Asymptotics, exact results, and analogies in p-adic random matrix theory

Abstract

This thesis is a compilation of exact results regarding p-adic random matrices and Hall-Littlewood polynomials, and asymptotic results proven using these tools. Many of the results of both types are motivated and guided by analogies to existing results in classical random matrix theory over R, C or H, but often exhibit probabilistic behaviors which differ markedly from these known cases. Specifically, we prove the following:

(1) We show exact relations between products and corners of random matrices over Qₚ and Hall-Littlewood processes, which are direct analogues of the classical relations between singular values of real or complex random matrices and type A Heckman-Opdam hypergeometric functions. (2) We prove that the boundary of the Hall-Littlewood t-deformation of the Gelfand-Tsetlin graph is parametrized by infinite integer signatures, extending results of Gorin and Cuenca on boundaries of related deformed Gelfand-Tsetlin graphs. (3) In the special case when 1/t is a prime p we combine this with the aforementioned relations between matrix corners and Hall-Littlewood polynomials to recover results of Bufetov-Qiu [BQ17] and Assiotis [Ass20] on infinite p-adic random matrices. (4) Using the above relation between matrix products and Hall-Littlewood polynomials, together with explicit formulas for the latter, we obtain exact product formulas for the joint distribution of the cokernels of products A₁,A₂A₁,A₃A₂A₁, . . . of independent additive-Haar-distributed matrices A subscript i over the p-adic integers Zₚ. This generalizes the explicit formula for the classical Cohen-Lenstra measure on abelian p-groups. (5) We give an exact sampling algorithm for products of corners of Haar GLₙ(Zₚ)-distributed matrices, and show by analyzing it that the singular numbers of such products obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions. (6) We classify left GLₙ(Qₚ)-invariant stochastic processes on the (discrete) homogeneous space GLₙ(Qₚ)/GLₙ(Zₚ) with independent increments. We consider the one with smallest jumps, a p-adic analogue of multiplicative Brownian motion on GLₙ(C)/U(N), and show that the singular numbers of the matrix evolve as independent Poisson jump processes which are forced to remain ordered by reflection off the walls of the type A Weyl chamber. (7) As N and time go to ∞, we show that this process converges to a stationary limit, with density explicitly expressed in terms of certain intricate exponential sums. The proof uses new Macdonald process computations, which feature a symmetric function incarnation of the explicit solution to the inverse moment problem for abelian p-groups shown recently by Sawin and Wood [SW22b]. (8) We prove that this reflected Poisson walk is universal, governing dynamical local limits for the singular numbers of p-adic random matrix products at both the bulk and edge, and may thus be viewed as a p-adic analogue of the extended sine and Airy processes. (9) Extrapolating this process to general real p > 1, we analyze the limit as p --> 1. We prove a law of large numbers, a central limit theorem relating it to stationary solutions of certain SDEs, and a bulk limit to a certain explicit stationary Gaussian process on R. Unlike most previously studied limits of Macdonald processes, the latter exhibits scaling exponents characteristic of the Edwards-Wilkinson universality class in (1+1) dimensions, which may be seen as a reflection of locality of interactions between singular numbers which differs markedly from classical random matrix theory.