[Paper Review] Approximating Partition Functions of Two-State Spin Systems

Jinshan Zhang, Heng Liang+1 more|ArXiv.org|Nov 29, 2009

Markov Chains and Monte Carlo Methods7 references4 citations

TL;DR

This paper establishes strong correlation decay for two-state spin systems on bounded-degree graphs when the absolute inverse temperature $ J $ is below a critical threshold $ J_d $, enabling an FPTAS for the partition function. Using self-avoiding trees and recursive marginal probability computation, the authors prove uniqueness of the Gibbs measure under the condition $ (d-1)\tanh J < 1 $, which generalizes Dobrushin's condition and is tight for the Ising model.

ABSTRACT

Two-state spin systems is a classical topic in statistical physics. We consider the problem of computing the partition function of the systems on a bounded degree graph. Based on the self-avoiding tree, we prove the systems exhibits strong correlation decay under the condition that the absolute value of "inverse temperature" is small. Due to strong correlation decay property, an FPTAS for the partition function is presented under the same condition. This condition is sharp for Ising model.

Motivation & Objective

• To establish strong correlation decay for two-state spin systems on bounded-degree graphs.
• To prove the uniqueness of the Gibbs measure under a generalized condition $ (d-1)\tanh J < 1 $, extending Dobrushin's criterion.
• To develop a fully polynomial-time approximation scheme (FPTAS) for the partition function under the strong correlation decay condition.
• To demonstrate the sharpness of the condition $ J < J_d $ for the Ising model using self-avoiding tree techniques.

Proposed method

• Uses the self-avoiding tree (SAW) technique to embed the graph into a tree-like structure for analysis.
• Applies recursive marginal probability computation on truncated SAW trees to estimate root spin probabilities.
• Employs a decay function $ f(t) = 4Jd((d-1)\tanh J)^{t-1} $ to bound the influence of boundary configurations on the root.
• Truncates the SAW tree at height $ t_j = \frac{\log(4nJd\epsilon^{-1})}{\log((d-1)\tanh J)^{-1}} + 1 $ to control approximation error.
• Constructs an FPTAS by combining conditional marginal probability estimates via recursive computation on truncated trees.
• Uses the identity $ Z(G) = Z(G, \Phi_{n+1}) \prod_{i=1}^n p_i^{-1} $, where $ p_i $ is the marginal probability of spin $ + $ at vertex $ i $.

Experimental results

Research questions

• RQ1 Does the two-state spin system exhibit strong correlation decay on bounded-degree graphs when $ |J| < J_d $?
• RQ2 Can the condition $ (d-1)\tanh J < 1 $ be used to prove uniqueness of the Gibbs measure for general two-state spin systems?
• RQ3 Is the FPTAS for the partition function valid under the strong correlation decay condition?
• RQ4 Is the threshold $ J_d $ sharp for the Ising model, particularly on $ d $-regular trees?

Key findings

• Strong correlation decay holds for two-state spin systems on bounded-degree graphs when $ |J| < J_d $, where $ J_d $ is the critical inverse temperature for uniqueness on the infinite $ d $-regular tree.
• The condition $ (d-1)\tanh J < 1 $ ensures uniqueness of the Gibbs measure on infinite bounded-degree graphs, generalizing Dobrushin’s condition $ d\tanh J < 1 $.
• The FPTAS for the partition function runs in time $ O(n^{O(1)} \epsilon^{-O(1)}) $, polynomial in $ n $ and $ \epsilon^{-1} $, under the strong correlation decay condition.
• The bound on the ratio of true to estimated marginal probabilities satisfies $ e^{-\epsilon/n} \leq \widehat{p_j}/p_j \leq e^{\epsilon/n} $, ensuring $ \epsilon $-relative error in the final partition function estimate.
• The threshold $ J_d $ is sharp for the Ising model, as it matches the critical point for uniqueness on the infinite $ d $-regular tree.
• The method extends to sparse graphs and Erdős–Rényi random graphs $ G(n, d/n) $, suggesting broad applicability beyond regular graphs.