Stochastic Dynamics of Growing Young Diagrams and Their Limit Shapes

TL;DR

This paper studies the growth and limit shapes of Young diagrams under specific constraints and dynamics, providing explicit formulas for their asymptotic shapes in various growth regimes.

Contribution

It introduces a detailed analysis of Young diagrams constrained by height differences and derives their limit shapes for different growth processes.

Findings

Explicit limit shapes for constrained Young diagrams.

Limit shapes for diffusively growing diagrams with addition and removal.

Asymptotic behavior characterized for various growth dynamics.

Abstract

We investigate a class of Young diagrams growing via the addition of unit cells and satisfying the constraint that the height difference between adjacent columns $\geq r$. In the long time limit, appropriately re-scaled Young diagrams approach a limit shape that we compute for each integer $r\geq 0$. We also determine limit shapes of `diffusively' growing Young diagrams satisfying the same constraint and evolving through the addition and removal of cells that proceed with equal rates.