to the real axis by gt1 , the curve defined by γt1(s),

17


with γt1(s) = gt1(γ(t1, s+t1]) touches the real axis at s = t2−t1. By conformal

invariance and domain Markov property (see below) γt1 and γ have the same

law. Therefore, in order to study the probability of γ having double points,

one studies the probability of γ to go back to the real line. Let x be on the

real line. If x is swallowed by Kt at a given time t, then gt(x) = Wt, and the

random process Xt = gt(x)−Wt√κ

hits 0. But dXt = 2κXt− dBt, by symmetry of

Bt and −Bt, and corresponds to a Bessel process of dimension n = 1 + 4κ. It

hits 0 an infinite number of times with probability 1 for n < 2, and does not

hit 0 with probability 1 for n ≥ 2. Therefore the curve γ is simple for κ ≤ 4

and has an infinite number of double points for κ > 4. Actually, for κ > 4

one can show that the hull Kt of the path grows towards H.

We have seen that SLEκ for κ ≤ 4 and κ > 4 exhibits different properties.

However there exists a duality relation between the outer boundary of SLEκ

and SLE16/κ for κ > 4. The outer boundary of SLEκ curves for κ > 4 looks

like SLE16/κ curves.

There is another characterization of SLEκ curves. If a family of random

curves is SLEκ, then it is conformally invariant and satisfies the Domain

Markov Property. And reciprocally, if a family of stochastic curves satisfies

conformal invariance and the Domain Markov property, then there exists a

κ ∈ R+ such that these curves are SLEκ.

Conformal invariance Given two domains D and D′ that are conformally

equivalent, one can transfer the probability from one domain to the other

one by simply mapping the curves into the new domain. Let us consider a

conformal map Φ from the domain D onto the domain D′, Φ : D → D′,

and two points on the boundary of D, a, b ∈ ∂D, which are mapped by Φ to

a′, b′ ∈ ∂D′. The probability measure PD,a,b on the curves γ in D from a to

b induces under Φ a probability measure Φ ∗ PD,a,b on curves in D′ = Φ(D)

from a′ = Φ(a) to b′ = Φ(b):

∀U ⊂ D, Φ ∗ PD,a,b (Φ(γ) ⊂ Φ(U)) := PΦ(D),Φ(a),Φ(b) (Φ(γ) ⊂ Φ(U))

= PD,a,b (γ ⊂ U) .(2.16)

18


In this equation, we just transfered the probability measure P between two

conformaly equivalent domains. The conformal invariance is taken in the

scaling limit.

The process is conformally invariant if, in the continuum limit, the mapped

probability measure Φ∗PD,a,b is the same as the probability measure PD′,a′,b′on the continuum limit of lattice curves generated in D′ going from a′ to b′:

∀U ⊂ D, Φ ∗ PD,a,b (γ ⊂ U) = PD′,a′,b′ (Φ(γ) ⊂ Φ(U)) . (2.17)

Domain Markov Property Consider a curve γ in D starting in a and

ending in b. Take a point c belonging to the curve, and consider the two

propability measures, see Fig. 2.6:

1. PD,a,b(·|γ[a,c]

)the probability measure in (D, a, b) on the curves starting

in a and ending in b in the domain D conditionned to start with γ[a,c],

2. and PD\γ[a,c],c,b (·) the probability measure in(D \ γ[a,c], c, b

)on the

curves γ[c,b] starting in c and ending in b in the domain D \ γ[a,c].

The Domain Markov Property (DMP) states that the two probability mea-

sures are equal:

PD,a,b(·|γ[a,c]

)= PD\γ[a,c],c,b (·) . (2.18)

The DMP combined to the conformal invariance property expressed in

Eq. (2.17) becomes,

PD,a,b(·|γ[a,c]

)= Φ ∗ PD,a,b (·) , (2.19)

where Φ : D \ γ[a, c] → D is a conformal map such that Φ(c) = a and

Φ(b) = b. It means that Φ(γ[c,b]) is independant of γ[a,c] and has the same

distribution as the original one for curves from a to b in D [42].

Let us consider a random curve γ, growing in the upper half-plane starting

at the origin and growing towards infinity. We define ft(z) = gt(z) −Wt =

z −Wt + 2tz

+ O(

1|z|2

), which is a conformal map that maps the tip of the

curve γ(t) back to the origin and infinity to infinity. Let t, s > 0. Then, by

19


b

a

c

PD,a,b(γ[ab]| γ[ac])

γ[ac]

γ[ab]

D b

a

c

γ[cb]

D\γ[ac]

PD\γ[ac],c,b(γ[cb])

Fig. 2.6: The two probability measures in the domains D and D \ γ[a,c]. If

the Domain Markov Property is satisfied, these two probabilities are equal.

the Domain Markov Property and the conformal invariance of SLEκ, one has

that γ[0, s] = ft(γ[t, t + s]) is equal in law to γ[t, t + s] and independant on

γ[0, t′] with t′ ≤ t, see Fig. 2.7. From this one deduces that fs is distributed

like ft+s ◦ f−1t , i.e. ft+s = fs ◦ ft in law. For all fixed t and s > 0, one has:

fs ◦ ft(z) = z − (Wt +Ws) +2(t+ s)

z+O

(1

|z|2

)and ft+s(z) = z −Wt+s +

2(t+ s)

z+O

(1

|z|2

),

(2.20)

and therefore for any fixed t ≥ 0, (Wt+s − Wt)s≥0 has the same law as

(Ws)s≥0, and independant of the past values of (Wt′)t′≤t. We conclude that

(Wt)t≥0 is a random process with independant and stationary increments.

Moreover it is continuous (not proven here, see [1] for a proof), and therefore

there exists κ ∈ R+ and α ∈ R such that Wt =√κBt + αt with (Bt)t≥0 a

standard one-dimensional Brownian motion6. By reflection symmetry around

the imaginary axis, α = 0.

6This comes from the theorem stating that a one dimensional Markov process with

continuous trajectory and stationary increments is a Brownian motion with a prossible

drift term.

20


ℍ

O

γ(s)~

ft

ℍγ(t+s)

γ(t)

O

γ[0,t]

γ[t,t+s]

ℍ\γ[0,t]γ(t+s)

γ(t)

O

γℍ\γ[0,t][t,t+s]

Domain Markovproperty

Conformal invariance

Fig. 2.7: Under the assumptions of conformal invariance and Domain Markov

Property, the curves γ(t, t + s] knowing γ[0, t] and γ(0, s] are equal in law.

From this one deduces that the driving function has stationary independant

identically distributed increments, i.e. (Wt+s −Wt)s≥0 has the same law as

(Ws)s≥0. 21


2.6 Mapping of curves generated in a rectan-

gle into the upper-half plane

In numerical simulations, curves are generated in a bounded domain. Thus,

to employ the chordal SLE formalism one needs to use a conformal map to

map it into the upper half-plane. In the context of this thesis, the curves

we generate numerically, are defined in a lattice enclosed in a rectangle of

size Lx×Ly, starting at the bottom boundary and ending at the upper one.

However, we will use results that are valid for chordal curves, like the left-

passage probability formula computed by Schramm [49], see Section 2.7.3

below, or the chordal Loewner Equation Eq. (2.18). Therefore we have to

map conformally the original curves into the upper half-plane using an inverse

Schwarz-Christoffel transformation [50], mapping one point of the boundary

to the origin and an other point of the boundary to infinity. One usually

supposes that the curve starts at the middle point of the bottom edge and

end at the middle point of the top edge. Then, by mapping the first point to

the origin and the second to infinity, the curves generated in the rectangle are

mapped conformally to chordal curves. But if the curves are generated with

“free” boundary conditions, i.e. without any constraints on the boundaries,

such that the paths have no fixed starting and ending points, we relocate

them, in order for them to start at the origin; the curves are now defined in

the rectangle [−Lx, Lx]× [0, Ly] in lattice units. We then make the approxi-

mation, that the curves are defined in the rectangle [−Lx, Lx]× [0, 2Ly], and

that the generated curve is part of a curve starting at the origin and ending

at the point (0, 2Ly). We then use an inverse Schwarz-Christoffel transfor-

mation that maps the rectangle [−Lx, Lx]× [0, 2Ly] onto the upper half plane

with the point (0, 2Ly) being mapped to infinity.

Applying the Schwarz-Christoffel formula for a half-plane to the rectangle

defined in the complex plane as R = (−K,K,K + iK ′,−K + iK ′), one

22

2.7. NUMERICAL TESTS OF SCHRAMM-LOEWNER EVOLUTION 23

obtains the conformal map f : z ∈ H→ w ∈ R [50, 51]:

w = f(z, k)

w =

∫ z

0

dζ√(1− ζ2) (1− k2ζ2)

w =

∫ sin−1 z

0

dθ√1− k2 sin2 θ

w = F(sin−1 z, k

),

(2.21)

with F (φ, k) the elliptic integral of the first kind, and k the elliptic modulus7.

Therefore, from the inversion of the elliptic integral, one gets that

z = f−1(w, k) = sn(w, k), (2.22)

where sn(·, k) is the sine Jacobi elliptic function of modulus k8. The obtained

mapping maps the points w1 = −K, w2 = K, w3 = K + iK ′ and w4 =

−K + iK ′ to the point z1 = −1, z2 = 1 and z3 = k−1 and z4 = −k−1,

with K = K(k) and K ′ = K(√

1− k2), where K(k) =∫ π/2

0dθ√

1−k2 sin2 θis the

complete elliptic integral of the first kind. One needs to adapt the parameter

k to the geometry of the rectangle, i.e. find k such that

K ′(k)

2K(k)=LyLx. (2.23)

Therefore, by solving Eq. (2.23) and applying the inverse Schwarz-Christoffel

transformation sn(·, k) from the rectangle to the upper half-plane, one maps

the generated curves into chordal ones. In the following we will work with

chordal curves.

2.7 Numerical tests of Schramm-Loewner

Evolution

In order to test numerically the agreement of the statistics of the random

curves obtained from physical models with the SLE theory, one uses some

7with 0 < k < 1.8Sometimes the elliptic integrals and Jacobi elliptic functions are denoted with the

parameter m = k2.

23


Ow1

w2

w3

w4

(a)

O-1 1-k-1 k-1

z1

z2

z3

z4

(b)

Ow1

w2

w3

w4

(c)

O-1 1-k-1 k-1

z1

z2

z3

z4

(d)

Ow1

w2

w3

w4

(e)

O-1 1-k-1 k-1

z1

z2

z3

z4

(f)

Fig. 2.8: Mapping from a rectangle into the upper half plane Rect-

angle mapped to the upper half-plane. (a) Rectangle of aspect ratio 1 and

(b) mapped points into the upper half-plane. (c) Rectangle of aspect ratio

2 and (d) mapped points into the upper half-plane. (e) Rectangle of aspect

ratio 4 and (f) mapped points into the upper half-plane.

24


properties of SLE curves to test the compatibility of the statistics of the

random curves with the predictions made by the SLE theory. We use in par-

ticular four tests: the fractal dimension, the winding angle, the left-passage

probability and the diffusion of the driving function.

2.7.1 The fractal dimension

The fractal dimension of a curve is defined as follows. If N(r) is the number

of balls of radius r needed to recover a fractal curve, then it scales as N(r) ∼r−df , where df is the fractal dimension of the curve. As has been proven by

Beffara [52], SLEκ curves are fractal with a fractal dimension df depending

on the value of the diffusion coefficient κ as:

df = min(

2, 1 +κ

8

). (2.24)

This constitutes the first way to relate SLE theory to universality classes

characterized through their fractal dimensions. If one wants to show the

SLE behavior of some curves, one uses this relation (2.24) to estimate the

value of κ.

There are several methods to compute the fractal dimension of a curve. We

used the yardstick method, where one approximates the curve by sticks of

different lengths r.

2.7.2 The winding angle

In SLEκ theory, the statistics of the winding of the curve is related to the

value of κ. This result is a consequence of the conformal invariance of the

SLEκ curves.

The winding angle θ between two points X and Y of a curve is defined as

the continuous angle along the curve. Duplantier and Saleur in Ref. [53]

computed the probability distribution of the winding angle of random curves

using conformal invariance and Coulomb gas techniques. As the Coulomb

gas constant is related to κ, one deduces the probability distribution of SLEκ

curves.

25


In the case of SLE, Schramm [29] proved the convergence of the winding angle

distribution of SLEκ to a Gaussian of variance κ ln(L). Using the relation

κ = 4g, the result obtained through non rigourous Coulomb gas arguments

in [53] is in agreement with the one obtained by Schramm.

In our numerical tests, we compute the variance of the winding angle along all

the points of the curve, instead of the endpoints only, following the relation

obtained by Wieland and Wilson in [54].

For each path, we have a discrete set of points zi on the lattice. We compute

the winding angle θi at each edge zi of the curve iteratively: θi+1 = θi + αi,

where αi is the turning angle between the two consecutive points zi and zi+1,

see Fig. 2.9. Then for SLEκ, the winding angle along all the edges of the

curve exhibits a Gaussian distribution of variance

〈θ2〉 − 〈θ〉2 = b+κ

4ln(Ly), (2.25)

where b is a constant and Ly is the vertical lattice size [54]. The mean value

of the winding angle 〈θ〉 is expected to be zero. Therefore κ is given by the

slope of the graph 〈θ2〉 vs ln(Ly) up to a factor 4.

This test is actually not rigorously a SLE test, as it is considered as a

conformal invariance test. It has for example been used to show the conformal

invariance of rocky shorelines [55]. Therefore, the winding angle test gives

an indication on conformal invariance of the problem, a necessary condition

for SLE. Given the simplicity of this test, we always start with it.

2.7.3 The left-passage probability

A chordal curve, starting at the origin and growing towards infinity, splits

the space into two parts: the points that are at the left of the curve and the

ones that are at the right. The curve is said to pass at the left of a given point

if this point belongs to the right side of the curve, see Fig. 2.10. For chordal

SLEκ curves, Schramm has computed the probability of a curve to go to the

left of a given point z = Reiφ, where R and φ are the polar coordinates of

z [49]. For a chordal SLEκ curve in H, the probability Pκ(φ) that it passes

26


θ1

αi

zi

zi+1

zi-1

Ly

Fig. 2.9: (Winding angle) The winding angle is defined iteratively, as the

sum of the turning angles αi between two consecutive segments (zi−1, zi) and

(zi, zi+1).

27


Fig. 2.10: (Left passage) Curve separating the space into the points belong-

ing to the right side of the curve (space marked in grey) and points belonging

to the left side of the curve (space left blank).

to the left of Reiφ depends only on φ and is given by Schramm’s formula,

Pκ(φ) =1

2+

Γ (4/κ)√πΓ(

8−κ2κ

) cot(φ)2F1

(1

2,

4

κ,3

2,− cot(φ)2

), (2.26)

where Γ is the Gamma function and 2F1 is the Gauss hypergeometric func-

tion. We define a set of sample points S in H for which we numerically com-

pute the probability P (z) that the curve passes to the left of these points

and compare it to the expected value Pκ(φ(z)).

To estimate κ, we minimize the mean square deviation Q(κ) defined as,

Q (κ) =1

|S|∑z∈S

[P (z)− Pκ(φ(z))]2 , (2.27)

where |S| is the cardinality of the set S.

The probabilities for different values of κ are plotted in Fig. (2.11). One sees

that the value of κ can be associated to the spacial spreading of the random

curves starting from 0. For smaller values of κ, the curves are statistically

28


Π

4

Π

2

3 Π

4Π

Φ

0.25

0.5

0.75

1.

PΚ

Κ=7.9

Κ=6

Κ=4

Κ=2

Κ=0.1

Fig. 2.11: (Left passage probability) Probability of a point belonging the

right side of the curve dependant on the angle φ. The left-passage probability

is independant on the value of the radius R of the point z ∈ H.

more concentrated around the imaginary axis, than for larger values of κ and

therefore the transition at φ = π/2 is sharper.

One advantage of this method is that the left-passage formula is a conse-

quence of the SLE theory as it relies on the Brownian motion property of the

driving function, see for example the computations in Refs. [41, 43].

2.7.4 The direct SLE algorithm

Let us consider a curve in the upper half-plane given by the set of points {z0 =

0, z1, . . . , zn, . . .}. We want to extract the corresponding driving function, i.e.

find the time serie {W0,Wt1 , . . . ,Wtn , . . .} that corresponds to the given set

of points. Therefore we will iteratively compute the conformal maps gtimapping the portion of the curve between two consecutive points zi and zi+1

to the real line, and compute the associated driving function by Wti = gti(zi)

at the Loewner times ti for i = 0, . . . , n, . . ..

More precisely, one maps the first part of the curve γ([0, t1]) between z0 and

29


z0=O

z1

z2

z3

ℍ

W0

ℍ

W1

gt1(z2)

gt1(z3)

gt1 gt2

W0

ℍ

W1

gt2○gt1(z3)

W2

Fig. 2.12: Iteration of the conformal maps gti : H\gti−1◦. . .◦g1(γ([ti−1, ti]))→

H and extraction of the underlying driving function.

z1 to the real axis by gt1 : H \ γ([0, t1]) → H, and compute Wt1 = gt1(z1).

One applies the conformal map gt1 to the rest of the curve γ[t1,∞). One

then apply the same procedure to the new mapped curve gt1(γ([t1,∞)), i.e

one computes the conformal map gt2 : H \ gt1(γ([t1, t2])) → H that maps

the upper half-plane minus a slit back to the upper half-plane, cf. Fig. 2.12.

Therefore one gets that:

gtk = gtk ◦ gtk−1◦ . . . gt1 (2.28)

with gti : H\gti−1◦. . .◦g1(γ([ti−1, ti]))→ H. Each gti satisfies the Loewner dif-

ferential equation (2.10), with the driving function starting at Wti−1. There-

fore at each iteration, one computes the increment of the driving function

Wti −Wti−1in the time interval δti = ti − ti−1, from which one deduces the

time evolution of the driving function (Wti)i=0,...,n,....

In order to compute numerically gti , one needs to approximate the original

curve. One might do so, whether by approximating the original curve by

a line segment between Wti−1and gti−1

(zi), i.e. by a ’tilted slit’, or by a

vertical segment between Re(gti−1(zi)) and gti−1

(zi), i.e. by a ’vertical slit’,

see Fig. 2.13. In [56], the author compares the two approximations, and

concludes that there is no significant difference between them. However, the

’vertical slit’ method is computationaly less expensive than the other method,

and we will use this method in the following.

Indeed the ’vertical slit’ approximation corresponds to the first order approx-

30


gti(γ[0,ti])

gti(γ(ti+1))

gti(γ[ti,ti+1])gti+1[gti(γ(ti+1))]

approx.

Uti+1=ξti+1

(4δt)1/2

Fig. 2.13: Approximation made in the ’vertical’ slit map algorithm

imation of the conformal map, as has been shown in Eq. (2.13). It is com-

puted by assuming that the driving function is constant over the small time

intervals δti. With this method, one obtains the slit map equation [41,56],

gt(z) = ξt +

√(z − ξt)2 + 4δt. (2.29)

We start with Wt = 0 at t = 0 and considers the initial points of the curve

{z00 = 0, z0

1 = z1, . . . , z0N = zN}. At each iteration i = 1, . . . , N , we ap-

ply the conformal map gti to the remaining points {zi−1i , . . . , zi−1

N } of the

curve. As gti sends zi−1i to the real axis by setting Wti = Re{zi−1

i } and

δti = ti − ti−1 =(Im{zi−1

i })2/4 in Eq. (2.29), one gets a new set of points

{zii+1 = gti(zi−1i+1), . . . , ziN = gti(z

i−1N )} shorter by one point, constituing the

remaining part of the curve. Re{} and Im{} are the real and imaginary

parts, respectively.

This method allows to observe directly the driving function, and therefore

test the consistency with the SLE theory. However the numerical results

obtained with the direct SLE method are usually less precise than with the

other analyses and, therefore, characterized by larger error bars, as is well

known in the literature [30,32,33,57–59]. Indeed at each time step one makes

an error due to the first order approximation of the conformal map gt [60].

31


2.8 Known results in SLE

Impressive progress has recently been made in the field of critical lattice

models using the SLE theory. Two mathematicians, Wendelin Werner in

2006 and Stanislav Smirnov in 2010 obtained the Fields Medal for their

important contributions to this field.

In the SLE framework, with the value of κ, one can obtain exactly several

probability distributions for the curve, allowing to compute, for example,

crossing probabilities and critical exponents [61–63]. SLE has been shown

to describe many conformally invariant scaling limits of contours of two-

dimensional critical models. In particular, SLE6 was first conjectured [29]

and later proved on the triangular lattice [61] to describe the hull in criti-

cal percolation [64]. It is believed to hold also for other lattice types. SLE

has been successfully used to compute rigorously other critical exponents of

percolation-related objects [63, 65] as, for example, the order parameter ex-

ponent β, the correlation length exponent ν, and the susceptibility exponent

γ [63].

The other models that have been proven rigorously to be SLE are the Loop

Erased Random Walk shown to be SLE2 [29, 48, 66], the interfaces of the

spin and Fortuin-Kasteleyn Ising clusters shown to be respectively SLE3

and SLE16/3 [67–70], the harmonic explorer [71] and the level lines of the

discreate and continuous Gaussian Free Field [35, 36] shown to be SLE4,

and the Uniform Spanning Tree shown to be SLE8 [48]. Some other models

are conjectured to be SLE, like the Self-Avoiding Walk, conjectured to be

SLE8/3, the Potts model in general and the Fortuin-Kasteleyn model.

Other processes have been shown numerically to be compatible with SLE

like domain walls in two-dimensional spin glasses with Gaussian disorder [57,

72], domain walls in the random-field Ising model with Gaussian distributed

random field [73], avalanche frontiers in the Abelian Sandpile model [74],

isoheight lines of the Kardar-Parisi-Zhang and Edwards-Wilkinson surfaces

[38] and watersheds in percolation [30]. SLE processes have also been seen in

32

2.8. KNOWN RESULTS IN SLE 33

natural physical processes like WO3 deposited surfaces [37], and turbulences

[32,33].

But not all the processes are SLE. The domain walls in the disordered Solid

On Solid model [75], the domain walls in the Edwards-Anderson spin glass

model with bimodal coupling [76] and negative weight percolation [77] have

been shown to display statistics incompatible with SLE.

In the following, we will focus on chordal SLE and apply this theory to

describe the shortest path in percolation in Chapter 3 and study its relation

to long-range correlated surfaces in Chapters 6 and 7.

33


34

Chapter 3

Shortest path in percolation

and Schramm-Loewner

Evolution

In this Chapter, we numerically show that the statistical properties of the

shortest path on critical percolation clusters are consistent with the ones

predicted for SLE curves for κ = 1.04± 0.02. The shortest path results from

a global optimization process. To identify it, one needs to explore an entire

area. Establishing a relation with SLE permits to generate curves statistically

equivalent to the shortest path from a Brownian motion, see Section 2.5. We

numerically analyze the winding angle, the left passage probability, and the

driving function of the shortest path and compare them to the distributions

predicted for SLE curves with the same fractal dimension. The consistency

with SLE opens the possibility of using a solid theoretical framework to

describe the shortest path and it raises relevant questions regarding conformal

invariance and domain Markov properties, which we also discuss.

This chapter is based on Ref. [31]:

N. Pose, K. J. Schrenk, N. A. M. Araujo and H. J. Herrmann, Shortest path

and Schramm-Loewner Evolution, Scientific Reports 4, 5495 (2014).

35

36CHAPTER 3. SHORTEST PATH IN PERCOLATION AND

SCHRAMM-LOEWNER EVOLUTION

3.1 Introduction

Percolation was first introduced by Flory to describe the gelation of polymers

[78] and later studied in the context of physics by Broadbent and Hammersley

[79]. This model is considered the paradigm of connectivity and has been

extensively applied in several different contexts, such as, conductor-insulator

or superconductor-conductor transitions, flow through porous media, sol-

gel transitions, random resistor network, epidemic spreading, and resilience

of network-like structures [80–87]. In the lattice version, lattice elements

(either sites or bonds) are occupied with probability p, and a continuous

phase transition is observed at a critical probability pc, where for p < pc, as

the correlation function decays exponentially, all clusters are of exponentially

small size, and for p > pc there is a spanning cluster. At pc, the spanning

cluster is fractal [88]. In this Chapter we focus on the shortest path, defined as

the minimum number of lattice elements which belong to the spanning cluster

and connect two opposite borders of the lattice [17, 89], see Fig. 3.1. The

shortest path is related with the geometry of the spanning cluster [23,89–92].

Thus, studies of the shortest path resonate in several different fields. For

example, the shortest path is used in models of hopping conductivity to

compute the decay exponent for superlocalization in fractal objects [93, 94].

It is also considered in the study of flow through porous media to estimate

the breakthrough time in oil recovery [95] and to compute the hydraulic

path of flows through rock fractures [96]. The shortest path has even been

analyzed in cold atoms experiments to study the breakdown of superfluidity

[97]. However, despite its relevance, the fractal dimension of the shortest

path is among the few critical exponents in two-dimensional percolation that

are not known exactly [20,98].

Let us consider critical site percolation on the triangular lattice, in a two-

dimensional strip geometry of width Lx and height Ly (Ly > Lx), in units of

lattice sites, see Fig. 3.1. Each site is occupied with probability p = pc. The

largest cluster spans the lattice with non-zero probability, and the average

shortest path length 〈l〉, defined as the number of sites in the path, scales

as 〈l〉 ∼ Ldminy , where dmin is the shortest path fractal dimension and its best

36

3.1. INTRODUCTION 37

Fig. 3.1: A spanning cluster on the triangular lattice in a strip of vertical

size Ly = 512. The shortest path is in red and all the other sites belonging

to the spanning cluster are in blue.

estimation is dmin = 1.13077(2) [20, 99]. There have been several attempts

to compute exactly this fractal dimension [18, 19, 100–104]. Most tentatives

were based on scaling relations, conformal invariance, and Coulomb gas the-

ory. But the existing conjectures have all been ruled out by precise numerical

calculations. For example, Ziff computed the critical exponent g1 of the scal-

ing function of the pair-connectiveness function in percolation using confor-

mal invariance arguments [105]. g1 has been conjectured to be related to the

fractal dimension of the shortest path [103]. In turn Deng et al. conjectured

a relation between dmin and the Coulomb gas coupling for the random-cluster

model [104]. Both conjectures were discarded by the latest numerical esti-

mates of dmin [20, 25]. Thence, as recognized by Schramm in his list of open

problems, a solid theory for the shortest path is still considered one of the

major unresolved questions in percolation [106]. Impressive progress has re-

cently been made in the field of critical lattice models using SLE theory, see

Section 2.8. Therefore, it is legitimate to ask if the SLE techniques can help

solving the long standing problem of the fractal dimension of the shortest

37



path.

Also, a possible description of the physical process through SLE gives in-

teresting insights in new ways of generating the shortest path curves. Once

SLEκ is established, the value of κ suffices to generate, from only a Brow-

nian motion, curves having the same statistical properties as the shortest

path [56,107,108]. This can be very useful in the case of problems involving

optimization processes like the shortest path, watersheds [30], or spin glass

problems [57,72,73], as traditional algorithms imply the exploration of large

areas. Using the relation with SLE, even if one does not compute the curve

resulting from the optimization process for each given configuration, which

is deterministic, one is able to generate random curves, that are statistically

equivalent to the original ones, by just applying iteratively the inverse of the

uniformizing map g−1t . This leads to new ways of studying numerically and

analytically the statistical properties of complex problems.

In this article, we will show that the numerical results are consistent with

SLE predictions with κ = 1.04± 0.02. SLEκ curves have a fractal dimension

df related to κ by df = min(2, 1 + κ

8

)[52], see Section 2.7.1. From the

estimate of the fractal dimension of the shortest path, one deduces the value

of the diffusion coefficient κ corresponding to an SLE curve of same fractal

dimension; κfract = 1.0462 ± 0.0002. In what follows, we compute three

different estimates of κ using different analyses and compare them to κfract.

In particular we consider the variance of the winding angle [29,53,54], the left

passage probability [49], and the statistics of the driving function [32,57]. All

estimates are in agreement with the one predicted from the fractal dimension,

and therefore constitute a strong numerical evidence for the possibility of an

SLE description of the shortest path.

3.2 Method

We generate random site percolation configurations on a rectangular lattice

Lx×Ly with triangular mesh, where Lx and Ly are respectively the horizontal

and vertical lattice sizes, in units of lattice sites. The sites of the lattice are

38

3.3. WINDING ANGLE METHOD 39

occupied randomly with the critical probability pc = 12. If the configuration

percolates, we obtain the spanning cluster and identify the shortest path

between the top and bottom layers using a burning method [17, 82, 91]. In

short, we burn the spanning cluster from the bottom sites, indexing the sites

by the first time they have been reached, and stop the burning when we reach

for the first time the top line. We then start a second burning from the sites

on the top line that have been reached by the first burning, burning only

sites with lower index. With this procedure, we identify all shortest paths

from the bottom line to the top one. We randomly choose one of these paths,

by choosing randomly one of the starting points of the shortest path on the

bottom line, and then at each brancing of the path, choosing randomly one of

the branches. The results presented in the paper are for Ly ranging from 16

to 16384 and an aspect ratio of Lx/Ly = 1/2. We generated 10000 samples

and discarded the paths touching the vertical borders.

3.3 Winding angle method

The first result related to SLE deals with the winding angle. For each shortest

path curve we compute the variance of the winding angle, see Section 2.7.2,

and test its Gaussian statistics of variance given by Eq. (2.25)

〈θ2〉 − 〈θ〉2 = b+κ

4ln(Ly),

where b is a constant and Ly is the vertical lattice size [54]. Figure 3.2

shows the dependance of the winding angle of the shortest path on Ly in

a lin-log plot. In the inset, one shows that the distribution of the winding

angle is a Gaussian with a variance consistent with Eq. (2.25). The estimate

κwinding = 1.046 ± 0.004 that we get from fitting the data with Eq. (2.25) is

in agreement with the value deduced from the fractal dimension.

3.4 Left passage probability method

In the following, we work with chordal SLE. But the curves that we generate

with the algorithm described above are defined in a stripe starting at the

39



0.5

1.0

1.5

2.0

101

102

103

104

<θ

2>

Ly

κ=1.046(4)

0.0

0.1

0.2

-8 -4 0 4 8

P( θ

)

θ

Fig. 3.2: Variance of the winding angle against the lattice size Ly. The

analysis has been done for Ly ranging from 16 to 16384. The statistics are

computed over 104 samples. The error bars are smaller than the symbol size.

By fitting the results with Eq. (2.25), one gets κwinding = 1.046 ± 0.004. In

the inset, the probability distribution of the winding angle along the curve is

compared to the predicted Gaussian distribution, drawn in green, of varianceκ4

ln(Ly) with κ = 1.046 and Ly = 16384.

bottom boundary and ending at the upper one, and generated with “free”

boundary conditions, i.e. without any constraints on the boundaries, such

that the shortest path has no fixed starting and ending points. Therefore we

have to map conformally the original curves into the upper half plane by an

inverse Schwarz-Christoffel transformation [50] as described in Section 2.6.

We numerically compute the left passage probability of the shortest path

curves for a given set of samples points S in the upper half-plane H and

compare the results to Schramm’s formula (2.26):

Pκ(φ) =1

2+

Γ (4/κ)√πΓ(

8−κ2κ

) cot(φ)2F1

(1

2,

4

κ,3

2,− cot(φ)2

).

We considered for convenience 400 points, regularly spaced in

[−0.1Lx, 0.1Lx] × [0.15Ly, 0.35Ly] which are then mapped through the

40

3.5. DIRECT SLE METHOD 41

inverse Schwarz-Christoffel mapping into H [50]. To estimate κ, we minimize

the weighted mean square deviation Q(κ) defined as

Q (κ) =1

|S|∑z∈S

[P (z)− Pκ(φ(z))]2

∆P (z)2,

where |S| is the cardinality of the set S, and ∆P (z)2 is defined as ∆P (z)2 =P (z)(1−P (z))

Ns−1, where Ns is the number of samples [77]. We define the error

bar ∆Q for the minimum of Q(κ) using the fourth moment of the binomial

distribution. The error ∆κ is defined such thatQ(κ±∆κ)−∆Q = Q(κ)+∆Q.

For a lattice size of Ly = 16384, the minimum of the mean square deviation

is observed for κLPP = 1.04 ± 0.02 as shown in Fig. 3.3. This value is in

agreement with the estimate of κ obtained from the fractal dimension and

the winding angle. As a remark, one might notice that the minimum of

Q deviates slightly from the expected value 1 but is of the same order of

magnitude. However one expects the SLE description of the shortest path to

be only valid in the scaling limit. The finite lattice spacing does not allow the

curve to go in all possible directions, but only in two distinct directions when

going from one site to another. Therefore, finite size effects might affect the

left-passage probability measurement and the value of the minimum of Q.

3.5 Direct SLE method

The winding angle and left passage analyses are indirect measurements of κ.

Therefore we also test the properties of the driving function directly in order

to see if it corresponds to a Brownian motion with the expected value of κ.

3.5.1 In chordal space

As for the left passage probability, we consider the chordal curves in the

upper half plane, starting at the origin and growing towards infinity. We

want to compute the driving function ξt underlying the process. For that,

we numerically solve the Loewner differential equation (2.10) using the slit

41



3.0

4.0

5.0

6.0

1 1.02 1.04 1.06 1.08

Q

κ0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

P(φ)

φ/π(a)

0

0.25

0.5

0.75

1

0 0.25 0.5 0.75 1

P(φ)

φ/π

-0.02

-0.01

0

0.01

0.02

0.25 0.5 0.75

P()-

P(

)

/

(b)

Fig. 3.3: Left passage probability test. (a) Weighted mean square deviation

Q(κ) as a function of κ, for Ly = 16384. The vertical blue line corresponds the

minimum ofQ(κ), and the green vertical line is a guide to the eye at κ = κfract.

The minimum of the mean square deviation is at κLPP = 1.038 ± 0.019.

The light blue area corresponds to the error bar on the value of κLPP . (b)

Computed left passage probability as a function of φ/π for R ∈ [0.70, 0.75]

and κ = 1.038. The blue line is a guide to the eye of Schramm’s formula

(2.26) for κ = 1.038.42


map equation (2.29)

gt(z) = ξt +

√(z − ξt)2 + 4δt.

In the case of SLEκ the extracted driving function gives a Brownian motion

of variance κ. The direct SLE test consists in verifying that the driving

function is a Brownian motion and compute its variance 〈ξ2t 〉−〈ξt〉2 to obtain

the value of κ. The variance should behave as 〈ξ2t 〉 − 〈ξt〉2 = κt.

Figure 3.4 shows the variance of the driving function as a function of the

Loewner time t. We observe a linear scaling of the variance with t. The

local slope κdSLE(t) is shown in the inset of Fig. 3.4a. In Fig. 3.4b, we plot

the mean correlation function c(τ) = 〈c(t, τ)〉t of the increments δξt of the

driving function, where the correlation function is defined as,

c(t, τ) =〈δξt+τδξt〉 − 〈δξt+τ 〉〈δξt〉√

(〈δξ2t+τ 〉 − 〈δξt+τ 〉2) (〈δξ2

t 〉 − 〈δξt〉2). (3.1)

One sees that the correlation function vanishes after a few time steps. The

initial decay is due to the finite lattice spacing, which introduces short range

correlations. But in the continuum limit, the process is Markovian, with a

correlation function dropping immediately to zero. In the inset of Fig. 3.4b,

we show the probability distribution of the driving function for different

t. This distribution is well fitted by a Gaussian, in agreement with the

hypothesis of a Brownian driving function. From this result and the estimates

of the diffusion coefficient computed for several lattice sizes, we obtain κ =

0.9± 0.2.

We note that the numerical results obtained with the direct SLE method

are less precise than with the other analyses and, therefore, characterized by

larger error bars, as explained in Section 2.7.4. The result we have obtained

for κ is in agreement with the ones obtained with the fractal dimension,

winding angle, and left-passage probability.

3.5.2 In dipolar space

We also extracted the driving function of the curves in dipolar space, i.e.

defining the curves as starting from the origin and growing in the strip. We

43



0

2

4

6

8

10

0.0 0.5 1.0

<ξ2

>-<

ξ>2

t

0.92

x10-3

x10-2

0.87

0.92

0.97

0.0 0.2 0.5

κd

SL

E(t

)

t

x10-2

(a)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.5 1.0 1.5 2.0

c(τ)

τ

x10-3

g(10**(-3)*x)f(x)

10-2

10-1

100

101

-0.4 -0.2 0.0 0.2 0.4

P(ξ

t)

ξt

1.2x10-3

3.7x10-3

9.95x10-3

(b)

Fig. 3.4: Driving function computed using the slit map algorithm. (a) Mean

square deviation of the driving function 〈ξ2t 〉 as a function of the Loewner

time t. The diffusion coefficient κ is given by the slope of the curve. In the

inset we see the local slope κdSLE(t). The thick green line is a guide to the

eye corresponding to κdSLE = 0.92. (b) Plot of the correlation c(t, τ) given

by Eq. (3.1), and averaged over 50 time steps. The averaged value is denoted

c(τ). In the inset are shown the probability distributions of the driving

function for three different Loewner times t1 = 1.2 × 10−3, t2 = 3.7 × 10−3

and t3 = 9.95 × 10−3. The solid lines are guides to the eye of the form

P (ξt) = 1√2πκti

exp(− ξ2t

2κti

), for i = 1, 2, 3.

44


also obtained a value of κ consistent with the fractal dimension.

Loewner’s Equation in dipolar space. Let us consider the case of a

dipolar curve growing in the strip S of height Ly, that starts at the origin

and stops the first time it hits the upper boundary. We again study the case

of simple curves. As described in Chapter 2, one expects the properties to

be the same in the strip S as in the upper half-plane H, but formulas and

especially the Loewner differential Equation (2.10) might change. Therefore

we will first develop the same formalism as in the case of chordal SLE but for

dipolar SLE leading to the Loewner differential Equation in dipolar space.

By the Riemann mapping theorem, there exist conformal maps from the strip

S minus the curve γ[0, t] into the strip S such that gt(∞) =∞ and gt(−∞) =

−∞. The map gt is then defined up to a translation by a real constant. It is

made unique by choosing the normalization limz→∞ gt(z) + gt(−z) = 0. One

parametrizes the curves such that limz→∞ gt(z)− z = t, where t is called the

Loewner time [109,110]. Dipolar SLE is defined as the collection of conformal

maps gt satisfying the following stochastic differential equation

∂gt(z)

∂t=

π/Lytanh (π (gt(z)− ξt) /2Ly)

, and g0(z) = z, (3.2)

where ξt =√κBt and Bt is a one dimensional Brownian motion starting at

the origin [109,111].

Slit map algorithm in dipolar space. Using the theory of dipolar SLE,

one can develop a numerical method to compute the driving function of

dipolar curves, as has been done in the case of chordal curves. Therefore one

has to solve Eq. (3.2). As in the chordal case, one supposes that the driving

function is constant over small time intervals δt. Under this assumption, one

can solve Eq. (3.2) and obtain the so-called slit map Equation in dipolar

space Eq. (3.3). The slit map algorithm becomes the following. Consider

a dipolar curve defined by the initial set of points {z00 , ..., z

0N}. One maps

recursively the sequence of points {zi−1i , ..., zi−1

N } of the mapped curve to the

45



0

250

500

750

1000

0 250 500 750 1000

<ξ2

>-<

ξ>2

t

0.9

Fig. 3.5: Driving function computed using the dipolar slit map given by

Eq. (3.3), for Ly = 16384. The mean square deviation of the driving function

〈ξ2t 〉 − 〈ξt〉2 is plotted as a function of the Loewner time t. The diffusion

coefficient κ is given by the slope of the curve.

shortened sequence {zii+1, ..., ziN} by the conformal map

gti(z) = ξti + 2Lyπ

cosh−1

(cosh (π(z − ξti)/2Ly)

cos(∆i)

), (3.3)

where ξti = Re{zi−1i } and δti = ti − ti−1 = −2(Ly/π)2 ln (cos(∆i)), with

∆i = πIm{zi−1i }/2Ly and Re{} and Im{} being respectively the real and

imaginary parts [57]. If the curve follows SLEκ statistics, then the driving

function is a one dimensional Brownian motion of variance 〈ξ2t 〉 − 〈ξt〉2 = κt.

In Fig. (3.5) we show the mean variance, averaged over different curves,

of the driving function of dipolar shortest path curves as a function of the

Loewner time. One obtains a value of κ in agreement with the value obtained

in the chordal case κ = 0.9± 0.2.

46

3.6. FINAL REMARKS 47

3.6 Final Remarks

All tests are consistent with SLE predictions. The numerical results ob-

tained with the winding angle, left-passage, and direct SLE analyses are in

agreement with the latest value of the fractal dimension. Being SLE implies

that the shortest path fulfills two properties: conformal invariance and do-

main Markov property (DMP), see Section 2.5. Thus, the agreement with

SLE predictions lends strong arguments in favor of conformal invariance and

DMP of the shortest path.

The DMP is related to the evolution of the curve in the domain of definition.

Let us consider the shortest path γ defined in a domain D, starting in a and

ending in b. We take a point c on the shortest path different from a and b.

Then if the DMP holds, one would have that

PD (γ[a, b]|γ[a, c]) = PD\γ[a,c] (γ[c, b]) , (3.4)

where γ[c, b] is the shortest path starting in c and ending in b in the domain

D except the curve γ[a, c], denoted as D \ γ[a, c], and PD and PD\γ[a,c] are the

probabilities in the domains D and D \ γ[a, c] respectively. One can classify

the models as the ones for which DMP holds already on the lattice, and the

ones for which it holds only in the scaling limit. Many classical models, like

the percolation hulls, the LERW, or the Ising model [42] for example, belong

to the first case. But some two-dimensional spin glass models with quenched

disorder [57, 73] are believed to only fulfill DMP in the scaling limit. Our

numerical results suggest that, for the shortest path, DMP holds at least in

the scaling limit. Further studies should be done to test the validity of DMP

on finite lattices.

The second result we can expect if SLE is established for the shortest path

is conformal invariance. Conformal invariance, being a powerful tool to com-

pute critical exponents, is of interest for the study of the shortest path.

Conformal invariance, associated to Coulomb gas theory for example, could

be useful to develop a field theoretical approach of the shortest path. There

is no proof of conformal invariance of the shortest path, but our numerical

results give strong support to this hypothesis. For example, the expression of

47



the winding angle is based on conformal invariance and agrees with the pre-

dictions based on the fractal dimension. Also the left passage probabilities

and the direct SLE measurements have been performed on curves conformally

mapped to the upper half plane and gave consistent results. In addition, we

obtained the same estimate of κ by extracting the driving function in chordal

and dipolar space. However, even if the scaling limit would not be confor-

mally invariant, our results suggest that one could still apply SLE techniques

to the study of this problem, as some SLE techniques have also been used to

study off-critical and especially non conformal problems [111–115].

Analyzing the shortest path in terms of an SLE process would give a deeper

understanding of probability distributions of the shortest path, allowing to

compute more quantities, like for example the hitting probability distribution

of the shortest path on the upper boundary segment [109].

48

Chapter 4

Long-range correlated

landscapes and correlated

percolation

In the previous chapter, we have studied the shortest path in usual perco-

lation, where each site is occupied randomly with a given probability pc,

independently of the state, empty or occupied, of the other sites. In the fol-

lowing we will be interested in the properties of correlated percolation, where

the state of each cell is not independent from the state of the others. Actually

correlated percolation can be related to long-range correlated landscapes in

the framework of ranked surfaces introduced recently in the work of Schrenk

et al. [116].

Since the seminal paper of Mandelbrot [117], real and artificial landscapes

have gained much interest [12,118]. Physicists have been especially interested

in the self-affine properties of random surfaces, in the fractal dimension of

their isoheight lines, and their critical properties. Landscapes can describe

several properties such as topography [119], energy [120], temperature [33]

or vorticity fields [32]. They have been employed in the study of water

basins [30, 121–125], electronic transport [120], deposition of particles on

layers [37,126,127] and turbulence [32,33]. Landscapes are characterized by

the so-called Hurst exponent. The features of the physical system are then

intimately connected to the properties of the landscape. In the following, we

49

50CHAPTER 4. LONG-RANGE CORRELATED LANDSCAPES AND

CORRELATED PERCOLATION

will define correlated percolation in the framework of ranked surfaces, study

the characterization of these surfaces and present a method to generate long-

range correlated surfaces.

4.1 Correlated percolation

To study correlated percolation on a lattice, we work with a landscape of ran-

dom heights h, where h(x) is the height of the landscape at position x in the

lattice [14,22,27,118,128,129]. One then constructs the ranked surface [116]

associated to this discrete landscape. It is constructed as follows: one first

ranks all sites in the landscape according to their height, from the smallest

to the largest value. Then the ranked surface is constructed by assigning

to each site a number corresponding to its position in the rank. One then

floods the landscape starting from the lowest to the heighest rank. A frac-

tion p of sites is flooded. If one considers the height hc of the last occupied

site, then the problem is equivalent to the one where one floods the land-

scape till height hc. As developed by Ziman and Weinrib [120, 130], there is

a simple mapping between correlated percolation and random surfaces with

−1 ≤ H < 0, where H is the Hurst exponent related to the height-height

correlation function through

c(r) = 〈h(x)h(x + r)〉x ∼ |r|2H . (4.1)

By identifying for example the sites above hc as occupied and the sites be-

low hc as empty, one can map this problem to a percolation problem with

occupation probability 1 − p. For example, if the heights are distributed

uniformly at random, classical percolation with a fraction of occupied sites

1−p is obtained [2,3]. In the case of long-range correlated percolation, there

exists a critical height hc or critical occupation probability pc, at which a

global connectivity appears in the system [99].

In the context of percolation, one will mostly speak about boundary of perco-

lation clusters, focusing on the occupation probability, whereas in the context

of SLE, one mostly speaks of isoheight lines at a critical height hc, but the

two problems are related, see Fig. 4.1.

50

4.1. CORRELATED PERCOLATION 51

Fig. 4.1: A random landscape and its percolation clusters. In blue are the

empty sites and in brown the occupied ones. The sites in green belong to the

percolating cluster. The paths in red represent the left and right accessible

perimeters and the paths in yellow the one site wide fjords.

51



4.2 The Hurst exponent of long-range corre-

lated landscapes

Let us consider random surfaces, i.e. a function that returns to each grid

point (x, y) in a lattice the corresponding random height h(x, y) = h(x).

They are parametrized by the so-called Hurst exponent H, and their statis-

tical properties depend on H.

The most general way to characterize the Hurst exponent of a given surface

is through its power-spectrum that is related to the Fourier transform of the

surface. It is defined as,

S(q) =1

(2π)2 |〈h(r)e−iq·r〉r|2 (4.2)

and is expected to scale as

S(q) ∼ |q|−2(H+1), (4.3)

where H is the Hurst exponent. In the following, we will be interested in

the cases H ∈ [−1, 0) and H ∈ (0, 1]. By the Wiener-Khintchine theorem

[131,132], this definition translates into scaling relations for the height-height

correlation function [14].

One usually considers the scaling of the height-height correlation function

c(r) = 〈h(x)h(x + r)〉x ∼ a(H) + b(H)|r|2H . (4.4)

For −1 < H < 0, one considers the decrease of the height-height correlation

function for large values of |r|, with a(H) = 0 and b(H) > 0 in Eq. (4.4),

see Eq. (4.1). Weinrib proposed to use these surfaces to develop the so-

called long-range correlated percolation framework, where the formalism of

percolation is applied [118,128].

In the case 0 < H < 1, one considers the decrease of the height-height

correlation function for small values of |r|, with a(H) > 0 and b(H) < 0

in Eq. (4.4). These surfaces with 0 < H < 1 are sometimes called rough

52

4.3. THE FOURIER FILTERING METHOD FOR CORRELATEDLANDSCAPES 53

surfaces, andH is called the roughness exponent. In particular, these surfaces

are self-affine [133] and their height profiles are scale invariant [134]

h(εx) ∼ εHh(x). (4.5)

Surfaces with values of H next to one are “smoother” than surfaces with

values of H next to zero, which look more rough1.

Rough self-affine surfaces [136], i.e. surfaces with Hurst exponent H ∈ (0, 1],

have been the object of many theoretical [12, 27] and experimental studies,

and have triggered the development of many models [126,127,137]. Most of

the real landscapes one encounters in nature have postive Hurst exponent.

Random surfaces with negative Hurst exponent, H ∈ [−1, 0), have been

studied in relation to correlated percolation [22,128,129,138,139].

4.3 The Fourier Filtering Method for corre-

lated landscapes

The simplest landscapes one can think of are uncorrelated landscapes, where

the height of each site is chosen uniformly at random. But real landscapes

are correlated and the heights typically are long-range correlated. There-

fore specific methods were developed to generate rough surfaces exhibiting

correlations. For example, growth models have been developed, like con-

tinuous growth models in the cases of the Edward-Wilkinson [126] and the

Kardar-Parisi-Zhang [127] models, and discrete growth models in the cases

of ballistic deposition [140] and Solid-On-Solid models [141]. Here we will

present one method to generate random surfaces, called the Fourier Filtering

Method, and present its relation to Gaussian random fields.

1In many real surfaces, there exists a correlation length ξ and the surface will

exhibit self-affinity only on a length scale smaller than the correlation length. In

this case one might write the height-height correlation function in the form H(r) =

w2

[1− exp

(−(|r|ξ

)2H)], which corresponds to an exponential decrease of the auto-

correlation function. Also the scaling law is expected to be valid for |q| >> ξ−1 [135].

53



Here, we are interested in the case where the heights h have long-range power-

law spatial correlations as described in Eq. (4.4). This kind of correlated

disorder can be generated using the Fourier filtering method (Ffm) [22, 123,

132, 142–149]. The Ffm is based on the Wiener-Khintchine theorem (WKt)

[131, 132] that states that the auto-correlation of a time serie equals the

Fourier transform of its power spectrum, i.e. of the absolute squares of

the Fourier coefficients. This fact is exploited in two dimensions by the Ffm.

Starting with Eq. (4.2), we impose the following power-law form of the power

spectrum S(q) of the disorder:

S(q) ∼ |q|−βc =(√

q2x + q2

y

)−βc, (4.6)

where βc is related to the Hurst exponent H via βc = 2(H + 1), see Eq. (4.3).

By the WKt, one recovers for −1 ≤ H < 0 the power-law correlation function

c(r) of the heights h,

c(r) = 〈h(x)h(x + r)〉x ∼ |r|2H , (4.7)

In simulations, for a desired value of H, one starts in the Fourier space

q = (qx, qy) of the original space x = (x, y). For each site (qx, qy) one

generates a random Fourier coefficient as the product of a complex Gaussian

random variable u(q) of mean zero and variance one and the square root

of the power spectrum√S(q). The random variables u(q) are independant

but satisfy the condition u(−q) = u(q) in order to get a real valued surface

when one goes back into the real space. One then apply an inverse fast

Fourier transform to obtain the correlated random heights h(x) in the real

space [22,123,132,142–145,147–149]

h (x) = F(S(q)1/2u(q)

)(4.8)

where F denotes the inverse Fourier transform.

In Fig. 4.2 we show the influence of the Hurst exponent H on the surfaces.

For increasing values of H, the correlations lead to the appearance of larger

“valleys” and “hills”. In Fig. 4.3 we show the power spectrum of the surface

54

4.3. THE FOURIER FILTERING METHOD FOR CORRELATEDLANDSCAPES 55

(a) (b)

(c) (d)

Fig. 4.2: Random landscapes generated with the Fourier filtering method

for different values of the Hurst exponent: (a) H = −1, (b) H = −0.5, (c)

H = 0, and (d) H = 0.5.

55



jqj100 101 102 103

Sjq

j

10-4

10-3

10-2

10-1

100

(a)

qx

1 256 512

q y

1

256

512

-3

-2

-1

0

1

2

3

(b)

Fig. 4.3: Properites of the Fourier transform of one random landscape of

Hurst exponent H = −0.5 generated with the Fourier Filtering Method.

(a) Power spectrum Sq = |h(q)|2 as a function of |q|, where h is the Fourier

transform of the surface h. The red line is a guide to the eye of the form |q|−βc

with βc = 2(H + 1). (b) Phase φ = arg(h(q)) as a function of q = (qx, qy).

56

4.4. CORRELATED SURFACES, FOURIER FILTERING METHODAND FRACTIONAL GAUSSIAN FIELDS 57

generated with the Fourier filtering method and the phase of its Fourier

components. The phases are uniformly random in (−π, π].

In the following, we will be especially interested in the range −1 ≤ H ≤ 0,

that is related to correlated percolation [22,118,128,129]. The case H = −1

corresponding to βc = 0 is such that the power spectrum in Eq. (4.6) is in-

dependent on the frequency, and the landscape profile is white noise. Since

H ≤ 0, as H increases towards zero, the correlation function decays more

slowly. For H = 0 one recovers the Gaussian Free Field (GFF) [150].

The discrete Gaussian Free Field Γ is a centered Gaussian random vector,

whose covariance matrix is the Green’s function of the random walk, i.e.

E(Γ(x)Γ(y)) = GD(x, y), where GD(x, y) is the Green’s function of the ran-

dom walk in the domain D [151]. It is defined as the mean number of times

a random walker (Xn)n≥0 starting at x spends at y before it exits the domain

D, i.e. GD(x, y) = Ex(∑τ−1

n=0 1Xn=y), where τ is the first exit time of D of the

random walker (Xn)n≥0, the index x in Ex indicating that the random walker

starts at x. Actually, GD(x, y) is symmetric in x and y, and is an admissible

covariance matrix, see [151]. It is also the Green’s function of the Laplacian

operator, see [151], and Section 4.4.

4.4 Correlated surfaces, Fourier Filtering

Method and fractional Gaussian fields

In this section we draw a parallel between random surfaces generated with the

Fourier filtering method and the mathematical theory of fractional Gaussian

fields that could be of interest to tackle long-range correlated percolation

theory from an analytical point of view. Here we do not enter in mathematical

details, but for rigorous proofs, the reader can refer to Refs. [39,150].

One can generalize the results obtained for the Gaussian Free Field, asso-

ciated to a Laplacian operator [150], to the so-called Fractional Gaussian

Fields (FGF) [39], associated to fractional Laplacians. The fractional Lapla-

cian is parametrized by a real valued parameter s, that plays a similar role

57



to H in the case we studied before. For long-range correlated percolation,

we are interested in the range s ∈ (0, 1), see below. The fractional Lapla-

cian for s ∈ (0, 1) is defined such that the eigenvectors of the fractional

Laplacian are the eigenvectors of the Laplacian with Dirichlet boundary con-

ditions. If (λn, fn) are the eigenvalues and eigenfunctions of the Laplace

operator with Dirichlet boundary conditions, then (λs/2n , fn) are the eigenval-

ues and eigenfunctions of the fractional Laplacian operator and the functions

(λ−s/2n fn) form an othonormal basis associated to an inner product related to

the fractional Laplacian (see [39] and especially section 9 on the eigenfunc-

tion FGF). But on the unit torus, see [150] paragraph 2.5, the eigenfunctions

of the Dirichlet Laplace operator are x 7→ e2πix.k with eigenvalues (2π|k|)2

and k ∈ Z2 \ {0}. Therefore using H = s− 1, one obtains that the functions

x 7→ 1(2π|k|)s e

2πix.k are an othornomal basis for the inner product associated

to the fractional Laplacian. By proposition 2.1 point 2 of [150], multiplying

each of these components by a Gaussian random variable uk, one obtains that

h(x) =∑

k uk(2π|k|)−se2πix.k is a Gaussian random variable with probability

measure µV = e−(v,v)∇sZ−1dv, where (. , . )∇s is the inner product associated

to the fractional Laplacian and Z is a normalizing constant. For a compar-

ison between the case of the GFF and the FGF, see [39] paragraph 12.2.

Using the relation s = βc/2 in the case of a lattice with periodic boundary

conditions one recovers the random function defined in the Fourier filtering

method, up to a multiplicative constant (2π)s−1/2.

One might also approach the problem in real space. We define in R2 the

function valued fractional gradient

∇sf(x) =

(y 7→ f(x+ y)− f(x)

|y|1+s

), (4.9)

and the real valued fractional Laplacian

(−∆)sf(x) =

∫R2

f(x+ y)− 2f(x) + f(x− y)

|y|2+2sdy, (4.10)

see [39] paragraph 12. These formulas are defined up to normalization con-

stants, see Proposition 2.2 in [39]. Actually, the easiest way to defined the

58


fractional Laplacian is by generalizing the usual derivation expressed using

the Fourier Transform. If (−∆)sg = h, then the Fourier transforms f , g

satisfy f(k) = |k|2sg(k).

One defines the inner product (. , . )∇s as

(f, g)∇s =

∫R2

(∇sf(x),∇sg(x)) dx (4.11)

and shows that it is equal to∫R2

(∇sf(x),∇sg(x)) dx =

∫R2

((−∆)sf(x))g(x)dx,

(f, g)∇s = ((−∆)sf, g),

(4.12)

see [39] proposition 12.1, where (·, ·) denotes the usual L2 inner product in

R2, i.e. (f, g) =∫R2 f(x)g(x)dx.

The covariance follows the expected scaling relation, see Eq. (4.7), i.e. scales

as |r|2H , cf. Theorem 3.3 in [39].

From this we see that the main difference between the cases H = 0 and

H ∈ (−1, 0), is that the random landscapes that we generate with the Fourier

filtering method are not related to a local operator anymore, the Laplacian,

but to a long-range operator, the fractional Laplacian operator (−∆)s.

This formalism is especially usefull if one wants to simulate long-range corre-

lated random fields in a domain D with fixed boundary conditions f |∂D. In

order to do so, one generates a Gaussian random field h(x) using the Fourier

filtering method. Then one computes the deterministic correction field h(x)

as the solution of:

(−∆)βc/2h = 0,

and h|∂D = f |∂D − h|∂D.(4.13)

Then the long-range correlated landscape hfinal in D with fixed boundary

conditions f |∂D is defined as:

hfinal(x) = h(x) + h(x), (4.14)

for x ∈ D.

59



Intuitively, if one computes (hfinal, hfinal)∇s with s = βc/2, by Eq. (4.12)

one gets that (hfinal, hfinal)∇s = (h, h)∇s + 2(h, h)∇s + (h, h)∇s = (h, h)∇s +

2((−∆)sh, h) + ((−∆)sh, h) = (h, h)∇s . Therefore the inner product, and as

a consequence the probability distribution, is not changed. For more details,

see [35] and Section 5 of [39].

Using this method, one generates long-range correlated Gaussian fields with

fixed boundary conditions and the desired probability density, see Fig. 4.4.

Fixing the boundary conditions might play a role in SLE studies, see for

example Ref. [35].

60


3530

2520

x15

105

00

5

10

15

20

y

25

30

0

10

20

30

40

35

z

(a)

3530

2520

x15

105

00

5

10

15

20

y

25

30

20

0

-20

-40

-60

-8035

z

(b)

Fig. 4.4: Generating long range correlated surfaces with fixed

boundary conditions. Square surface of lattice size L = 32 and Hurst

exponent H = −0.1. The boundary conditions are L on one half of the

boundary and −L on the other half. (a) Random surface h(x) generated

with free boundary counditions. (b) Correction field h satisfying the bound-

ary value problem Eq. (4.13).

61



3530

2520

x15

105

00

5

10

15

20

y

25

30

-40

-20

0

20

40

35

z

(c)

Fig. 4.4: Generating long range correlated surfaces with fixed

boundary conditions. (continued) Square surface of lattice size L = 32

and Hurst exponent H = −0.1. The boundary conditions are L on half of the

boundary and −L on the other half. (c) Summing the two fields one obtains

the random surface hfinal(x) = h(x) + h(x) with the desired fixed boundary

conditions.

62

Chapter 5

Long-range correlated

percolation

As we have seen in Chapter 4, and especially in Fig. 4.2, the long-range corre-

lations change the topography of the landscapes and therefore the percolation

properties associated to these surfaces.

In this Chapter, we investigate quantitatively, using Monte-Carlo simula-

tions, the influence of the power-law correlation of the site occupation proba-

bilities on the long-range correlated percolation problem. We compute several

critical exponents as function of the Hurst exponent H which characterizes

the spatial correlations among the occupation of sites as described in Sec-

tion 4.2. In particular, we study the fractal dimension of the largest cluster

and of its complete and accessible perimeters. We also discuss the cluster

size distribution at criticality. Concerning the inner structure and transport

properties of the largest cluster, we analyze its shortest path, backbone, and

cluster conductivity. We propose expressions for the functional dependence

of the critical exponents associated to these quantities on the Hurst exponent

H.

This chapter is based on Ref. [99]:

K. J. Schrenk, N. Pose, J. J. Kranz, L. V. M. van Kessenich, N. A. M.

Araujo and H. J. Herrmann, Percolation with long-range correlated disorder,

Physical Review E 88, 052102 (2013).

63

64 CHAPTER 5. LONG-RANGE CORRELATED PERCOLATION

5.1 Introduction

Percolation theory and related models have been applied to study trans-

port and geometrical properties of disordered systems [118,152] as described

in Section 3.1. Frequently the disorder in the system under study exhibits

power-law long-range spatial correlations. This fact has motivated some

studies of percolation models where the sites of the lattice are not occupied

independently, but with some correlation between the occupation probabili-

ties, leading to correlated percolation as described in Section 5.2. As one can

guess from Fig. 4.2 and from previous work [22,118,128,129,139,146,152–159],

in the presence of long-range correlations, percolation clusters become more

compact and their transport properties change accordingly.

The critical exponents for uncorrelated percolation in two dimensions are

known rigorously for the triangular lattice [63]. In addition, at the critical

point, the correlation-length diverges and universality holds, i.e., critical ex-

ponents and amplitude ratios do not depend on short-range details, such as

lattice specifics [82, 83, 88, 118, 152]. This statement has been made precise

by renormalization group theory, which predicts that the scaling functions

within a universality class are the same, while the lattice structure only in-

fluences the non-universal metric factors [160, 161]. If, by contrast, infinite-

range, power-law correlations are present, according to the extended Harris

criterion, the critical exponents can change, depending on how the correla-

tions decay with spatial distance [128,129,146,162,163].

In this Chapter, we investigate a two-dimensional percolation model where

the sites of a lattice are occupied based on power-law correlated disorder, see

Section 4.1, parametrized by the Hurst exponent H and generated with the

Fourier filtering method, see Section 4.3. We find that the fractal dimension

of the largest cluster, its complete and accessible perimeters, as well as the

dimension of its shortest path depend on H. Strong dependence on H is

also found for the fractal dimension of the backbone and the electrical con-

ductivity exponent. For two-dimensional critical phenomena, conformal field

theory has been used to obtain exact values of critical exponents in the form

of simple rational numbers [164–166]. Therefore, we make proposals for the

64

5.2. CORRELATED PERCOLATION AND THE EXTENDED HARRISCRITERION 65

functional dependence of all measured exponents on the Hurst exponent H,

as being the simplest rational expressions that fit the numerical data.

In Sec. 5.2, we discuss the influence of the long-range correlations on the

percolation problem in terms of the extended Harris criterion. In Sec. 5.3,

we study the percolation threshold of the used lattice. In Sec. 5.4 we measure

the fractal dimension of the largest cluster and the cluster size distribution at

the percolation threshold. In Sec. 5.5 the complete and accessible perimeters

of the largest cluster are investigated. Section 5.6 discusses the shortest path

of the largest cluster, its backbone and cluster conductivity at the percolation

threshold. Finally, in Sec. 5.7, we present some concluding remarks.

5.2 Correlated percolation and the extended

Harris criterion

In correlated percolation, the usual uncorrelated percolation model is mod-

ified by long-range correlations of the occupation probabilities. One might

wonder if the correlations are strong enough to change the critical behav-

ior of the system. For short-range correlated systems, the Harris criterion

states that short-range correlations are relevant if dν − 2 < 0 [167]. As this

is always positive for percolation, short-range correlations have no influence

on the critical behavior of percolation. But this is not always the case for

long-range power-law correlations, for which the Harris criterion has been

extended. In a system of lateral size L, where the control parameter is the

height h of the landscape, for −1 ≤ H < 0, the height-height correlation

function looks like c(r) ∼ |r|2H , as described in Section 4.1 discussing the re-

lation between correlated landscapes and correlated percolation. Therefore

if one considers a lattice of size L at criticality, the correlation length ξ scales

as ξ ∼ L. Therefore if one considers the variance of the mean value of the

heights ∆2 = 〈[1/ξ2

∫ξ×ξ dxh(x)

]2

〉, where the average is over the disorder

65


realisations, in a square of lateral size ξ, then it scales as:

〈[1/ξ2

∫ξ×ξ

dxh(x)

]2

〉 = 1/ξ4

∫(ξ×ξ)2

dxdy〈h(x)h(y)〉,

∆2 ∼ 1/ξ2

∫ ξ

0

c(r)rdr,

∆2 ∼ ξ2H ,

∆2 ∼ L2H .

(5.1)

But by definition of the correlation length critical exponent L ∼ ξ ∼|h∗ − hc|−νuncorr , where h∗ is the effective critical height, hc is the infinite

size critical height, and νuncorr the correlation-length critical exponent in 2D

in the uncorrelated case. The effect of the long-range correlations is neg-

ligeable in the thermodynamic limit if the local fluctuations of the control

parameter due to the long-range correlations are small compared to the dis-

tance to the critical point. Therefore the effect of the long-range correlations

is negligeable in the thermodynamic limit if ∆2/|h∗−hc|2 ∼ L2H+2/νuncorr → 0

when L→∞, leading to H < −1/νuncorr.

Therefore, the extended Harris criterion, as formulated in Refs. [128, 129,

146, 162, 163], states that for the range −d/2 < H < 0 the correlations

do not affect the critical exponents of the percolation transition if H ≤−1/νuncorr, where νuncorr is the correlation-length critical exponent and for

d = 2, νuncorr2D = 4/3 [63,82]. Whereas for −1/νuncorr

2D < H < 0 the critical ex-

ponents are expected to depend on the value of H. The quantitative depen-

dence of the critical exponents on H, in this regime, is not yet entirely clear.

Concerning the correlation-length critical exponent for the correlated case

νH , the analytical works in Refs. [128, 129, 162] predict that νH = −1/H.

In the case of Weinrib and Halperin [128, 162] this is a conjecture based

on renormalization group calculations; Schmittbuhl et al. [129] found the

same result by analyzing hierarchical networks. Therefore, in both analyti-

cal approaches, it is not certain that νH actually behaves as conjectured and

there is some controversy regarding this question, as discussed, e.g., in the

field-theoretical work of Prudnikov et al. [168, 169]. For correlated percola-

tion, the relation νH = −1/H has been supported by the numerical work in

66

5.3. PERCOLATION THRESHOLD 67

Refs. [146,170,171]. Agreement has also been reported by Prakash et al. [22],

however only approximately for the range −1/νuncorr2D ≤ H ≤ −0.5. Finally,

for H > 0 there is no percolation transition [129, 172]. In the following, we

consider values of the Hurst exponent in the range −1 ≤ H ≤ 0.

5.3 Percolation threshold

We consider the correlated percolation model defined in Section 4.1 on tri-

angular lattice stripes of vertical length Ly = L and aspect ratio A = Lx/Ly,

consisting of N = AL2 sites. To investigate critical correlated percolation,

one first needs to identify the percolation threshold pc, which is pc = 1/2 for

site percolation on the triangular lattice [82]. To compute the percolation

threshold of a percolation problem on a given lattice, one can find their cor-

responding matching lattice. The most visual explanation of the concept of

matching lattice is the following [118]: Suppose that for a lattice G1 there ex-

ists a different lattice G2, such that each site in lattice G1 is uniquely related

to one site in G2 and the other way around. Also, assume that if a site is oc-

cupied in one of the lattices, its partner in the other one can not be occupied.

Now, if the presence of a cluster spanning G2 in one direction prevents any

cluster spanning G1 in the perpendicular direction and, conversely, there can

only be a percolating cluster in G1 if there is no percolation in G2, then G1

and G2 are matching lattices. For example, the triangular lattice is its own

matching lattice, called self-matching, while the square lattice is matched by

the star lattice [173]. Sykes and Essam argued, based on the uniqueness of

the threshold pc [173–175], that for any lattice G1 and its matching one G2,

the sum of the thresholds of both equals unity:

pG1c + pG2

c = 1. (5.2)

Then, since the triangular lattice is self-matching, one has pG1c = pG2

c and it

follows that pc = 1/2. The question of which pairs of lattices match each

other is independent on the statistical properties of the heights h that deter-

mine the cluster properties. Therefore, the site percolation threshold of the

triangular lattice is pc = 1/2, also for correlated percolation. We also checked

67


10-4

10-3

10-2

10-1

101

102

103

104

p

c,J

−pc

Lattice size L

0.75±0.02

0.73±0.03

0.68±0.05

0.41±0.05

0.13±0.05

-1

-0.85

-0.7

-0.4

-0.1

Fig. 5.1: Convergence of the percolation threshold estimator pc,J . The differ-

ence between the estimator and the threshold |pc,J − 1/2| is shown as function

of the lattice size L for H = −1, H = −0.85, −0.7, −0.4, and −0.1. The

data is shifted vertically to improve visibility. Results are averages over at

least 104 samples. We keep track of the cluster properties with the labeling

method proposed by Newman and Ziff [2, 3], as in Ref. [4].

this statement numerically by measuring pc for different values of the Hurst

exponent H, finding that it is compatible with 1/2, within error bars.

As a first check of the theory presented in Refs. [128,129,162] regarding the

dependence of νH on H, we consider here the convergence of a threshold

estimator, namely the value pc,J at which the maximum change in the size

of the largest cluster smax occurs [176–183]. The expected scaling behavior

[99,178] is

|pc,J(L)− pc| ∼ L−1/νH , (5.3)

where pc = 1/2. Figure 5.1 shows |pc,J(L)− pc| as function of the lattice size

L for different values of H. Within error bars, the data is compatible with

1/νH = −H for the considered values of H.

68

5.4. MAXIMUM CLUSTER SIZE AND CLUSTER SIZEDISTRIBUTION 69

5.4 Maximum cluster size and cluster size

distribution

At the threshold, p = pc, the largest cluster is a fractal of fractal dimension

df , i.e., its size smax scales with the lattice size L as

smax ∼ Ldf . (5.4)

This is also related to the order parameter P∞ of the percolation transition,

which is defined as the fraction of sites in the largest cluster,

P∞ = smax/N, (5.5)

and is expected to scale at p = pc as

P∞ ∼ L−β/ν = Ldf−d, (5.6)

where β is the order parameter critical exponent and d = 2 is the spatial di-

mension [82]. For uncorrelated percolation, β = 5/36 and ν = νuncorr2D = 4/3,

such that df = 91/48 ≈ 1.8958 [82]. To measure df as function of H, we con-

sidered the scaling of the size of the largest cluster smax with the lattice size,

see Fig. 5.2 and Eq. (5.4). For different values of H, we measured smax(L)

and calculated the local slopes df (L) of the data (see e.g. Ref. [25]),

df (L) = log[smax(2L)/smax(L/2)]/ log(4). (5.7)

Finally, df (L) is extrapolated to the thermodynamic limit, L→∞, to obtain

df (H), as displayed in Fig. 5.3. The fractal dimension is, within error bars,

independent on H, for H . −1/3. For H approaching zero, the value of

df does increase. While this behavior is in agreement with Ref. [22], it is

in strong contrast to the behavior of all other fractal dimensions considered

in this work, whose values depend strongly on H. Based on the data, we

propose the following dependence of df on H (in the range −1/3 ≤ H ≤ 0)

as being the simplest rational expression that fits the numerical data:

df (H) =91

48+

13

80

(1

3+H

). (5.8)

69


0.2

0.5

101

102

103

104L

arg

est

clu

ster

fra

ctio

n s

max

/N

Lattice size L

-1-0.85-0.7

-0.55-0.4

-0.25-0.1

0

Fig. 5.2: Fraction of sites in the largest cluster smax/N as function of the

lattice size L for different values of H. The data is shifted vertically to

improve visibility. Solid black lines are guides to the eye. Results are averages

over at least 104 samples.

70


1.75

1.80

1.85

1.90

1.95

2.00

-1.0 -0.8 -0.6 -0.4 -0.2 0.0Hurst exponent H

df

Fig. 5.3: Fractal dimension of the largest cluster df as function of the Hurst

exponent H. For H > −1/3, the solid line show the expressions of Eq. (5.7)

and and for H < −1/3 the constant value duncorrf = 91/48 is assumed.

71


10-8

10-4

100

104

108

101

102

103

104

105

106

107

Clu

ste

r siz

e n

s

Lattice size L

-1

-0.85

-0.7

-0.55

-0.4

-0.25

-0.1

0

Fig. 5.4: Cluster size distribution ns as a function of the number of occupied

sites s, for different values of the Hurst exponent H. The data is shifted

vertically to improve visibility. Solid black lines are guides to the eye. Results

are averages over 104 samples.

We also consider the distribution of cluster sizes ns, for different values of

the Hurst exponent H. The average number ns of clusters with s occupied

sites, usually rescaled by the number of sites of the lattice, is expected to

scale as ns ∼ s−τ at criticality, where τ is the so-called Fisher exponent. We

computed the cluster size distribution at pc using the Hoshen-Koppelman

algorithm [4] and found the expected power-law scaling behavior, see Fig. 5.4.

Based on the data, we propose the following dependence of τ on H in the

range −1/3 ≤ H ≤ 0 (as in the case of the fractal dimension df ), as being

the simplest rational expression that fits the numerical data based on the

values 187/91 for H = −1 and 2 for H = 0:

τ =187

91− 15

91

(H +

1

3

). (5.9)

72


1.95

2.00

2.05

2.10

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Fis

her

exponen

t τ

Hurst exponent H

Fig. 5.5: Fisher exponent t as function of the Hurst exponent H. For

H > −1/3, the solid line show the expressions of Eq. (5.9) and and for

H < −1/3 the constant value τuncorr = 187/91 is assumed.

73


5.5 Cluster perimeters

Here, we consider triangular lattice stripes of aspect ratio A = 8. For

every largest cluster that spans the lattice vertically (between the long

sides of the lattice) and does not touch its vertical boundaries, there are

two contours that can be defined: the complete and accessible perime-

ters [5–7, 13–15, 62, 184–188]. Figure 5.6 shows the definition of the two

perimeters, which live on the dual of the original lattice, i.e. the honeycomb

lattice in the case of the triangular lattice. The complete perimeter consists

of all bonds of the honeycomb lattice that separate sites belonging to the

spanning cluster from unoccupied sites that can be reached from the verti-

cal boundaries of the lattice without crossing sites belonging to the largest

cluster. If, in addition, fjords of the perimeter with diameter less than√

3/3

(lattice units) are inaccessible, the accessible perimeter is obtained. Figure

5.7 shows the left hand side complete and accessible perimeters of a perco-

lating cluster on a lattice of size L = 128. In the upper inset of Fig. 5.8, the

length of the complete perimeter Mcp is observed to scale with the lattice

size L as,

Mcp ∼ Ldcp , (5.10)

where for the uncorrelated case, given by H = −1, it is known that dcp = 7/4

[5, 6, 62, 186]. In addition to considering the scaling of Mcp with L, we also

determined the fractal dimension dcp using the yardstick method [189, 190].

There, one measures the number of sticks S(m) of size m needed to follow the

perimeter from one end to the other. Figure 5.9 shows that, for intermediate

stick lengths, S(m) scales as

S ∼ m−dcp . (5.11)

We measured the value of the fractal dimension with this method for different

lattice sizes L, see Fig. 5.9, and then extrapolated the results to L→∞ to

obtain dcp. The fractal dimension dcp(H) determined by this method is com-

patible with the one obtained from the scaling of the length of the perimeter,

see Eq. (5.10), and we combined both measurements for the final estimates.

74

5.5. CLUSTER PERIMETERS 75

Fig. 5.6: Complete and accessible perimeter. The blue (filled) sites of the

triangular lattice are part of the largest cluster, while the white (empty) sites

are unoccupied. Bonds of the dual lattice are shown as dashed lines. Assume

that the largest cluster percolates in the vertical direction and does not touch

the left or right boundaries of the lattice. Consider a walker starting on the

left-bottom side of the lattice, which never visits a bond twice and traces out

the complete perimeter, turning left or right depending on which of the two

available bonds separates an occupied from an empty site. The complete

perimeter is fully determined when the top side of the lattice is reached.

Performing the same walk, but with the additional constraint that fjords with

diameter ≤√

3/3 (in lattice units) are not accessible, yields the accessible

perimeter. The solid green (thick) lines on the honeycomb lattice form the

accessible perimeter, while dashed green (thick) lines indicate bonds that are

part of the complete perimeter but not of the accessible one. A similar walk

yields the two perimeters on the right hand side of the cluster.

75


(a) H = −1 (b) H = −0.5

(c) H = −0.25 (d) H = 0

Fig. 5.7: Snapshots of typical complete and accessible perimeters. The ac-

cessible perimeter is shown in bold solid blue lines. In addition, the parts

of the complete perimeter that do not belong to the accessible perimeter are

drawn with thin black lines. The snapshots are taken for (a) H = −1, (b)

−0.5, (c) −0.25, and (d) 0, on a lattice of (vertical) length L = 128.

76

5.5. CLUSTER PERIMETERS 77

1.3

1.4

1.5

1.6

1.7

1.8

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Per

imet

er f

ract

al d

imen

sion

Hurst exponent H

CompleteAccessibleFit

101

102

101

102

103

Acc

essi

ble

/L

Lattice size L

101

102

101

102

103

Com

ple

te/L

Lattice size L

-1-0.85-0.7

-0.55-0.4

-0.25-0.1

0

Fig. 5.8: Main plot: Fractal dimension of the complete perimeter dcp and

of the accessible perimeter dap as function of the Hurst exponent H. For

H = −1 (uncorrelated), our results, dcp = 1.75± 0.02 and dap = 1.34± 0.02,

are in agreement with values previously reported [5–11]. With increasing

H, both fractal dimensions seem to approach 3/2, compatible with the data

of Kalda et al. [12–15]. In the range −1/νuncorr2D ≤ H ≤ 0, the solid lines

show the expressions dcp = 3/2−H/3 and dap = (9− 4H)/(6− 4H). Insets:

Length of the complete and of the accessible perimeters as function of the

lattice size L for the values of H shown in the main plot.

77


100

101

102

103

104

105

100

101

102

103

Num

ber

of

stic

ks

S

Stick length m

H=0

−1.49±0.03

163264

128256512

1024

Fig. 5.9: Yardstick method to measure the fractal dimension of the complete

perimeter. The number of sticks needed to follow the perimeter S is shown

as function of the stick length m, for different lattice sizes L, and H = 0. The

numerical value of the complete perimeter fractal dimension dcp(H) obtained

with the yardstick method, dcp(0) = 1.49± 0.03, agrees, within error bars,

with the results of the analysis of the local slopes of the perimeter length,

see Fig. 5.8, as well as with the literature [13–15].

In Fig. 5.8, one sees the fractal dimension of the complete perimeter as func-

tion of the Hurst exponent H. For H approaching zero, dcp decreases and

finally converges towards 3/2, in agreement with previous results [13–15].

The fractal dimension of the accessible perimeter dap is defined by the scal-

ing of the length of the accessible perimeter Map with L, see lower inset of

Fig. 5.8,

Map ∼ Ldap . (5.12)

For uncorrelated percolation the fractal dimension of the accessible perimeter

is known to be dap = 4/3 [8,9,62,186]. Figure 5.8 shows dap(H), determined

using the scaling of Map and the yardstick method.

For the critical Q-state Potts model [191], Duplantier [11, 16] established

78

5.6. TRANSPORT PROPERTIES: SHORTEST PATH, BACKBONEAND CLUSTER CONDUCTIVITY 79

the following duality relation between the fractal dimension of the complete

perimeter dcp and of the accessible perimeter dap:

(dap − 1)(dcp − 1) = 1/4. (5.13)

The case Q = 1 corresponds to uncorrelated percolation [192]. This relation

has been extended to other lattice models in the case of SLE, see Section 2.5,

relating the diffusion coefficients of the complete and accessible perimeters to

each other in the case the complete perimeter follows SLE statistics. Having

measured dcp and dap as functions of H, we see in Fig. 5.10 that the duality

relation of Eq. (5.13) holds, within error bars, for −1 ≤ H ≤ 0. Therefore,

taking the known results for H = −1 and H = 0 into account, we propose the

following functional dependence of the complete perimeter fractal dimension

on H (in the range −1/νuncorr2D ≤ H ≤ 0, see Ref. [15]):

dcp =3

2− H

3, (5.14)

which, assuming the validity of the duality relation also for correlated percola-

tion, implies the following form of the accessible perimeter fractal dimension:

dap =9− 4H

6− 4H. (5.15)

5.6 Transport properties: shortest path,

backbone and cluster conductivity

For uncorrelated percolation, the shortest path between two sites in the

largest cluster is a fractal of dimension dsp ≈ 1.131, see Chapter 3. For a

given configuration, it can be identified using the burning method [17] de-

scribed in Section 3.2. On the cluster spanning the lattice vertically, with

aspect ratio A = 1, we select one cluster site in the top row and one in the

bottom row, such that their Euclidean distance is minimized and find the

79


0.20

0.25

0.30

0.35

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

(dap

-1)(

d cp-1

)

Hurst exponent H

DataDuality

Fig. 5.10: Left hand side of the duality relation for cluster perimeters,

(dap − 1)(dcp − 1) = 1/4 [11,16], as function of the Hurst exponent H.

number of sites Msp in the shortest path between them. The following scal-

ing of the length with the lattice size L is observed:

Msp ∼ Ldsp , (5.16)

which can be used to determine the fractal dimension dsp(H) using the local

slopes, see Eq. (5.7), as displayed in Fig. 5.11. For increasing correlation,

dsp decreases and is compatible with unity for H = 0, as also reported in

Ref. [21]. Using this observation and the literature results for uncorrelated

percolation [17–20], we propose the following dependence of dsp on the Hurst

exponent H (in the range −1/νuncorr2D ≤ H ≤ 0):

dsp(H) =147

130− 3/4 +H

195/34 +H. (5.17)

In order to get even more insights into the geometrical properties of the

percolating cluster, one can study the backbone and cluster conductivity

between two points. These two last quantities are related to the following

80


0.95

1.00

1.05

1.10

1.15

1.20

1.25

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Short

est

pat

h d

sp

Hurst exponent H

DataFit

101

102

103

104

101

102

103

Pat

h l

ength

Lattice size L

-1-0.85-0.7

-0.55-0.4

-0.25-0.1

0

Fig. 5.11: Fractal dimension of the shortest path dsp of the largest clus-

ter as function of the Hurst exponent H. The inset shows the number

of sites in the shortest path as function of the lattice size L for the same

value of H as in the main plot. For uncorrelated disorder, i.e. H = −1,

we find dsp = 1.130± 0.005, in agreement with the literature [17–20]. With

increasing Hurst exponent, dsp approaches unity [21]. The solid line is

the graph of the proposed behavior of the shortest path fractal dimen-

sion: dsp(H) = 147/130− (3/4 +H)/(195/34 +H), for −3/4 ≤ H ≤ 0, and

dsp(−1 ≤ H ≤ −1/νuncorr2D ) = dsp(−1/νuncorr

2D ) = 147/130.

81


question. If one supposes that the occupied sites of the percolating cluster

are resistors and a potential difference is applied between these two sites,

which sites would carry non-zero current? This subset of sites of the largest

cluster is called effective backbone [193] and is related to the geometrical

backbone defined as the union of all non-self-crossing paths between these

two sites [17,19,23–26,65,82]. Actually some sites, called perfectly balanced

sites [193], belong to the geometrical backbone but do not carry current,

because the potentials are perfectly balanced, and therefore do not belong to

the effective backbone. Algorithmically, for a given cluster, the geometrical

backbone can be found with the burning method described in Ref. [17]. It

works as follows. One first burns the sites of the spanning cluster starting

from the lower site and indexes the burned sites according to the time they

have been reached, till the higher edge is reached. Then one does a second

burning, starting from the higher edge, but only burning sites with a lower

index as defined in the first burning. This identifies the so-called elastic

backbone, which is the union of all the shortest paths between the two points.

One then grows the backbone as follows, starting from the elastic backbone.

One starts a third burning. For each site where a loop was closed in the

first burning, one burns the sites with a lower index, with the condition that

the growning backbone cannot burn. One reaches the growning backbone

whether at a single point or at several points. Only in the case where the

growing backbone is reached at several points, all the points that have been

burned are added to the growing backbone. One repeats this procedure, till

no more part can be attached to the growing backbone and one obtains the

full backbone.

Other methods based on the conductivity properties rather than the geo-

metrical properties of the backbone have been developed [193, 194]. They

actually compute the current passing through every node of the largest clus-

ter. The advantage of using the geometrical properties of the backbone over

methods solving Kirchoff’s law to determine which bonds carry current or

not is that the burning algorithm presented above is fast (there is no matrix

to invert, see later) and computes exactly the geometrical backbone, whereas

the other methods computes only the effective backbone. In the following

82


we are interested in the geometrical backbone computed by the burning al-

gorithm [17].

The total number of sites in the backbone Mbb scales with the lattice size L,

Mbb ∼ Ldbb , (5.18)

where dbb is the backbone fractal dimension, see inset of Fig. 5.12. With in-

creasing H, dbb increases and is compatible with the fractal dimension of the

largest cluster for H approaching zero. Similarly to Ref. [22], for the func-

tional dependence of dbb on H, we propose to interpolate linearly between the

best known value for uncorrelated percolation, dbb(−1) = 1.6434± 0.0002

[26] and the fractal dimension of the largest cluster for H = 0, see Eq. (5.8):

dbb(H) =39

20(1 +H)− 166

101H. (5.19)

The backbone becomes more compact with increasing correlation, which is

also compatible with the fact that the shortest path fractal dimension is

decreasing in this limit, see Fig. 5.11.

At the percolation threshold, the backbone of the largest cluster is a fractal

and the conductivity C(r) between its ends has a power-law dependence on

the Euclidean distance r of the end sites,

C(r) ∼ r−tH/νH , (5.20)

where tH is the conductivity exponent and we call tH/νH the reduced

conductivity exponent [22, 25, 92, 195–200]. For uncorrelated percolation,

tuncorr2D /νuncorr

2D = 0.9826±0.0008 [25]. As the backbone becomes more compact

with increasing correlation, see Fig. 5.12, one might expect the conductivity

to decay more slowly with the spatial separation, and, consequently, that

tH/νH decreases [22,201].

To measure the conductivity C(r) of the backbone, we solve Kirchhoff’s laws

to obtain for every site i in the backbone:∑k

Cik(Vi − Vk) = 0, (5.21)

83


1.6

1.7

1.8

1.9

2.0

2.1

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Bac

kbone

d bb

Hurst exponent H

DataFit

101

102

103

104

105

106

107

101

102

103

Bac

kb

on

e si

ze

Lattice size L

-1-0.85

-0.7-0.55

-0.4-0.25

-0.10

Fig. 5.12: Fractal dimension of the backbone dbb as function of the Hurst

exponent H. With increasing H, the backbone becomes more compact and,

consequently, dbb increases, while the fractal dimension of the shortest path,

see Fig. 5.11, decreases [22]. For uncorrelated disorder, H = −1, we measure

dbb = 1.64± 0.02, compatible with the results reported in Refs. [17,19,23–26].

The solid line is the graph of the following interpolation: dbb(H) = 39/20(1+

H) − 166/101H. Inset: Backbone size as function of the lattice size L for

the same values of H as in the main plot.

84


0.3

0.5

0.7

0.9

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Red

uce

d e

xponen

t t H

/νH

Hurst exponent H

DataFit

10-3

10-2

10-1

100

101

101

102

103

Co

nd

uct

ivit

y C

Lattice size L

-1-0.85

-0.7-0.55

-0.4-0.25

-0.10

Fig. 5.13: Reduced conductivity exponent tH/νH as function of the Hurst

exponent H. For uncorrelated disorder, we find tH/νH(−1) = −0.992±0.027

in agreement with Ref. [25]. The solid line corresponds to the expression

tH/νH = 16/41−H−7H2/25 in the range −1/νuncorr2D ≤ H ≤ 0 and tH/νH =

t/νuncorr2D for −1 ≤ H ≤ −1/νuncorr

2D . Inset: Conductivity C as function of the

lattice size L, for the same values of the Hurst exponent H as in the main

plot.

85


where the sum runs over the nearest neighbors k and the conductivity Cik

is unity if the site k also belongs to the backbone and zero otherwise. The

boundary conditions are chosen such that Vtop = N on the top end of the

backbone and Vbottom = 0 on its bottom end. In order to solve this linear

system of equations, we invert the sparse conductivity matrix C of elements

Cik, using a sparse matrix solver1 and obtain the value of the potential Vi at

each site i of the backbone2. From this we deduce the global conductivity as

follows [92]. As one knows the potentials at each site and the resistors are set

to unity, one is able to compute the currents in the system and especially the

total current Itot entering the top end of the backbone. From this one deduce

the global conductivity Ctot through the relation Ctot = Itot/(Vtop− Vbottom).

The inset of Fig. 5.13 shows the conductivity C as function of the lattice

size L, for different values of H. Since in our setup the distance between the

end points r ∼ L, we use this scaling to determine the reduced conductivity

exponent tH/νH , see Fig. 5.13. Our result for uncorrelated percolation agrees

with the literature and one observes that tH/νH decreases with increasing H.

We propose the following functional dependence of the reduced conductivity

exponent on H (in the range −1/νuncorr2D ≤ H ≤ 0):

tHνH

=16

41−H − 7H2

25. (5.22)

5.7 Final remarks

Concluding, we studied percolation with long-range correlation in the site

occupation probabilities, as characterized by the Hurst exponent H. The

site percolation threshold of the triangular lattice was argued to be 1/2,

independent of H. For H approaching zero the fractal dimension of the

largest cluster was found to increase, meaning that the largest cluster gets

more dense for increasing values of H. Also the cluster size distribution was

1To invert the sparse matrix C we use the Intel MKL Direct Sparse Solver.2Solving Kirchoff’s law only for the backbone and not for the full largest cluster, as

done in Ref. [194], is computationally more effective as the conductivity matrix C to be

inverted is much smaller.

86

5.7. FINAL REMARKS 87

shown to follow a power law dependence with its critical exponent decreasing

towards 2 for H going to zero. The fractal dimensions of the complete and

accessible perimeters of the largest cluster were observed to approach 3/2 for

H → 0, while the duality relation between both fractal dimensions seems to

hold independently of the value of H. As H increased, the fractal dimension

of the shortest path was observed to decrease towards 1 for H going to 0. This

is in agreement with the backbone becoming more compact with increasing

value of H, as does the largest cluster. This has also an influence on the

cluster conductivity, such that the reduced conductivity exponent decreases

with H. We also proposed quantitative relations for the dependence of the

studied critical exponents of the percolation transition on H, as being the

simplest rational expressions that fit the numerical data.

87


88

Chapter 6

Schramm-Loewner Evolution

on long-range correlated

landscapes

In the previous Chapter we have seen how the power-law correlations modify

the geometrical properties of the largest cluster, and conjectured quantitative

relations for the dependency of the critical exponents on the Hurst exponent

H ∈ [−1, 0]. In this Chapter, we go beyond the description in terms of crit-

ical exponents and study the paths at criticality in the framework of SLE.

We analyze the statistical properties of the zero isoheight lines for corre-

lated landscapes for H ∈ [−1, 1]. We show numerically that for H ≤ 0 the

statistics of these lines is compatible with SLEκ for κ ∈ [8/3, 4], and that

established analytical results are recovered for H = −1 and H = 0. This

result suggests that for negative Hurst exponent in spite of the long-range na-

ture of correlations, surprisingly the statistics of isolines is fully encoded in a

Markovian Brownian motion with a single parameter in the continuum limit.

However, for strictly positive Hurst exponent, we find that the underlying

driving function is not Markovian and therefore not consistent with SLEκ.

We also analyze the shortest path at criticality and the watersheds, but, in

contrast to the the case of the isoheight lines, we do not find agreement with

SLE statistics in the case of correlated landscapes, i.e. for H > −1.

89

90CHAPTER 6. SCHRAMM-LOEWNER EVOLUTION ON

LONG-RANGE CORRELATED LANDSCAPES

This Chapter is based on:

N. Pose, K. J. Schrenk, N. A. M. Araujo, and H. J. Herrmann, Schramm-

Loewner Evolution on correlated landscapes, in progress.

6.1 Introduction

In this Chapter we map isoheight lines of long-range correlated landscapes on

uncorrelated random walks. These lines are extracted from the landscapes

by following paths at a given height [55]. In the case of land scenes, the lines

correspond to the boundary of lakes or seas [55] but one can also consider

energy landscapes where isoheight lines are paths of constant energy [120,

130], or vorticity fields in turbulence where these lines correspond to the

vortex lines [32]. These isoheight lines are usually scale invariant [12, 189]

and empirical results on rocky shorelines suggest that isoheight lines are even

conformally invariant [55]. The long-range correlations of the landscapes

are parametrized by their Hurst exponent H ∈ [−1, 1] and we show that

in the case H ≤ 0 the statistical properties of the zero isoheight lines are

compatible with SLE [29]. This allows to map the long-range correlated lines

of landscapes characterized by their Hurst exponent onto one-dimensional

Markovian processes.

This is done by finding the conformal maps gt that iteratively map the com-

plement of the curve in the upper half-plane H back onto H and satisfies

Loewner’s differential equation Eq. (2.10),

∂gt(z)

∂t=

2

gt(z)− ξt,

with g0(z) = z and ξt a real continuous function. In the case of SLEκ the

driving function ξt is equal to ξt =√κBt with Bt a standard one-dimensional

Brownian motion, and κ a diffusion constant. Therefore studying the iso-

height lines for H ∈ [−1, 0] and relating them to the SLE theory, shows

that the statistics of the isoheight lines is encoded in a Markovian process,

consisting of a Brownian motion of diffusivity κ. As the fractal dimension

of the isoheight lines depends on the value of the Hurst exponent H, see

90

6.2. ISOHEIGHT LINES ON CORRELATED LANDSCAPES WITHNEGATIVE HURST EXPONENT 91

Section 5.5, and the fractal dimension of SLEκ curves depends on the value

of κ, see Section 2.7.1, one expects a relation between the value of κ and H

for the isoheight lines. Below we focus on the accessible perimeter of zero

isoheight lines on triangular lattices of size Lx × Ly with Lx = 8Ly [99].

6.2 Isoheight lines on correlated landscapes

with negative Hurst exponent

We consider random landscapes on a lattice. To each site x = (x, y) on the

lattice one associates a random height h(x). We suppose that the heights

have long-range correlations of the form:

c(r) = 〈h(x)h(x + r)〉x ∼ |r|2H , (6.1)

where H is the Hurst exponent. Using the Wiener-Khintchine theorem [132]

relating the height-height correlation function c(r) to the power spectrum,

one generates random landscapes on a finite lattice using the Fourier filtering

method, see Section 4.3. As mentionned in Chapter 4, one recovers uncor-

related landscapes for H = −1, and the discrete Gaussian Free Field (GFF)

for H = 0 [39]. To identify the complete perimeter of the isoheight lines

one extracts the interface on the dual lattice between the sites lower and the

sites higher than a given height hc. One can map these isoheight lines to the

perimeters of percolation clusters, like in Fig. 4.1 , where the sites above hc

are marked as occupied and the sites below hc as empty. One can then define

the accessible perimeter as the complete perimeter from which one suppresses

the fjords of neck size equal to the lattice unit of the dual lattice, i.e.√

3/3,

see Fig. 5.6. It has been shown in Section 5.3, that on the triangular lattice

a spanning cluster emerges at a critical height hc = 0. Therefore we consider

the percolating clusters at hc = 0 and extract their complete and accessible

perimeters that span from the bottom line to the top line. In order to study

the SLE properties of these isoheight lines using the chordal SLE formalism,

we map them into the upper half-plane using the methodology described in

Section 2.6.

91



The zero isoheight lines on the triangular lattice in the case of percolation

and of the GFF are analytically tractable and it has been proven that they

are SLE [29,35,61].

6.2.1 SLE and fractal dimension

As proven by Beffara [52], SLEκ curves are fractals of a fractal dimension df

that is related to the diffusion coefficient κ as:

df = min(

2, 1 +κ

8

).

The fractal dimension df of the accessible perimeter of the isoheight lines

for different values of H was numerically estimated and even a conjecture

was proposed for its dependence on H in Section 5.5. Using the equation

above, one gets a first estimate of the expected values of κ for SLE curves

of the same fractal dimension, see Table 6.2. For H = −1 and H = 0, the

values of the fractal dimension and diffusivity are known exactly: df = 4/3,

κ = 8/3, and df = 3/2, κ = 4, respectively. We will compare this result to

the estimates of κ obtained with two indirect methods, the winding angle

and the left passage probability, and with the one obtained from the direct

SLE method.

6.2.2 Winding angle

The winding angle of SLEκ curves follows a Gaussian distribution and the

variance scales with κ as,

〈θ2〉 − 〈θ〉2 = σ2θ = b+

κ

4ln(Ly),

where b is a constant and Ly is the vertical lattice size [29, 53, 54]. κ/4

corresponds to the slope of σ2θ vs ln(Ly). The estimates of κ are displayed in

Table 6.2.

For values near H = 0, one has less precision on the results, as the system

is strongly influenced by finite-size effects, see Fig. 5.1 in Chapter 4. The

result we obtain from the winding angle measurement is, within error bars,

92


Ly

101 102 103

4(<

2 3!

b)=5

2

3

4

5

6

70-0.1-0.25-0.4-0.55-0.7-0.85-1

3=<3-5 0 5

<!

13

P(3

)

0

0.2

0.4

Fig. 6.1: Rescaled variance of the winding angle along the curve for different

Hurst exponents H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. The rela-

tionσ2θ−bκ/4

= lnLy is displayed by the black solid line. In the inset, the rescaled

probability distributions are plotted and compared to a normal distribution

for H = −1,−0.55, 0.

93



in agreement with the previous estimates from the fractal dimension of the

accessible perimeter of zero isoheight lines. Indeed, the winding angle result

gives insights into the conformal invariance of the problem, see Section 2.7.2.

From our results in Fig. 6.1, we have strong numerical evidence that for

−1 ≤ H ≤ 0 the accessible zero isoheight lines display conformal invariance

at criticality, which is a prerequisit for SLE.

6.2.3 Left Passage Probability

As we simulate the curves in a bounded rectangular domain, we map confor-

mally the isoheight lines into the upper half-plane, using an inverse Schwarz-

Christoffel transformation, see Section 2.6, in order to obtain the chordal

curve. For chordal SLEκ curves, Schramm has computed the probability

Pκ(φ) that a given point z = Reiφ in the upper half-plane H is on the right-

hand side of the curve [49]. Pκ(φ) is given by Schramm’s formula (2.26):

Pκ(φ) =1

2+

Γ (4/κ)√πΓ(

8−κ2κ

) cot(φ)2F1

(1

2,

4

κ,3

2,− cot(φ)2

).

We define a set of sample points S in H for which we compute the left-passage

probability in order to compare it to the values predicted by Schramm’s

formula (2.26). To estimate κ, we minimize the mean square deviation Q(κ)

between the computed and predicted probabilities. The estimated value of

κ corresponds to the point where the minimum of Q(κ) is observed.

As shown in Fig. 6.2, the minimum of Q is less pronouced for higher values

of H, as it is expected for functions of the form of Pκ(φ) with values of κ

increasing towards four, see Fig. 2.11. In the inset, we plot the difference

between the computed left-passage probbilities and the one predicted by

Schramm’s formula. One sees that the computed left-passage probabilities

are well fitted by the form of Pκ(φ) predicted for SLE curves. The estimates

of κ we obtain are in agreement with the ones predicted from the fractal

dimension and confirmed by the winding angle analysis.

94


5=5min

0.9 0.95 1 1.05 1.1

Q(5

)=Q

(5m

in)

1

2

3

4

5

60-0.1-0.25-0.4-0.55-0.7-0.85-1

3=:0 0.5 1

P(3

)!

P5(3

)

-0.1

0

0.1

Fig. 6.2: Measured rescaled mean square deviation Q(κ)/Q(κmin) as

a function of κ/κmin with κmin the value of κ where the min-

imum of Q(κ) is attained, for different Hurst exponents H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. In the inset, the difference be-

tween the measured left passage probabilities P (θ) and the left passage prob-

abilities Pκ(θ) predicted by Schramm’s formula of Eq. (2.26) with κ = κmin,

for H = −0.85,−0.55,−0.25, 0. For convenience, we chose 502 points in the

range [−0.025Lx, 0.025Lx]× [0.15Ly, 0.25Ly] with Ly = 1024 and Lx = 8Ly,

which are then mapped to the upper half-plane through an inverse Schwarz-

Christoffel transformation, see Section 2.6.

95



t #10-30 1 2 3

h92 ti=5

#10-3

0

1

2

3

0-0.1-0.25-0.4-0.55-0.7-0.85-1

x=p5t

-5 0 5

p5t!

1P

(x)

0

0.2

0.4

=0 2

c(=)

-1

0

1

#10!3

Fig. 6.3: Rescaled variance of the driving functions for different values of

H = −1,−0.85,−0.7,−0.55,−0.25,−0.1, 0. In the upper inset, we present

the rescaled probability distributions of the driving functions and compare

them to a Gaussian distribution for H = −1,−0.55, 0. In the lower one, one

sees the auto-correlation function c(τ) = 〈c(t, τ)〉τ of the increments for the

same values of H, averaged over 50 time steps.

6.2.4 Direct SLE

To test if the curves are SLEκ, one has to check that the statistics of the

driving function ξt is consistent with a one-dimensional Brownian motion

with variance κt. This can be done by solving Loewner’s Equation (2.10)

numerically. In order to do so, we use the so-called vertical slit map algo-

rithm, where one considers the driving function to be constant over small

time intervals δt, as described in Section 2.7.4.

We extract the driving function of all paths, and compute the diffusion co-

efficient κ from the variance of the driving function, see values in Table 6.2,

96


Hurst exponent H-1 -0.8 -0.6 -0.4 -0.2 0

Est

imate

d5

1.5

2

2.5

3

3.5

4

4.5

5fract. dim.winding anglelppdSLE

Fig. 6.4: Estimated diffusion coefficients κ from the fractal dimension (fract.

dim.), the winding angle, the left-passage probability (lpp), and the direct

SLE (dSLE) methods for H ≤ 0. The red crosses correpond to the analyti-

cally known results.

and test its Gaussian distribution at a given Loewner time t, see Fig. 6.3.

We also test the Markovian property of the driving function by computing

the auto-correlation function c(τ) = 〈c(t, τ)〉t of the increments, with

c(t, τ) =〈δξt+τδξt〉 − 〈δξt+τ 〉〈δξt〉√

(〈δξ2t+τ 〉 − 〈δξt+τ 〉2) (〈δξ2

t 〉 − 〈δξt〉2). (6.2)

The correlation function drops to zero after few time steps, see lower inset

of Fig. 6.3. Our results are in agreement with the driving function being a

Brownian motion and the estimated values of κ agree with the other results

and with the analytically computed values of κ = 8/3 and κ = 4 for H = −1

and H = 0, respectively.

97



6.3 Isoheight lines on correlated landscapes

with positive Hurst exponent

We apply the same tests for Gaussian random landscapes with positive Hurst

exponent H > 0, but contrary to the case H < 0 we do not find agreement

with SLEκ.

6.3.1 Winding angle

We apply the winding angle test to the zero isoheight lines of landscapes

with positive H to estimate κ, and find that the estimate of κ deduced from

the winding angle measurement strongly depends on the value of H ≥ 0. We

get an agreement between the estimates of κ from the fractal dimension of

the curves and of the winding angle measurement, see Fig. 6.5. However for

increasing values of H, one sees a slower convergence towards a linear behavior

of the variance, and a larger difference of the probability distribution from a

Gaussian distribution. This could be due to finite size effects.

6.3.2 Left-Passage probability

We now apply the left-passage probability test to the isoheights in the case

of positive Hurst exponent. The estimates of κ are displayed in Fig. 6.6,

where we plot the reduced mean square deviation Q(κ)/Q(κmin) as a function

of κ. One sees that contrary to what one expects from the winding angle

measurement, we do not find a strong dependence of the estimate of κ on the

value of H. In particular, this is in contradiction with the strong dependence

of the value of the fractal dimension of the isoheight lines on the value of H.

6.3.3 Direct SLE method

We extract the driving function from the isoheight lines and compute the

variance, the probability distribution and the autocorrelation function of the

increments. From the variance of the driving function, see Fig. 6.7, one ob-

tains that the estimate of κ depends strongly on H, but that it is not in

98

6.3. ISOHEIGHT LINES ON CORRELATED LANDSCAPES WITHPOSITIVE HURST EXPONENT 99

Ly

101 102 103

4(<

2 3!

b)=5

1

2

3

4

5

6

700.10.250.550.70.95

3=<3-5 0 5

<!

13

P(3

)

0

0.5

Fig. 6.5: Rescaled variance of the winding angle along the curve for different

Hurst exponents H = 0, 0.1, 0.25, 0.55, 0.7, 0.95. The relationσ2θ−bκ/4

= lnLy

is displayed by the black solid line. In the inset, the rescaled probability

distributions are plotted and compared to a normal distribution for the same

values of H.

99



53 3.2 3.4 3.6 3.8 4

Q(5

)=Q

(5m

in)

1

1.2

1.4

1.6

1.8

2 00.10.250.550.70.95

Fig. 6.6: Measured rescaled mean square deviation Q(κ)/Q(κmin) as a func-

tion of κ, with κmin the value of κ where the minimum of Q(κ) is attained, for

different Hurst exponents H = 0, 0.1, 0.25, 0.55, 0.7, 0.95. For convenience,

we chose 502 points in the range [−0.025Lx, 0.025Lx]× [0.15Ly, 0.25Ly] with

Ly = 1024 and Lx = 8Ly, which are then mapped to the upper half-plane

through an inverse Schwarz-Christoffel transformation, see Section 2.6.

100

6.3. ISOHEIGHT LINES ON CORRELATED LANDSCAPES WITHPOSITIVE HURST EXPONENT 101

t #10-30 1 2 3

h92 ti=5

#10-3

0

1

2

3

00.10.250.550.70.95

x=p5t

-4 -2 0 2 4

p5t!

1P

(x)

0

0.2

0.4

Fig. 6.7: Rescaled variance of the driving functions for different values of

H = 0, 0.1, 0.25, 0.55, 0.7, 0.95. In the upper inset, we present the rescaled

probability distributions of the driving functions and compare them to a

Gaussian distribution for the same values of H.

agreement with the estimates from the fractal dimension, therefore being

in contradiction with the SLE theory. Also, if one studies the autocorrela-

tion function, one sees increasing time correlations in the increments with

increasing values of the Hurst exponent, see Fig. 6.9.

The summary of the results obtained for the isoheight lines on landscapes

with positive Hurst exponent is displayed in Fig. 6.8.

101



Hurst exponent H0 0.2 0.4 0.6 0.8 1

Est

imate

d5

0

2

4

6




SLE (dSLE) methods for H ≥ 0. The red cross correponds to the analytically

known result for H = 0.

102

6.4. MARKOVIAN PROPERTIES OF THE DRIVING FUNCTIONS 103

6.4 Markovian properties of the driving func-

tions

In order to have a closer look at the Markovian property of the increments,

we study the auto-correlation function c(τ) defined above in Eq. (6.2) for the

driving function extracted from the isoheight lines for negative and positive

Hurst exponent, and display it in logarithmic scale, see Fig. 6.9. One sees a

transition in the type of correlation from short-range, correlations over one

decade for H ≤ 0, to long-range correlations, correlations over three decades

for H > 0, therefore confirming that the SLE description applies for isoheight

lines on surfaces with H ≤ 0 but not for surfaces with strictly positive Hurst

exponent. In the case of strictly positive Hurst exponent, the driving function

is not Markovian anymore and displays long-range correlations characterized

by power-law time correlations.

The case H = 0, corresponding to the Gaussian Free Field, is analytically

known to be SLE, therefore the driving function in the scaling limit is a

Brownian motion characterized by independant increments.

6.5 Other critical curves

We also tested if other paths defined on long-range correlated landscapes

display statistics compatible with SLE. We focused on the shortest path and

watersheds. The shortest path is a typical object studied in percolation

problems and is related to the geometry of the spanning cluster [89, 90]. It

is defined as the path connecting the two opposite borders of the lattice

with the minimum number of lattice elements belonging to the spanning

cluster. The watershed is defined as the line separating adjacent drainage

basins [121, 122, 202, 203]. Here we only consider two drainage basins, with

the left and right boundaries of the lattice belonging to two different sinks.

Then the watershed line spans from the bottom to the top and divides the

lattice into two basins.

In recent works [30,31] it has been shown that these paths display statistics

103



=10-4 10-3 10-2

c(=)

10-4

10-3

10-2

10-1

-0.7-0.55-0.2500.250.550.7

Fig. 6.9: Auto-correlation function c(τ) = 〈c(t, τ)〉τ , averaged over 50

time steps, displayed in log-log scale for values of the Hurst exponent

H = −0.7,−0.55,−0.25, 0, 0.25, 0.55, 0.7.

104

6.5. OTHER CRITICAL CURVES 105

compatible with SLE for usual uncorrelated percolation in the case of the

shortest path, see Chapter 3, and uncorrelated landscapes in the case of

watersheds [30]. We want to check if these results still hold in the case

of long-range correlated percolation and long-range correlated landscapes.

Therefore, we compared the results obtained with the fractal dimension, the

winding angle, the direct SLE and in the case of watersheds also the left-

passage probability.

6.5.1 Winding angle measurements for the shortest

path and the watersheds

In Refs. [28, 99, 124, 147], the authors studied the dependence of the fractal

dimension of the shortest path and the watersheds on the value of the Hurst

exponent. Therefore we first test the compatibility between the variance of

the winding angle and the fractal dimension of the shortest path and the

watersheds for different values of the Hurst exponent. In the case of the

shortest path, the results are obtained on a triangular lattice with aspect

ratio one and H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. In the case

of the watersheds, they are obtained on a square lattice of aspect ratio one

for H = −1, two for H = −0.8,−0.6,−0.5,−0.4,−0.2 and four for H = 0. In

Figs. 6.10 and 6.11, we show the rescaled variance of the winding angle for the

shortest path and the watersheds respectively. In both cases, the variance

σ2θ strongly depends on the value of the Hurst exponent H, see insets of

Figs. 6.10 and 6.11. In the case of the shortest path, for increasing values of

H, the linear regime is reached only for larger values of the vertical lattice

size Ly. This might be due to finite size effects, as the fractal dimension

of the curve is going to one, see Section 5.6, and therefore the variance of

the winding angle should converge to a constant, as displayed in the inset of

Fig. 6.10.

Using the relation df = 1 + κ8

and σ2θ = b + κ

4ln(Ly), one gets that σ2

θ =

b + 2(df − 1) ln(Ly) and obtains an estimate of the fractal dimension from

the variance of the winding angle. The results obtained by studying σ2θ give

estimates of the fractal dimension df in agreement with the directly measured

105



Ly

101 102 103

4(<

2 3!

b)=5

-2

0

2

4

6

8

0-0.1-0.25-0.4-0.55-0.7-0.85-1Ly

102

<2 3

0

1

2

Fig. 6.10: Rescaled variance 4(σ2θ − b)/κ of the winding angle for the

shortest path in the percolating cluster for values of the Hurst exponent

H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. In the inset, the variance

σ2θ is shown.

106


Ly

101 102 103

4(<

2 3!

b)=5

2

4

6

8

10

0-0.2-0.4-0.5-0.6-0.8-1

Ly

102

<2 3

0

2

4

Fig. 6.11: Rescaled variance 4(σ2θ − b)/κ of the winding angle for

the watershed lines in random surfaces of Hurst exponents H =

−1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. In the inset, the variance σ2θ

is shown.

107



1.0

1.1

1.2

-1.0 -0.8 -0.6 -0.4 -0.2 0.0

Fra

ctal

dim

ensi

on

df

Hurst exponent H

watershed w.a.watershed y.s.

shortest path w.a.shortest path y.s.

Fig. 6.12: Fractal dimension computed using the variance of the winding

angle (w.a.) compared with the fractal dimension measured using the yard-

stick method (y.s.) for the watershed lines and the shortest path in the cases

H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. The black lines are the

conjectured dependence of the fractal dimension of the watersheds [28] and

the shortest path Eq. (5.17) on the value of the Hurst exponent H.

fractal dimension using the yardstick method, see Fig. 6.12.

6.5.2 Direct SLE test for the shortest path

In order to test if the results obtained with the winding angle measurements

are in agreement with SLE theory, we compute the variance of the driv-

ing function of the shortest path using the vertical slit map algorithm, see

Section 2.7.4. In Fig. 6.13, we see that, as expected from [31], in the uncor-

related case the variance of the driving function displays a linear behavior,

whereas in the correlated cases it is not linear anymore, but corresponds to

a superdiffusive process. Also, if one rescales the variance through the val-

108


t0 0.1 0.2 0.3 0.4 0.5 0.6

h92 ti

0

0.5

1

1.5

2

2.50-0.1-0.25-0.4-0.55-0.7-0.85-1

t0 0.5h9

2 ti=5

0

5

10

Fig. 6.13: Variance of the driving function for the shortest path in the cases

H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0 and Ly = 1024. In the

inset, one rescales the variance by dividing the variance through the value of

κ estimated by the winding angle. The data do not collapse.

ues of κ estimated by the winding angle method, we see that the curves do

not collapse on a single line. Therefore the statistics of the shortest path in

correlated percolation are not compatible with SLE measurements.

For H = 0, one expects the shortest path to have a fractal dimension of 1

according to Fig. 5.11 and the conjecture of Eq. (5.17), which would corre-

spond to κ = 0 in the SLE theory. However, even if the largest cluster gets

more and more dense as the Hurst exponent approaches zero see Figs. 5.2

and 5.3 and conjecture of Eq.(5.8), it has some holes, that the shortest path

has to circumvent. Therefore, it cannot be a vertical straight line and its

statistics are incompatible with SLE.

109



t0 0.002 0.004 0.006 0.008 0.01

h92 ti

0

0.005

0.01

0.015

0-0.2-0.4-0.5-0.6-0.8-1

x=p

t-5 0 5

pt!

1P

(x)

0

0.2

Fig. 6.14: Variance of the driving function for the watershed lines in the cases

H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0 with Ly = 8192. In the

inset, the probability distribution of the driving function for a fixed Loewner

time is displayed for H = −1,−0.5, 0.

6.5.3 Direct SLE and left-passage probability for wa-

tersheds

We also compared the results obtained with the winding angle for watersheds

with the results obtained from the direct SLE and left-passage probability

measurements. In Fig. 6.14, the variance of the driving function is displayed,

along with the probability distribution of the driving function for a fixed

Loewner time. Contrary to the results obtained for the winding angle, in

Fig. 6.14 we do not see any clear dependence of the results of the variance of

the driving function on the value of the Hurst exponent. However, the auto-

correlation functions of the different driving functions displayed in Fig. 6.15

reveals that as soon as the landscape displays some correlations, i.e. H > −1,

the driving function displays long-range power-law time correlations, whereas

it is not the case for H = −1. Our results show that only in the case of

110


t10-4 10-3 10-2

c(t)

10-4

10-2

100

0-0.2-0.4-0.5-0.6-0.8-1

Fig. 6.15: Auto-correlation function of the driving function extracted

from the watershed lines c(τ) = 〈c(t, τ)〉τ , averaged over 50 time

steps, displayed in log-log scale for values of the Hurst exponent H =

−1,−0.8,−0.6,−0.5,−0.4,−0.2, 0.

111



51.6 1.62 1.64 1.66 1.68 1.7 1.72

Q(5

)=Q

(5m

in)

1

2

3

4

5

60-0.2-0.4-0.5-0.6-0.8-1

Fig. 6.16: Rescaled mean square deviation of the left-passage probability of

watersheds for H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0 and Ly =

4096.

112


Hurst exponent H-1 -0.8 -0.6 -0.4 -0.2 0

Est

imate

d5

1

1.2

1.4

1.6

1.8

2

fract. dim.winding anglelppdSLE

Fig. 6.17: Estimated values of κ obtained with the fractal dimension (fract.

dim.), winding angle, left-passage probability (lpp) and direct SLE (dSLE)

measurements for H = −1,−0.85,−0.7,−0.55,−0.4,−0.25,−0.1, 0. The

black solid line corresponds to the value of κ deduced from the conjectured

dependence of the fractal dimension df on H [28], using κ(H) = 8(df (H)−1).

113



uncorrelated landscapes, i.e. H = −1 watersheds have statistics compatible

with SLE, therefore recovering the result of Ref. [30].

We also did the left-passage probability measurement to estimate the value of

κ. We did not find a clear dependence of κ on the value of the Hurst exponent

H, see Fig. 6.16, as already seen for the direct SLE test. However we see

that the minimum is more pronouced for the value H = −1 corresponding

to the uncorrelated case. The results regarding watersheds are summarized

in Fig. 6.17.

6.6 Conclusion

We numerically showed that the statistics of the accessible perimeter of the

zero isoheight lines of Gaussian random landscapes are consistent with SLE

only for the range −1 ≤ H ≤ 0. Results for the fractal dimension, winding

angle and direct SLE are in agreement within error bars, see Fig. 6.4. How-

ever, for values of H near 0, we see strong finite-size effects, as expected from

the results presented in Section 5.3. This means that one can describe these

curves with a Brownian motion parametrized by a diffusivity κ.

In the two analytical limits H = −1 and H = 0, the complete perimeters are

SLE6 [29, 61] and SLE4 [35] respectively. From the duality relation, if the

complete perimeters are SLEκ the corresponding accessible perimeters are

SLEκ∗ with κκ∗ = 16 [11], see Section 2.5. Thus, in the analytical limits,

the accessible perimeters are SLE8/3 and SLE4 for H = −1 and H = 0

respectively, results that we recover in our numerical analysis.

To our knowledge, this is the first time that for an entire range of values of

the Hurst exponent H, a family of curves coupled to random landscapes is

shown to be SLE. This gives new insights in the field of fractional Gaussian

Fields in two dimensions [39] and suggests that this recent framework might

be helpful to understand correlated landscapes, see Section 4.4.

One might also wonder if these isoheight lines can be related to some random

walk process, as in the case of uncorrelated random landscapes and GFF.

For example, the isoheight lines for H = −1 and H = 0 are two specific

114

6.6. CONCLUSION 115

cases of the overruled harmonic walker [204] called respectively the critical

percolation exploration path [49, 205] and the harmonic explorer [71]. Our

work might provide insight for a future generalization of such walkers.

For isoheight lines in landscapes with positive Hurst exponent H > 0, we did

not find agreement with an SLE description of the curves. Even if we found

numerically that the driving function scales linearly in time, the driving

function displays power-law correlations, which contradicts the Markovian

properties of the driving function in the SLE theory. The results for both

positive and negative Hurst exponents are displayed in Fig. 6.18.

This work opens the possibility of applying SLE to the study of landscapes

with negative Hurst exponent H. As we have shown, many systems can be

considered from the point of view of a landscape, where isoheights play an

important role. Especially, it has been shown that zero vorticity isolines in

two-dimensional turbulence are SLE [32]. There have been also attempts to

extend this result to isolines in a generalized Navier-Stokes equation [34] to

study the conformal invariance of a larger class of turbulence problems. It

would be interesting to see if a relation between this problem and our results

can be drawn for the accessible perimeters of these contour lines.

We showed that, contrary to the case of the accessible perimeter, the shortest

path and the watershed line do not display statistics compatible with SLE

for H > −1, i.e when the system displays long-range correlations. In the

case of the shortest path, for −1 < H ≤ −3/4, from the numerical results,

it seems that the curve is no longer SLE, although there is no dependence of

the fractal dimension on the Hurst exponent H in this range.

115



Hurst exponent H-1 -0.5 0 0.5 1

Est

imate

d5

0

2

4

6




SLE (dSLE) methods for H ∈ [−1, 1]. The red crosses correpond to the

analytically known result for H = −1, i.e. percolation, and H = 0, i.e.

the GFF. We see that the results are compatible with SLE for H ≤ 0 but

incompatible for H > 0.

116

6.6. CONCLUSION 117

Pro

ble

mW

ind

ing

angl

eL

eft

Pass

age

Pro

b.

Dir

ect

SL

E

〈θ2〉

∼κ 4

ln(L

y)

Gau

ssia

nP

(θ)

κm

in=

minκQ

(κ)

Sch

ram

m’s

Pκ(φ

)

〈ξ2 t〉=

κt

Mark

ovia

nδξt

Gau

ssia

nξ t

WSH

=−

1Y

esX

Yes

Yes

Yes

Yes

Yes

SPH

=−

1Y

esY

esY

esY

esY

esY

esY

es

isoh

eigh

t

lin

esac

-

cess

ible

per

imet

er

H∈

[−1,

0]

Yes

Yes

Yes

Yes

Yes

Yes

Yes

isoh

eigh

t

lin

esac

-

cess

ible

per

imet

er

H∈

(0,1

]

Yes

,b

ut

slow

er

conve

rgen

cefo

r

larg

erva

lues

of

H

Yes

,b

ut

larg

er

dev

iati

onfo

r

larg

erva

lues

of

H

No,

no

stro

ng

dep

end

ence

on

Hfo

rκ

min

Yes

,b

ut

larg

erd

evi-

ati

on

sfr

om

Sch

ram

m’s

form

ula

for

incr

easi

ngH

No,

lin

ear

vari

an

ceb

ut

κdSLE

>4

inco

mp

ati

ble

wit

hdf<

3/2

No,

pow

er-l

aw

dec

rease

ofc(τ)

Yes

WS

H∈

(−1,

0]Y

esX

No,

no

dep

en-

den

ceonH

XN

o,

lin

ear

vari

-

an

ceb

ut

no

de-

pen

den

ceonH

No,

pow

er-l

aw

dec

rease

ofc(τ)

Yes

SP

H∈

(−1,

0]Y

esX

XX

No,

no

lin

ear

vari

an

ce

XX

Tab

le6.

1:S

um

mar

yof

the

resu

lts

obta

ined

for

the

wate

rsh

eds

(WS

),th

esh

ort

est

path

(SP

)an

dth

eis

oh

eight

lin

esw

ith

the

diff

eren

t

SL

Ete

sts:

the

win

din

gan

gle,

the

left

-pas

sage

pro

bab

ilit

yan

dth

ed

irec

tS

LE

test

s.F

or

the

win

din

gan

gle

test

,w

ete

sted

the

scali

ng

of

the

vari

ance

ofth

ew

ind

ing

angl

ean

dco

mp

ared

its

pro

bab

ilit

yd

istr

ibu

tion

wit

ha

Gau

ssia

nd

istr

ibu

tion

.F

or

the

left

-pass

age

pro

bab

ilit

y

test

,w

ees

tim

ated

theκ

from

the

min

imu

mof

the

mea

nsq

uare

dev

iati

on

an

dco

mp

are

dth

ele

ft-p

ass

age

pro

bab

ilit

ies

for

diff

eren

tp

oin

ts

inth

eu

pp

erh

alf-

pla

ne

wit

hS

chra

mm

’sfo

rmu

laE

q.

(2.2

6).

For

the

dir

ect

SL

Ete

st,

we

com

pu

ted

the

vari

an

ceof

the

dri

vin

gfu

nct

ion

,

its

Gau

ssia

nd

istr

ibu

tion

and

the

ind

epen

dan

ceof

its

incr

emen

ts.

Th

ete

sts

com

pati

ble

wit

hS

LE

are

den

ote

dby

’Yes

’(d

isp

laye

din

gree

n)

and

the

one

that

are

not

com

pat

ible

wit

ha

SL

Ed

escr

ipti

on

by

’No’

(dis

pla

yed

inre

d).

117



H κfrac κθ κLPP κdSLE

−1 2.76± 0.16 2.66± 0.01 2.69± 0.08 2.66± 0.12

−0.85 2.80± 0.24 2.76± 0.02 2.80± 0.07 2.79± 0.10

−0.7 2.96± 0.24 2.94± 0.03 2.95± 0.11 2.97± 0.17

−0.55 3.20± 0.24 3.14± 0.03 3.13± 0.15 3.29± 0.32

−0.4 3.35± 0.31 3.32± 0.03 3.27± 0.17 3.46± 0.26

−0.25 3.64± 0.28 3.45± 0.09 3.40± 0.21 3.67± 0.26

−0.1 3.90± 0.18 3.49± 0.04 3.50± 0.24 3.84± 0.20

0 3.88± 0.20 3.44± 0.18 3.52± 0.25 3.98± 0.19

0.1 3.44± 0.40 3.34± 0.27 3.56± 0.26 4.07± 0.13

0.25 3.04± 0.24 3.07± 0.07 3.59± 0.29 4.26± 0.13

0.55 1.92± 0.24 2.22± 0.08 3.62± 0.36 4.84± 0.29

0.7 1.36± 0.16 1.65± 0.15 3.62± 0.41 5.35± 0.25

0.95 0.56± 0.16 0.62± 0.31 3.63± 0.50 5.97± 0.51

Table 6.2: For the isoheight lines: Diffusion coefficient κ computed from

the fractal dimension κfrac using data from [99] for H ≤ 0 and measured

using the yardstick method for H > 0, from the winding angle κθ, from

the left-passage probability κLPP and the direct SLE method κdSLE, for the

different values of the Hurst exponent H.

118

Chapter 7

Conformal Invariance in

Graphene

In the previous chapter, we have seen that isoheight lines in random surfaces

can display statistics compatible with SLE, but that for rough surfaces they

are not necessarily SLE. We studied the case of Gaussian random surfaces

with negative and positive Hurst exponent generated by the Fourier filtering

method.

In this Chapter we are interested in real surfaces simulated using molecu-

lar dynamics. In particular, the analysis of isoheight lines in the framework

of SLE might give insights into the conformal invariant properties of the

isoheight lines in real surfaces. In this Chapter we apply the SLE frame-

work to isoheight lines in Graphene. Due to thermal fluctuations, suspended

graphene sheets exhibit correlated random deformations. On these surfaces,

one can identify isoheight lines. We show that those lines, and the area

enclosed by them are fractal and that their fractal dimensions are indepen-

dant of temperature and system size, providing evidence for scale invariance

in graphene membranes. Furthermore, we provide numerical evidence that

these contour lines are conformally invariant by showing that they have the

same statistical properties as SLEκ curves with κ = 2.24± 0.07.

This Chapter in based on:

I. Giordanelli, N. Pose, M. Mendoza, and H. J. Herrmann, Conformal invari-

ance in graphene, in progress.

119

120 CHAPTER 7. CONFORMAL INVARIANCE IN GRAPHENE

7.1 Introduction

Graphene, consisting of literally a single carbon monolayer, represents the

first instance of a truly two-dimensional material [206–208]. The existence

of strictly two-dimensional crystals was first studied theoretically by Peierls

[209,210] and Landau [211,212], who demonstrated that, in the standard har-

monic approximation [213], thermal fluctuations should destroy long-range

order, resulting in melting of any 2D lattice at any finite temperature. Fur-

thermore, it was shown that crystalline order cannot exist in two dimen-

sions [214], due to the fact that long-wavelength fluctuations destroy the

long-range order of two-dimensional crystals, and suspended two-dimensional

membranes have a tendency to be crumpled [215]. Although the theory does

not permit the existence of perfect two-dimensional crystals, it allows the

existence of nearly perfect ones embedded in the three-dimensional space,

see Fig. 7.1. Indeed, a detailed analysis of the 2D crystal problem, beyond

the harmonic approximation, has led to the conclusion that atomically thin

membranes can be stabilised through their deformation in the third dimen-

sion [216–218].

Recently, it has been possible to construct experimentally suspended

graphene membranes, exhibiting random elastic deformations involving all

three dimensions [219]. The study of the structure of graphene membranes is

of great interest for the understanding of its electronic and mechanical prop-

erties. For instance, it has been shown that reducing the standard deviation

of the heights increases the electronic mobility, and consequently increases the

electrical conductivity [220]. Furthermore, the structure of the membranes

can lead to fluctuations of the charge density and thus to the formation of

electron and hole valleys (puddles) in globally neutral samples. The existence

of stable rippled graphene membranes and the interplay between thermal

fluctuations and anharmonic coupling between bending and stretching modes

open up the question whether suspended graphene can be studied within the

theory of critical phenomena, where scale and conformal invariance play an

important role. Conformal invariance is a very powerful property to classify

theoretical models and gain insights into physical systems [165,221–223]. In

120

7.1. INTRODUCTION 121

Fig. 7.1: Graphene membrane after thermalisation. Inset: The blue points

represent carbon atoms that are above the isoheight plane. The red line

shows the extracted path along the intersection between the membrane and

the isoheight plane.

121


recent years, the SLE theory has become a very useful numerical tool to in-

vestigate conformal invariance of many physical systems. This is due to the

fact that random curves that are SLE are necessarily conformally invariant.

Although SLE was primarily developed to describe analytically curves in sta-

tistical models on lattices at criticality [29–31, 35, 48, 60, 61, 69, 70, 224, 225],

it has been successfully applied to other systems such as two-dimensional

turbulence [32], isoheight lines in growth models [226], and experimentally

grown surfaces [37]. Conformal invariant curves also possess special statisti-

cal properties of their winding angle, see Section 2.7.2. Our goal is to find out

whether scale and conformal invariance are properties of isoheight contour

lines of graphene membranes.

7.2 Methodology

Here we simulate graphene membranes using molecular dynamics with the

adaptive intermolecular reactive bond-order (AIREBO) potential [227]. The

AIREBO potential can be represented by a sum over pairwise interactions,

which includes the covalent bonding interactions, a torsion term, guarantee-

ing the correct dihedral angles, and a Lennard-Jones term, which describes

the non-bonded interaction between the atoms. This many-body potential

describes properly graphene membranes [228].

We have performed simulations of graphene membranes of size 200A×200A,

400A × 400A, and 800A × 800A, at different temperatures, i.e. 100 K, 300

K, and 600 K. The simulation time step has been set to one fs, which is

sufficiently small to capture the carbon-carbon interactions. We let the sam-

ples evolve for over 1.4 ns, 3 ns, and 13 ns, for system sizes 800A × 800A,

400A×400A, and 200A×200A, respectively. In order to control the temper-

ature, we have applied the Nose-Hoover thermostat. Each simulation starts

with a flat graphene membrane located in the x−y plane with a small random

perturbation in z-direction, to break the symmetry in the z-direction, and

zero initial velocity, i.e. zero temperature. For each temperature and sys-

tem size we performed three independent simulations with different random

122

7.3. SCALE INVARIANCE IN GRAPHENE 123

seeds for the initial fluctuations. In order to accelerate the thermalisation

of the graphene membrane, we heat up the membrane gradually. We use an

equilibration time of 0.2 ns to reach the desired temperature.

From each simulation, we extract graphene sheets in intervals of 5 ps, to avoid

correlated samples. By using this procedure, we obtain for each temperature

up to 720, 1680, and 7680 graphene sheets for 800A × 800A, 400A × 400A,

and 200A× 200A, respectively.

7.3 Scale invariance in graphene

Once the samples have reached the thermal equilibrium, we first study the

height-height correlations, see Section 4.2. For numerical convenience, in-

stead of working with the height-height correlation function c(r) = 〈h(x +

r)h(x)〉, one works with the function H(r) = 〈(h(x + r)− h(x))2〉 =

2〈h(x)2〉 − 2c(r), where h denotes the local height of the graphene mem-

brane. For self-affine surfaces, the function H(r) exhibits a power-law be-

haviour, i.e. H(|r|) ∝ |r|2H , where the exponent H is the so-called Hurst

exponent and characterises the roughness of the surface, see Subsection 4.2.

We have performed the statistical averages 〈...〉 by taking the carbon atoms

inside the inner half of the graphene membrane to avoid boundary effects.

In particular, this allows us to neglect the curvature of the sheet that might

arrise at the boundary of the sheet, where some crumpling can appear. All

the measurements have been done on the inner half of the surface.

The results presented in Fig. 7.2 indicate that the Hurst exponent is both

independent of system size and independent of temperature within error bars.

The obtained Hurst exponent is H = 0.72 ± 0.01. Our results are in good

agreement with the Hurst exponent measured by Adedpour et al. [229], where

it was found 0.6 < H < 0.74. The existence of a Hurst exponent suggests

that graphene membranes are self-affine surfaces.

The contour lines of self-affine surfaces often exhibit scale-invariance, as it is

the case for random Gaussian surfaces [12]. Assuming that we only consider

atoms of the graphene membrane that are above a certain height, and starting

123


101

102

10−1

100

101

r

H(r)

101

102

100

101

10−1

r

H(r)

100 K

300 K

600 K

1.44 ± 0.02

200 Å

400 Å

800 Å

Fig. 7.2: The function H(r), as defined in the main text, computed for the

inner carbon atoms of the graphene membrane. Main panel: The function

H(r) for different system sizes at T = 300 K. Inset: The function H(r) for

different temperatures for a system of 800A× 800A.

124

7.3. SCALE INVARIANCE IN GRAPHENE 125

100

101

10210

0

101

102

103

Stick length m

Num

ber

of s

ticks

S

200 Å400 Å800 Å

100

10210

0

102

Stick length m

Num

ber

of s

ticks

S

100 K300 K600 K

Fig. 7.3: Main panel: Fractal dimension of the isoheight contour lines com-

puted with the yardstick method for a fixed temperature T = 300 K and dif-

ferent system sizes. We find a universal fractal dimension of df = 1.28±0.05.

Inset: Fractal dimension df for different temperatures and fixed system size

of 800A× 800A. The solid line denotes the slope −df = −1.28.

from the highest point of the membrane, one can systematically lower the

height until a cluster, formed by carbon atoms, meets both opposite sides.

This gives us a spanning cluster. We extract the contour lines and the area

of these formed clusters for each graphene sheet, see Fig. 7.1, as it leads to

paths connecting two opposite sides of the membrane.

In Fig. 7.3, we show that the isoheight contour lines present a fractal dimen-

sion of df = 1.28 ± 0.05. By collapsing the data for different temperatures

and system sizes we find that the fractal dimension is universal within error

bars for temperatures up to 600 K, see Figs. 7.3.

The fractal dimension da of the area of the spanning clusters has also been

measured, see Fig. 7.4, finding a value of da = 1.82 ± 0.01, which is also

independent on the system size and temperature. Our results show that the

125


101

102

103

104

r

N

101

102

102

104

103

r

N

100 K

300 K

600 K

1.82 ± 0.01

200 Å

400 Å

800 Å

Fig. 7.4: Fractal dimension of the area enclosed by the isoheight contour

lines of the spanning clusters, computed with the box counting method. The

number N of boxes used to cover the atoms belonging to the spanning cluster

is displayed as a function of the lateral size r of the boxes. Main panel: Box

counting method for different system sizes at T = 300 K. Inset: Box counting

method for different temperatures for a system size of 800A× 800A.

isoheight contour lines of the spanning cluster and the area enclosed by them

possess scale invariant properties.

7.4 Conformal invariance and SLE properties

in graphene

Let us study the conformal invariance of the isoheight contour lines. A family

of random curves is said to be conformally invariant if their statistics is con-

served by conformal transformations, see Section 2.5. This property allows to

make analytical developments toward the solution of complex problems [165].

126

7.4. CONFORMAL INVARIANCE AND SLE PROPERTIES INGRAPHENE 127

One of the properties deduced from conformal invariance is that the winding

angle of the curves has a Gaussian distribution, see Section 2.7.2. The wind-

ing angle distribution is expected to be Gaussian of mean zero and variance

〈θ2〉 = b+ 2 (df − 1) lnL, (7.1)

where df is the fractal dimension of the curves, and b is a constant [53, 54].

We computed the variance of the winding angle, see Fig. 7.5 and found

a value df = 1.23 ± 0.03 for the fractal dimension of the curves by using

Eq.(7.1). This result is independent of the temperature and is in agreement

with the fractal dimension obtained with the yardstick method within error

bars. Furthermore, for a given length L, the distribution of the winding angle

displays the expected Gaussian behaviour, see inset of Fig. 7.5.

Since the statistics of the isoheight lines satisfy Eq. (7.1), one has a first

hint of conformal invariance. However, to get a further numerical evidence,

we will show that the system follows SLE statistics. We therefore consider

spanning paths and make the approximation that they are defined in the

upper half-plane. By the properties of SLE processes, see Paragraph 2.5, if

one can show that the isoheight lines are SLE, we have a strong numerical

indication of conformal invariance of the system.

The SLE theory relates the fractal dimension df of SLEκ curves to the value

of κ by: df = min (2, 1 + κ/8). Therefore to test the consistency of the iso-

height contour lines of the graphene sheets with SLE, one solves numerically

the Loewner differential equation, and compares the fractal dimension pre-

dicted by the diffusivity κ of the driving function with the fractal dimension

of the curves. We use the so-called slit map algorithm, see Section 2.7.4, to

extract numerically the driving function ξt, from which we study the statis-

tical properties, as displayed in Fig. 7.6. If the random curves follow SLEκ

statistics, ξt is a Brownian motion of variance κt. In Fig. 7.6, we plot the

variance of the driving function for three different temperatures. We find that

it evolves linearly in time, and that the increments of the driving function

are independent Gaussian random variables, see insets of Fig. 7.6. From this,

we conclude that our results are in agreement with an SLE description of the

isoheight contour lines. We compute the value of the parameter κ from the

127


Endpoint distance L10

1

0

0.5

1

1.5

2

〈θ〉

〈θ2〉

100 K

300 K

600 K

θ/〈θ2〉1/2-2 0 2

〈θ2〉−

1/2P(θ)

10-2

10-1

Fig. 7.5: Main panel: Mean 〈θ〉 and variance 〈θ2〉 of the winding angle θ

of the isoheight contour lines for three different temperatures T = 100 K,

T = 300 K, and T = 600 K, in a semi-log plot. The mean is approximately

zero and the variance linear in lnL. The solid line denotes the best fitting.

Inset: Rescaled probability distribution of the winding angle for L = 13.37

A, compared to a Gaussian distribution of variance one (solid line).

128

7.4. CONFORMAL INVARIANCE AND SLE PROPERTIES INGRAPHENE 129

t

0 50 100 150 200

〈ξ2 t〉−

〈ξt〉2

0

100

200

300

400

500

100K

300K

600K

x

-20 0 20

P(x)

0

0.2

t0 100

C(t)

-1

0

1

Fig. 7.6: Main panel: Variance of the driving function 〈ξ2t 〉 for three different

temperatures T = 100 K, 300 K, 600 K. The solid line shows the linear depen-

dence of the variance of a driving function with diffusivity κ = 2.24. Upper-

left inset: the probability distribution of the driving function at t = 29. The

solid line is the probability distribution of a Gaussian random variable of zero

mean and variance 2.24. Bottom-right inset: The autocorrelation function

of the increments of the driving function, averaged over the range t = 29 to

49.

129


51 1.5 2 2.5 3 3.5 4

Q(5

)=Q

min

2

4

6

8

10

100K300K600K

3=:0 0.5 1

P5(3

)

0

0.5

1

Fig. 7.7: Measured rescaled mean square deviation Q(κ)/Qmin as a function

of κ with Qmin the minimum value of Q, for temperatures T = 100K, T =

300K, and T = 600K. Inset: the measured left-passage probabilities are

compared with Schramm’s formula for κ = 2.24 (displayed as the solid line).

variance of the driving function for three different temperatures obtaining

κ = 2.24 ± 0.07. This result implies a fractal dimension of ∼ 1.28, which is

in good agreement with the fractal dimension of the curves measured with

both, the yardstick method and the winding angle. Note that the value of

the diffusion coefficient κ is independent of the temperature. We also checked

that the left-passage probability of the curves can be described by Schramm’s

formula, see Fig. 7.7, and estimated the value of κ to be κ = 2.27 ± 0.08.

In the inset of Fig. 7.7, we show that the left-passage probability satisfies

Schramm’s formula (2.26).

130

7.5. DIFFERENCE BETWEEN GRAPHENE SHEETS AND GAUSSIANRANDOM SURFACES 131

7.5 Difference between graphene sheets and

Gaussian random surfaces

As we have shown before, graphene sheets are rough surfaces characterized

by a positive Hurst exponent H = 0.72±0.01. However, as we have shown in

Section 6.3, Gaussian random surfaces with positive Hurst exponent and free

boundary conditions do not display statistics compatible with SLE. Also their

isoheights have different fractal dimensions. This suggests that graphene

sheets are not Gaussian random surfaces, therefore we will need to further

study the difference between graphene sheets and the random surfaces gen-

erated with the Fourier filtering method.

First, from a qualitative point of view, Gaussian random surfaces and

graphene sheets look differently. As characterized by Kalda in Ref. [14],

rough surfaces with H ∈ [0, 1) display peaks of different sizes and the heights

of the surfaces are expected to be unbounded, see for example Fig. 4.2(d),

whereas for graphene sheets you do not expect the surface do display peaks

of unbounded sizes, see Fig. 7.1, as the number of atoms is finite and very

strong peaks are expected to crumple the surface.

From a quantitative point of view, the difference of the fractal dimensions

of the isoheight lines between graphene and the Gaussian random surface

shows that these two surfaces display different statistical properties. In order

to further study this point, we make a Fourier analysis of graphene surfaces

and study them within the framework of random surfaces. In particular, we

investigate the properties of the Fourier coefficients uq = |uq|eiφq , and in

particular their phase φq, and modulus |uq|. By studying the distribution of

the modulus, see Fig. 7.8, we have found that graphene sheets and Gaussian

random surfaces generated with the Ffm display different distribution of the

modulus, even though both phases are uniformly distributed. This might

lead to the property that isoheight lines display SLE statistics although we

would need further studies to conclude. For example, one has to study the

independence of the Fourier coefficients, especially that the phases of the

131


juqj0 10 20 30 40 50

P(ju

qj)

0

0.05

0.1

0.15graphene sheetGaussian random sheet

Fig. 7.8: Probability distribution of the modulus |uq| of the Fourier coeffi-

cients of the Fourier transforms in the case of Gaussian random surfaces and

graphene sheets for a fixed q.

coefficients are independent and that the phases and modulus are mutually

independent.

Finally, it is worth to mention that usually random surfaces are characterized

by their Hurst exponent H. However, as we can conclude from this study, the

Hurst exponent might not be enough to describe completely the statistics of

general random surfaces as we have shown that two different surfaces with the

same Hurst exponent display different fractal dimensions of their isoheight

lines.

7.6 Conclusion

Summarising, we have shown that the isoheight contour lines of the span-

ning cluster and the area enclosed by them present scale invariant properties

132

7.6. CONCLUSION 133

independent of the temperature within the range of 100-600 K. Furthermore

using four different numerical tests, we have shown the the isoheight contour

lines of the spanning cluster are conformal invariant and can be described

by SLEκ with κ = 2.24 ± 0.08. Also we have shown that the Hurst expo-

nent is not enough to characterize fully the statistics of the isoheight lines

of random surfaces. The influence of other parameters on the statistics of

the random surfaces should be further studied, like the influence of proba-

bility distribution of the modulus and the correlations between the Fourier

coefficients.

The fact that the isoheight lines are conformal invariant relates the study

of graphene with other critical phenomena in two dimensions, like percola-

tion, and other phenomenas like turbulence [32], and opens up the question

whether there are additional shared properties that can help to understand

the physical properties of graphene.

The consequences of our findings to the electronic mobility in graphene will

be a subject of future research. Furthermore, an open question remains,

whether the scale and conformal invariant properties are typical for graphene

or are also exhibited by other two-dimensional crystals. We have made tests

on suspended silicene, a membrane made with silicon atoms, and we have

found that it crumples, preventing us to perform the same analysis.

133


134

Chapter 8

Discussion and Outlook

Schramm-Loewner Evolution is a very recent and successful mathematical

theory to describe Statistical Physics models, which has allowed to compute

exactly critical exponents. The SLE theory gives not only insights into crit-

ical exponents but also into the probability distribution of curves, allowing

for example to compute the probability for given points to be at the left or

at the right of a random curve obeying SLE statistics. In this thesis, we

studied this theory and applied it to different problems like the shortest path

in percolation, long-range correlated percolation, and graphene.

Some complex optimisation problems like computing the shortest path, as

we have studied in Chapter 3, have been shown to be analytically difficult

to track using common Statistical Physics tools. Many conjectures have

been proposed, which have all been ruled out. Therefore, the consistency

of the statistics of the shortest path in percolation with SLE opens a new

possibility to tackle the long-standing problem of its fractal dimension by

using this theory.

Recently, physicists and mathematicians have been interested in the coupling

between isoheight lines and random surfaces. It is a very promising field and

has triggered a lot of studies on the SLE properties of the isoheight lines on

these surfaces. Even if mathematically only two surfaces have been proven to

have isoheight lines obeying SLE properties, the uniformly random surface

and the Gaussian Free Field (GFF), physicists have shown empirically or nu-

135

136 CHAPTER 8. DISCUSSION AND OUTLOOK

merically that isoheight lines in real surfaces can display statistics compatible

with SLE.

In Chapter 5 we studied the critical properties of long-range correlated per-

colation characterized by the Hurst exponent H. We studied the dependence

of the fractal dimension of the percolating cluster and showed that it be-

comes more compact for increasing values of H. We also studied the cluster

size distribution at criticality and its dependence on H. Moreover the fractal

dimension of the complete and accessible perimeters of the largest cluster

were studied and found to satisfy the duality relation for every value of H.

For increasing values of H, the fractal dimension of the shortest path was

found to be decreasing towards 1, and the conductivity exponent was found

to sharply decrease for H going to 0. These results are in agreement with the

backbone of the largest cluster getting more compact for increasing H. We

also proposed quantitative relations on the dependence of the studied critical

exponents on H.

In Chapter 6, we numerically showed that the statistics of the accessible

perimeter of the zero isoheight lines of Gaussian random landscapes are con-

sistent with SLE only for −1 ≤ H ≤ 0 and not for 0 < H < 1, and recovered

the analytical results in the limiting cases H = −1 and H = 0. To our

knowledge, this is the first time that for an entire range of values of the

Hurst exponent H, a family of curves coupled to their random landscapes

is shown to be compatible with SLE. This result opens the possibility of

applying SLE to the study of landscapes with negative Hurst exponent H

within the framework of fractional Gaussian fields. Contrary to the case of

the accessible perimeter, for watersheds and the shortest path, we did not

find agreement with SLE statistics as soon as correlations were introduced

in the system, i.e. for H > −1.

In Chapter 7, we studied the properties of graphene sheets as rough sur-

faces. This allowed us to show that, for a wide range of temperatures, the

sheets display not only scale invariance but also conformal invariance. We

computed their Hurst exponent and the fractal dimension of their isoheight

136

137

lines. We also showed numerically that the statistics of the winding angle are

compatible with predictions from conformal invariance and that the isoheight

lines even display statistics compatible with SLE. Comparing with results ob-

tained in Chapter 6, we have shown that the Hurst exponent is not enough

to characterize fully the statistical properties of isoheight lines on general

rough surfaces, which opens up the question of the best characterization of

rough surfaces to describe fully the statistics of random landscapes.

As the analytical analysis of random processes from Statistical Physics using

SLE is difficult to tackle, numerical analysis of random processes like the

shortest path in percolation gives interesting insights into the field of appli-

cations of SLE. Numerical results and conjectures show that SLE has more

applications than the few rigorously proven cases, and might successfully de-

scribe scaling limits of the Loop-Erased Random Walk, Potts model, spin

glasses, watersheds, the shortest path in percolation and accessible hulls in

correlated percolation. Therefore further studies in these fields could lead

to interesting results, and in the case of the shortest path lead to the ex-

act computation of one of the last critical exponent, whose exact value is

unknown.

SLE related to random surfaces has gained an increasing interest, from a

numerical and theoretical points of view. The coupling between SLE and

random surfaces has many implications in physical systems like graphene,

turbulence, or grown surfaces. Much work can be done in characterizing ran-

dom surfaces and their coupling with SLE. Here we focused on Gaussian ran-

dom surfaces with negative Hurst exponent and suspended Graphene sheets.

But other kind of surfaces, especially displaying non Gaussian statistics, or

with other type of correlations, like correlations between the phases in the

Fourier coefficients of the Fourier transformed surface, could be studied to

better understand the coupling between SLE and real surfaces. As we have

seen in the case of Graphene sheets, real surfaces are not always Gaussian

random surfaces. Also, the Hurst exponent is not be enough to characterize

fully the properties of the random surfaces, and other parameters should be

taken into account to describe better the properties of the surface. Also,

137

138 CHAPTER 8. DISCUSSION AND OUTLOOK

the interplay between random walkers and random surfaces could lead to

interesting characterizations of surfaces. In the case of the GFF, a relation

between the GFF and a random walker has been made. Studying this relation

for other surfaces could lead to interesting developments.

Interesting results have been proven by extending the SLE theory to other

driving functions in order to describe new kinds of processes like biased or

branching processes for example. This can be done by changing the driving

function, adding a drift term to the usual Brownian motion in the SLE(κ, ρ)

theory [59, 230] or replacing for example the Brownian motion by Levy pro-

cesses [231, 232] to describe branching, or even by changing the derivative

operator in the Loewner Equation [233]. However we have seen that in some

of the studied cases, like in the case of watersheds for correlated surfaces, the

underlying driving process displays correlations. Studying correlated pro-

cesses might give interesting insights into some common Statistical Physics

problems and lead to new results.

138

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