Law of Large Numbers for Infinite Random Matrices over a Finite Field

Abstract

Asymptotic representation theory of general linear groups GL(n,q) over a finite field leads to studying probability measures \rho on the group U of all infinite uni-uppertriangular matrices over F_q, with the condition that \rho is invariant under conjugations by arbitrary infinite matrices. Such probability measures form an infinite-dimensional simplex, and the description of its extreme points (in other words, ergodic measures \rho) was conjectured by Kerov in connection with nonnegative specializations of Hall-Littlewood symmetric functions. Vershik and Kerov also conjectured the following Law of Large Numbers. Consider an n by n diagonal submatrix of the infinite random matrix drawn from an ergodic measure coming from the Kerov's conjectural classification. The sizes of Jordan blocks of the submatrix can be interpreted as a (random) partition of n, or, equivalently, as a (random) Young diagram \lambda(n) with n boxes. Then, as n goes to infinity, the rows and columns of \lambda(n) have almost sure limiting frequencies corresponding to parameters of this ergodic measure. Our main result is the proof of this Law of Large Numbers. We achieve it by analyzing a new randomized Robinson-Schensted-Knuth (RSK) insertion algorithm which samples random Young diagrams \lambda(n) coming from ergodic measures. The probability weights of these Young diagrams are expressed in terms of Hall-Littlewood symmetric functions. Our insertion algorithm is a modified and extended version of a recent construction by Borodin and the second author (arXiv:1305.5501). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).