Random Matrix Theory over Integers of Local Fields

Abstract

This thesis is meant to bring together several different strands of existing work on random matrix theory over the ring of integers $R$ of a non-Archimedean local field and over finite fields. It attempts to give some insight into the relation of these two theories to one another and to classical random matrix theory, while also presenting a few new results. We discuss the number-theoretic background motivating the study of random matrices over $\Z_p$ via the Cohen-Lenstra heuristics, as well as the analogies between eigenvalues of random matrices over $\C$ and cokernels of random matrices over $R$, which include Horn's (ex-)conjecture for Hermitian matrices and an analogous result for cokernels. We prove that the distribution of the cokernel of a product of Haar-distributed random matrices over $R$ is described by products of Hall-Littlewood polynomials, in analogy with the known relation between the distribution of eigenvalues of a sum of random Hermitian matrices and multivariate Bessel functions. On the finite field side, we present a modified version of Fulman's construction of random partitions from random matrices over $\F_q$. Reinterpreting these partitions in terms of cokernels of random matrices over $\F_q[T]$, we prove that the same random partitions may also be obtained using a different distribution on $M_n(\F_q[T])$ which is more natural from the point of view of random matrices over rings of integers of global fields, of which the case of $\Z$ is well-studied. Finally, we define a plane partition associated to any matrix over $R$, which includes the information of both the matrix's cokernel and the partitions studied by Fulman; related to the distribution of this object is an interesting Markov chain on filtered vector spaces.