Arctic curves phenomena for bounded lecture hall Tableaux

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.

Concavity analysis of the tangent method

“…It was then elevated to a universal geometric principle and successfully checked in a number of situations where it indeed reproduced known results [12][13][14][15][16][17]. It has been subsequently used to make predictions in cases the analytic shape of the arctic curve was not known, see [12,14,[17][18][19][20].…”

Arctic curves of the $6$V model with partial DWBC and double Aztec rectangles

“…In the thermodynamic limit, obtained by sending the mesh size to 0 while keeping the size of the domain unchanged, this separation is the so-called arctic curve of the model. Without being exhaustive, such an arctic phenomenon is observed in lozenge tilings of hexagons [1], in domino tilings of Aztec rectangles with defects [2], in bounded lecture hall tableaux [3] or in configuations of the six-vertex (6V) with various boundary conditions [4].…”

“…The EFP technique yields an arctic curve whose shape is the solution of F(x, y, z) = 0 and ∂ z F(x, y, z) = 0, which exactly describes the envelope of a family of curves parametrized by z, which moreover turned out to be straight lines. This observation was then elevated to a universal geometric principle by the development of the tangent method [13], which provides an alternative derivation of the arctic curve and which has been applied to a large class of models [2,3,[14][15][16][17][18][19][20][21][22][23].…”

“…The computation of the residue at z = ∞ requires to consider a clockwise-oriented contour whose interior englobes all the other residues. The change of variables z = 1/w transforms this contour into a counterclockwise oriented contour the origin, excluding all the other poles 3. Under this change of variables, we have dydu = 1−u (1−v) 2 dudv.…”