Location of zeros for the partition function of the Ising model on bounded degree graphs

TL;DR

This paper precisely characterizes the location of zeros of the Ising model's partition function on bounded degree graphs, extending Lee-Yang results and using complex dynamics analysis.

Contribution

It provides an exact description of zero locations for bounded degree graphs in both ferromagnetic and anti-ferromagnetic cases, based on degree and temperature.

Findings

Zeros lie on specific curves depending on degree and temperature

Exact zero locations are determined for graphs with bounded degree

Analysis involves complex dynamical systems

Abstract

The seminal Lee-Yang theorem states that for any graph the zeros of the partition function of the ferromagnetic Ising model lie on the unit circle in $\mathbb C$. In fact the union of the zeros of all graphs is dense on the unit circle. In this paper we study the location of the zeros for the class of graphs of bounded maximum degree $d\geq 3$, both in the ferromagnetic and the anti-ferromagnetic case. We determine the location exactly as a function of the inverse temperature and the degree $d$. An important step in our approach is to translate to the setting of complex dynamics and analyze a dynamical system that is naturally associated to the partition function.