2023/07/21 by Borga, Jacopo, Boutillier, Cédric, Féray, Valentin +1
#Combinatorics (math.CO) #FOS: Mathematics #Probability (math.PR) #Representation Theory (math.RT)
paper · doi:10.48550/arxiv.2307.11885
We obtain scaling and local limit results for large random Young tableaux of fixed shape λ0 via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). More precisely, we prove: (1) an explicit description of the limiting surface of a uniform random Young tableau of shape λ0, based on solving a complex-valued polynomial equation; (2) a simple criteria to determine if the limiting surface is continuous in the whole domain; (3) and a local limit result in the bulk of a random Poissonized Young tableau of shape λ0. Our results have several consequences, for instance: they lead to explicit formulas for the limiting surface of L-shaped tableaux, generalizing the results of Pittel and Romik (2007) for rectangular shapes; they imply that the limiting surface for L-shaped tableaux is discontinuous for almost-every L-shape; and they give a new one-parameter family of infinite random Young tableaux, constructed from the so-called random infinite bead process.