Strong Spatial Mixing and Approximating Partition Functions of Two-State Spin Systems without Hard Constrains

TL;DR

This paper proves strong spatial mixing for two-state spin systems on trees and extends the results to sparse graphs, enabling efficient approximation of partition functions under certain conditions on temperature and external field.

Contribution

It introduces new conditions for strong spatial mixing in two-state spin systems, including the first consideration of external field effects in terms of maximum average degree and interaction energy.

Findings

Strong spatial mixing holds under small inverse temperature or large external field.

Extension of results to sparse graphs using Weitz's self-avoiding tree.

Development of an FPTAS for partition functions in general graph families.

Abstract

We prove Gibbs distribution of two-state spin systems(also known as binary Markov random fields) without hard constrains on a tree exhibits strong spatial mixing(also known as strong correlation decay), under the assumption that, for arbitrary `external field', the absolute value of `inverse temperature' is small, or the `external field' is uniformly large or small. The first condition on `inverse temperature' is tight if the distribution is restricted to ferromagnetic or antiferromagnetic Ising models. Thanks to Weitz's self-avoiding tree, we extends the result for sparse on average graphs, which generalizes part of the recent work of Mossel and Sly\cite{MS08}, who proved the strong spatial mixing property for ferromagnetic Ising model. Our proof yields a different approach, carefully exploiting the monotonicity of local recursion. To our best knowledge, the second condition of `external field' for strong spatial mixing in this paper is first considered and stated in term of `maximum average degree' and `interaction energy'. As an application, we present an FPTAS for partition functions of two-state spin models without hard constrains under the above assumptions in a general family of graphs including interesting bounded degree graphs.