Truncation of Haar random matrices in $\mathrm{GL}_N(\mathbb{Z}_m)$

Yanqi Qiu

doi:10.1214/16-ecp7

Abstract

The asymptotic law of the truncated $S\times S$ random submatrix of a Haar random matrix in $\mathrm{GL}_N(\mathbb{Z}_m)$ as $N$ goes to infinity is obtained. The same result is also obtained when $\mathbb{Z}_m$ is replaced by any commutative compact local ring whose maximal ideal is topologically closed.

Highlights

In the theory of random matrices, some particular attention is payed recently to the asymptotic distributions of the truncated S × S upper-left corners of a large N × N random matrices from different matrix ensembles (CUE, COE, Haar Unitary Ensembles, Haar Orthogonal Ensembles), see [6, 4, 2, 1].In the present paper, we consider the truncation of a Haar random matrix in GLN (Zm) with Zm = Z/mZ
The asymptotic law of the truncated S × S random submatrix of a Haar random matrix in GLN (Zm) as N goes to infinity is obtained
The same result is obtained when Zm is replaced by any commutative compact local ring whose maximal ideal is topologically closed

Summary

Introduction

In the theory of random matrices, some particular attention is payed recently to the asymptotic distributions of the truncated S × S upper-left corners of a large N × N random matrices from different matrix ensembles (CUE, COE, Haar Unitary Ensembles, Haar Orthogonal Ensembles), see [6, 4, 2, 1]. We consider the truncation of a Haar random matrix in GLN (Zm) with Zm = Z/mZ. This research is motivated by its application in a forthcoming paper on the classification of ergodic measures on the space of infinite p-adic matrices, where the asymptotic law of a fixed size truncation of the Haar random matrix from the group of N × N invertible matrices over the ring of p-adic integers is essentially used and is derived from a particular case of our main result, Theorem 3.1.

Notation

Main result

A generalization