On Hydrodynamic Limits of Young Diagrams

TL;DR

This paper investigates the dynamic evolution of two-dimensional Young diagrams under stochastic models, revealing that their hydrodynamic limits are governed by different parabolic PDEs based on the energy configurations.

Contribution

It introduces a framework for understanding the dynamical scaling limits of Young diagrams, extending static results to time-dependent behaviors with PDE characterizations.

Findings

Hydrodynamic limits depend on energy structures

Different parabolic PDEs describe the shape evolution

Static limits are well-understood, dynamic limits are newly characterized

Abstract

We consider a family of stochastic models of evolving two-dimensional Young diagrams, given in terms of certain energies, with Gibbs invariant measures. `Static' scaling limits of the shape functions, under these Gibbs measures, have been shown by several over the years. The purpose of this article is to study corresponding `dynamical' limits of which less is understood. We show that the hydrodynamic scaling limits of the diagram shape functions may be described by different types parabolic PDEs, depending on the energy structure.