Ergodic measures and infinite matrices of finite rank

TL;DR

This paper simplifies the classification of ergodic measures related to infinite orthogonal and unitary groups acting on matrices, making the results more accessible to probabilists by using the Vershik-Kerov method.

Contribution

It introduces a straightforward approach using the Vershik-Kerov method to classify ergodic measures, replacing complex asymptotic representation theory proofs.

Findings

Simplified classification of ergodic measures for infinite groups

Application of Vershik-Kerov method to matrix actions

Accessible proof technique for probabilists

Abstract

Let $O(\infty)$ and $U(\infty)$ be the inductively compact infinite orthogonal group and infinite unitary group respectively. The classifications of ergodic probability measures with respect to the natural group action of $O(\infty)\times O(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{R})$ and that of $U(\infty)\times U(m)$ on $\mathrm{Mat}(\mathbb{N}\times m, \mathbb{C})$ are due to Olshanski. The original proofs for these results are based on the asymptotic representation theory. In this note, by applying the Vershik-Kerov method, we propose a simple method for obtaining these two classifications, making it accessible to pure probabilists.