Markov dynamics on the dual object to the infinite-dimensional unitary group

TL;DR

This paper explores how representation theory of large groups can be applied to probabilistic models, using the example of Markov dynamics on the dual object to the infinite-dimensional unitary group, as part of a mini-course.

Contribution

It demonstrates the application of representation theory concepts to probabilistic problems through a specific Markov dynamics model related to the infinite-dimensional unitary group.

Findings

Illustrates the connection between representation theory and probability

Provides a detailed example of Markov dynamics on dual objects

Serves as educational material for advanced probabilistic methods

Abstract

These are notes for a mini-course of 3 lectures given at the St. Petersburg School in Probability and Statistical Physics (June 2012). My aim was to explain, on the example of a particular model, how ideas from the representation theory of big groups can be applied in probabilistic problems. The material is based on the joint paper arXiv:1009.2029 by Alexei Borodin and myself; a broader range of topics is surveyed in the lecture notes by Alexei Borodin and Vadim Gorin arXiv:1212.3351.