Riemann-Hilbert Approach to the Six-Vertex Model

TL;DR

This paper reviews the application of the Riemann-Hilbert method to analyze the large-scale asymptotics of the six-vertex model with domain wall boundary conditions, connecting integrable models with random matrix theory.

Contribution

It demonstrates how the Riemann-Hilbert approach can be used to derive asymptotic results for the six-vertex model, building on the connection to random matrix models.

Findings

Asymptotic behavior of the six-vertex model in different phase regions

Connection between the model's partition function and random matrix theory

Application of Riemann-Hilbert techniques to integrable models

Abstract

The six-vertex model, or the square ice model, with domain wall boundary conditions (DWBC) has been introduced and solved for finite $n$ by Korepin and Izergin. The solution is based on the Yang-Baxter equations and it represents the free energy in terms of an $n\times n$ Hankel determinant. Paul Zinn-Justin observed that the Izergin-Korepin formula can be re-expressed in terms of the partition function of a random matrix model with a nonpolynomial interaction. We use this observation to obtain the large $n$ asymptotics of the six-vertex model with DWBC. The solution is based on the Riemann-Hilbert approach. In this paper we review asymptotic results obtained in different regions of the phase diagram.