Infinite-dimensional diffusions as limits of random walks on partitions

TL;DR

This paper constructs a family of infinite-dimensional diffusion processes on the Thoma simplex, derived as scaling limits of Markov chains on partitions, with each process having a z-measure as its invariant distribution.

Contribution

It introduces a novel class of diffusions on the Thoma simplex linked to z-measures, connecting finite symmetric group approximations to infinite-dimensional stochastic dynamics.

Findings

Diffusions are ergodic and reversible with respect to z-measures.

Spectrum of the generator is explicitly described.

Dirichlet form associated with the diffusions is computed.

Abstract

The present paper originated from our previous study of the problem of harmonic analysis on the infinite symmetric group. This problem leads to a family {P_z} of probability measures, the z-measures, which depend on the complex parameter z. The z-measures live on the Thoma simplex, an infinite-dimensional compact space which is a kind of dual object to the infinite symmetric group. The aim of the paper is to introduce stochastic dynamics related to the z-measures. Namely, we construct a family of diffusion processes in the Toma simplex indexed by the same parameter z. Our diffusions are obtained from certain Markov chains on partitions of natural numbers n in a scaling limit as n goes to infinity. These Markov chains arise in a natural way, due to the approximation of the infinite symmetric group by the increasing chain of the finite symmetric groups. Each z-measure P_z serves as a unique invariant distribution for the corresponding diffusion process, and the process is ergodic with respect to P_z. Moreover, P_z is a symmetrizing measure, so that the process is reversible. We describe the spectrum of its generator and compute the associated (pre)Dirichlet form.