Rectangular Young tableaux and the Jacobi ensemble

TL;DR

This paper investigates the fluctuation behavior of large rectangular Young tableaux, revealing Gaussian fluctuations at the corners and Tracy-Widom distribution elsewhere, using a connection with the Jacobi ensemble.

Contribution

It establishes the fluctuation distributions of Young tableaux edges, linking them to Gaussian and Tracy-Widom laws via the Jacobi ensemble.

Findings

Gaussian fluctuations at the corners

Tracy-Widom fluctuations away from corners in square cases

Connection between Young tableaux and Jacobi ensemble

Abstract

It has been shown by Pittel and Romik that the random surface associated with a large rectangular Young tableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle. We show that in the corner, these fluctuations are gaussian wheras, away from the corner and when the rectangle is a square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with the Jacobi ensemble.