Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary

TL;DR

This paper constructs a family of Markov processes on infinite Gelfand-Tsetlin schemes that preserve Gibbs measures and connect to harmonic analysis on the infinite-dimensional unitary group, revealing their invariant measures and determinantal structure.

Contribution

It introduces a four-parameter family of Markov processes on Gelfand-Tsetlin schemes that extend to the boundary, linking harmonic analysis, determinantal processes, and representation theory.

Findings

Processes preserve Gibbs measures

Invariant measure is a determinantal point process

Connection to harmonic analysis on U(infinity)

Abstract

We construct a four-parameter family of Markov processes on infinite Gelfand-Tsetlin schemes that preserve the class of central (Gibbs) measures. Any process in the family induces a Feller Markov process on the infinite-dimensional boundary of the Gelfand-Tsetlin graph or, equivalently, the space of extreme characters of the infinite-dimensional unitary group U(infinity). The process has a unique invariant distribution which arises as the decomposing measure in a natural problem of harmonic analysis on U(infinity) posed in arXiv:math/0109193. As was shown in arXiv:math/0109194, this measure can also be described as a determinantal point process with a correlation kernel expressed through the Gauss hypergeometric function.