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

Abstract

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).

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Notes

  1. 1.

    The conjecture was originally formulated in equivalent terms of nonnegative specializations of Hall–Littlewood symmetric functions. Another equivalent formulation involves coherent probability measures on the Young branching graph with formal edge multiplicities depending on \(\mathfrak {q}\) (cf. Sect. 5.4). See [6, 34], Thm. 2.3], [37, Prop. 4.7] for details of these equivalences.

  2. 2.

    Our notation of parameters \((\varvec{\alpha };\varvec{\beta };\gamma )\) borrowed from Borodin and Corwin [8] and also used in Borodin and Petrov [16] differs from the one of Kerov [45], see also Gorin et al. [37]. Details are explained in Remark 2.6 below.

  3. 3.

    \(P_\lambda \) is a constant multiple of \(Q_\lambda \). We use the standard notation of [50].

  4. 4.

    Here and below \({\mathbf {1}}_{A}\) means the indicator of \(A\).

  5. 5.

    See also Aissen et al. [1].

  6. 6.

    However, note that a horizontal or a vertical strip is allowed to be empty.

  7. 7.

    This means that \(\mathsf {Sym}\) is the projective limit (in the category of graded algebras) of algebras of symmetric polynomials in growing number of variables.

  8. 8.

    Lexicographic order means that, for example, \(x_1^{2}\) is higher than \(\mathrm {const}\cdot x_1x_2\) which is in turn higher than \(\mathrm {const}\cdot x_2^{2}\).

  9. 9.

    Sometimes we will also use the term Macdonald-nonnegative, cf. [8, §2.2.1].

  10. 10.

    Sometimes to emphasize that we are working with dual variables, we will use the hat notation. For example, a dual variable equal to \(1\) will be denoted by \(\hat{1}\).

  11. 11.

    However, in the present paper, we restrict ourselves to \(t\in [0,1)\), which excludes cases (3) and (4) from the consideration.

  12. 12.

    Note that because \(P_\lambda \) is a multiple of \(Q_\lambda \), (3.1) reduces to (1.4) when \(q=0\), \(t=\mathfrak {q}^{-1}\), and \(p_1(\varvec{\alpha };{\varvec{\beta }};\mathbf {Pl}_\gamma )=1\).

  13. 13.

    The reason for the multiplication of \(\tau \) by \(p_1(\varvec{\alpha };{\varvec{\beta }};\mathbf {Pl}_\gamma )\) is the future convenience of certain formulas. Note that for now we are not assuming that \(p_1(\varvec{\alpha };{\varvec{\beta }};\mathbf {Pl}_\gamma )=1\) (see Remark 3.2).

  14. 14.

    Note that the expression \(\varPi (u;\mathbf {A})\) in (2.4) is a particular case of (4.2) corresponding to \(\mathbf {B}=(u)\), a specialization into a single usual variable.

  15. 15.

    Note that the condition \(\big [{\begin{matrix} \lambda \\ \bar{\lambda } \end{matrix}}\big ]\in \mathbb {Y}^{(2)}(\mathbf {A};\mathbf {B})\) implies that \(\bar{\lambda }\subseteq \lambda \).

  16. 16.

    These dynamics may be viewed as discrete \((q,t)\)-analogs of the classical Dyson Brownian motion [22] from random matrix theory, e.g., see [17].

  17. 17.

    This actually is the first place when we drop the nonnegativity assumption.

  18. 18.

    We are using probabilistic terms despite the fact that some ‘probabilities’ can be negative (see the beginning of Sect. 4 for more detail). When we speak about conditioning on an event which possibly can have negative probability, this should be understood as an intuitive appeal to the product rule (4.15) defining the jump rates via the quantities (5.15)–(5.17).

  19. 19.

    Setting \(h=+\infty \) means that the last (i.e., the leftmost) particle jumps independently.

  20. 20.

    We assume that the lower level particles evolve according to the univariate dynamics \(\mathsf {Q}_{\mathbf {A}}\). The functions \((W_{(\alpha )}^{h},V_{(\alpha )}^{h})\) then provide the necessary “induction step” leading to the upper univariate dynamics \(\mathsf {Q}^{(2)}_{\mathbf {A};(\alpha )}\).

  21. 21.

    It is possible to develop randomized “sampling” algorithms (i.e., formal Markov “dynamics” with negative probabilities of certain elements in a “transition matrix”) for the general parameters \((q,t)\) by analogy, but we will not pursue this direction here.

  22. 22.

    Thus, the lower Young diagram \(\bar{\lambda }(k)\) has a random number of boxes, but \(\lambda (k)\) has exactly \(k\) boxes.

  23. 23.

    In fact, these particles evolve according to Dynamics 8 in [16], up to renaming \(t\) by \(q\) and considering the evolution in continuous time, cf. Remark 6.17.

  24. 24.

    Of course, particles in the bulk of the interlacing array (i.e., all particles except the leftmost ones \(\lambda ^{(i)}_i\)) will move in some way, but this cannot break the desired inequalities. This remark applies to other two cases as well.

  25. 25.

    Note that once a particle \(\tau ^{(m)}_{1}\), for some \(m>i\) was not pushed, all upper particles \(\tau ^{(j)}_{1}\) (\(j>m\)) also cannot be pushed, see Remark 6.20.

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Acknowledgments

This work was started at the 2013 Cornell Probability Summer School, and we would like to thank the organizers for the invitation and warm hospitality. We are very grateful to Alexei Borodin, Jason Fulman, Vadim Gorin, Grigori Olshanski, and Anatoly Vershik for helpful discussions. We also would like to thank the anonymous referee for extremely valuable suggestions on improving the presentation of our results. A.B. was partially supported by Simons Foundation—IUM scholarship, by Moebius Foundation for Young Scientists, by “Dynasty” foundation, and by the RFBR Grant 13-01-12449.

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Correspondence to Leonid Petrov.

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To Grigori Olshanski on the occasion of his 65th birthday.

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Bufetov, A., Petrov, L. Law of Large Numbers for infinite random matrices over a finite field. Sel. Math. New Ser. 21, 1271–1338 (2015). https://doi.org/10.1007/s00029-015-0179-9

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Keywords

  • 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