Six-Vertex Model and Random Matrix Distributions

TL;DR

This paper explores the deep connections between the six-vertex model in statistical mechanics and random matrix theory, highlighting universal distributions and providing simplified proofs of key asymptotic results.

Contribution

It offers shorter, more transparent proofs of known asymptotic theorems linking the six-vertex model to GUE and Tracy-Widom distributions, and introduces essential tools like the Yang-Baxter equation.

Findings

GUE-corners process appears in the six-vertex model

Tracy-Widom distribution $F_2$ emerges asymptotically

Shorter proofs for key asymptotic theorems

Abstract

We survey the connections between the six-vertex (square ice) model of 2d statistical mechanics and random matrix theory. We highlight the same universal probability distributions appearing on both sides, and also indicate related open questions and conjectures. We present full proofs of two asymptotic theorems for the six-vertex model: in the first one the Gaussian Unitary Ensemble and GUE-corners process appear; the second one leads to the Tracy-Widom distribution $F_2$. While both results are not new, we found shorter transparent proofs for this text. On our way we introduce the key tools in the study of the six-vertex model, including the Yang-Baxter equation and the Izergin-Korepin formula.