The limit shape of large alternating sign matrices

TL;DR

This paper investigates the asymptotic shape of large alternating sign matrices by analyzing the emptiness formation probability in a related statistical model, proposing a conjectural limit shape based on saddle-point analysis.

Contribution

It introduces a conjectural formula for the limit shape of large ASMs derived from saddle-point analysis of integral representations, connecting combinatorics and statistical physics.

Findings

Derived a conjectural limit shape for large ASMs

Analyzed the EFP in the six-vertex model context

Extended the analysis to 3-enumerated ASMs

Abstract

The problem of the limit shape of large alternating sign matrices (ASMs) is addressed by studying the emptiness formation probability (EFP) in the domain-wall six-vertex model. Assuming that the limit shape arises in correspondence to the `condensation' of almost all solutions of the saddle-point equations for certain multiple integral representation for EFP, a conjectural expression for the limit shape of large ASMs is derived. The case of 3-enumerated ASMs is also considered.