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Large deviations of Schramm-Loewner evolutions: A survey
Yilin Wang
Massachusetts Institute of [email protected]
January 22, 2021
Abstract
These notes survey the first and recent results on large
deviations of Schramm-Loewner evolutions (SLE) with the emphasis on
interrelations among rate functionsand applications to complex
analysis. More precisely, we describe the large deviationsof SLEκ
when the κ parameter goes to zero in the chordal and multichorcal
caseand to infinity in the radial case. The rate functions, namely
Loewner energy andLoewner-Kufarev energy, are of interest from the
geometric function theory viewpointand closely related to the
Weil-Petersson class of quasicircles and real rational
functions.
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Contents
1 Introduction 31.1 Large deviation principle . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 41.2 Chordal Loewner chain
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3
Chordal SLE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 7
2 Large deviations of chordal SLE0+ 82.1 Chordal Loewner energy
and large deviations . . . . . . . . . . . . . . . . . 82.2
Reversibility of Loewner energy . . . . . . . . . . . . . . . . . .
. . . . . . . 92.3 Loop energy and Weil-Petersson quasicircles . .
. . . . . . . . . . . . . . . . 10
3 Cutting, welding, and flow-lines 133.1 Cutting-welding
identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
133.2 Flow-line identity . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 163.3 Applications . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 18
4 Large deviations of multichordal SLE0+ 194.1 Multichordal SLE
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
194.2 Real rational functions and Shapiro’s conjecture . . . . . .
. . . . . . . . . 214.3 Large deviations of multichordal SLE0+ . .
. . . . . . . . . . . . . . . . . . 224.4 Minimal potential . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5 Large deviations of radial SLE∞ 265.1 Radial SLE . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.2
Loewner-Kufarev equations . . . . . . . . . . . . . . . . . . . . .
. . . . . . 275.3 Loewner-Kufarev energy and large deviations . . .
. . . . . . . . . . . . . . 28
6 Foliation of Weil-Petersson quasicircles 296.1 Whole-plane
Loewner evolution . . . . . . . . . . . . . . . . . . . . . . . . .
306.2 Energy duality . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 31
7 Summary 33
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1 Introduction
The Schramm-Loewner evolution (SLE) is a model for conformally
invariant random fractalcurves in the plane, introduced by Schramm
[Sch00] by combining Loewner’s classicaltheory [Loe23] for the
evolution of planar slit domains with stochastic analysis.
Schramm’sSLE is a one-parameter family of probability measures on
non-self-crossing curves, indexedby κ ≥ 0, and thus denoted by
SLEκ. When κ > 0, these curves are fractal, andthe parameter κ
reflects the curve’s roughness. SLEs play a central role in the
recentdevelopment of 2D random conformal geometry. For instance,
they describe interfaces inconformally invariant systems arising
from scaling limits of discrete models in statisticalphysics, which
was also Schramm’s original motivation, see, e.g.,
[LSW04,Sch07,Smi06,SS09].Through their relationship with critical
statistical physics models, SLEs are also closelyrelated to
conformal field theory (CFT) whose central charge is a function of
κ, see,e.g., [BB03,Car03,FW03,FK04,Dub15,Pel19].Large deviation
principle describes the probability of rare events of a given
family ofprobability measures on an exponential scale. The
formalization of the general frameworkof large deviation was
developed by (or commonly attributed to) Varadhan [Var66]. Agreat
deal of mathematics has been developed since then. Large deviations
estimates haveproved to be the crucial tool required to handle many
questions in statistics, engineering,statistical mechanics, and
applied probability.In these notes, we give a minimalist account of
basic definitions and ideas from boththe SLE and large deviation
theory, only sufficient for considering the large deviations ofSLE.
We by no means attempt to give a thorough reference to the
background of thesetwo theories. Our approach focuses on how large
deviation consideration propels to thediscovery (and rediscovery)
of interesting deterministic objects, including Loewner
energy,Loewner-Kufarev energy, Weil-Petersson quasicircles, real
rational functions, foliations, etc.,and the interplay among them.
Unlike the fractal or discrete nature of objects consideredin
random conformal geometry, these deterministic objects, arising
from the κ → 0+ or∞ large deviations (on which the rate function is
finite), are more regular. However, theystill capture the essence
of many coupling results in random conformal geometry. As forthe
large deviation literature, these results also show new ways to
apply this powerfulmachinery. We summarize and compare the
quantities and theorems from both randomconformal geometry and the
finite energy world in the last section. Impatient readers mayskip
to the end to get a flavor about the closeness of the analogy.The
main theorems presented here are collected from
[Wan19a,Wan19b,RW19,VW20a,PW20,APW20,VW20b]. Compared to those
research papers, we choose to outline theintuition behind the
theorems and sometimes omit proofs or only present the proof in
asimpler case to illustrate the idea. These lecture notes are
written based on two lectureseries that I gave at the joint webinar
of Tsinghua-Peking-Beijing Normal Universities andRandom Geometry
and Statistical Physics online seminars during the Covid-19 time.
Iwould like to thank the organizers for the invitation and the
online lecturing experienceunder the pandemic’s unusual situation
and is supported by the NSF grant DMS-1953945.
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1.1 Large deviation principle
We consider first a simple example to illustrate the concept of
large deviations. LetX ∼ N (0, σ2) be a real, centered Gaussian
random variable of variance σ2. The densityfunction of X is given
by
pX(x) =1√
2πσ2exp(− x
2
2σ2 ).
Let ε > 0,√εX ∼ N (0, σ2ε). As ε → 0+,
√εX converges almost surely to 0, so the
probability measure p√εX on R converges to the Dirac measure
δ0.Let M > 0. We are interested in the rare event {
√εX ≥M} which has probability
P(√εX ≥M) = 1√
2πσ2ε
∫ ∞M
exp(− x2
2σ2ε)dx.
To quantify how rare this event happens when ε→ 0+, we have
ε logP(√εX ≥M) = ε log
(1√
2πσ2ε
∫ ∞M
exp(− x2
2σ2ε)dx)
= −12ε log(2πσ2ε) + ε log
∫ ∞M
exp(− x2
2σ2ε)dx
ε→0+−−−−→ −M2
2σ2 =: −IX(M) = − infx∈[M,∞)IX(x).
The function IX : x 7→ x2/2σ2 is called the large deviation rate
function of the family ofrandom variables (
√εX)ε>0.
Now let us state the large deviation principle more precisely.
Let X be a Polish space, Bits Borel σ-algebra, {µε}ε>0 a family
of probability measures on (X ,B). To reduce possiblemeasurability
questions, all probability spaces are assumed to have been
completed.
Definition 1.1. A rate function is a lower semicontinuous
mapping I : X → [0,∞], namely,for all α ≥ 0, the sub-level set {x :
I(x) ≤ α} is a closed subset of X . A good rate functionis a rate
function for which all the sub-level sets are compact subsets of X
.
Definition 1.2. We say that a family of probability measures
{µε}ε>0 on (X ,B) satisfiesthe large deviation principle of rate
function I if for all open sets O ∈ B and closed setF ∈ B,
limε→0+
ε logµε(O) ≥ − infx∈O
I(x); limε→0+
ε logµε(F ) ≤ − infx∈F
I(x).
Note that if a large deviation rate function exists then it is
unique, see, e.g., [Din93, Lem. 1.1].
Remark 1.3. If A ∈ B and infx∈Ao I(x) = infx∈A I(x) (we call
such Borel set a continuityset of I), then the large deviation
principle gives
limε→0+
ε logµε(A) = − infx∈A
I(x).
Remark 1.4. It is not hard to check that the distribution of
{√εX}ε>0 satisfies the large
deviation principle with good rate function IX .
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The reader should mind carefully that large deviation results
depend on the topologyinvolved which can be a subtle point. On the
other hand, it follows from the definitionthat the large deviation
principle transfers nicely through continuous functions:
Theorem 1.5 (Contraction principle [DZ10, Thm. 4.2.1]). If X ,Y
are two Polish spaces,θ : X → Y a continuous function {µε}ε>0 a
family of probability measures on X , satisfyingthe large deviation
principle with good rate function I : X → [0,∞]. Let I ′ : Y →
[0,∞] bedefined as
I ′(y) := infx∈θ−1{y}
I(x).
Then the family of pushforward probability measures (θ∗µε)ε>0
on Y satisfies the largedeviation principle with good rate function
I ′.
The next result is the large deviation principle of the scaled
Brownian path. Let T ∈ (0,∞].We recall that the Dirichlet energy of
a real-valued function (t 7→Wt) ∈ C0[0, T ] is
IT (W ) :=12
∫ T0|∂tWt|2 dt, (1.1)
if W is absolutely continuous, and set to equal ∞ otherwise.
Equivalently, we can write
IT (W ) = supΠ
k−1∑i=0
(Wti+1 −Wti)2
2(ti+1 − ti)(1.2)
where the supremum is taken over all k ∈ N and all partitions {0
= t0 < t1 < · · · < tk ≤ T}.Note that the sum on the
right-hand side of (1.2) is the Dirichlet energy of the
linearinterpolation of W from its values at (t0, · · · , tk) which
is set to be constant on [tk, T ].
Theorem 1.6. (Schilder; see, e.g., [DZ10, Ch. 5.2]) Fix T ∈
(0,∞). The family of processes{(√εBt)t∈[0,T ]}ε>0 satisfies the
following large deviation principle in (C0[0, T ], ‖·‖∞) with
good rate function IT .
Remark 1.7. We note that Brownian path has almost surely
infinite Dirichlet energy, i.e.,IT (B) =∞. In fact, W has finite
Dirichlet energy implies that W is 1/2-Hölder, whereasBrownian
motion is only a.s. (1/2− δ)-Hölder for δ > 0. However,
Schilder’s theorem showsthat Brownian motion singles out the
Dirichlet energy which quantifies, as ε → 0+, thedensity of
Brownian path around a deterministic function W . In fact, let Oδ(W
) be the δneighborhood of W in C0[0, T ]. From the monotonicity of
infW̃∈Oδ(W ) IT (W̃ ) in δ, Oδ(W )is a continuity set for IT with
exceptions for at most countably many δ. Hence, we have
−ε logP(√εB ∈ Oδ(W ))
ε→0+−−−−→ infW̃∈Oδ(W )
IT (W̃ )δ→0+−−−−→ IT (W ). (1.3)
Or, written with some abuse:
P(√εB stays close to W ) ∼ε→0+ exp(−IT (W )/ε). (1.4)
We now give some heuristics to show that the Dirichlet energy
appears naturally as thelarge deviation rate function of the scaled
Brownian motion. Fix 0 = t0 < t1 < . . . < tk ≤ T .The
finite dimensional marginals of the Brownian motion (Bt0 , . . . ,
Btk) gives a family
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of independent Gaussian random variables (Bti+1 −Bti)0≤i≤k−1
with variances (ti+1 − ti)respectively. Multiplying the Gaussian
vector by
√ε, we obtain the large deviation principle
of the finite dimensional marginal with rate function∑k−1i=0
(Wti+1−Wti )2
2(ti+1−ti) from Remark 1.4,Theorem 1.5, and the independence of
the family of increments. Approximating theBrownian motion on the
finite interval [0, T ] by the linearly interpolated function from
itsvalue at times ti, it suggests that the scaled Brownian paths
satisfy the large deviationprinciple of rate function the supremum
of the rate function of all of its finite dimensionalmarginals
which then turns out to be the Dirichlet energy by (1.2).A rigorous
proof of Schilder’s theorem uses the Cameron-Martin theorem which
allowsgeneralization to any abstract Wiener space. Namely, the
associated family of Gaussianmeasures scaled by
√ε satisfies the large deviation principle with rate function
1/2 times
its Cameron-Martin norm. See, e.g., [DS89, Thm. 3.4.12].
Remark 1.8. Dawson-Gärtner theorem (see, e.g., [DZ10, Thm.
4.6.1]) on the large devia-tion principle of projective limits
shows that Schilder’s theorem also holds for the infinitetime
Brownian motion, when C0[0,∞) is endowed with the topology of
uniform convergenceon compact sets (since it is the projective
limit of C0[0, T ] as T →∞).
1.2 Chordal Loewner chain
The description of SLE is based on the Loewner transform, a
deterministic procedure thatencodes a non-self-intersecting curve
on a 2-D domain into a driving function. In thissurvey, we use two
types of Loewner chain: the chordal Loewner chain in (D;x, y),
whereD is a simply connected domain with two distinct boundary
points x (starting point) andy (target point); and later in Section
5, the radial Loewner chain in D targeting at aninterior point. The
definition is invariant under conformal maps (biholomorphic
functions).Hence by Riemann mapping theorem it suffices to describe
in the chordal case when(D;x, y) = (H; 0,∞), and in the radial case
when D = D, targeting at 0. Throughout thearticle, H = {z ∈ C :
Im(z) > 0} is the upper halfplane, H∗ = {z ∈ C : Im(z) < 0}
is thelower halfplane, D = {z ∈ C : |z| < 1} is the unit disk,
and D∗ = {z ∈ C : |z| > 1} ∪ {∞}.Let us start with this chordal
Loewner description of a continuous simple curve γ from0 to ∞ in H.
We parameterize the curve by the halfplane capacity. More
precisely, γis continuously parametrized by R+, with γ0 = 0, γt → ∞
as t → ∞, in the way suchthat the unique conformal map gt from H r
γ[0,t] onto H with the expansion at infinitygt(z) = z + o(1)
satisfies
gt(z) = z + 2t/z + o(1/z). (1.5)
The coefficient 2t is the halfplane capacity of γ[0,t]. One
shows that gt can be extendedby continuity to the boundary point γt
and that the real-valued function Wt := gt(γt) iscontinuous. This
function W is called the driving function of γ.Conversely, the
chordal Loewner chain in (H; 0,∞) driven by a continuous
real-valuedfunction W ∈ C0[0,∞) is the family of conformal maps
(gt)t≥0, obtained by solving theLoewner equation for each z ∈
H,
∂tgt(z) =2
gt(z)−Wtwith initial condition g0(z) = z. (1.6)
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The solution t 7→ gt(z) to (1.6) is defined up to the swallowing
time
τ(z) := sup{t ≥ 0: infs∈[0,t]
|gs(z)−Ws| > 0}
of z, and we obtain a family of growing hulls Kt = {z ∈ H : τ(z)
≤ t}. The solution gt of(1.6) satisfies the expansion (1.5) near ∞
and is the unique conformal map from HrKtonto H with expansion z+
o(1) as z →∞. Moreover, Kt has halfplane capacity 2t. ClearlyKt and
gt uniquely determine each other. We list a few properties of the
Loewner chain.
• If W is the driving function of a simple chord γ, then we have
Kt = γ[0,t], and thesolution gt of (1.6) is exactly the conformal
map constructed from γ as in (1.5).• The imaginary axis iR+ is
driven by W ≡ 0.• (Additivity) Let (Kt)t≥0 be the family of hulls
generated by the driving function W .
Fix s > 0, the driving function generating
(gs(Kt+srKs)−Ws)t≥0 is t 7→Ws+t−Ws.• (Scaling) Fix λ > 0, the
driving function generating the scaled (and reparameterized)family
of hulls (λKλ−2t)t≥0 is t 7→ λWλ−2t.• Not every continuous driving
function arises from a simple chord. It is unknownhow to
characterize analytically the class of functions which generate
simple curves.Sufficient conditions exist, such as when W is
1/2-Hölder with Hölder norm strictlyless than 4 [MR05,Lin05].
1.3 Chordal SLE
We now very briefly review the definition and relevant
properties of chordal SLE. Forfurther SLE background, we refer the
readers to, e.g., [Law05,Wer04b]. The chordalSchramm-Loewner
evolution of parameter κ in (H; 0,∞), denoted by SLEκ, is the
randomnon-self-intersecting curve tracing out the hulls (Kt)t≥0
generated by
√κB via the Loewner
transform, where B is the standard Brownian motion and κ ≥ 0.
SLEs describe theinterfaces in the scaling limit of various
statistical mechanics models, e.g.,
• SLE2 ↔ Loop-erased random walk [LSW04];• SLE8/3 ↔
Self-avoiding walk (conjecture);• SLE3 ↔ Critical Ising model
interface [Smi10];• SLE4 ↔ Level line of the Gaussian free field
[SS09];• SLE6 ↔ Critical independent percolation interface
[Smi06];• SLE8 ↔ Contour line of uniform spanning tree [LSW04].
The reason that SLE curves describe those interfaces arising
from conformally invariantsystems is that they are the unique
random Loewner chain that are scaling-invariant andsatisfy the
domain Markov property. More precisely, for λ > 0, the law is
invariant underthe scaling transformation (then reparametrize by
capacity)
(Kt)t≥0 7→ (Kλt := λKλ−2t)t≥0
and for all s ∈ [0,∞), if one defines K(s)t = gs(Ks+trKs)−Ws,
where (Kt) is driven by W ,then (K(s)t )t≥0 has the same
distribution as (Kt)t≥0 and is independent of σ(Wr : r ≤ s).
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In fact, these two properties on (Kt) translate into the
independent stationary incrementsproperty (i.e., being a continuous
Lévy process) and the invariance under Wt λWλ−2tof the driving
function. Multiples of Brownian motions are the only continuous
processessatisfying these two properties.The scaling-invariance
makes it possible to define SLE in other simply connected
domains(D;x, y) as the preimage of SLE in (H; 0,∞) by a conformal
map ϕ : D → H takingrespectively the boundary points x, y to 0,∞
(another choice of ϕ̃ equals λϕ for someλ > 0). The chordal SLE
is therefore conformally invariant from the definition.
Remark 1.9. The SLE0 in (H; 0,∞) is simply the Loewner chain
driven by W ≡ 0,namely the imaginary axis iR+. It implies that the
SLE0 in (D;x, y) equals ϕ(iR+) (i.e.,the hyperbolic geodesic in D
connecting x and y).
SLE curves exhibit phase transitions depending on the value of
κ:
Theorem 1.10 ([RS05]). For κ ∈ [0, 4], SLEκ is almost surely a
simple curve starting at0. For κ ∈ (4, 8), SLEκ is almost surely
traced out by a self-touching curve. For κ ∈ [8,∞),SLEκ is almost
surely a space-filling curve. Moreover, for all κ ≥ 0, the trace of
SLE goesto ∞ as t→∞ almost surely.
2 Large deviations of chordal SLE0+
2.1 Chordal Loewner energy and large deviations
We first specify the topology on simple chords in our large
deviation result (Theorem 2.4).
Definition 2.1. The Hausdorff distance dh of two compact sets
F1, F2 ⊂ D is defined as
dh(F1, F2) := inf{ε ≥ 0
∣∣∣ F1 ⊂ ⋃x∈F2
Bε(x) and F2 ⊂⋃x∈F1
Bε(x)},
where Bε(x) denotes the Euclidean ball of radius ε centered at x
∈ D. We then define theHausdorff metric on the set of closed
subsets of D via the pullback by a conformal mapD → D. Although the
metric depends on the choice of the conformal map, the
topologyinduced by dh is canonical, as conformal automorphisms of D
are fractional linear functions,that are uniformly continuous on D.
We endow the space X (D;x; y) of unparametrizedsimple chords in
(D;x; y) with the relative topology induced by the Hausdorff
metric.
Definition 2.2. The Loewner energy of a chord γ ∈ X (D;x, y) is
defined as the Dirichletenergy (1.1) of its driving function,
ID;x,y(γ) := IH;0,∞(ϕ(γ)) := I∞(W ), (2.1)
where ϕ is any conformal map from D to H such that ϕ(x) = 0 and
ϕ(y) =∞, W is thedriving function of ϕ(γ) and I∞(W ) is the
Dirichlet energy as defined in (1.1) and (1.2).
Note that the definition of ID;x,y(γ) does not depend on the
choice of ϕ. In fact, since twochoices differ only by
post-composing by a scaling factor. From the scaling property of
the
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Loewner driving function,W changes to t 7→ λWλ−2t which has the
same Dirichlet energy asW . The Loewner energy ID;x,y(γ) is
non-negative and minimized by the hyperbolic geodesicsince the
driving function of ϕ(η) is just the constant function W ≡ 0, so
ID;x,y(η) = 0.
Theorem 2.3 ([Wan19a, Prop. 2.1]). If I∞(W ) 0, with ε = κ.
Indeed, the following result is proved in [PW20] which strengthensa
similar result in [Wan19a]. As we are interested in the 0+ limit,
we only consider small κwhere the trace γκ := ∪t≥0Kt of SLEκ is
almost surely in X (D;x, y).
Theorem 2.4 ([PW20, Thm. 1.4]). The family of distributions
{Pκ}κ>0 on X (D;x, y) ofthe chordal SLEκ curves satisfies the
large deviation principle with good rate function ID;x,y.That is,
for any open set O and closed set F of X (D;x, y), we have
limκ→0+
κ logPκ[γκ ∈ O] ≥ − infγ∈O
ID;x,y(γ),
limκ→0+
κ logPκ[γκ ∈ F ] ≤ − infγ∈F
ID;x,y(γ),
and the sub-level set {γ ∈ X (D;x, y) | ID;x,y(γ) ≤ c} is
compact for any c ≥ 0.
We note that the Loewner transform mapping continuous driving
function to the union ofhulls it generates, endowed with the
Hausdorff metric, is not continuous and the contractionprinciple
(Theorem 1.5) does not apply. Therefore Schilder’s theorem does not
implytrivially the large deviation principle of SLE0+. This result
thus requires some work,see [PW20] for more details.
Remark 2.5. As Remark 1.7, we emphasize that finite energy
chords are more regularthan SLEκ curves for any κ > 0. In fact,
as we will see in Theorem 2.14, finite energy chordis part of a
Weil-Petersson quasicircle which is rectifiable, see Remark 2.17.
On the otherhand, Beffara [Bef08] shows that for 0 < κ ≤ 8, SLEκ
has Hausdorff dimension 1 + κ/8 > 1,thus even the length of a
piece of SLEκ curve is not defined when κ > 0.
2.2 Reversibility of Loewner energy
Given that SLEκ curves arise as scaling limits of interfaces in
lattice models from statisticalphysics, it was natural to
conjecture that chordal SLE is reversible, i.e., the time-reversal
ofa chordal SLEκ from a to b in D is a chordal SLEκ from b to a in
D after reparametrization,at least for specific κ values. It was
first settled by Zhan [Zha08b] in the case of the simplecurves κ ∈
[0, 4] via rather non-trivial couplings of both ends of the SLE
path. See alsoDubédat’s commutation relations [Dub07], and Miller
and Sheffield’s approach based on theGaussian Free Field
[MS16a,MS16b,MS16c] that also provides a proof in the
non-simplecurve case when κ ∈ (4, 8].
Theorem 2.6 (SLE reversibility [Zha08b]). For κ ∈ [0, 4], the
distribution of the trace γκof SLEκ in (H, 0,∞) coincides with that
of its image under ι : z → −1/z.
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We deduce from Theorem 2.4 and Theorem 2.6 the following
result.
Theorem 2.7 (Energy reversibility [Wan19a]). We have ID;x,y(γ) =
ID,y,x(γ) for anychord γ ∈ X (D;x, y).
Proof. Without loss of generality, we assume that (D;x, y) = (H;
0,∞). We want to showthat IH;0,∞(γ) = IH;0,∞(ι(γ)).Let δn be a
sequence of numbers converging to 0+, such that Oδn(γ) = {γ̃ ∈ X
(H; 0,∞) :θ∗dh(γ, γ̃) < δn} is a continuity set for IH;0,∞. The
sequence exists since there are at mosta countable number of
exceptions of δ such that Oδ(γ) is not a continuity set. In fact,
forδ < δ′, we have the inclusion Oδ(γ) ⊂ Oδ(γ) ⊂ Oδ′(γ) and
infγ̃∈Oδ′ (γ)
IH;0,∞(γ̃) ≤ infγ̃∈Oδ(γ)
IH;0,∞(γ̃) ≤ infγ̃∈Oδ(γ)
IH;0,∞(γ̃).
Therefore the function inf γ̃∈Oδ(γ) IH;0,∞(γ̃) has a jump at
points δ > 0 for which Oδ(γ) isnot a continuity set. The
monotonicity of δ 7→ inf γ̃∈Oδ(γ) IH;0,∞(γ̃) shows that we have
atmost countably many such discontinuity points. From Remark 1.3
and Theorem 2.6, wehave
limκ→0+
κ logPκ(γκ ∈ Oδn(γ)) = − infγ̃∈Oδn (γ)
IH;0,∞(γ̃), (2.2)
which tends to −IH;0,∞(γ) as n→∞ from the lower-semicontinuity
of IH;0,∞.We now specify the conformal map ϑ : H→ D sending i to 0
to define the Hausdorff metricin H as in Definition 2.1. This
choice makes ϑ ◦ ι ◦ ϑ−1 = − IdD. In other words, ι inducesan
isometry on closed sets of H. Theorem 2.6 then shows that
Pκ(γκ ∈ Oδn(γ)) = Pκ(ι(γκ) ∈ ι(Oδn(γ))) = Pκ(γκ ∈
Oδn(ι(γ))).
We obtain the claimed energy reversibility by applying (2.2) to
ι(γ).
Remark 2.8. This proof presented above is different from
[Wan19a] but very close in thespirit. We used here Theorem 2.4 from
the recent work [PW20], whereas the original proofin [Wan19a] used
the more complicated left-right passing events without the strong
largedeviation result at hand.
Remark 2.9. The energy reversibility is a result about
deterministic chords. However, fromthe definition alone, the
reversibility is not obvious as the Loewner evolution is
directional.To illustrate it, consider the example of a driving
function W with finite Dirichlet energythat is constant after time
1 (and contributes 0 energy). From the additivity property
ofdriving function, γ[1,∞) is the hyperbolic geodesic in Hr γ[0,1].
The reversed curve ι(γ) is achord starting with an analytic curve
which deviates from the imaginary axis. Thereforeunlike γ, the
energy of ι(γ) typically spreads over the whole time interval
R+.
2.3 Loop energy and Weil-Petersson quasicircles
We now generalize the Loewner energy to Jordan curves (simple
loops) on the Riemannsphere Ĉ = C ∪ {∞}. This generalization will
show that the Loewner energy exhibits more
10
-
0γ
z 7→ z2
0∞0
γ
H
C \ R+
Figure 1: From chord in (H; 0,∞) to a Jordan curve.
symmetries (Theorem 2.10). Moreover, an equivalent description
(Theorem 2.14) of theloop energy will give an analytic proof of
those symmetries including the reversibility.Let γ : [0, 1]→ Ĉ be
a parametrized Jordan curve with the marked point γ(0) = γ(1).
Forevery ε > 0, γ[ε, 1] is a chord connecting γ(ε) to γ(1) in
the simply connected domainĈ r γ[0, ε]. The rooted loop Loewner
energy of γ rooted at γ(0) is defined as
IL(γ, γ(0)) := limε→0
IĈrγ[0,ε],γ(ε),γ(0)(γ[ε, 1]).
The loop energy generalizes the chordal energy. In fact, let η
be a simple chord in(Cr R+, 0,∞) and we parametrize γ = η ∪ R+ in a
way such that γ[0, 1/2] = R+ ∪ {∞}and γ[1/2, 1] = η. Then from the
additivity of chordal energy,
IL(γ,∞) =ICrR+,0,∞(η) + limε→0 IĈrγ[0,ε],γ(ε),γ(0)(γ[ε,
1/2])
=ICrR+,0,∞(η),
since γ[ε, 1/2] is contained in the hyperbolic geodesic between
γ(ε) and γ(0) in Ĉ r γ[0, ε]for all 0 < ε < 1/2, see Figure
1. Steffen Rohde and I proved the following result.
Theorem 2.10 ([RW19]). The loop energy does not depend on the
root.
We do not present the original proof of this theorem since it
will follow immediate fromTheorem 2.14, see Remark 2.16.
Remark 2.11. From the definition, the loop energy IL is
invariant under the Möbiustransformations of Ĉ, and IL(γ) = 0 if
and only if γ is a circle (or a line).
Remark 2.12. The loop energy is presumably the large deviation
rate function of SLE0+loop measure on Ĉ constructed in [Zha20]
(see also [Wer08,BD16] for the constructionwhen κ = 8/3 and 2).
However, the conformal invariance of the SLE loop measures
impliesthat they have infinite total mass and has to be normalized
properly for considering largedeviations. We do not claim it here
and think it is an interesting question to work out.However, these
ideas will serve as heuristics to deduce results for finite energy
loops inSection 3.
In [RW19] we also showed that if a Jordan curve has finite
energy, then it is a quasicircle,namely the image of a circle or a
line under a quasiconformal map of C (and a quasiconformalmap is a
homeomorphism that maps locally small circles to ellipses with
uniformly bounded
11
-
eccentricity). However, not all quasicircles have finite energy
since they may have Hausdorffdimension larger than 1. The natural
question is then to identify the family of finite
energyquasicircles. The answer is surprisingly a family of
so-called Weil-Petersson quasicircles,which has been studied by
both physicists and mathematicians since the eighties,
includingBowick, Rajeev, Witten, Nag, Verjovsky, Sullivan, Cui,
Tahktajan, Teo, Shen, Sharon,Mumford, Pommerenke, González, Bishop,
etc., and is still an active research area. See theintroduction of
the recent preprint [Bis19] for a summary and a list of (currently)
morethan twenty equivalent definitions of very different nature.We
will use the following definition. The class of Weil-Petersson
quasicircles is preservedunder Möbius transformation, so without
loss of generality, we only spell out the definitionof a bounded
Weil-Petersson quasicircle. Let γ be a bounded Jordan curve, and
write Ωand Ω∗ be the connected components of Ĉ r γ. Let f be a
conformal map from D onto thebounded component Ω, and h a conformal
map from D∗ onto the unbounded componentΩ∗ fixing ∞.
Definition 2.13. The bounded Jordan curve γ is a Weil-Petersson
quasicircle if and onlyif the following equivalent conditions
hold:
1. DD(log |f ′|) := 1π∫D |∇ log |f ′| (z)|
2 dA(z) = 1π∫D |f ′′(z)/f ′(z)|
2 dA(z)
-
then shows that Weil-Petersson quasicircles are asymptotically
smooth, namely, chord-arcwith local constant 1: for all x, y on the
curve, the shorter arc γx,y between x and y satisfies
lim|x−y|→0
length (γx,y)/|x− y| = 1.
(Chord-arc means length(γx,y)/|x− y| is uniformly bounded.)
The connection between Loewner energy and Weil-Petersson
quasicircles goes further: Notonly the Loewner energy identifies
Weil-Petersson quasicircles from its finiteness, but isalso closely
related to the Kähler structure on T0(1), the Weil-Petersson
Teichmüller space,naturally associated to the class of
Weil-Petersson quasicircles. In fact, the right-handside of (2.3)
coincides with the universal Liouville action introduced by
Takhtajan andTeo [TT06] and shown by them to be a Kähler potential
of the Weil-Petersson metric, whichis the unique homogeneous Kähler
metric on T0(1) up to a scaling factor. To summarize,we obtain:
Corollary 2.18. The Loewner energy is a Kähler potential of the
Weil-Petersson metricon T0(1).
We do not enter into further details as it goes beyond the scope
of large deviations that wechoose to focus on here. Another reason
is that this unexpected link still lacks a betterexplanation and we
definitely need to understand further the relation between SLEs
andthe Kähler structure on T0(1).
3 Cutting, welding, and flow-lines
Pioneering works [Dub09b,She16,MS16a] on couplings between SLEs
and Gaussian freefield (GFF) have led to many remarkable
applications in the study of 2D random conformalgeometry. These
couplings are often speculated from the link to the discrete
models.In [VW20a], Viklund and I provided another view on these
couplings through the lens oflarge deviations by showing the
interplay between Loewner energy of curves and Dirichletenergy of
functions defined in the complex plane (which is the large
deviation rate functionof scaled GFF). These results are analogous
to the SLE/GFF couplings, but the proofs areremarkably short and
use only analytic tools without any of the probabilistic
models.
3.1 Cutting-welding identity
To state the result, we write E(Ω) for the space of real
functions on a domain Ω ⊂ C withweak first derivatives in L2(Ω) and
recall the Dirichlet energy of ϕ ∈ E(Ω) is
DΩ(ϕ) :=1π
∫Ω|∇ϕ|2dA(z).
Theorem 3.1 (Cutting [VW20a, Thm. 1.1]). Suppose γ is a Jordan
curve through ∞ andϕ ∈ E(C). Then we have the identity:
DC(ϕ) + IL(γ) = DH(u) +DH∗(v), (3.1)
13
-
whereu = ϕ ◦ f + log
∣∣f ′∣∣ , v = ϕ ◦ h+ log ∣∣h′∣∣ , (3.2)where f and h map
conformally H and H∗ onto, respectively, H and H∗, the two
componentsof Cr γ, while fixing ∞.
The function ϕ ∈ E(C) implies that it has vanishing mean
oscillation. The John-Nirenberginequality (see, e.g., [Gar07,
Thm.VI.6.4]) shows that e2ϕ is locally integrable and
strictlypositive. In other words, e2ϕdA(z) defines a σ-finite
measure supported on C, absolutelycontinuous with respect to
Lebesgue measure dA(z). The transformation law (3.2) ischosen such
that e2udA(z) and e2vdA(z) are the pullback measures by f and h of
e2ϕdA(z),respectively.We explain first why we consider this theorem
as a deterministic analog of an SLE/GFFcoupling. Note that we do
not make rigorous statement here and only argue heuristically.In
fact, Sheffield’s quantum zipper theorem couples SLEκ curves with
quantum surfacesvia a cutting operation and as welding curves
[She16,DMS14]. A quantum surface is adomain equipped with a
Liouville quantum gravity (
√κ-LQG) measure, defined using
a regularization of e√κΦdA(z), where
√κ ∈ (0, 2), and Φ is a Gaussian field with the
covariance of a free boundary GFF. The analogy is outlined in
the table below.
SLE/GFF with κ� 1 Finite energySLEκ loop Jordan curve γ with
IL(γ)
-
On the other hand the independence between Φ1 and Φ2 gives
“ limκ→0−κ logP(
√κΦ1 stays close to 2u,
√κΦ2 stays close to 2v)
= DH(u) +DH∗(v)”.
We obtain the identity (3.1) heuristically.We now present our
short proof of Theorem 3.1 in the case where γ is smooth andϕ ∈ C∞c
(C) to illustrate the idea. The general case follows from an
approximationargument, see [VW20a] for the complete proof.
Proof. From Remark 2.15, if γ passes through ∞, then
IL(γ) = 1π
∫H|∇σf |2 dA(z) +
1π
∫H∗|∇σh|2 dA(z),
where σf and σh are the shorthand notation for log |f ′| and log
|h′|. The conformal invarianceof Dirichlet energy gives
DH(ϕ ◦ f) +DH∗(ϕ ◦ h) = DH(ϕ) +DH∗(ϕ) = DC(ϕ).
To show (3.1), after expanding the Dirichlet energy terms, it
suffices to verify the crossterms that∫
H〈∇σf (z),∇(ϕ ◦ f)(z)〉 dA(z) +
∫H∗〈∇σh(z),∇(ϕ ◦ h)(z)〉dA(z) = 0. (3.3)
By Stokes’ formula, the first term on the left-hand side
equals∫R∂nσf (x)ϕ(f(x))dx =
∫Rk(f(x))
∣∣f ′(x)∣∣ϕ(f(x))dx = ∫∂H
k(z)ϕ(z) |dz|
where k(z) is the geodesic curvature of γ = ∂H at z using the
identity ∂nσf (x) =|f ′(x)|k(f(x)) (this follows from an elementary
differential geometry computation, see,e.g., [Wan19b, Appx.A]). The
geodesic curvature at the same point z ∈ γ, considered asa point of
∂H∗, equals −k(z). Therefore the sum in (3.3) cancels out and
completes theproof in the smooth case.
The following result is on the converse operation of the
cutting, which shows that we canalso recover γ and ϕ from u and v
by conformal welding. More precisely, an increasinghomeomorphism w
: R→ R is said to be a (conformal) welding homeomorphism of a
Jordancurve γ through ∞, if there are conformal maps f, h of the
upper and lower half-planesonto the two components of Cr γ,
respectively, such that w = h−1 ◦ f |R. Suppose H andH∗ are each
equipped with an infinite and positive boundary measure defining a
metricon R such that the distance between x < y equals the
measure of [x, y]. Under suitableassumptions, the increasing
isometry w : R = ∂H→ ∂H∗ = R fixing 0 is well-defined and awelding
homeomorphism of some curve γ. In this case, we say that the
corresponding tuple(γ, f, h) solves the isometric welding problem
for the given measures.
Theorem 3.2 (Isometric conformal welding [VW20a, Thm. 1.2]).
Suppose u ∈ E(H) andv ∈ E(H∗) are given. The isometric welding
problem for the measures eudx and evdx on Rhas a solution (γ, f, h)
and the welding curve γ has finite Loewner energy. Moreover,
thereexists a unique ϕ ∈ E(C) such that (3.2) is satisfied.
15
-
In the statement, the measures eudx and evdx are defined using
the traces of u, v on R,that are H1/2(R) functions. We say that u ∈
H1/2(R) if
‖u‖2H1/2 :=1
2π2∫∫
R×R
|u(x)− u(y)|2
|x− y|2dx dy
-
Theorem 3.4 (Flow-line identity [VW20a, Thm. 1.4]). Let ϕ ∈ E(C)
∩ C0(Ĉ). Any flow-line γ of the vector field eiϕ is a Jordan curve
through ∞ with finite Loewner energy andwe have the formula
DC(ϕ) = IL(γ) +DC(ϕ0). (3.6)
Remark 3.5. The assumption of ϕ ∈ C0(Ĉ) is for technical reason
to consider the flow-lineof eiϕ in the classical differential
equation sense. One may also drop this assumption bydefining a
flow-line to be a chord-arc curve γ passing through ∞ on which ϕ =
τ arclengthalmost everywhere. We will further explore these ideas
in a setting adapted to boundedcurves (see Theorem 6.13).
This identity is analogous to the flow-line coupling between SLE
and GFF, of criticalimportance, e.g., in the imaginary geometry
framework of Miller-Sheffield [MS16a]: veryloosely speaking, an
SLEκ curve may be coupled with a GFF Φ and thought of as a
flow-lineof the vector field eiΦ/χ, where χ = 2/γ − γ/2. As γ → 0,
we have eiΦ/χ ∼ eiγΦ/2.Let us finally remark that by combining the
cutting-welding (3.1) and flow-line (3.6)identities, we obtain the
following complex identity. See also Theorem 6.13 the
complexidentity for a bounded Jordan curve.
Corollary 3.6 (Complex identity [VW20a, Cor. 1.6]). Let ψ be a
complex-valued functionon C with finite Dirichlet energy and whose
imaginary part is continuous in Ĉ. Let γ be aflow-line of the
vector field eψ. Then we have
DC(ψ) = DH(ζ) +DH∗(ξ),
where ζ = ψ ◦ f + (log f ′)∗, ξ = ψ ◦ h+ (log h′)∗ and z∗ is the
complex conjugate of z.
Remark 3.7. A flow-line γ of the vector field eψ is understood
as a flow-line of ei Imψ, asthe real part of ψ only contributes to
a reparametrization of γ.
Proof. From the identity arg f ′ = P[Imψ] ◦ f , we have
ζ =(Reψ ◦ f + log |f ′|
)+ i(Imψ ◦ f − arg f ′
)= u+ i Imψ0 ◦ f ;
ξ = v + i Imψ0 ◦ h,
where u := Reψ ◦ f + log |f ′| ∈ E(H), v := Reψ ◦ h+ log |h′| ∈
E(H∗) and ψ0 = ψ−P [ψ|γ ].From the cutting-welding identity (3.1),
we have
DC(Reψ) + IL(γ) = DH(u) +DH∗(v).
On the other hand, the flow-line identity gives DC(Imψ) = IL(γ)
+DC(Imψ0). Hence,
DC(ψ) = DC(Reψ) +DC(Imψ) = DC(Reψ) + IL(γ) +DC(Imψ0)= DH(u)
+DH∗(v) +DC(Imψ0)= DH(ζ) +DH∗(ξ)
as claimed.
17
-
Remark 3.8. From Corollary 3.6 we can easily recover the
flow-line identity (Theorem 3.4),by taking Imψ = ϕ and Reψ = 0.
Similarly, the cutting-welding identity (3.1) followsfrom taking
Reψ = ϕ and Imψ = P[τ ] where τ is the winding function along the
curve γ.Therefore, the complex identity is equivalent to the union
of cutting-welding and flow-lineidentities.
3.3 Applications
We now show that these identities between Loewner and Dirichlet
energies inspired byprobabilistic couplings, have interesting
consequences in geometric function theory.Suppose γ1, γ2 are
locally rectifiable Jordan curves of the same length (possibly
bothinfinite) bounding two domains Ω1 and Ω2 and we mark a point on
each curve. Let wbe an arclength isometry γ1 → γ2 matching the
marked points. We obtain a topologicalsphere from Ω1 ∪ Ω2 by
identifying the matched points. Following Bishop [Bis90],
theisometric welding problem is to find a Jordan curve γ ⊂ Ĉ, and
conformal equivalencesf1, f2 from Ω1 and Ω2 to the two connected
components of Ĉr γ, such that f−12 ◦ f1|γ1 = w.The welding problem
is in general a hard question and have many pathological
examples.For instance, the mere rectifiability of γ1 and γ2 does
not guarantee the existence northe uniqueness of γ, but the
chord-arc property does. However, chord-arc curves are notclosed
under isometric conformal welding: the welding curve can have
Hausdorff dimensionarbitrarily close to 2, see [Dav82,Sem86,Bis90].
Rather surprisingly, our Theorem 3.1 andTheorem 3.2 imply that
Weil-Petersson quasicircles are closed under isometric
welding.Moreover, IL(γ) ≤ IL(γ1) + IL(γ2).We describe this result
more precisely in the case when both γ1 and γ2 are Jordan
curvesthrough ∞ with finite energy (see [VW20a, Sec. 3.2] for the
bounded curve case). LetHi, H
∗i be the connected components of Cr γi.
Corollary 3.9 ([VW20a, Cor. 3.4]). Let γ (resp. γ̃) be the
arclength isometric weldingcurve of the domains H1 and H∗2 (resp.
H2 and H∗1 ). Then γ and γ̃ have finite energy.Moreover,
IL(γ) + IL(γ̃) ≤ IL(γ1) + IL(γ2).
Proof. For i = 1, 2, let fi be a conformal equivalence H → Hi,
and hi : H∗ → H∗i bothfixing ∞. By (2.4),
IL(γi) = DH(log |f ′i |
)+DH∗
(log |h′i|
).
Set ui := log |f ′i |, vi := log |h′i|. Then γ is the welding
curve obtained from Theorem 3.2with u = u1, v = v2 and γ̃ is the
welding curve for u = u2, v = v1. Then (3.4) implies
IL(γ) + IL(γ̃) ≤ DH (u1) +DH∗ (v2) +DH (u2) +DH∗ (v1) = IL(γ1) +
IL(γ2)
as claimed.
The flow-line identity has the following corollary that we omit
the proof. When γ is abounded finite energy Jordan curve (resp.
finite energy Jordan curve passing through ∞),we let f be a
conformal map from D (resp. H) to one connected component of Cr
γ.
18
-
Corollary 3.10 ([VW20a, Cor. 1.5]). Consider the family of
analytic curves γr := f(rT),where 0 < r < 1 (resp. γr := f(R
+ ir), where r > 0). For all 0 < s < r < 1 (resp.0 <
r < s), we have
IL(γs) ≤ IL(γr) ≤ IL(γ), (resp. IL(γs) ≤ IL(γr) ≤ IL(γ), )
and equalities hold if only if γ is a circle (resp. γ is a
line). Moreover, IL(γr) (resp. IL(γr))is continuous in r and
IL(γr)r→1−−−−−→ IL(γ); IL(γr)
r→0+−−−−→ 0
(resp. IL(γr) r→0+−−−−→ IL(γ); IL(γr) r→∞−−−→ 0).
Remark 3.11. Both limits and the monotonicity is consistent with
the fact that theLoewner energy measures the “roundness” of a
Jordan curve. In particular, the vanishingof the energy of γr as r
→ 0 expresses the fact that conformal maps asymptotically takesmall
circles to circles.
4 Large deviations of multichordal SLE0+
4.1 Multichordal SLE
We now consider the multichordal SLEκ, that are families of
random curves (multichords)connecting pairwise distinct boundary
points of some planar domain. Constructions formultichordal SLEs
have been obtained by many groups
[Car03,Wer04a,BBK05,Dub07,KL07,Law09,MS16a,MS16b,BPW18,PW19], and
models the interfaces in 2D statisticalmechanics models with
alternating boundary condition.As in the single-chord case, we
include the marked boundary points to the domain data(D;x1, . . . ,
x2n), assuming that they appear in counterclockwise order along the
boundary∂D. The objects will be defined in a conformally invariant
or covariant way, so withoutloss of generality, we assume that ∂D
is smooth in a neighborhood of the marked points.Due to the
planarity, there exist Cn different possible pairwise non-crossing
connections forthe curves, where
Cn =1
n+ 1
(2nn
)(4.1)
is the n:th Catalan number. We enumerate them in terms of n-link
patterns
α = {{a1, b1}, {a2, b2}, . . . , {an, bn}}, (4.2)
that is, partitions of {1, 2, . . . , 2n} giving a non-crossing
pairing of the marked points.Now, for each n ≥ 1 and n-link pattern
α, we let Xα(D;x1, . . . , x2n) ⊂
∏j X (D;xaj , xbj )
denote the set of multichords γ = (γ1, . . . , γn) consisting of
pairwise disjoint chords whereγj ∈ X (D;xaj , xbj ) for each j ∈
{1, . . . , n}. We endow Xα(D;x1, . . . , x2n) with the
relativeproduct topology and recall that X (D;xaj , xbj ) is
endowed with the relative topologyinduced from a Hausdorff metric
as in Section 2. Multichordal SLEκ is a random multichordγ = (γ1, .
. . , γn) in Xα(D;x1, . . . , x2n), characterized in two equivalent
ways, when κ > 0.
19
-
By re-sampling property: From the statistical mechanics model
viewpoint, the naturaldefinition of multichordal SLE is such that
for each j, the law of the random curve γj isthe chordal SLEκ in
D̂j , conditioned on the other curves {γi | i 6= j}. Here, D̂j is
thecomponent of D r
⋃i 6=j γi containing γj , highlighted in grey in Figure 2. In
[BPW18], the
authors proved that multichordal SLEκ is the unique stationary
measure of a Markov chainon Xα(D;x1, . . . , x2n) defined by
re-sampling the curves from their conditional laws. Thisidea was
already introduced and used earlier in [MS16a,MS16b], where Miller
& Sheffieldstudied interacting SLE curves coupled with the
Gaussian free field (GFF) in the frameworkof “imaginary
geometry”.
x1
x2
x2n
xaj
xbjγj
Figure 2: Illustration of a multichord and the domain D̂j
containing γj .
By Radon-Nikodym derivative: We assume1 that 0 < κ < 8/3.
Multichordal SLE canbe obtained by weighting n independent SLEκ
by
exp(c(κ)
2 mD(γ1, . . . , γn)), where c(κ) := (3κ− 8)(6− κ)2κ < 0,
(4.3)
is the central charge associated to SLEκ. The quantity mD(γ) is
defined using the Brownianloop measure µloopD introduced by Lawler,
Schramm, and Werner [LSW03,LW04]:
mD(γ) :=n∑p=2
µloopD
({`∣∣ ` ∩ γi 6= ∅ for at least p chords γi})
=∫
max(#{chords hit by `} − 1, 0
)dµloopD (`)
which is positive and finite whenever the family (γi)i=1...n is
disjoint. In fact, the Brownianloop measure is an infinite measure
on Brownian loops, that is conformally invariant, andfor D′ ⊂ D,
µloopD′ is simply µ
loopD restricted to loops contained in D′. When D has
non-polar
boundary, the infinity of total mass of µloopD comes only from
the contribution of smallloops. In particular, the summand {`
∣∣ ` ∩ γi 6= ∅ for at least p chords γi} is finite if p ≥ 2and
the chords are disjoint. For n independent chordal SLEs connecting
(x1, . . . , x2n), theymay intersect each other. However, in this
case mD is infinite and the Radon-Nikodymderivative (4.3) vanishes
since c < 0. We point out that mD(γ) = 0 if n = 1 (which is
notsurprising since no weighting is needed for the single SLE).
1The same result holds for 8/3 ≤ κ ≤ 4, if one includes into the
exponent the indicator function of theevent that all γj are
pairwise disjoint.
20
-
Remark 4.1. Notice that when κ = 0, c = −∞. The second
characterization does notapply and we show its existence and
uniqueness by making links to rational functions inthe next
section.
4.2 Real rational functions and Shapiro’s conjecture
From the re-sampling property, the multichordal SLE0 in Xα(D;x1,
. . . , x2n) as a determin-istic multichord η = (η1, . . . , ηn)
with the property that each ηj is the SLE0 curve in its
owncomponent (D̂j ;xaj , xbj ). In other words, each ηj is the
hyperbolic geodesic in (D̂j ;xaj , xbj ),see Remark 1.9. We call a
multichord with this property a geodesic multichord. By
theconformal invariance of the geodesic property, we assume that D
= H without loss ofgenerality.The existence of geodesic multichord
for each α follows by characterizing them as minimizersof a lower
semicontinuous Loewner energy which is the large deviation rate
function ofmultichordal SLE0+, to be discussed in the next section.
Assuming the existence, theuniqueness is a consequence of the
following algebraic result.
Theorem 4.2 ([PW20, Thm. 1.2, Prop. 4.1]). If η̄ ∈ Xα(H;x1, . .
. , x2n) is a geodesicmultichord. The union of η̄, its complex
conjugate, and the real line is the real locus of arational
function hη of degree n+ 1 with critical points {x1, . . . , x2n}.
The rational functionis unique up to post-composition by PSL(2,R)
and by the map H→ H∗ : z 7→ −z.
A rational function is an analytic branched covering h of Ĉ
over Ĉ, or equivalently, theratio of two polynomials. The degree
of h is the number of preimages of any regular value.A point x0 ∈
Ĉ is a critical point (equivalently, a branched point) with index
k (wherek ≥ 2) if
h(x) = h(x0) + C(x− x0)k +O((x− x0)k+1)
for some constant C 6= 0. In other words, the function is
locally a k-to-1 branched cover inthe neighborhood of x0. By the
Riemann-Hurwitz formula, a rational function of degreen + 1 has 2n
critical points if and only if all of its critical points are of
index two. Thegroup
PSL(2,R) ={A =
(a b
c d
): a, b, c, d ∈ R, ad− bc = 1
}/A∼−A
acts on H by A : z 7→ az+bcz+d , which is also called a Möbius
transformation of H.We prove Theorem 4.2 by constructing the
rational function associated to a geodesicmultichord η.
Proof. The complement Hrη has n+1 components that we call faces.
We pick an arbitraryface F and consider a uniformizing conformal
map hη from F onto H. Without loss ofgenerality, we consider the
former case and assume that F is adjacent to η1. We call F ′
theother face adjacent to η1. Since η1 is a hyperbolic geodesic in
D̂1, the map hη extends byreflection to a conformal map on D̂1. In
particular, this extension of hη maps F ′ conformallyonto H∗. By
iterating the analytic continuation across all of the chords ηk, we
obtain a
21
-
meromorphic function hη : H→ Ĉ. Furthermore, hη also extends to
H, and its restrictionhη|R takes values in R. Hence, Schwarz
reflection allows us to extend hη to Ĉ by settinghη(z) := hη(z∗)∗
for all z ∈ H∗.Now, it follows from the construction that hη is a
rational function of degree n + 1, asexactly n+ 1 faces are mapped
to H and n+ 1 faces to H∗. Moreover, h−1η (R ∪ {∞}) isprecisely the
union of η, its complex conjugate and R∪ {∞}. Finally, another
choice of theface F we started with yields the same function up to
post-composition by PSL(2,R) andz 7→ −z. This concludes the
proof.
To find out all the geodesic multichords connecting {x1, . . . ,
x2n}, it thus suffices to classifyall the rational functions with
this set of critical points. The following result is due
toGoldberg.
Theorem 4.3 ([Gol91]). Let z1, . . . , z2n be 2n distinct
complex numbers. There are atmost Cn rational functions (up to
post-composition by a Möbius map of Ĉ in PSL(2,C)) ofdegree n+ 1
with critical points z1, . . . , z2n.
Assuming the existence of geodesic multichord in Xα(H;x1, . . .
, x2n) and observing thattwo rational functions constructed in
Theorem 4.2 are PSL(2,C) equivalent if and only ifthey are
span〈PSL(2,R), z 7→ −z〉 equivalent, we obtain:
Corollary 4.4. There exists a unique geodesic multichord in
Xα(D;x1, . . . , x2n) for each α.
The multichordal SLE0 is therefore well-defined. We also obtain
a by-product of this result:
Corollary 4.5. If all critical points of a rational function are
real, then it is a real rationalfunction up to post-composition by
a Möbius transformation of Ĉ.
This corollary is a special case of the Shapiro conjecture
concerning real solutions toenumerative geometric problems on the
Grassmannian, see [Sot00]. In [EG02], Eremenkoand Gabrielov first
proved this conjecture for the Grassmannian of 2-planes, when
theconjecture is equivalent to Corollary 4.5. See also [EG11] for
another elementary proof.
4.3 Large deviations of multichordal SLE0+
We now introduce the Loewner potential and energy and discuss
the large deviations ofmultichordal SLE0+.
Definition 4.6. Let γ := (γ1, . . . , γn). The Loewner potential
of γ is given by
HD(γ) :=112
n∑j=1
ID(γj) +mD(γ)−14
n∑j=1
logPD;xaj ,xbj , (4.4)
where ID(γj) = ID,xaj ,xbj (γj) is the chordal Loewner energy of
γj (Definition 2.2) andPD;x,y is the Poisson excursion kernel,
defined via
PD;x,y := |ϕ′(x)||ϕ′(y)|PH;ϕ(x),ϕ(y), and PH;x,y := |y −
x|−2,
and where ϕ : D → H is a conformal map such that ϕ(x), ϕ(y) ∈
R.
22
-
When n = 1,HD(γ) =
112ID(γ)−
14 logPD;a,b.
We denote the minimal potential by
MαD(x1, . . . , x2n) := infγHD(γ) ∈ (−∞,∞), (4.5)
with infimum taken over all multichords γ ∈ Xα(D;x1, . . . ,
x2n).One important property of the Loewner potential is that it
satisfies the following cascaderelation reducing the number of
chords by one, see [PW20, Lem. 3.7, Cor. 3.8].
Lemma 4.7 (Cascade relation [PW20, Lem. 3.7]). For each j ∈ {1,
. . . , n}, we have
HD(γ) = HD̂j (γj) +HD(γ1, . . . , γj−1, γj+1 . . . , γn).
(4.6)
In particular, any minimizer of HD in Xα(D;x1, . . . , x2n) is a
geodesic multichord, andHD(γ)
-
since c(κ)/2 ∼ −12/κ. The density of multichordal SLE is thus
given by
exp(c(κ)
2 mD(γ))∏
j exp(− ID(γ)κ
)Eind exp
(c(κ)
2 mD) ∼κ→0+ exp(−IαD(γ)
κ
).
4.4 Minimal potential
To define the energy IαD, one could have added to the potential
HD an arbitrary constantthat depends only on the boundary data (x1,
. . . , x2n;α), e.g., one may drop the Poissonkernel terms in HD
which also alters the value of the minimal potential. The
advantageof introducing the Loewner potential (4.4) is that it
allows comparing the potential ofgeodesic multichords of different
boundary data. This becomes interesting when n ≥ 2 asthe moduli
space of the boundary data is non-trivial. In this section we
discuss equationssatisfied by the minimal potential based on [PW20]
and the more recent [AKM20].We first use Loewner’s equation to
describe the geodesic multichord, whose driving functionscan be
expressed in terms of the minimal potential. We describe the result
when D = H.
Theorem 4.13 ([PW20, Prop. 1.6]). Let η be the geodesic
multichord in Xα(H;x1, . . . , x2n).For each j ∈ {1, . . . , n},
the Loewner driving function W of the chord ηj from xaj to xbjand
the evolution V it = gt(xi) of the other marked points satisfy the
differential equations
∂tWt = −12∂ajMαH(V 1t , . . . , Vaj−1t ,Wt, V
aj+1t , . . . , V
2nt ), W0 = xaj ,
∂tVit =
2V it −Wt
, V i0 = xi, for i 6= aj ,(4.7)
for 0 ≤ t < T , where T is the lifetime of the solution and
(gt)t∈[0,T ] is the Loewner flowgenerated by ηj.
Here again, the idea of SLE large deviations enables us to
speculate the form of Loewnerdifferential equations of the geodesic
multichord. In fact, for each n-link pattern α, oneassociates to
the multichordal SLEκ a (pure) partition function Zα defined as
Zα(H;x1, . . . , x2n) :=( n∏j=1
PH;xaj ,xbj
)(6−κ)/2κ× Eindκ exp
(c(κ)
2 mD(γ)),
where Eindκ denotes the expectation with respect to the law of
the independent SLEκmultichord γ. As a consequence of the large
deviation principle, we obtain that
κ logZα(H;x1, . . . , x2n)κ→0+−→ −12MαH(x1, . . . , x2n).
(4.8)
The marginal law of the chord γκj in the multichordal SLEκ in
Xα(H;x1, . . . , x2n) is givenby the stochastic Loewner equation
derived from Zα:
dWt =√κdBt + κ∂aj logZα
(V 1t , . . . , V
aj−1t ,Wt, V
aj+1t , . . . , V
2nt
)dt, W0 = xaj ,
dV it =2dt
V it −Wt, V i0 = xi, for i 6= aj .
See [PW19, Eq. (4.10)]). Replacing naively κ logZα by −12MαH, we
obtain (4.7).
24
-
To prove Theorem 4.13 rigorously, we analyse the geodesic
multichords and the minimalpotential directly and do not need to go
through SLE, which might be tedious to controlthe errors when
interchanging derivatives and limits.
Proof of Theorem 4.13. Let us illustrate the case when n = 1.
For n ≥ 2, we use aninduction and a conformal restriction formula
which compares the driving function of asingle chord under
conformal maps. See [PW20, Sec. 4.2] for the proof.When n = 1, the
minimal potential has an explicit formula:
MH(x1, x2) =12 log |x2 − x1| =⇒ ∂1MH(x1, x2) =
12(x1 − x2)
. (4.9)
The hyperbolic geodesic in (H;x1, x2) is the semi-circle η with
endpoints x1 and x2. Wecheck directly that ∂tWt|t=0 = 6/(x2 − x1).
Since hyperbolic geodesic is preserved underits own Loewner flow,
i.e., gt(η[t,T ]) is the semi-circle with end points Wt and V 2t =
gt(x2),we obtain
∂tWt =6
V 2t −Wt, W0 = x1,
∂tV2t =
2V 2t −Wt
, V 20 = x2.
By (4.9), this is exactly Equation (4.7).
Similarly, using the level two null-state
Belavin-Polyakov-Zamolodchikov equation satisfiedby the SLE
partition function:κ
2∂2xj +
∑i 6=j
(2
xi − xj∂xi −
(6− κ)/κ(xi − xj)2
)Zα = 0, j = 1, . . . , 2n, (4.10)the same idea prompts us to
find the following result, see also [BBK05,AKM20].
Theorem 4.14 ([PW20, Prop. 1.7]). Let Uα := −12MαH. For j ∈ {1,
. . . , 2n}, we have12(∂jUα(x1, . . . , x2n))
2 +∑i 6=j
2xi − xj
∂iUα(x1, . . . , x2n) =∑i 6=j
6(xi − xj)2
. (4.11)
The recent work [AKM20] proves further an explicit expression of
Uα(x1, · · · , x2n) in termsof the rational function hη associated
to the geodesic multichord in Xα(H;x1, . . . , x2n) asconsidered in
Section 4.2. More precisely, following [AKM20], we normalize the
rationalfunction such that hη(∞) =∞ by possibly post-composing hη
by an element of PSL(2,R)and denote the other n poles (ζα,1, · · ·
, ζα,n) of hη.
Theorem 4.15 ([AKM20]). For the boundary data (x1, . . . ,
x2n;α), we have
exp Uα = C∏
1≤j
-
5 Large deviations of radial SLE∞
We now turn to the large deviations of SLE∞, namely, when ε :=
1/κ using the notation inDefinition 1.2. From (1.6), one can easily
show that in the chordal setup, for any fixed t,the conformal map
ft = g−1t : H→ HrKt converges uniformly on compacts to the
identitymap as κ→∞ almost surely. In other words, the complement of
the SLEκ hull convergesfor the Carathéodory topology towards H
which is not interesting for the large deviations.The main hurdle
is that the driving function can be arbitrarily close to the
boundary point(i.e. ∞) where we normalize the conformal maps in the
Loewner evolution. We thereforeswitch to the radial version of
SLE.
5.1 Radial SLE
We now describe the radial SLE on the unit disk D targeting at
0. The radial Loewnerdifferential equation driven by a continuous
function R+ → S1 : t 7→ ζt is defined as follows:for all z ∈ D,
∂tgt(z) = gt(z)ζt + gt(z)ζt − gt(z)
. (5.1)
As in the chordal case, the solution t 7→ gt(z) to (5.1) is
defined up to the swallowing time
τ(z) := sup{t ≥ 0: infs∈[0,t]
|gs(z)− ζs| > 0},
and the growing hulls are given by Kt = {z ∈ D : τ(z) ≤ t}. The
solution gt is the conformalmap from Dt := DrKt onto D satisfying
gt(0) = 0 and g′t(0) = et.Radial SLEκ is the curve γκ tracing out
the growing family of hulls (Kt)t≥0 driven by aBrownian motion on
the unit circle S1 = {ζ ∈ C : |ζ| = 1} of variance κ, i.e.,
ζt := βκt = eiBκt , (5.2)
where Bt is a standard one dimensional Brownian motion. Radial
SLEs exhibit the samephase transitions as in the chordal case as κ
varies. In particular, when κ ≥ 8, γκ is almostsurely space-filling
and Kt = γκ[0,t].We now argue heuristically to give an intuition
about our results on the κ→∞ limit andlarge deviations in [APW20].
During a short time interval [t, t + ∆t] where the flow
iswell-defined for a given point z ∈ D, we have gs(z) ≈ gt(z) for s
∈ [t, t+ ∆t]. Hence, writingthe time-dependent vector field (z(ζt +
z)(ζt − z)−1)t≥0 generating the Loewner chain as(∫S1 z(ζ + z)(ζ −
z)−1δβκt (dζ))t≥0, where δβκt is the Dirac measure at β
κt , we obtain that
∆gt(z) is approximately∫ t+∆tt
∫S1gt(z)
ζ + gt(z)ζ − gt(z)
δβκs (dζ)ds =∫S1gt(z)
ζ + gt(z)ζ − gt(z)
d(`κt+∆t(ζ)− `κt (ζ)), (5.3)
where `κt is the occupation measure (or local time) on S1 of βκ
up to time t. As κ→∞,the occupation measure of βκ during [t, t+ ∆t]
converges to the uniform measure on S1 oftotal mass ∆t. Hence the
radial Loewner chain converges to a measure-driven Loewner
26
-
chain (also called Loewner-Kufarev chain) with the uniform
probability measure on S1 asdriving measure, i.e.,
∂tgt(z) =1
2π
∫S1gt(z)
ζ + gt(z)ζ − gt(z)
|dζ| = gt(z).
This implies gt(z) = etz. Similarly, (5.3) suggests that the
large deviations of SLE∞ canalso be obtained from the large
deviations of the process (`κt )t≥0.
5.2 Loewner-Kufarev equations
The heuristic outlined above leads naturally to consider the
Loewner-Kufarev chain drivenby measures that we now define more
precisely. LetM(Ω) (resp. M1(Ω)) be the space ofBorel measures
(resp. probability measures) on Ω. We define
N+ = {ρ ∈M(S1 × R+) : ρ(S1 × I) = |I| for all intervals I ⊂
R+}.
From the disintegration theorem (see e.g. [Bil95, Theorem
33.3]), for each measure ρ ∈ N+there exists a Borel measurable map
t 7→ ρt from R+ toM1(S1) such that dρ = ρt(dζ) dt.We say (ρt)t≥0 is
a disintegration of ρ; it is unique in the sense that any two
disintegrations(ρt)t≥0, (ρ̃t)t≥0 of ρ must satisfy ρt = ρ̃t for
a.e. t. We denote by (ρt)t≥0 one suchdisintegration of ρ ∈ N+.For z
∈ D, consider the Loewner-Kufarev ODE
∂tgt(z) = gt(z)∫S1
ζ + gt(z)ζ − gt(z)
ρt(dζ), g0(z) = z. (5.4)
Let τ(z) be the supremum of all t such that the solution is
well-defined up to time t withgt(z) ∈ D, and Dt := {z ∈ D : τ(z)
> t} is a simply connected open set containing 0. Thefunction gt
is the unique conformal map of Dt onto D such that gt(0) = 0 and
g′t(0) > 0.Moreover, it is straightforward to check that ∂t log
g′t(0) = |ρt| = 1. Hence, g′t(0) = et,namely, Dt has conformal
radius e−t seen from 0. We call (gt)t≥0 the Loewner-Kufarevchain
driven by ρ ∈ N+.It is also convenient to use its inverse (ft :=
g−1t )t≥0, which satisfies the Loewner PDE :
∂tft(z) = −zf ′t(z)∫S1
ζ + zζ − z
ρt(dζ), f0(z) = z. (5.5)
We write L+ for the set of Loewner-Kufarev chains defined for
time R+. An element of L+can be equivalently represented by (ft)t≥0
or (gt)t≥0 or the evolution family of domains(Dt)t≥0 or the
evolution family of hulls (Kt = DrDt)t≥0.
Remark 5.1. In terms of the domains (Dt), according to a theorem
of Pommerenke [Pom65,Satz 4] (see also [Pom75, Thm. 6.2] and
[RR94]), L+ consists exactly of those (Dt)t≥0 suchthat Dt ⊂ D has
conformal radius e−t and for all 0 ≤ s ≤ t, Dt ⊂ Ds.
We now restrict the Loewner-Kufarev chains to the time interval
[0, 1] for the topology andsimplicity of notation. The results can
be easily generalized to other finite intervals [0, T ]or to R+ as
the projective limit of chains on all finite intervals. Define
N[0,1] = {ρ ∈M1(S1 × [0, 1]) : ρ(S1 × I) = |I| for all intervals
I ⊂ [0, 1]},
27
-
endowed with the Prokhorov topology (the topology of weak
convergence) and the corre-sponding set of restricted Loewner
chains L[0,1]. Identifying an element (ft)t∈[0,1] of L[0,1]with the
function f defined by f(z, t) = ft(z) and endow L[0,1] with the
topology of uniformconvergence of f on compact subsets of D× [0,
1]. (Or equivalently, viewing L[0,1] as theset of processes
(Dt)t∈[0,1], this is the topology of uniform Carathéodory
convergence.) Thefollowing result allows us to study the limit and
large deviations with respect to the topologyof uniform
Carathéodory convergence.
Theorem 5.2 ([MS16d, Prop. 6.1], [JVST12]). The Loewner
transform N[0,1] → L[0,1] :ρ 7→ f is a homeomorphism.
By showing that the random measure δβκt (dζ) dt ∈ N[0,1]
converges almost surely to theuniform measure (2π)−1|dζ|dt on S1 ×
[0, 1] as κ→∞, we obtain:
Theorem 5.3 ([APW20, Prop. 1.1]). As κ→∞, the complement of
hulls (Dt)t∈[0,1] of theradial SLEκ converges almost surely to
(e−tD)t∈[0,1] for the uniform Carathéodory topology.
5.3 Loewner-Kufarev energy and large deviations
From the contraction principle Theorem 1.5 and Theorem 5.2, the
large deviation principleof radial SLEκ as κ→∞ boils down to the
large deviation principle of δβκt (dζ) dt ∈ N[0,1]for the Prokhorov
topology. For this, we approximate δβκt (dζ) dt by
ρκn :=2n−1∑i=0
µκn,i(dζ)1t∈[i/2n,(i+1)/2n) dt,
where µκn,i ∈M1(S1) is the average of the measure δβκt on the
time interval [i/2n, (i+1)/2n):
µκn,i = 2n(`κ(i+1)/2n − `κi/2n).
We start with the large deviation principle for µκn,i as κ → ∞.
Let `κt := t−1`κt be the
average occupation measure of βκ up to time t. From the Markov
property of Brownianmotion, we have µκn,i = `
κ2−n in distribution up to a rotation (by βκi/2n). The following
result
is a special case of a theorem of Donsker and Varadhan.Define
the functional IDV :M(S1)→ [0,∞] by
IDV (µ) = 12
∫S1|v′(ζ)|2 |dζ|, (5.6)
if µ = v2(ζ)|dζ| for some function v ∈W 1,2(S1) and ∞
otherwise.
Remark 5.4. Note that IDV is rotation-invariant and IDV (cµi) =
cIDV (µi) for c > 0.
Theorem 5.5 ([DV75, Thm. 3]). Fix t > 0. The average
occupation measure {`κt }κ>0admits a large deviation principle
as κ→∞ with good rate function tIDV . Moreover, IDVis convex.
28
-
The above κ→∞ large deviation principle is in the sense of
Definition 1.2 with ε = 1/κ,i.e., for any open set O and closed set
F ⊂M1(S1),
limκ→∞
1κ
logP[`κt ∈ O] ≥ − infµ∈O
tIDV (µ);
limκ→∞
1κ
logP[`κt ∈ F ] ≤ − infµ∈F
tIDV (µ).
Theorem 5.5 implies that the 2n-tuple (µκn,0, . . . .µκn,2n−1)
satisfies the large deviation prin-ciple with rate function
IDVn (µ0, . . . , µ2n−1) := 2−n2n−1∑i=0
IDV (µi).
Taking the n→∞ limit, we obtain the following definition.
Definition 5.6. We define the Loewner-Kufarev energy on L[0,1]
(or equivalently on N[0,1])
S[0,1]((Dt)t∈[0,1]) := S[0,1](ρ) :=∫ 1
0IDV (ρt) dt
where ρ is the driving measure generating (Dt) and IDV .
Theorem 5.7 ([APW20, Thm. 1.2]). The measure δβκt (dζ) dt ∈
N[0,1] satisfies the largedeviation principle with good rate
function S[0,1] as κ→∞.
Remark 5.8. Here we note that if ρ ∈ N[0,1] has finite
Loewner-Kufarev energy, thenρt is absolutely continuous with
respect to the Lebesgue measure for a.e. t with densitybeing the
square of a function in W 1,2(S1). In particular, ρt is much more
regular than aDirac measure. We see once more the regularizing
phenomenon from the large deviationconsideration.
From the contraction principle Theorem 1.5 and Theorem 5.2, we
obtain immediately:
Corollary 5.9 ([APW20, Cor. 1.3]). The family of SLEκ on the
time interval [0, 1] satisfiesthe κ→∞ large deviation principle
with the good rate function S[0,1].
6 Foliation of Weil-Petersson quasicircles
SLE processes enjoy a remarkable duality [Dub09a,Zha08a,MS16a]
coupling SLEκ to theouter boundary of SLE16/κ for κ < 4. It
suggests that the rate functions of SLE0+ (Loewnerenergy) and SLE∞
(Loewner-Kufarev energy) are also dual to each other. Let us
firstremark that when S[0,1](ρ) = 0, the generated family
(Dt)t∈[0,1] consists of concentric disks.In particular (∂Dt)t∈[0,1]
are circles and thus have zero Loewner energy. This trivial
examplesupports the guess that some form of energy duality
holds.Viklund and I [VW20b] investigated the duality between these
two energies, and moregenerally, the interplay with the Dirichlet
energy of a so-called winding function. We nowdescribe briefly
those results. While our approach is originally inspired by SLE
theory,they are of independent interest from the analysis
perspective and the proofs are alsodeterministic.
29
-
6.1 Whole-plane Loewner evolution
To describe our results in the most generality, we consider the
Loewner-Kufarev energy forLoewner evolutions defined for t ∈ R,
namely the whole-plane Loewner chain. We define inthis case the
space of driving measures to be
N := {ρ ∈M(S1 × R) : ρ(S1 × I) = |I| for all intervals I}.
The whole-plane Loewner chain driven by ρ ∈ N , or equivalently
by its measurable familyof disintegration measures R→M1(S1) : t 7→
ρt, is the unique family of conformal maps(ft : D→ Dt)t∈R such
that
(i) For all s < t, 0 ∈ Dt ⊂ Ds.(ii) For all t ∈ R, ft(0) = 0
and f ′t(0) = e−t (i.e., Dt has conformal radius e−t).(iii) For all
s ∈ R, (f (s)t := f−1s ◦ ft : D→ D
(s)t )t≥s is the Loewner chain driven by (ρt)t≥s,
which satisfies (5.5) with the initial condition f (s)s (z) = z
as in Section 5.2.
See, e.g., [VW20b, Sec. 7.1] for a proof of the existence and
uniqueness of such family.
Remark 6.1. If ρt is the uniform probability measure for all t ≤
0, then ft(z) = e−tz fort ≤ 0 and (ft)t≥0 is the Loewner chain
driven by (ρt)t≥0 ∈ N+. Indeed, we check directlythat (ft)t∈R
satisfy the three conditions above. The Loewner chain in D is
therefore aspecial case of the whole-plane Loewner chain.
Note that the condition (iii) is equivalent to for all t ∈ R and
z ∈ D,
∂tft(z) = −zf ′t(z)∫S1
ζ + zζ − z
ρt(dζ). (6.1)
We remark that for ρ ∈ N , then ∪t∈RDt = C. Indeed, Dt has
conformal radius e−t,therefore contains the centered ball of radius
e−t/4 by Koebe’s 1/4 theorem. Since {t ∈ R :z ∈ Dt} 6= ∅, we define
for all z ∈ C,
τ(z) := sup{t ∈ R : z ∈ Dt} ∈ (−∞,∞].
We say that ρ ∈ N generates a foliation (γt := ∂Dt)t∈R of Cr {0}
if
1. For all t ∈ R, γt is a chord-arc Jordan curve (see Remark
2.17).2. It is possible to parametrize each curve γt by S1 so that
the mapping t 7→ γt is
continuous in the supremum norm.3. For all z ∈ Cr {0}, τ(z)
-
0z
ϕ(z)
Figure 3: Illustration of the winding function ϕ.
6.2 Energy duality
The following result gives a qualitative relation between finite
Loewner-Kufarev energymeasures and finite Loewner energy curves
(i.e., Weil-Petersson quasicircle by Theorem 2.14).
Proposition 6.3 (Weil-Petersson leaves [VW20b, Thm. 1.1]).
Suppose ρ is a measure onS1 × R with finite Loewner-Kufarev energy.
Then ρ generates a foliation (γt = ∂Dt)t∈R ofCr {0} in which all
leaves are Weil-Petersson quasicircles.
Every ρ with S(ρ) < ∞ thus generates a foliation of
Weil-Petersson quasicircles thatcontinuously sweep out the Riemann
sphere, starting from ∞ and moving towards 0, as tranges from −∞ to
∞. This class of measures seems to be a rare case for which a
completedescription of the geometry of the generated non-smooth
interfaces is possible.We also show a quantitative relation among
energies by introducing a real-valued windingfunction ϕ associated
to a foliation (γt)t∈R as follows. Let gt = f−1t : Dt → D. Since γt
isby assumption chord-arc, thus rectifiable, γt has a tangent
arclength-a.e. Given z at whichγt has a tangent, we define
ϕ(z) = limw→z
argwg′t(w)gt(w)
(6.2)
where the limit is taken inside Dt and approaching z
non-tangentially. We choose thecontinuous branch of arg which
vanishes at 0. Monotonicity of (Dt)t∈R implies that there isno
ambiguity in the definition of ϕ(z) if z ∈ γt ∩ γs. See [VW20b,
Sec. 2.3] for more details.
Remark 6.4. Geometrically, ϕ(z) equals the difference of the
argument of the tangent ofγt at z and that of the tangent to the
circle centered at 0 passing through z modulo 2π,see Figure 3.
In the trivial example when the measure ρ has zero energy, the
generated foliation consistsof concentric circles centered at 0
whose winding function is identically 0. The following isour main
theorem.
Theorem 6.5 (Energy duality [VW20b, Thm. 1.2]). Assume that ρ ∈
N generates afoliation and let ϕ be the associated winding function
on C. Then DC(ϕ)
-
Remark 6.6. We have speculated the stochastic counterpart of
this theorem in [VW20b,Sec. 10], that we call radial mating of
trees, analogous but different from the mating of treestheorem of
Duplantier, Miller, and Sheffield [DMS14] (see also [GHS19] for a
recent survey).We do not enter into further details as there is no
substantial theorem at this point.
Remark 6.7. A subtle point in defining the winding function is
that in the general caseof chord-arc foliation, a function defined
arclength-a.e. on all leaves need not be definedLebesgue-a.e., see,
e.g., [Mil97]. Thus to consider the Dirichlet energy of ϕ, we
consider thefollowing extension to W 1,2loc . A function ϕ defined
arclength-a.e. on all leaves of a foliation(γt = ∂Dt) is said to
have an extension φ in W 1,2loc if for all t ∈ R, the
Jonsson-Wallin traceφ|γt of φ on γt defined by the following limit
of averages on balls B(z, r) = {w : |w−z| < r}
φ|γt(z) := limr→0+
∫B(z,r)
φ dA (6.3)
coincides with ϕ arclength-a.e on γt. We also show that if such
extension exists then it isunique. The Dirichlet energy of ϕ in the
statement of Theorem 6.5 is understood as theDirichlet energy of
this extension.
Theorem 6.5 has several applications which show that the
foliation of Weil-Peterssonquasicircles generated by ρ with finite
Loewner-Kufarev energy exhibits several remarkablefeatures and
symmetries.The first is the reversibility of the Loewner-Kufarev
energy. Consider ρ ∈ N and thecorresponding family of domains
(Dt)t∈R. Applying z 7→ 1/z to Ĉ r Dt, we obtain anevolution family
of domains (D̃t)t∈R upon time-reversal and reparametrization, which
maybe described by the Loewner equation with an associated driving
measure ρ̃. While thereis no known simple description of ρ̃ in
terms of ρ, energy duality implies remarkably thatthe
Loewner-Kufarev energy is invariant under this transformation.
Theorem 6.8 (Energy reversibility [VW20b, Thm. 1.3]). We have
S(ρ) = S(ρ̃).
Proof sketch. The map z 7→ 1/z is conformal and the family of
circles centered at 0is preserved under this inversion. We obtain
from the geometric interpretation of ϕ(Remark 6.4) that the winding
function ϕ̃ associated to the foliation (∂D̃t) satisfies ϕ̃(1/z)
=ϕ(z) (one needs to work a bit as Remark 6.4 shows that the
equality holds modulo 2π a priori).Since Dirichlet energy is
invariant under conformal mappings, we obtain DC(ϕ̃) = DC(ϕ)and
conclude with Theorem 6.5.
Remark 6.9. It is not known whether whole-plane SLEκ for κ >
8 is reversible. (Forκ ≤ 8, reversibility was established in
[Zha15,MS17].) Therefore Theorem 6.8 cannot bepredicted from the
SLE point of view by considering the κ→∞ large deviations as we
didfor the reversibility of chordal Loewner energy in Theorem 2.7.
This result on the otherhand suggests that reversibility for
whole-plane radial SLE might hold for large κ as well.
From Proposition 6.3, being a Weil-Petersson quasicircle
(separating 0 from ∞) is anecessary condition to be a leaf in the
foliation generated by a finite energy measure. Thenext result
shows that this is also a sufficient condition and we can relate
the Loewner
32
-
energy of a leaf with the Loewner-Kufarev energy of measures
generating it as a leaf. Weobtain a new and quantitative
characterization of Weil-Petersson quasicircles.Let γ be a Jordan
curve separating 0 from ∞, f (resp. h) a conformal map from D
(resp.D∗) to the bounded (resp. unbounded) component of Cr γ fixing
0 (resp. fixing ∞).
Theorem 6.10 (Characterization of Weil-Petersson quasicircles
[VW20b, Thm. 1.4]). Thecurve γ is a Weil-Petersson quasicircle if
and only if γ can be realized as a leaf in thefoliation generated
by a measure ρ with S(ρ)
-
the finite energy/large deviation world. Although some results
involving the finite energyobjects are interesting on their own
from the analysis perspective, we choose to omit fromthe table
those without an obvious stochastic counterpart such as results in
Section 3.3.We hope that the readers are now convinced that the
ideas around large deviations of SLEare great sources generating
exciting results in the deterministic world. Vice versa, as
finiteenergy objects are more regular and easier to handle,
exploring their properties would alsoprovide a way to speculate new
conjectures in random conformal geometry. Rather thanan end, we
hope it to be the starting point of new development along those
lines and thissurvey can serve as a guide.
SLE/GFF with κ� 1 Finite energySection 1, 2
Chordal SLEκ in (D,x, y) A chord γ with ID,x,y(γ)
-
Multichordal SLEκ pure partition function Minimal potentialMαDZα
of link pattern α (Definition 4.6, Equation (4.8))Loewner evolution
of a chord in Loewner evolution of a chord in geodesicmultichordal
SLEκ multichord (Theorem 4.13)Level 2 null-state BPZ equation
Semiclassical null-state equationsatisfied by Zα satisfied byMαH
(Theorem 4.14)
Section 5, 6Loewner chain (Dt)t∈R driven by ρ ∈ N
Whole-plane radial SLE16/κ with Loewner-Kufarev energy S(ρ)
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