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Preprints, Working Papers, ... Year : 2023

A determinantal point process approach to scaling and local limits of random Young tableaux

Abstract

We obtain scaling and local limit results for large random Young tableaux of fixed shape $\lambda^0$ 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 $\lambda^0$, 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 $\lambda^0$. 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.
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hal-04171056 , version 1 (26-07-2023)

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Jacopo Borga, Cédric Boutillier, Valentin Féray, Pierre-Loïc Méliot. A determinantal point process approach to scaling and local limits of random Young tableaux. 2023. ⟨hal-04171056⟩
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