Dynamic Model

Abstract

In this chapter, we discuss a dynamical aspect of the limit shape problem for random Young diagrams. In a microscopic point of view, a continuous time Markov chain is introduced on the Young diagrams of size n which keeps the Plancherel measure invariant and has an initial distribution admitting the concentration at a profile as n tends to $\infty$. Our model is built on such a canonical setting. By considering a diffusive scaling limit in time versus space, we derive a macroscopic time evolution of the limit profile. The resulting evolution is described through the Kerov transition measure in terms of free-probabilistic notions.