In this paper, we study the distribution of the cokernel of a general random Hermitian matrix over the ring of integers O of a quadratic extension K of Qp. For each positive integer n, let Xn be a random n × n Hermitian matrix over 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 Xn always converges to the same distribution which does not depend on the choices of Xn as n→∞and provide an explicit formula for the limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture note of Wood [Probability theory for random groups arising in number theory, 2022] in the case of the ring of integers of a quadratic extension of Qp.
Received by the editors August 29, 2022, and, in revised form, February 19, 2023, May 4, 2023, and June 26, 2023. 2020 Mathematics Subject Classification. Primary 15B52, 60B20; Secondary 11E39. Key words and phrases. Random p-adic matrices, Hermitian matrices, universality. The author was supported by a KIAS Individual Grant (SP079601) via the Center for Mathematical Challenges at Korea Institute for Advanced Study.
