Selecta Mathematica

, Volume 21, Issue 4, pp 1271–1338

Law of Large Numbers for infinite random matrices over a finite field



Asymptotic representation theory of general linear groups \(\hbox {GL}(n,F_\mathfrak {q})\) over a finite field leads to studying probability measures \(\rho \) on the group \(\mathbb {U}\) of all infinite uni-uppertriangular matrices over \(F_\mathfrak {q}\), 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 infinite random matrix drawn from an ergodic measure coming from the Kerov’s conjectural classification and its \(n \times n\) submatrix formed by the first rows and columns. 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 \(n\rightarrow \infty \), 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 Petrov (2013). On the other hand, our randomized RSK insertion generalizes a version of the RSK insertion introduced by Vershik and Kerov (SIAM J. Algebr. Discret. Math. 7(1):116–124, 1986) in connection with asymptotic representation theory of symmetric groups (which is governed by nonnegative specializations of Schur symmetric functions).


Asymptotic representation theory General linear groups over a finite field Kerov’s conjecture Hall–Littlewood symmetric functions Law of Large Numbers for rows and columns of random Young diagrams Randomized Robinson–Schensted insertion 

Mathematics Subject Classification

Primary 05E10 Secondary 20G40 60J10 82C22 


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© Springer Basel 2015

Authors and Affiliations

  1. 1.International Laboratory of Representation Theory and Mathematical Physics, Department of MathematicsHigher School of EconomicsMoscowRussia
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.Department of MathematicsNortheastern UniversityBostonUSA

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