Limits and fluctuations of p-adic random matrix products

Image
January 14, 2021
10:20AM - 11:15AM
Location
Zoom info to follow

Date Range
Add to Calendar 2021-01-14 10:20:00 2021-01-14 11:15:00 Limits and fluctuations of p-adic random matrix products Speaker:  Roger Van Peski (MIT) Abstract:  Any nonsingular matrix A over the p-adic integers Z_p may, by Smith normal form, be written as U diag(p^{\lambda_1},\ldots,p^{\lambda_n}) V for U, V in GL_n(Z_p), for some integers \lambda_i called singular numbers (also known as invariant factors or elementary divisors). The distributions of singular numbers for random A are quite well-studied, largely because the equivalent data of the abelian p-group coker(A) provides useful models for many random groups in number theory and combinatorics, collectively called Cohen-Lenstra heuristics. By contrast, in this talk we consider a setting in which the (rescaled) fluctuations of singular numbers converge to continuous probability distributions on R. Namely, we show that the singular numbers of successive products A_k \cdots A_1, where A_i are corners of independent Haar-distributed p-adic matrices, obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions with k playing the role of time. These asymptotics rely on new exact expressions for distributions of singular numbers of products and corners of random matrices over Q_p in terms of Hall-Littlewood polynomials, which allow us to re-express the above matrix product process as a simple interacting particle system. We also indicate parallels between our results--both exact and asymptotic--and known results on singular values of complex matrices.   Zoom info to follow Department of Mathematics math@osu.edu America/New_York public
Description

Speaker:  Roger Van Peski (MIT)

Abstract:  Any nonsingular matrix A over the p-adic integers Z_p may, by Smith normal form, be written as U diag(p^{\lambda_1},\ldots,p^{\lambda_n}) V for U, V in GL_n(Z_p), for some integers \lambda_i called singular numbers (also known as invariant factors or elementary divisors). The distributions of singular numbers for random A are quite well-studied, largely because the equivalent data of the abelian p-group coker(A) provides useful models for many random groups in number theory and combinatorics, collectively called Cohen-Lenstra heuristics.

By contrast, in this talk we consider a setting in which the (rescaled) fluctuations of singular numbers converge to continuous probability distributions on R. Namely, we show that the singular numbers of successive products A_k \cdots A_1, where A_i are corners of independent Haar-distributed p-adic matrices, obey a law of large numbers and their fluctuations converge dynamically to independent Brownian motions with k playing the role of time. These asymptotics rely on new exact expressions for distributions of singular numbers of products and corners of random matrices over Q_p in terms of Hall-Littlewood polynomials, which allow us to re-express the above matrix product process as a simple interacting particle system. We also indicate parallels between our results--both exact and asymptotic--and known results on singular values of complex matrices.
 

Events Filters: