A Diffusive Limit for the Profiles of Random Young Diagrams by Way of Free Probability

Abstract

Concentration phenomena in statistical ensembles of Young diagrams have been investigated as static models first for the Plancherel ensemble by Vershik–Kerov and Logan–Shepp in 1970s and later in some other group-theoretical setting by Biane. On the other hand, a dynamical model of concentration for Young diagrams, which is not directly connected with group representations, was shown by Funak–{Sasada in the framework of hydrodynamic limit. The aim of this paper is to present a new dynamical model of concentration for Young diagrams featuring the group-theoretical sense. Starting from an initial state yielding concentration and a microscopic dynamics keeping the Plancherel measure invariant, we derive an evolution of the profiles of Young diagrams under a diffusive scaling limit. The resulting evolution along macroscopic time is described in terms of the notions of Voiculescu's free probability theory such as free compression and free convolution of Kerov transition measures.