From Alternating Sign Matrices to the Gaussian Unitary Ensemble

TL;DR

This paper proves that the boundary fluctuations of uniformly random alternating sign matrices follow the Gaussian Unitary Ensemble, linking combinatorial models to random matrix theory.

Contribution

It establishes a rigorous connection between alternating sign matrices and the GUE-corners process, a novel result in the study of combinatorial probability.

Findings

Boundary fluctuations are described by the GUE-corners process.

Alternating sign matrices exhibit universal behavior akin to random matrix ensembles.

Abstract

The aim of this note is to prove that fluctuations of uniformly random alternating sign matrices (equivalently, configurations of the six-vertex model with domain wall boundary conditions) near the boundary are described by the Gaussian Unitary Ensemble and the GUE-corners process.