Random Young diagrams in a Rectangular Box

TL;DR

This paper determines the typical shape of random Young diagrams constrained within a rectangle, showing how their fluctuations relate to Ornstein-Uhlenbeck bridges, thus advancing understanding of their probabilistic geometry.

Contribution

It introduces the limit shape for Young diagrams with exponential area distribution in a rectangle and links fluctuations to Ornstein-Uhlenbeck bridges, a novel probabilistic connection.

Findings

Identified the limit shape of constrained Young diagrams.

Connected fluctuations to Ornstein-Uhlenbeck bridges.

Provided a probabilistic description of the diagrams' geometry.

Abstract

We exhibit the limit shape of random Young diagrams having a distribution proportional to the exponential of their area, and confined in a rectangular box. The Ornstein-Uhlenbeck bridge arises from the fluctuations around the limit shape.