The Prokofiev-Svistunov-Ising process is rapidly mixingusers.monash.edu/~gfarr/research/slides/Garoni-main.pdfIntroduction Main Theorem Proof Discussion The Prokoﬁev-Svistunov-Ising

Introduction Main Theorem Proof Discussion

The Prokofiev-Svistunov-Ising process is

rapidly mixing

Tim Garoni

School of Mathematical SciencesMonash University


Collaborators

◮ Andrea Collevecchio (Monash University)

◮ Tim Hyndman (Monash University)

◮ Daniel Tokarev (Monash University)


Ising model - Motivation

◮ Introduced in 1920 as a model for ferromagnetism

◮ Hope was to explain the Curie transition:◮ Place iron in a magnetic field◮ Increase field to high value◮ Slowly reduce field to zero◮ Exists critical temperature Tc below which iron remains magnetized





◮ During mid 1930s it was realized the Ising model also describesphases transitions in other physical systems

◮ Gas-liquid critical phenomena (lattice gas)◮ Binary alloys







◮ Now a paradigm for order/disorder transitions with applications to

◮ Economics (opinion formation)◮ Finance (stock price dynamics)◮ Biology (hemoglobin, DNA, . . . )◮ Image processing (archetypal Markov random field)







◮ Now a paradigm for order/disorder transitions with applications to

◮ Economics (opinion formation)◮ Finance (stock price dynamics)◮ Biology (hemoglobin, DNA, . . . )◮ Image processing (archetypal Markov random field)

◮ Continues to play fundamental role in theoretical/mathematical

studies of phase transitions and critical phenomena


The Ising model

◮ Finite graph G = (V,E)

◮ Configuration space ΣG = {−1,+1}V

◮ Measure

P(σ) =1

Zexp

β∑

ij∈E

σiσj + h∑

i∈V

σi

◮ Inverse temperature β

◮ External field h

◮ Z is the partition function


The Ising model

◮ Finite graph G = (V,E)

◮ Configuration space ΣG = {−1,+1}V

◮ Measure

P(σ) =1

Zexp

β∑

ij∈E

σiσj + h∑

i∈V

σi

◮ Inverse temperature β

◮ External field h

◮ Z is the partition function

◮ Main physical interest is in certain expectations such as:◮ Two-point correlation cov(σu, σv) = E(σuσv)− E(σu)E(σv)

◮ Susceptibility χ =1

|V |

∑

u,v∈V

cov(σu, σv)


Phase transition

Let Λn = {−n, . . . , n}d ⊂ Zd, and consider

Σ+Λn = {σ ∈ {−1, 1}Zd

: σi = +1 for all i 6∈ Λn}

Sequence of Gibbs measures on Λn converges: P+Λn,β,h

⇒ P+β,h


Phase transition


Σ+Λn = {σ ∈ {−1, 1}Zd



⇒ P+β,h

Analogous construction for “minus” boundary conditions


Phase transition


Σ+Λn = {σ ∈ {−1, 1}Zd



⇒ P+β,h


Theorem (Aizenman, Duminil-Copin, Sidoravicius (2014))

1. If d = 1, then for any (β, h) ∈ [0,∞)× R there is a uniqueinfinite-volume Gibbs measure


Phase transition


Σ+Λn = {σ ∈ {−1, 1}Zd



⇒ P+β,h




2. If d ≥ 2 and h 6= 0, then for any β ∈ [0,∞) there is a uniqueinfinite-volume Gibbs measure


Phase transition


Σ+Λn = {σ ∈ {−1, 1}Zd



⇒ P+β,h




2. If d ≥ 2 and h 6= 0, then for any β ∈ [0,∞) there is a uniqueinfinite-volume Gibbs measure

3. If d ≥ 2 and h = 0, there exists βc(d) ∈ (0,∞) such that:

a. If β ≤ βc, there is a unique infinite-volume Gibbs measureb. If β > βc, then P

+

β,0 6= P−

β,0


Exact solutions

◮ Gn = Zn solved by Ising (1925)


Exact solutions


◮ Gn = Kn solved by Husimi (1953) and Temperley (1954)


Exact solutions



◮ Gn = Z2n solved by Peierls (1936), Kramers & Wannier (1941),

Onsager (1944), Yang (1952), Kasteleyn (1963), Fisher (1966)


Exact solutions





Smirnov (2006) proved critical interfaces between+/− components have conformally invariant limit

◮ SLE(3) - Schramm-Löwner Evolution


Exact solutions







K-F related Ising partition function to perfect matchings


Exact solutions







K-F related Ising partition function to perfect matchings◮ Elegant solution on planar graphs in terms of Pfaffians


Exact solutions







K-F related Ising partition function to perfect matchings◮ Elegant solution on planar graphs in terms of Pfaffians◮ Only tractable on graphs of bounded (small) genus


Exact solutions







K-F related Ising partition function to perfect matchings◮ Elegant solution on planar graphs in terms of Pfaffians◮ Only tractable on graphs of bounded (small) genus◮ Method not tractable for Gn = Z

dn with d ≥ 3


Computational Complexity

◮ PARTITION:◮ Input: Finite graph G = (V,E), and parameters β, h◮ Output: Ising partition function

◮ CORRELATION:◮ Input: Finite graph G = (V,E), a pair u, v ∈ V , and parameters β, h◮ Output: Ising two-point correlation function

◮ SUSCEPTIBILITY:◮ Input: Finite graph G = (V,E), and parameters β, h◮ Output: Ising susceptibility


Computational Complexity

◮ PARTITION:◮ Input: Finite graph G = (V,E), and parameters β, h◮ Output: Ising partition function

◮ CORRELATION:◮ Input: Finite graph G = (V,E), a pair u, v ∈ V , and parameters β, h◮ Output: Ising two-point correlation function

◮ SUSCEPTIBILITY:◮ Input: Finite graph G = (V,E), and parameters β, h◮ Output: Ising susceptibility

Proposition (Jerrum-Sinclair 1993; Sinclair-Srivastava 2014 )

PARTITION, SUSCEPTIBILITY and CORRELATION are #P-hard.


Markov-chain Monte Carlo

◮ Construct a transition matrix P on Ω which:◮ Is ergodic◮ Has stationary distribution π(·)

◮ Generate random samples with (approximate) distribution π

◮ Estimate π expectations using sample means






d(t) := maxx∈Ω

‖P t(x, ·)− π(·)‖ ≤ Cαt, for α ∈ (0, 1)

◮ Mixing time quantifies the rate of convergence

tmix(δ) := min {t : d(t) ≤ δ}






d(t) := maxx∈Ω

‖P t(x, ·)− π(·)‖ ≤ Cαt, for α ∈ (0, 1)



◮ How does tmix depend on size of Ω?






d(t) := maxx∈Ω

‖P t(x, ·)− π(·)‖ ≤ Cαt, for α ∈ (0, 1)



◮ How does tmix depend on size of Ω?◮ For Ising the size of a problem instance is n = |V |◮ If tmix = O(poly(n)) we have rapid mixing






d(t) := maxx∈Ω

‖P t(x, ·)− π(·)‖ ≤ Cαt, for α ∈ (0, 1)



◮ How does tmix depend on size of Ω?◮ For Ising the size of a problem instance is n = |V |◮ If tmix = O(poly(n)) we have rapid mixing◮ |Ω| = 2n so rapid mixing implies only logarithmically-many states

need be visited to reach approximate stationarity


Markov chains for the Ising model

◮ Glauber process (arbitrary field) 1963◮ Lubetzky & Sly (2012): Rapidly mixing on boxes in Z2 at h = 0 iff

β ≤ βc◮ Levin, Luczak & Peres (2010): Precise asymptotics on Kn for h = 0◮ Not used by computational physicists





◮ Swendsen-Wang process (zero field) 1987◮ Simulates coupling of Ising and Fortuin-Kasteleyn models◮ Long, Nachmias, Ding & Peres (2014): Precise asymptotics on Kn◮ Ullrich (2014): Rapidly mixing on boxes in Z2 at all β 6= βc◮ Empirically fast. State of the art 1987 – recently






◮ Jerrum-Sinclair process (positive field) 1993◮ Simulates high-temperature graphs for h > 0◮ Proved rapidly mixing on all graphs at all temperatures for all h > 0◮ No empirical results - not used by computational physicists






◮ Jerrum-Sinclair process (positive field) 1993◮ Simulates high-temperature graphs for h > 0◮ Proved rapidly mixing on all graphs at all temperatures for all h > 0◮ No empirical results - not used by computational physicists

◮ Prokofiev-Svistunov worm process (zero field) 2001◮ No rigorous results currently known◮ Empirically, best method known for susceptibility (Deng, G., Sokal)◮ Widely used by computational physicists


Mixing time bound for PS process

Theorem (Collevecchio, G., Hyndman, Tokarev 2014+)

For any temperature, the mixing time of the PS process on graph

G = (V,E) satisfies

tmix(δ) = O(∆(G)m2n5)

with n = |V |, m = |E| and ∆(G) the maximum degree.




For any temperature, the mixing time of the PS process on graph

G = (V,E) satisfies

tmix(δ) = O(∆(G)m2n5)


Only Markov chain for the Ising model currently known to be rapidly

mixing at the critical point for boxes in Zd


High-temperature expansions and the PS measure

◮ Let Ck = {A ⊆ E : (V,A) has k odd vertices}




◮ Let CW = {A ⊆ E : W is the set of odd vertices in (V,A) }





◮ PS measure defined on the configuration space C0 ∪ C2

π(A) ∝ x|A|

{

n, A ∈ C0,

2, A ∈ C2.






π(A) ∝ x|A|

{

n, A ∈ C0,

2, A ∈ C2.

◮ If x = tanhβ then:

◮ Ising susceptibility χ =1

π(C0)◮ Ising two-point correlation function E(σuσv) =

n

2

π(Cuv)

π(C0)






π(A) ∝ x|A|

{

n, A ∈ C0,

2, A ∈ C2.

◮ If x = tanhβ then:



n

2

π(Cuv)

π(C0)

◮ PS measure is stationary distribution of PS process


Fully-polynomial randomized approximation schemes

Definition

An fpras for an Ising property f is a randomized algorithm such thatfor given G, T , and ξ, η ∈ (0, 1) the output Y satisfies

P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η

and the running time is bounded by a polynomial in n, ξ−1, η−1.



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η


Combine rapid mixing of PS process with general fpras construction

of Jerrum-Sinclair (1993):



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η




◮ Let A ⊆ C0 ∪ C2 with π(A) ≥ 1/S(n) for some polynomial S(n)



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η





◮ The following defines an fpras for π(A):



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η





◮ The following defines an fpras for π(A):◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)]



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η





◮ The following defines an fpras for π(A):◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)]◮ Let Y = 1A



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η





◮ The following defines an fpras for π(A):◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)]◮ Let Y = 1A◮ Run the PS process T = R(G) time steps and record YT



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η





◮ The following defines an fpras for π(A):◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)]◮ Let Y = 1A◮ Run the PS process T = R(G) time steps and record YT◮ Independently generate 72ξ−2S(n) such samples and take the

sample mean



Definition


P[(1− ξ)f ≤ Y ≤ (1 + ξ)f ] ≥ 1− η





◮ The following defines an fpras for π(A):◮ Let R(G) be our upper bound for tmix(δ) with δ = ξ/[8S(n)]◮ Let Y = 1A◮ Run the PS process T = R(G) time steps and record YT◮ Independently generate 72ξ−2S(n) such samples and take the

sample mean◮ Repeat 6 lg⌈1/η⌉+ 1 such experiments and take the median


Prokofiev-Svistunov process

PS proposals:



PS proposals:

◮ If A ∈ C0:



PS proposals:

◮ If A ∈ C0:◮ Pick uniformly random u ∈ V



PS proposals:

◮ If A ∈ C0:◮ Pick uniformly random u ∈ V◮ Pick uniformly random v ∼ u



PS proposals:

◮ If A ∈ C0:◮ Pick uniformly random u ∈ V◮ Pick uniformly random v ∼ u◮ Propose A → A△uv



PS proposals:


◮ If A ∈ C2:



PS proposals:


◮ If A ∈ C2:◮ Pick random odd u ∈ V



PS proposals:


◮ If A ∈ C2:◮ Pick random odd u ∈ V◮ Pick uniformly random v ∼ u



PS proposals:


◮ If A ∈ C2:◮ Pick random odd u ∈ V◮ Pick uniformly random v ∼ u◮ Propose A → A△uv



PS proposals:


◮ If A ∈ C2:◮ Pick random odd u ∈ V◮ Pick uniformly random v ∼ u◮ Propose A → A△uv

Metropolize proposals with respect to PS measure π(·)


Proof of rapid mixing◮ We use the path method◮ Consider the transition graph G = (V, E) of the PS process

◮ V = C0 ∪ C2◮ E = {AA′ : P (A,A′) > 0}



◮ V = C0 ∪ C2◮ E = {AA′ : P (A,A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F



◮ V = C0 ∪ C2◮ E = {AA′ : P (A,A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F◮ If no transition is used too often, the process is rapidly mixing



◮ V = C0 ∪ C2◮ E = {AA′ : P (A,A′) > 0}

◮ Specify paths γI,F in G between pairs of states I, F◮ If no transition is used too often, the process is rapidly mixing◮ In the most general case, one specifies paths between each pair

I, F ∈ C0 ∪ C2



◮ V = C0 ∪ C2◮ E = {AA′ : P (A,A′) > 0}


I, F ∈ C0 ∪ C2◮ For PS process, convenient to only specify paths from C2 to C0

C2 C0



◮ V = C0 ∪ C2◮ E = {AA′ : P (A,A′) > 0}


I, F ∈ C0 ∪ C2◮ For PS process, convenient to only specify paths from C2 to C0

I

F

C2 C0


Lemma (Jerrum-Sinclair-Vigoda (2004))

Consider MC with state space Ω and stationary distribution π.Let S ⊂ Ω, and specify paths Γ = {γI,F : I ∈ S, F ∈ S

c}. Then

tmix(δ) ≤ log

1

δminA∈Ω

π(A)

[

2 + 4

(

π(S)

π(Sc)+

π(Sc)

π(S)

)]

ϕ(Γ)

where

ϕ(Γ) :=

(

max(I,F )∈S×Sc

|γI,F |

)

maxAA′∈E

∑

(I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F )

π(A)P (A,A′)




c}. Then

tmix(δ) ≤ log

1

δminA∈Ω

π(A)

[

2 + 4

(

π(S)

π(Sc)+

π(Sc)

π(S)

)]

ϕ(Γ)

where

ϕ(Γ) :=

(

max(I,F )∈S×Sc

|γI,F |

)

maxAA′∈E

∑

(I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F )

π(A)P (A,A′)

◮ We choose S = C2.




c}. Then

tmix(δ) ≤ log

1

δminA∈Ω

π(A)

[

2 + 4

(

π(S)

π(Sc)+

π(Sc)

π(S)

)]

ϕ(Γ)

where

ϕ(Γ) :=

(

max(I,F )∈S×Sc

|γI,F |

)

maxAA′∈E

∑

(I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F )

π(A)P (A,A′)

◮ We choose S = C2. Elementary to show:2

n

mx

mx+ 1≤

π(C2)

π(C0)≤ n−1, and π(A) ≥

(x

8

)m

for all A




c}. Then

tmix(δ) ≤ log

1

δminA∈Ω

π(A)

[

2 + 4

(

π(S)

π(Sc)+

π(Sc)

π(S)

)]

ϕ(Γ)

where

ϕ(Γ) :=

(

max(I,F )∈S×Sc

|γI,F |

)

maxAA′∈E

∑

(I,F )∈S×Sc

γI,F ∋AA′

π(I)π(F )

π(A)P (A,A′)

◮ We choose S = C2. Elementary to show:2

n

mx

mx+ 1≤

π(C2)

π(C0)≤ n−1, and π(A) ≥

(x

8

)m

for all A

◮ Therefore:

tmix(δ) ≤

(

log

(

8

x

)

−log δ

m

)(

3 +1

mx

)

2mnϕ(Γ)


Choice of Canonical Paths

◮ To transition from I to F◮ Flip each e ∈ I△F

I F

I△F




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

I F

I△F




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)

◮ A0 is a path◮ Ai disjoint cycles

I F

I△F




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F

A0




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F

A1




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F

A2




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F

◮ γI,F defined by:◮ Traverse A0◮ . . . then A1◮ . . . then A2◮ . . .

A2




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F


I




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F





◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F


F




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F


◮ Γ = {γI,F }

F




◮ If (I, F ) ∈ C2 × C0 thenI△F ∈ C2

◮ I△F = A0 ∪(

⋃

i≥1 Ai

)


I F


◮ Γ = {γI,F }

FOne can show that ϕ(Γ) ≤ ∆(G)n4m/2


Bounding the congestion

◮ Define measure Λ

Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)

If T = AA′ is a maximally congested transition

ϕ(Γ) =∑

(I,F )∈P(T )

π(I)π(F )

π(A)P (A,A′)max

(I,F )∈S×Sc|γI,F |




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


ϕ(Γ) ≤∑

(I,F )∈P(T )

π(I)π(F )

π(A)P (A,A′)m




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


ϕ(Γ) ≤∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)

m

Λ(C0 ∪ C2)




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)

◮ Define for each T = AA′ ∈ E◮ ηT : C2 × C0 → C0 ∪ C2 ∪ C4◮ ηT (I, F ) = I△F△(A ∪A

′)


ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


′)

◮Λ(I)Λ(F )

Λ(A)P (A,A′)≤ 4∆(G)nΛ(ηT (I, F ))


ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


′)

◮Λ(I)Λ(F )

Λ(A)P (A,A′)≤ 4∆(G)nΛ(ηT (I, F ))


ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)

≤ 4∆(G)nm1

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(ηT (I, F ))




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


′)

◮Λ(I)Λ(F )

Λ(A)P (A,A′)≤ 4∆(G)nΛ(ηT (I, F ))

◮ ηT is an injection


ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)

≤ 4∆(G)nm1

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(ηT (I, F ))




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


′)

◮Λ(I)Λ(F )

Λ(A)P (A,A′)≤ 4∆(G)nΛ(ηT (I, F ))



ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)

≤ 4∆(G)nm1

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(ηT (I, F ))

≤ 4∆(G)nmΛ(C0 ∪ C2 ∪ C4)

Λ(C0 ∪ C2)




Λ(A) = x|A|

n, A ∈ C0

2, A ∈ C2

1, A ∈ C4

◮ π(A) =Λ(A)

Λ(C0 ∪ C2)


′)

◮Λ(I)Λ(F )

Λ(A)P (A,A′)≤ 4∆(G)nΛ(ηT (I, F ))



ϕ(Γ) ≤m

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(I)Λ(F )

Λ(A)P (A,A′)

≤ 4∆(G)nm1

Λ(C0 ∪ C2)

∑

(I,F )∈P(T )

Λ(ηT (I, F ))

≤ 4∆(G)nmΛ(C0 ∪ C2 ∪ C4)

Λ(C0 ∪ C2)= 4∆(G)nm

(

1 +Λ(C4)

Λ(C0) + Λ(C2)

)


Bounding the congestion cont. . .

The final step is to note that the high-temperature expansion impliesΛ(C4)

Λ(C0)≤

1

n

(

n

4

)

so that

ϕ(Γ) ≤ 4∆(G)nm

(

1 +Λ(C4)

Λ(C0)

)




Λ(C0)≤

1

n

(

n

4

)

so that

ϕ(Γ) ≤ 4∆(G)nm

(

1 +Λ(C4)

Λ(C0)

)

≤ 4∆(G)nm

(

1 +1

n

(

n

4

))




Λ(C0)≤

1

n

(

n

4

)

so that

ϕ(Γ) ≤ 4∆(G)nm

(

1 +Λ(C4)

Λ(C0)

)

≤ 4∆(G)nm

(

1 +1

n

(

n

4

))

≤ 4∆(G)nmn3

8




Λ(C0)≤

1

n

(

n

4

)

so that

ϕ(Γ) ≤ 4∆(G)nm

(

1 +Λ(C4)

Λ(C0)

)

≤ 4∆(G)nm

(

1 +1

n

(

n

4

))

≤ 4∆(G)nmn3

8

=∆(G)n4 m

2




Λ(C0)≤

1

n

(

n

4

)

so that

ϕ(Γ) ≤ 4∆(G)nm

(

1 +Λ(C4)

Λ(C0)

)

≤ 4∆(G)nm

(

1 +1

n

(

n

4

))

≤ 4∆(G)nmn3

8

=∆(G)n4 m

2

�


Discussion

◮ Can we obtain sharper results if we focus on special families of

graphs, such as G = ZdL?


Discussion



◮ The PS process is closely related to a modification of the

“lamplighter walk” in which lamps are always switched when

visited. Can we use this similarity to say something more precise

when G = ZdL?


Discussion



◮ The PS process is closely related to a modification of the

“lamplighter walk” in which lamps are always switched when

visited. Can we use this similarity to say something more precise

when G = ZdL?

◮ Study related spin models using similar methods?



The mixing time of the PS process on graph G = (V,E) withparameter x ∈ (0, 1) satisfies

tmix(δ) ≤

(

log

(

8

x

)

−log δ

m

)(

3 +1

mx

)

∆(G)m2n5,


High-temperature expansions and the PS measure◮ Let ∂A = {v ∈ V : v has odd degree in (V,A)}

High-temperature expansions and the PS measure◮ Let ∂A = {v ∈ V : v has odd degree in (V,A)}◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V

High-temperature expansions and the PS measure◮ Let ∂A = {v ∈ V : v has odd degree in (V,A)}◮ Let CW := {A ⊆ E : ∂A = W} for W ⊆ V◮ Let Ck :=

⋃

W⊆V|W |=k

CW for integer 1 ≤ k ≤ |V |


⋃

W⊆V|W |=k


◮ λ(·) defined by λ(S) =∑

A∈S x|A| for S ⊆ {A ⊆ E}


⋃

W⊆V|W |=k




If x = tanhβ then

E(Ising)G,β

(

∏

v∈W

σv

)

=λ(CW )

λ(C0)


⋃

W⊆V|W |=k




If x = tanhβ then

E(Ising)G,β

(

∏

v∈W

σv

)

=λ(CW )

λ(C0)


π(A) =λ(A)

nλ(C0) + 2λ(C2)

{

n, A ∈ C0,

2, A ∈ C2.


⋃

W⊆V|W |=k




If x = tanhβ then

E(Ising)G,β

(

∏

v∈W

σv

)

=λ(CW )

λ(C0)


π(A) =λ(A)

nλ(C0) + 2λ(C2)

{

n, A ∈ C0,

2, A ∈ C2.



n

2

π(Cuv)

π(C0)


⋃

W⊆V|W |=k




If x = tanhβ then

E(Ising)G,β

(

∏

v∈W

σv

)

=λ(CW )

λ(C0)


π(A) =λ(A)

nλ(C0) + 2λ(C2)

{

n, A ∈ C0,

2, A ∈ C2.



n

2

π(Cuv)

π(C0)◮ PS measure is stationary distribution of PS process

Proof

The Prokofiev-Svistunov-Ising process is rapidly mixingusers.monash.edu/~gfarr/research/slides/Garoni-main.pdfIntroduction Main Theorem Proof Discussion The Prokoﬁev-Svistunov-Ising

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