Markov dynamics on the Thoma cone: a model of time-dependent determinantal processes with infinitely many particles

TL;DR

This paper constructs a family of continuous-time Markov processes on the Thoma cone, revealing their properties and connections to determinantal point processes and random matrix theory, with explicit spectral and correlation kernel descriptions.

Contribution

It introduces new Markov processes on the Thoma cone with explicit generators and correlation structures, inspired by representation theory and random matrix models.

Findings

Processes are Feller and have explicit generators.

Finite-dimensional distributions are determinantal point processes.

Time-dependent kernels resemble those in random matrix theory.

Abstract

The Thoma cone is an infinite-dimensional locally compact space, which is closely related to the space of extremal characters of the infinite symmetric group. In another context, the Thoma cone appears as the set of parameters for totally positive, upper triangular Toeplitz matrices of infinite size. The purpose of the paper is to construct a family of continuous time Markov processes on the Thoma cone, depending on two continuous parameters. Our construction largely exploits specific properties of the Thoma cone related to its representation-theoretic origin, although we do not use representations directly. On the other hand, we were inspired by analogies with random matrix theory coming from models of Markov dynamics related to orthogonal polynomial ensembles. We show that our processes possess a number of nice properties, namely: (1) every process X is a Feller process; (2) the infinitesimal generator of X, its spectrum, and the eigenfunctions admit an explicit description; (3) in the equilibrium regime, the finite-dimensional distributions of X can be interpreted as (the laws of) infinite-particle systems with determinantal correlations; (4) the corresponding time-dependent correlation kernel admits an explicit expression, and its structure is similar to that of time-dependent correlation kernels appearing in random matrix theory.