Interlaced processes on the circle

TL;DR

This paper constructs couplings for commuting Markov processes on conjugacy classes of the unitary group, inspired by the RSK algorithm, to explore connections in random matrix theory and determinantal measures.

Contribution

It introduces explicit couplings for commuting Markov processes on the circle, linking algebraic, combinatorial, and probabilistic methods in random matrix theory.

Findings

Constructed couplings for commuting Markov processes on conjugacy classes.

Identified determinantal structure of Gibbs measures on bead configurations.

Computed the space-time correlation kernel for these measures.

Abstract

When two Markov operators commute, it suggests that we can couple two copies of one of the corresponding processes. We explicitly construct a number of couplings of this type for a commuting family of Markov processes on the set of conjugacy classes of the unitary group, using a dynamical rule inspired by the RSK algorithm. Our motivation for doing this is to develop a parallel programme, on the circle, to some recently discovered connections in random matrix theory between reflected and conditioned systems of particles on the line. One of the Markov chains we consider gives rise to a family of Gibbs measures on `bead configurations' on the infinite cylinder. We show that these measures have determinantal structure and compute the corresponding space-time correlation kernel.