Zeros of ferromagnetic 2-spin systems

TL;DR

This paper investigates the zeros of partition functions in ferromagnetic 2-spin systems, introducing new zero-free regions that lead to improved deterministic algorithms for approximate counting, independent of graph degree.

Contribution

It refines existing contraction methods to identify zero-free regions without degree dependence and develops superior deterministic approximation algorithms.

Findings

New zero-free regions for ferromagnetic 2-spin systems

Deterministic algorithms outperform MCMC and correlation decay in certain regimes

Results are independent of the maximum degree of the underlying graph

Abstract

We study zeros of the partition functions of ferromagnetic 2-state spin systems in terms of the external field, and obtain new zero-free regions of these systems via a refinement of Asano's and Ruelle's contraction method. The strength of our results is that they do not depend on the maximum degree of the underlying graph. Via Barvinok's method, we also obtain new efficient and deterministic approximate counting algorithms. In certain regimes, our algorithm outperforms all other methods such as Markov chain Monte Carlo and correlation decay.