Approximating the Partition Function of Two-Spin Systems

Years and Authors of Summarized Original Work

• 1993; Jerrum, Sinclair

• 2003; Goldberg, Jerrum, Paterson

• 2006; Weitz

• 2012; Sinclair, Srivastava, Thurley

• 2013; Li, Lu, Yin

• 2015; Sinclair, Srivastava, Štefankovič, Yin

Problem Definition

Spin systems are well-studied objects in statistical physics and applied probability. An instance of a spin system is an undirected graphG = (V, E) of n vertices. A configuration of a two-state spin system, or simply just two-spin system on G, is an assignment $\sigma : V \rightarrow \{ 0,1\}$ of two spin states “0” and “1” (sometimes called “−” and “+” or seen as two colors) to the vertices of G. Let $\mathbf{A} = \left [\begin{array}{*{10}c} A_{0,0}&A_{0,1} \\ A_{1,0}&A_{1,1} \end{array} \right ]$ be a nonnegative symmetric matrix which specifies the local interactions between adjacent vertices and $\mathbf{b} = \left [\begin{array}{*{10}c} b_{0} \\ b_{1} \end{array} \right ]$a nonnegative vector which specifies preferences of individual vertices...