GICAR algebras and dynamics on determinantal point processes: discrete orthogonal polynomial ensemble case

TL;DR

This paper explores the connection between determinantal point processes and operator algebras, focusing on dynamics in processes related to discrete orthogonal polynomials and $z$-measures, with implications for representation theory.

Contribution

It extends the operator algebra framework to include dynamical aspects of determinantal point processes associated with orthogonal polynomials and $z$-measures.

Findings

Analysis of unitary dynamics on these point processes

Investigation of stochastic dynamics within the operator algebra framework

Enhanced understanding of the algebraic structure underlying these processes

Abstract

It is known that determinantal point processes have an intimate relation to operator algebras. In the paper, we extend this relationship to encompass dynamical aspects. Especially, we delve into two types of determinantal point processes. One is related to discrete orthogonal polynomials of hypergeometric type, and another is $z$-measures, which arise in the asymptotic representation theory of the symmetric groups. Within the framework of operator algebras, we investigate both unitary and stochastic dynamics on these point processes.