Random Walks on Strict Partitions

TL;DR

This paper studies a sequence of random walks on strict partitions, proving their convergence to a continuous-time Markov process on an infinite-dimensional simplex, with a focus on differential operators and shifted Young diagrams.

Contribution

It introduces a new class of random walks on strict partitions and characterizes their limit as a Markov process with a differential operator, also generalizing Kerov interlacing coordinates.

Findings

Random walks on strict partitions converge to a continuous-time Markov process.

The limit process is described by a second order differential operator.

Generalization of Kerov interlacing coordinates to shifted Young diagrams.

Abstract

We consider a certain sequence of random walks. The state space of the n-th random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to a continuous-time Markov process. The state space of this process is the infinite-dimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The main result about the limit process is the expression of its the pre-generator as a formal second order differential operator in a polynomial algebra. Of separate interest is the generalization of Kerov interlacing coordinates to the case of shifted Young diagrams.