Zero-Freeness of the Hard-Core Model with Bounded Connective Constant

TL;DR

This paper extends zero-free region results for the hard-core model's partition function from maximum degree bounds to the more precise connective constant, ensuring analyticity of free energy on infinite graphs.

Contribution

It introduces a new definition of the connective constant for finite graphs and proves zero-freeness in complex neighborhoods based on this measure.

Findings

Zero-free regions are established up to the connective constant threshold.

The results imply the analyticity of free energy density on infinite lattices.

A block contraction technique is used to extend correlation decay to complex neighborhoods.

Abstract

We study the zero-free regions of the partition function of the hard-core model on finite graphs and their implications for the analyticity of the free energy on infinite lattices. Classically, zero-freeness results have been established up to the tree uniqueness threshold $\lambda_c(\Delta-1)$ determined by the maximum degree $\Delta$. However, for many graph classes, such as regular lattices, the connective constant $\sigma$ provides a more precise measure of structural complexity than the maximum degree. While recent approximation algorithms based on correlation decay and Markov chain Monte Carlo have successfully exploited the connective constant to improve the threshold to $\lambda_c(\sigma)$, analogous results for complex zero-freeness have been lacking. In this paper, we bridge this gap by introducing a proper definition of the connective constant for finite graphs based on a lower bound on the number of $k$-depth self-avoiding walks. We prove that for any graph family with a lower connective constant $\mu$, the partition function is zero-free in a complex neighborhood of the interval $[0, \lambda]$ for all $\lambda < \lambda_c(\mu)$. As a direct consequence, we establish the uniqueness and analyticity of the free energy density for infinite lattices up to the connective constant threshold, extending the known regions derived from maximum degree bounds. Our proof utilizes a block contraction technique that lifts the correlation decay property from a real interval to a strip-like complex neighborhood.