Asymptotics of Bounded Lecture-Hall Tableaux

Abstract

We study weighted bounded \(n\)-lecture hall tableaux with bound \(t_n\) in the joint limit \(n,t_n\to\infty\) with \(t_n/n\to\alpha\in(0,\infty)\). Using their non-intersecting-path representation, we construct a Schur generating function adapted to this model and derive moment formulas for the limiting counting measures. When the normalized tail sums of the weights have a \(C^2\) macroscopic profile with negative derivative, the rescaled height functions converge in probability in \(L^1_{\mathrm{loc}}(\mathbb R\times(0,\alpha))\) to a deterministic limit shape whose complex slope satisfies a Burgers equation with an explicit source term. In the uniformly weighted model of \cite{SKN21}, the source term vanishes, proving Conjecture~6.1 of that work. Under an additional interval-structure assumption on the limiting bottom boundary, and only continuity and strict monotonicity of the tail profile, the unrescaled height fluctuations converge to the Gaussian free field in the sense of joint moments of horizontal polynomial observables. The method applies although the particle configurations are not Gelfand--Tsetlin patterns and the associated dimer model is not doubly periodic.