Selecta Mathematica

, Volume 21, Issue 4, pp 1271–1338 | Cite as

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

  • Alexey Bufetov
  • Leonid PetrovEmail author


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 



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.


  1. 1.
    Aissen, M., Edrei, A., Schoenberg, I.J., Whitney, A.: On the generating functions of totally positive sequences. Proc. Natl. Acad. Sci. USA 37, 303–307 (1951)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aissen, M., Schoenberg, I.J., Whitney, A.: On the generating functions of totally positive sequences I. J. Anal. Math. 2, 93–103 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Baik, J., Deift, P., Johansson, K.: On the distribution of the length of the longest increasing subsequence of random permutations. J. Am. Math. Soc. 12(4), 1119–1178 (1999). arXiv:math/9810105 [math.CO]
  4. 4.
    Berele, A., Regev, A.: Hook Young diagrams with applications to combinatorics and representations of Lie superalgebras. Adv. Math. 64(2), 118–175 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borodin, A.: Limit Jordan normal form of large triangular matrices over a finite field. Funct. Anal. Appl. 29(4), 279–281 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Borodin, A.: The law of large numbers and the central limit theorem for the jordan normal form of large triangular matrices over a finite field. J. Math. Sci. 96(5), 3455–3471 (1999)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Borodin, A.: Schur dynamics of the Schur processes. Adv. Math. 228(4), 2268–2291 (2011). arXiv:1001.3442 [math.CO]
  8. 8.
    Borodin, A., Corwin, I.: Macdonald processes. Prob. Theory Rel. Fields 158, 225–400 (2014). arXiv:1111.4408 [math.PR]
  9. 9.
    Borodin, A., Corwin, I., Gorin, V., Shakirov, S.: Observables of Macdonald processes. (2013). arXiv:1306.0659 [math.PR]
  10. 10.
    Borodin, A., Ferrari, P.: Anisotropic growth of random surfaces in 2 + 1 dimensions. (2008). arXiv:0804.3035 [math-ph] (to appear in Comm. Math. Phys)
  11. 11.
    Borodin, A., Gorin, V.: Markov processes of infinitely many nonintersecting random walks. Probab. Theory Rel. Fields 155(3–4), 935–997 (2013). arXiv:1106.1299 [math.PR]
  12. 12.
    Borodin, A., Olshanski, G.: Harmonic functions on multiplicative graphs and interpolation polynomials. Electron. J. Comb. 7, R28 (2000). arXiv:math/9912124 [math.CO]
  13. 13.
    Borodin, A., Olshanski, G.: Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary. J. Funct. Anal. 263(1), 248–303 (2012). arXiv:1009.2029 [math.PR]
  14. 14.
    Borodin, A., Olshanski, G.: The boundary of the Gelfand–Tsetlin graph: a new approach. Adv. Math. 230, 1738–1779 (2012). arXiv:1109.1412 [math.CO]
  15. 15.
    Borodin, A., Olshanski, G.: The Young bouquet and its boundary. Mosc. Math. J. 13(2), 193–232 (2013). arXiv:1110.4458 [math.RT]
  16. 16.
    Borodin, A., Petrov, L.: Nearest neighbor Markov dynamics on Macdonald processes. Adv. Math. (2013). arXiv:1305.5501 [math.PR] (to appear)
  17. 17.
    Borodin, A., Petrov, L.: Integrable probability: from representation theory to Macdonald processes. Probab. Surv. 11, 1–58 (2014). arXiv:1310.8007 [math.PR]
  18. 18.
    Boyer, R.: Infinite traces of AF-algebras and characters of \(U(\infty )\). J. Oper. Theory 9, 205–236 (1983)zbMATHGoogle Scholar
  19. 19.
    Bufetov, Al: The central limit theorem for extremal characters of the infinite symmetric group. Funct. Anal. Appl. 46(2), 83–93 (2012). arXiv:1105.1519 [math.RT]
  20. 20.
    Corwin, I., Petrov, L.: The q-PushASEP: a new integrable model for traffic in 1 + 1 dimension. J. Stat. Phys. (2013). arXiv:1308.3124 [math.PR] (to appear)
  21. 21.
    Diaconis, P., Fill, J.A.: Strong stationary times via a new form of duality. Ann. Probab. 18, 1483–1522 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dyson, F.J.: A Brownian motion model for the eigenvalues of a random matrix. J. Math. Phys. 3(6), 1191–1198 (1962)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Edrei, A.: On the generating functions of totally positive sequences. II. J. Anal. Math. 2, 104–109 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Edrei, A.: On the generating function of a doubly infinite, totally positive sequence. Trans. Am. Math. Soc. 74, 367–383 (1953)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Féray, V., Méliot, P.-L.: Asymptotics of q-plancherel measures. Probab. Theory Rel. Fields 152(3–4), 589–624 (2012). arXiv:1001.2180 [math.RT]
  26. 26.
    Fomin, S.: Two-Dimensional Growth in Dedekind Lattices. Master’s thesis, Leningrad State University (1979)Google Scholar
  27. 27.
    Fomin, S.: Generalized Robinson–Schnested–Knuth correspondence. Zapiski Nauchn. Sem. LOMI 155, 156–175 (1986) (in Russian)Google Scholar
  28. 28.
    Fomin, S.: Duality of graded graphs. J. Algebr. Comb. 3(4), 357–404 (1994)CrossRefzbMATHGoogle Scholar
  29. 29.
    Fomin, S.: Schensted algorithms for dual graded graphs. J. Algebr. Comb. 4(1), 5–45 (1995)CrossRefzbMATHGoogle Scholar
  30. 30.
    Forrester, P.J., Rains, E.M.: Interpretations of some parameter dependent generalizations of classical matrix ensembles. Prob. Theory Rel. Fields 131(1), 1–61 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Fulman, J.: Probabilistic measures and algorithms arising from the Macdonald symmetric functions (1997). arXiv:math/9712237 [math.CO]
  32. 32.
    Fulman, J.: A probabilistic approach toward conjugacy classes in the finite general linear and unitary groups. J. Algebr. 212(2), 557–590 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Fulman, J.: The eigenvalue distribution of a random unipotent matrix in its representation on lines. J. Algebr. 228(2), 497–511 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Fulman, J.: Random matrix theory over finite fields. Bull. Am. Math. Soc. 39(1), 51–85 (2001). arXiv:math/0003195 [math.GR]
  35. 35.
    Fulman, J.: Cohen–Lenstra heuristics and random matrix theory over finite fields. J. Group Theory 17(4), 619–648 (2014). arXiv:1307.0879 [math.NT]
  36. 36.
    Gorin, V.: The q-Gelfand–Tsetlin graph, Gibbs measures and q-oeplitz matrices. Adv. Math. 229(1), 201–266 (2012). arXiv:1011.1769 [math.RT]
  37. 37.
    Gorin, V., Kerov, S., Vershik, A.: Finite traces and representations of the group of infinite matrices over a finite field. Adv. Math. 254, 331–395 (2014). arXiv:1209.4945 [math.RT]
  38. 38.
    Gorin, V., Panova, G.: Asymptotics of symmetric polynomials with applications to statistical mechanics and representation theory. DMTCS Proceedings, 25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013), pp. 37–48 (2013). arXiv:1301.0634 [math.RT]
  39. 39.
    Ivanov, V.: The dimension of Skew Shifted Young diagrams, and projective characters of the infinite symmetric group. J. Math. Sci., 96(5), 3517–3530 (1999) (in Russian: Zap. Nauchn. Sem. POMI 240, 115–135 (1997)). arXiv:math/0303169 [math.CO]
  40. 40.
    Jack, H.: A class of symmetric functions with a parameter. Proc. R. Soc. Edinb. A 69, 1–18 (1970)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Jack, H.: A surface integral and symmetric functions. Proc. R. Soc. Edinb. A 69, 347–363 (1972)MathSciNetzbMATHGoogle Scholar
  42. 42.
    Johansson, K.: Shape fluctuations and random matrices. Commun. Math. Phys. 209(2), 437–476 (2000). arXiv:math/9903134 [math.CO]
  43. 43.
    Kerov, S.: Combinatorial examples in the theory of AF-algebras. Zapiski Nauchn. Semin. LOMI, 172, 55–67 (English translation. J. Soviet Math. 59(1992), 1063–1071) (1989)Google Scholar
  44. 44.
    Kerov, S.: Generalized Hall–Littlewood symmetric functions and orthogonal polynomials. Adv. Sov. Math. 9, 67–94 (1992)MathSciNetGoogle Scholar
  45. 45.
    Kerov, S.: Asymptotic Representation Theory of the Symmetric Group and Its Applications in Analysis, vol. 219. AMS, Translations of Mathematical Monographs, Providence (2003)Google Scholar
  46. 46.
    Kerov, S., Okounkov, A., Olshanski, G.: The boundary of Young graph with Jack edge multiplicities. Int. Math. Res. Not. 4, 173–199 (1998). arXiv:q-alg/9703037
  47. 47.
    Kingman, J.F.C.: Random partitions in population genetics. Proc. R. Soc. Lond. A 361, 1–20 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kirillov, A.A.: Variations on the triangular theme. Trans. Am. Math. Soc. 169, 43–74 (1995)MathSciNetGoogle Scholar
  49. 49.
    Littlewood, D.E.: On certain symmetric functions. Proc. Lond. Math. Soc. 43, 485–498 (1961)MathSciNetCrossRefGoogle Scholar
  50. 50.
    Macdonald, I.G.: Symmetric Functions and Hall Polynomials, 2nd edn. Oxford University Press, Oxford (1995)zbMATHGoogle Scholar
  51. 51.
    Meliot, P.-L.: A central limit theorem for the characters of the infinite symmetric group and of the infinite Hecke algebra. (2011). arXiv:1105.0091 [math.RT]
  52. 52.
    Nazarov, M.L.: Projective representations of the infinite symmetric group. In: Vershik, A.M. (ed.) Representation Theory and Dynamical Systems. Advances in Soviet Mathematics, vol. 9, pp. 115–130. American Mathematical Society (1992)Google Scholar
  53. 53.
    O’Connell, N.: A path-transformation for random walks and the Robinson–Schensted correspondence. Trans. Am. Math. Soc. 355(9), 3669–3697 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  54. 54.
    O’Connell, N.: Conditioned random walks and the RSK correspondence. J. Phys. A 36(12), 3049–3066 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    O’Connell, N., Pei, Y.: A q-weighted version of the Robinson–Schensted algorithm. Electron. J. Probab. 18(95), 1–25 (2013) arXiv:1212.6716 [math.CO]
  56. 56.
    Okounkov, A., Olshanski, G.: Asymptotics of Jack polynomials as the number of variables goes to infinity. Int. Math. Res. Not. 1998(13), 641–682 (1998). arXiv:q-alg/9709011
  57. 57.
    Olshanski, G., Vershik, A.: Ergodic unitarily invariant measures on the space of infinite Hermitian matrices. In: Contemporary Mathematical Physics. F.A. Berezi’s Memorial Volume. American Mathematical Society Translations (Advances in the Mathematical Sciences—31), vol. 175 of 2, pp. 137–175. (1996). arXiv:math/9601215v1 [math.RT]
  58. 58.
    Pei, Y.: A symmetry property for q-weighted Robinson–Schensted algorithms and other branching insertion algorithms. J. Algebr. Comb. 40, 743–770 (2013) arXiv:1306.2208 [math.CO]
  59. 59.
    Petrov, L.: The Boundary of the Gelfand–Tsetlin graph: new proof of Borodin–Olshanski’s formula, and its q-analogue. (2012). arXiv:1208.3443 [math.CO] (to appear in Moscow Math. J.)
  60. 60.
    Petrov, L.: \(\mathfrak{sl}(2)\) operators and Markov processes on branching graphs. J. Algebr. Comb. 38(3), 663–720 (2013). arXiv:1111.3399 [math.CO]
  61. 61.
    Romik, D., Sniady, P.: Jeu de taquin dynamics on infinite Young tableaux and second class particles. Ann. Probab. 43(2), 682–737 (2015). arXiv:1111.0575 [math.PR]
  62. 62.
    Sagan, B.E.: The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions. Springer, Berlin (2001)CrossRefGoogle Scholar
  63. 63.
    Skudlarek, H.-L.: Die unzerlegbaren Charaktere einiger discreter Gruppen. Math. Ann. 223, 213–231 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  64. 64.
    Sniady, P.: Robinson–Schensted–Knuth Algorithm, Jeu de Taquin, and Kerov–Vershik Measures on Infinite Tableaux. SIAM J. Discret. Math. 28(2), 598–630 (2014). arXiv:1307.5645 [math.CO]
  65. 65.
    Stanley, R.: Enumerative Combinatorics, vol. 2. Cambridge University Press, Cambridge [with a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin (2001)]Google Scholar
  66. 66.
    Thoma, E.: Die unzerlegbaren, positive-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe. Math. Z. 85, 40–61 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  67. 67.
    Thoma, E.: Characters of the group \(GL(\infty , q)\). In: Lecture Notes in Mathematics, vol. 266, pp. 321–323. Springer, New York (1972)Google Scholar
  68. 68.
    Vershik, A., Kerov, S.: Asymptotic theory of the characters of the symmetric group. Funktsional. Anal. i Priloz. 15(4), 15–27 (1981)MathSciNetGoogle Scholar
  69. 69.
    Vershik, A., Kerov, S.: Characters and factor representations of the infinite symmetric group. Dokl. Akad. Nauk. SSSR 257(5), 1037–1040 (1981)MathSciNetGoogle Scholar
  70. 70.
    Vershik, A., Kerov, S.: Characters and factor-representations of the infinite unitary group. Dokl. Akad. Nauk. SSSR 267(2), 272–276 (1982)MathSciNetGoogle Scholar
  71. 71.
    Vershik, A., Kerov, S.: The characters of the infinite symmetric group and probabiliy properties of the Robinson–Shensted–Knuth algorithm. SIAM J. Algebr. Discret. Math. 7(1), 116–124 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  72. 72.
    Vershik, A., Kerov, S.: Characters and realizations of infinite-dimensional Hecke algebra and knot invariants. Sov. Math. Dokl. 38, 134–137 (1989)MathSciNetzbMATHGoogle Scholar
  73. 73.
    Vershik, A., Kerov, S.: On a infinite-dimensional group over a finite field. Funct. Anal. Appl. 32(3), 147–152 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  74. 74.
    Vershik, A., Kerov, S.: Four drafts of the representation theory of the group of infinite matrices over a finite field. J. Math. Sci. 147(6), 7129–7144 (2007). arXiv:0705.3605 [math.RT]
  75. 75.
    Voiculescu, D.: Representations factorielles de type \(II_1\) de \(U(\infty )\). J. Math. Pures Appl. 55, 1–20 (1976)MathSciNetzbMATHGoogle Scholar

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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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