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 $n\times n$ 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).