Hydrodynamic limit for an evolutional model of two-dimensional Young diagrams

TL;DR

This paper constructs a dynamic model for two-dimensional Young diagrams, demonstrating that their scaled height profiles converge to a nonlinear PDE, with stationary solutions linked to the Vershik curve.

Contribution

It introduces a new dynamics for Young diagrams associated with grandcanonical ensembles and proves hydrodynamic limits converging to a specific PDE.

Findings

Height variables converge to a nonlinear PDE as size diverges.

Stationary solutions correspond to the Vershik curve.

Results hold under both uniform and restricted uniform statistics.

Abstract

We construct dynamics of two-dimensional Young diagrams, which are naturally associated with their grandcanonical ensembles, by allowing the creation and annihilation of unit squares located at the boundary of the diagrams. The grandcanonical ensembles, which were introduced by Vershik, are uniform measures under conditioning on their size (or equivalently, area). We then show that, as the averaged size of the diagrams diverges, the corresponding height variable converges to a solution of a certain non-linear partial differential equation under a proper hydrodynamic scaling. Furthermore, the stationary solution of the limit equation is identified with the so-called Vershik curve. We discuss both uniform and restricted uniform statistics for the Young diagrams.