Arctic Curves Phenomena for Bounded Lecture Hall Tableaux

Abstract

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 octagons. We use the tangent method to investigate the arctic curve in the model of non-intersecting lattice paths with fixed starting points and ending points distributed according to some arbitrary piecewise differentiable function. We then study the dimer model and use an ansatz to guess the asymptotics of the inverse of the Kasteleyn, which confirm the arctic curve computed with the tangent method for two examples.