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

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.