Gaussian universality of $p$-adic random matrix products via corners

TL;DR

This paper proves the universality of singular number distributions in products of random matrices over p-adic groups, extending previous work and establishing laws of large numbers and CLTs for these matrix products.

Contribution

It introduces a broad class of distributions called 'split' and extends universality results to $ ext{GL}_n(Q_p)$, $ ext{GSp}_{2n}$, and potentially other split reductive groups.

Findings

Universality of singular number asymptotics for p-adic matrix products.

Strong law of large numbers for i.i.d. matrix products.

Central limit theorem for singular numbers.

Abstract

We establish the universality of the singular numbers in random matrix products over $\mathrm{GL}_n(\mathbb{Q}_p)$ as the number of products approaches infinity, with a fixed $n\ge 1$. We demonstrate that, under a broad class of distributions, which we term as ``split", the asymptotics of matrix products align with the sum of the singular numbers of the matrix corners. Specifically, when matrices are independent and identically distributed, we derive the strong law of large numbers and the central limit theorem. Our approach is inspired by Van Peski's work (arXiv:2011.09356), which examines products of $n\times n$ corners of Haar-distributed elements $\mathrm{GL}_N(\mathbb{Z}_p)$. We extend the method so that the criterion now works as long as the measures are left- and right-invariant under the multiplication of $\mathrm{GL}_n(\mathbb{Z}_p)$. Building on this approach for $\mathrm{GL}_n$, we further demonstrate similar universality for $\mathrm{GSp}_{2n}$ and discuss potential extensions to general split reductive groups.