The five-vertex model as a discrete log-gas

TL;DR

This paper analyzes the five-vertex model on a rectangular lattice with specific boundary conditions, reformulating its partition function as a discrete log-gas to evaluate free-energy density in the scaling limit, and providing explicit resolvent forms.

Contribution

It introduces a reformulation of the five-vertex model's partition function as a discrete log-gas and derives explicit resolvent forms across regimes, advancing understanding of its thermodynamic behavior.

Findings

Reproduces previous free-energy results using a new approach.

Provides explicit resolvent expressions in all regimes.

Lays groundwork for studying limit shape phenomena.

Abstract

We consider the five-vertex model on a rectangular domain of the square lattice, with the so-called `scalar-product' boundary conditions. We address the evaluation of the free-energy density of the model in the scaling limit, that is when the number of sites is sent to infinity and the mesh of the lattice to zero, while keeping the size of the domain constant. To this aim, we reformulate the partition function of the model in terms of a discrete log-gas, and study its behaviour in the thermodynamic limit. We reproduce previous results, obtained by using a differential equation approach. Moreover, we provide the explicit form of the resolvent in all possible regimes. This work is preliminary to further studies of limit shape phenomena in the model.