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

Abstract

In this paper, we establish the universality property for the distribution of the cokernel of a random Hermitian matrix over the ring of integers O of a quadratic extension K of Qp.More precisely, for each positive integer n let Xn be a random n × n Hermitian matrix over O whose upper triangular entries are independent and not too concentrated. We show that the distribution of the cokernel of Xn is asymptotically universal as n → ∞ and provide the formula for the limiting distribution. This is an analogue of the universality of random symmetric matrices over Zp proved by Wood (J. Amer. Math. Soc. 30 (2017), 915-958).

Hall–Littlewood Polynomials, Boundaries, and p-Adic Random Matrices

“…p-adic random matrix theory. Many works have considered discrete random matrices over the integers Z or p-adic integers Z p , which yield distributions on abelian groups appearing in arithmetic statistics and the theory of random graphs: see Friedman-Washington [39], Ellenberg-Venkatesh-Westerland [26,27], Bhargava-Kane-Lenstra-Poonen-Rains [7], Clancy-Kaplan-Leake-Payne-Wood [23], Wood [68,66,67], Kovaleva [49], Lipnowski-Sawin-Tsimmerman [52], Mészáros [53], Cheong-Huang [20] -Kaplan [21] and -Yu [22], Nguyen-Wood [56,55], and Lee [50,51]. The p-adic case is simpler than the integer case, while retaining most interesting features.…”

Limits and fluctuations of p-adic random matrix products

“…an abelian group. Many works such as Clancy-Kaplan-Leake-Payne-Wood [30], Wood [85,83,84], Mészáros [66,67], Nguyen-Wood [70,69], and Lee [58,59] study cokernels of random matrices over the integers 2 Z, motivated by problems in number theory, graph theory and topology where random or pseudorandom integral matrices appear.…”