Asymptotics of Bounded Lecture-Hall Tableaux

TL;DR

This paper analyzes the asymptotic behavior of bounded lecture hall tableaux, establishing limit shapes, proving a conjecture about slope equations, and demonstrating Gaussian free field fluctuations.

Contribution

It proves a conjecture on the complex Burgers equation for scaled height functions and introduces new methods for analyzing non-Gelfand-Tsetlin particle configurations.

Findings

Limit shapes form with linear bounds.

Rescaled height functions satisfy a complex Burgers equation.

Fluctuations converge to the Gaussian free field.

Abstract

We study the asymptotics of bounded lecture hall tableaux. Limit shapes form when the bounds of the lecture hall tableaux go to infinity linearly in the lengths of the partitions describing the large-scale shapes of these tableaux. We prove Conjecture 6.1 in \cite{SKN21}, stating that the slopes of the rescaled height functions in the scaling limit satisfy a complex Burgers equation. We also show that the fluctuations of the unrescaled height functions converge to the Gaussian free field. The proof is based on new construction and analysis of Schur generating functions for the lecture hall tableaux, whose corresponding particle configurations do not form a Gelfand-Tsetlin scheme; and the corresponding dimer models are not doubly periodic.