GOE fluctuations for the maximum of the top path in alternating sign matrices

TL;DR

This paper proves that the maximum of the top level line in a random alternating sign matrix follows the GOE Tracy-Widom distribution, revealing new edge fluctuation behavior in the six-vertex model at a specific parameter.

Contribution

It establishes the GOE Tracy-Widom distribution for the maximum level line in ASMs at  = 1/2, a novel fluctuation result outside the free fermion case.

Findings

Maximum of top level line follows GOE Tracy-Widom distribution

First edge fluctuation result away from tangency points for this model

Connects ASM combinatorics with random matrix theory

Abstract

The six-vertex model is an important toy-model in statistical mechanics for two-dimensional ice with a natural parameter $\Delta$. When $\Delta = 0$, the so-called free-fermion point, the model is in natural correspondence with domino tilings of the Aztec diamond. Although this model is integrable for all $\Delta$, there has been very little progress in understanding its statistics in the scaling limit for other values. In this work, we focus on the six-vertex model with domain wall boundary conditions at $\Delta = 1/2$, where it corresponds to alternating sign matrices (ASMs). We consider the level lines in a height function representation of ASMs. We show that the maximum of the topmost level line for a uniformly random ASMs has the GOE Tracy--Widom distribution after appropriate rescaling. A key ingredient in our proof is Zeilberger's proof of the ASM conjecture. As far as we know, this is the first edge fluctuation result away from the tangency points for the domain-wall six-vertex model when we are not in the free fermion case.