Note on the zero-free region of the hard-core model

TL;DR

This paper establishes a new zero-free region for the partition function of the hard-core model, expanding understanding of its analytic properties for graphs with bounded degree.

Contribution

It proves a novel zero-free region for the independence polynomial of graphs, improving previous bounds and covering a specific half disk in the complex plane.

Findings

Defines a new zero-free domain including a half disk in the complex plane.

Extends the understanding of the partition function's analytic behavior.

Provides bounds relevant for statistical physics and combinatorics.

Abstract

In this paper we prove a new zero-free region for the partition function of the hard-core model, that is, the independence polynomials of graphs with largest degree $\Delta$. This new domain contains the half disk $$D=\left\{ \lambda \in \mathbb{C}\ |\ \mathrm{Re}(\lambda)\geq 0, |\lambda|\leq \frac{7}{8}\tan \left( \frac{\pi}{2(\Delta-1)}\right)\right\}.$$