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

TL;DR

This paper analyzes the asymptotic behavior of large random Young tableaux using determinantal point processes, providing explicit descriptions of their limiting surfaces and local limits, with applications to specific shapes and infinite tableaux.

Contribution

It introduces a new approach using determinantal point processes to derive explicit scaling and local limit results for random Young tableaux of fixed shape.

Findings

Explicit formula for the limiting surface of Young tableaux.

Criteria for the continuity of the limiting surface.

Local limit results in the bulk of Poissonized 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.