# Arctic curves phenomena for bounded lecture hall Tableaux

## TL;DR

This paper explores the asymptotic behavior of bounded lecture hall tableaux through combinatorial models and the tangent method, revealing arctic curves and confirming their shape in various models.

## Contribution

It introduces a novel analysis of lecture hall tableaux asymptotics using nonintersecting paths and dimer models, applying the tangent method to determine arctic curves.

## Key findings

- Identification of arctic curves in nonintersecting path models
- Confirmation of arctic curve shapes via Kasteleyn matrix analysis
- Application of the tangent method to complex combinatorial models

## Abstract

Recently the first author and Jang Soo Kim introduced lecture hall tableaux in their study of multivariate little q-Jacobi polynomials. They then enumerated bounded lecture hall tableaux and showed that their enumeration is closely related to standard and semistandard Young tableaux. In this paper we study the asymptotic behavior of these bounded tableaux thanks to two other combinatorial models: non intersecting paths on a graph whose faces are squares and pentagons and dimer models on a lattice whose faces are hexagons and octogons. We use the tangent method to investigate the arctic curve in the model of nonintersecting lattice paths with fixed starting points and ending points distributibuted according to some arbitrary piecewise differentiable function. We then study the dimer model and use some ansatz to guess the asymptotics of the inverse of the Kasteleyn matrix confirm the arctic curve computed with the tangent method for two examples.