Fisher zeros and correlation decay in the Ising model

Jingcheng Liu 1Alistair Sinclair

1Piyush Srivastava

2

1University of California, Berkeley

2Tata Institute of Fundamental Research

ITCS 2019

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 1 / 11

Motivation

In the past few years:

Phase transitions in statistical physics→ algorithms

In this work, we study the converse:

Can we study phase transitions in statistical physics via algorithmic techniques?

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 2 / 11

Ising model

Configuration: σ ∈ {+,−}V

Edge potentials: ϕe(σu, σv) =

{β if σu 6= σv

1 otherwise

+ +

+

β

β β-

A spin configuration with weight β3

Ising model as cut generating polynomial

ZG(β) =∑S⊆V

β|E(S,V\S)| =

|E|∑k=0

γkβk

where γk := number of k-edge cuts

Gibbs distribution: Pr[(S,V \ S)] = 1ZG(β) · β

|E(S,V\S)|

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 3 / 11

Two notions of phase transition in statistical physics

Definition I. Decay of long range correlations (informal)

Let e and f be any edges that are “far apart”. Then in a random cut,

Pr[edge e is cut | edge f is cut] ≈ Pr[edge e is cut]

The study of algorithms based on correlation decay (notably, Weitz’s algorithm) has been fruitful

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 4 / 11

Two notions of phase transition in statistical physics

Definition II. Analyticity of free energy (informal)

The “free energy” log Z is analytic in a complex neighborhood.

Analyticity ≈ continuity of observables: the average cut size is precisely β · d log Zdβ

O

F(β)

β O

F(β)

β

Analyticity of log Z ≡ absence of zeros in ZEven when only positive real-valued parameters make physical sense, one needs to study

complex-valued parameters

Algorithmic use of location of zeros originated only recently in the work of Barvinok

What relationship, if any, do the two notions (decay of correlations and zero-freeness) have?

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 5 / 11

Prior works, and Lee-Yang zeros versus Fisher zeros

Fisher zeros has been studied classically, but li�le is known for general graphs

Fisher zeros (1965): view β as variable

ZG(β) =∑S⊆V

β|E(S,V\S)|

Lee-Yang zeros (1952): view λ as variable

ZβG (λ) =∑S⊆V

β|E(S,V\S)|λ|S|

For general Fisher zeros, Barvinok and Soberón: ZG(β) 6= 0 if |β − 1| < c/∆, for c ≈ 0.34Recently Peters and Regts: in the hard-core model, zero-free regions can be extended to the

entire correlation decay regime

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 6 / 11

Our result: correlation decay implies zero-freeness for the Ising model

β = ∆−2∆

β = 1

β = ∆∆−2

Correlation decay

[Zhang-Liang-Bai’11]

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 7 / 11

Our result: correlation decay implies zero-freeness for the Ising model

β = ∆−2∆

β = 1

β = ∆∆−2

Correlation decay

[Zhang-Liang-Bai’11]

Barvinok and Soberon

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 7 / 11

Our result: correlation decay implies zero-freeness for the Ising model

β = ∆−2∆

β = 1

β = ∆∆−2

Correlation decay

[Zhang-Liang-Bai’11]

Barvinok and SoberonOur work

Theorem

ZG(β) does not vanish in a complex open region containing the entire correlation decay intervalB :=

(∆−2

∆ , ∆∆−2

).

By-product: algorithms to approximate ZG(β) in the same region.

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 7 / 11

Our technique: Weitz’s algorithm

Our proof crucially exploits the correlation decay property

Choose any vertex, say u, then

ZG(β) =∑S⊆V

β|E(S,V\S)| =∑S⊆Vu∈S

β|E(S,V\S)| +∑S⊆Vu 6∈S

β|E(S,V\S)| = Σ+ + Σ−

Consider the ratio RG,u(β) := Σ+

Σ−.

To show ZG(β) 6= 0, it su�ices if Σ− 6= 0 and RG,u(β) 6= −1

Weitz’s algorithm provides a formal recurrence F(·) for computing the ratio RG,u(β)

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 8 / 11

Our technique (Weitz’s algorithm cont’d)

· · ·

G

u

v1 v2 vk

Given the ratios at v1, · · · , vk , then the ratio at u is given by RG,u = F(RG1,v1 , · · · , RGk,vk), where

Fβ,k,s(~x) := βsk∏

i=1

β + xiβxi + 1

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Proof sketch

To show RG,u 6= −1, it su�ices to design a complex neighbor-

hood D such that

1 F(Dk) ⊆ D2 −1 6∈ D3 D contains all the “starting points” of Weitz’s algorithm

RG,u = F(RG1,v1 , · · · , RGk,vk )

To find such a set, the key steps are:

For “convex” region D, the univariate recurrence f (·) satisfies f (D) = F(Dk)

For a suitable choice of ϕ, we show that ϕ ◦ f ◦ ϕ−1 approximately contracts every

rectanglular region that contains the fixed point ϕ(1)←− Correlation decay!We choose a “convex” D so that ϕ(D) ≈ a rectangular region

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 10 / 11

Proof sketch

To show RG,u 6= −1, it su�ices to design a complex neighbor-

hood D such that

1 F(Dk) ⊆ D2 −1 6∈ D3 D contains all the “starting points” of Weitz’s algorithm

RG,u = F(RG1,v1 , · · · , RGk,vk )

To find such a set, the key steps are:

For “convex” region D, the univariate recurrence f (·) satisfies f (D) = F(Dk)

For a suitable choice of ϕ, we show that ϕ ◦ f ◦ ϕ−1 approximately contracts every

rectanglular region that contains the fixed point ϕ(1)←− Correlation decay!We choose a “convex” D so that ϕ(D) ≈ a rectangular region

Jingcheng Liu (UC Berkeley) Fisher zeros ITCS 2019 10 / 11

Discussion and Open problems

Open problem

Is “correlation decay implies absence of zeros” a general phenomenon in spin systems and graphical

models?

Open problem

Connections of locations of zeros, to algorithms such as MCMC and the correlation decay

approach?

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