Stochastic dynamics related to Plancherel measure on partitions

TL;DR

This paper studies a Markov chain of Young diagrams derived from a Poisson process, analyzing their correlation functions and scaling limits in the context of the Plancherel measure.

Contribution

It introduces a continuous-time Markov chain model for Young diagrams associated with Poissonized Plancherel measure and computes their correlation functions and scaling limits.

Findings

Derived explicit formulas for dynamical correlation functions.

Identified bulk and edge scaling limits of the Markov chains.

Established connections to asymptotic representation theory.

Abstract

Consider the standard Poisson process in the first quadrant of the Euclidean plane, and for any point (u,v) of this quadrant take the Young diagram obtained by applying the Robinson-Schensted correspondence to the intersection of the Poisson point configuration with the rectangle with vertices (0,0), (u,0), (u,v), (0,v). It is known that the distribution of the random Young diagram thus obtained is the poissonized Plancherel measure with parameter uv. We show that for (u,v) moving along any southeast-directed curve in the quadrant, these Young diagrams form a Markov chain with continuous time. We also describe these chains in terms of jump rates. Our main result is the computation of the dynamical correlation functions of such Markov chains and their bulk and edge scaling limits.