Infinite p-adic random matrices and ergodic decomposition of p-adic Hua measures

TL;DR

This paper studies the ergodic decomposition of p-adic Hua measures on infinite matrices, providing an explicit description linking it to Hall-Littlewood measures and employing Markov chain techniques.

Contribution

It explicitly characterizes the ergodic decomposition of p-adic Hua measures, connecting it to Hall-Littlewood measures and advancing understanding of invariant measures on p-adic matrix spaces.

Findings

Explicit ergodic decomposition measure derived

Connection established with Hall-Littlewood measures

Uses Markov chains in the analysis

Abstract

Neretin constructed an analogue of the Hua measures on the infinite $p$-adic matrices $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$. Bufetov and Qiu classified the ergodic measures on $Mat\left(\mathbb{N},\mathbb{Q}_p\right)$ that are invariant under the natural action of $GL(\infty,\mathbb{Z}_p)\times GL(\infty,\mathbb{Z}_p)$. In this paper we solve the problem of ergodic decomposition for the $p$-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.