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

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 $n\to \infty$ 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 ${\mathbb{Q}}_p$.

arXiv comment31 pages, to appear in TAMS

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Cited by in corpus (3)

• Mixed moments and the joint distribution of random groups
• Joint distribution of the cokernels of random $p$-adic matrices II
• The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue