Universality of the cokernels of random $p$-adic Hermitian matrices

TL;DR

This paper proves that the distribution of cokernels of large random Hermitian matrices over the integers of quadratic extensions of p-adic fields converges to a universal limit, providing explicit formulas and solving an open problem.

Contribution

It establishes the universal limiting distribution of cokernels for random p-adic Hermitian matrices over quadratic extensions, extending previous work and solving an open problem.

Findings

Distribution of cokernels converges as matrix size grows

Limiting distribution is universal, independent of matrix entries

Explicit formula for the limiting distribution provided

Abstract

In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers $\mathcal{O}$ of a quadratic extension $K$ of $\mathbb{Q}_p$. For each positive integer $n$, let $X_n$ be a random $n \times n$ Hermitian matrix over $\mathcal{O}$ whose upper triangular entries are independent and their reductions are not too concentrated on certain values. We show that the distribution of the cokernel of $X_n$ always converges to the same distribution which does not depend on the choices of $X_n$ as $n \rightarrow \infty$ and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood in the case of the ring of integers of a quadratic extension of $\mathbb{Q}_p$.