Fisher zeros and correlation decay in the Ising model · 2020-02-07 · Pr[edge e is cut jedge f is...
Transcript of Fisher zeros and correlation decay in the Ising model · 2020-02-07 · Pr[edge e is cut jedge f is...
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)|
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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?
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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
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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)
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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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