Six-vertex models and the GUE-corners process

TL;DR

This paper introduces new operators for six-vertex models, enabling correlation function analysis and asymptotic behavior characterization, revealing connections to the GUE-corners process near the boundary.

Contribution

It develops operators inspired by Macdonald difference operators to analyze correlation functions in six-vertex models, linking their asymptotics to the GUE-corners process.

Findings

Correlation functions expressed as contour integrals.

Asymptotic analysis via steepest descent.

Boundary behavior described by GUE-corners process.

Abstract

In this paper we consider a class of probability distributions on the six-vertex model from statistical mechanics, which originate from the higher spin vertex models of https://arxiv.org/abs/1601.05770. We define operators, inspired by the Macdonald difference operators, which extract various correlation functions, measuring the probability of observing different arrow configurations. The development of our operators is largely based on the properties of a remarkable family of symmetric rational functions, which were previously studied in https://arxiv.org/abs/1410.0976. For the class of models we consider, the correlation functions can be expressed in terms of multiple contour integrals, which are suitable for asymptotic analysis. For a particular choice of parameters we analyze the limit of the correlation functions through a steepest descent method. Combining this asymptotic statement with some new results about Gibbs measures on Gelfand-Tsetlin cones and patterns, we show that the asymptotic behavior of our six-vertex model near the boundary is described by the GUE-corners process.