Counterexamples to a Weitz-Style Reduction for Multispin Systems

TL;DR

This paper demonstrates fundamental obstacles to extending Weitz's correlation decay reduction from two-state to multispin systems, highlighting nonconvexity issues in belief propagation for models like the ferromagnetic Potts model.

Contribution

It identifies key barriers to applying Weitz-style reductions to multispin systems and provides insights into the convexity properties of belief propagation in these models.

Findings

Nonconvexity of belief propagation in multispin systems

Obstacles to extending Weitz's reduction to multispin models

Convexity evidence in the antiferromagnetic Potts model

Abstract

In a seminal paper, Weitz showed that for two-state spin systems, such as the Ising and hardcore models from statistical physics, correlation decay on trees implies correlation decay on arbitrary graphs. The key gadget in Weitz's reduction has been instrumental in recent advances in approximate counting and sampling, from analysis of local Markov chains like Glauber dynamics to the design of deterministic algorithms for estimating the partition function. A longstanding open problem in the field has been to find such a reduction for more general multispin systems like the uniform distribution over proper colorings of a graph. In this paper, we show that for a rich class of multispin systems, including the ferromagnetic Potts model, there are fundamental obstacles to extending Weitz's reduction to the multispin setting. A central component of our investigation is establishing nonconvexity of the image of the belief propagation functional, the standard tool for analyzing spin systems on trees. On the other hand, we provide evidence of convexity for the antiferromagnetic Potts model.