# Lee-Yang Zeros of the antiferromagnetic Ising Model

## TL;DR

This paper analyzes the distribution of zeros of the partition function in the anti-ferromagnetic Ising Model, revealing their density and sparsity on the unit circle for different graph classes, using dynamical systems methods.

## Contribution

It provides a precise characterization of zeros on Cayley trees and shows their density in certain graph classes, contrasting with the sparsity on Cayley trees.

## Key findings

- Zeros are nowhere dense on certain arcs of the unit circle for Cayley trees.
- Zeros are dense in a circular sub-arc for graphs with bounded degree.
- Dynamical systems describe ratios of partition functions on recursive trees.

## Abstract

We investigate the location of zeros for the partition function of the anti-ferromagnetic Ising Model, focusing on the zeros lying on the unit circle. We give a precise characterization for the class of rooted Cayley trees, showing that the zeros are nowhere dense on the most interesting circular arcs. In contrast, we prove that when considering all graphs with a given degree bound, the zeros are dense in a circular sub-arc, implying that Cayley trees are in this sense not extremal. The proofs rely on describing the rational dynamical systems arising when considering ratios of partition functions on recursively defined trees.