Zeros of the Potts Model Partition Function in the Large-$q$ Limit

TL;DR

This paper investigates the distribution of zeros of the Potts model partition function in the large-q limit, revealing they lie on the unit circle under certain conditions, with implications for understanding phase transitions.

Contribution

It provides a detailed analysis of the zeros' locations in the complex plane for large q, extending previous results to more general lattice sections and boundary conditions.

Findings

Zeros lie on the unit circle in the complex plane for large q

Zeros' loci are determined near the unit circle for finite lattice sections

Results apply to various boundary conditions and lattice types

Abstract

We study the zeros of the $q$-state Potts model partition function $Z(\Lambda,q,v)$ for large $q$, where $v$ is the temperature variable and $\Lambda$ is a section of a regular $d$-dimensional lattice with coordination number $\kappa_\Lambda$ and various boundary conditions. We consider the simultaneous thermodynamic limit and $q \to \infty$ limit and show that when these limits are taken appropriately, the zeros lie on the unit circle $|x_\Lambda|=1$ in the complex $x_\Lambda$ plane, where $x_\Lambda=v q^{-2/\kappa_\Lambda}$. For large finite sections of some lattices we also determine the circular loci near which the zeros lie for large $q$.