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

### 2014/02/06

(with

Alexey Bufetov)

*Selecta Math. 21 (2015), no. 4, 1271–1338* •

arXiv:1402.1772 [math.PR]

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).