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

TL;DR

This paper investigates the asymptotic behavior of submatrices of Haar random matrices in general linear groups over finite rings, extending results to a broad class of commutative compact local rings as N approaches infinity.

Contribution

It provides the first asymptotic law for truncated Haar random matrices in $ ext{GL}_N(Z_m)$ and generalizes to other commutative compact local rings.

Findings

Asymptotic distribution of truncated Haar matrices in $ ext{GL}_N(Z_m)$

Extension of results to commutative compact local rings

Behavior characterized as matrix size N tends to infinity

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.