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

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 $d\ge 3$, 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.

24 pages, 3 figures. Made a number of small clarifications, corrections and changes in notation. Results remain unchanged. To appear in the Journal of the London Mathematical Society

References in corpus (3)

• Fisher zeros and correlation decay in the Ising model
• Limiting Measure of Lee--Yang Zeros for the Cayley Tree
• Lee-Yang Zeros of the antiferromagnetic Ising Model

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• Ising model with Curie-Weiss perturbation
• On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs
• Statistical physics approaches to Unique Games
• Dichotomy for Graph Homomorphisms with Complex Values on Bounded Degree Graphs