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

Selecta Mathematica, New Series. 2015. Vol. 21. No. 4. P. 1271-1338.
Bufetov A., Петров Л.
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  was conjectured by Kerov in connection with nonnegative specializations of Hall-Littlewood symmetric functions. Vershik and Kerov also conjectured the Law of Large Numbers for random Young diagrams distributed according to these measures.

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.