Electronic Journal of Probability

Monotonous subsequences and the descent process of invariant random permutations

Mohamed Slim Kammoun

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Abstract

It is known from the work of Baik, Deift and Johansson [3] that we have Tracy-Widom fluctuations for the longest increasing subsequence of uniform permutations. In this paper, we prove that this result holds also in the case of the Ewens distribution and more generally for a class of random permutations with distribution invariant under conjugation. Moreover, we obtain the convergence of the first components of the associated Young tableaux to the Airy Ensemble as well as the global convergence to the Vershik-Kerov-Logan-Shepp shape. Using similar techniques, we also prove that the limiting descent process of a large class of random permutations is stationary, one-dependent and determinantal.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 118, 31 pp.

Dates
Received: 11 June 2018
Accepted: 11 November 2018
First available in Project Euclid: 27 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1543287754

Digital Object Identifier
doi:10.1214/18-EJP244

Subjects
Primary: 60C05: Combinatorial probability

Keywords
descent process determinantal point processes longest increasing subsequence random permutations Robinson-Schensted correspondence Tracy-Widom distribution

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kammoun, Mohamed Slim. Monotonous subsequences and the descent process of invariant random permutations. Electron. J. Probab. 23 (2018), paper no. 118, 31 pp. doi:10.1214/18-EJP244. https://projecteuclid.org/euclid.ejp/1543287754


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References

  • [1] David J. Aldous, Exchangeability and related topics, École d’été de probabilités de Saint-Flour, XIII—1983, Lecture Notes in Math., vol. 1117, Springer, Berlin, 1985, pp. 1–198.
  • [2] Richard Arratia, A. D. Barbour, and Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003.
  • [3] Jinho Baik, Percy Deift, and Kurt Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc. 12 (1999), no. 4, 1119–1178.
  • [4] Jean Bertoin, Random fragmentation and coagulation processes, Cambridge Studies in Advanced Mathematics, vol. 102, Cambridge University Press, Cambridge, 2006.
  • [5] Alexei Borodin, Persi Diaconis, and Jason Fulman, On adding a list of numbers (and other one-dependent determinantal processes), Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 639–670.
  • [6] Alexei Borodin and Vadim Gorin, Lectures on integrable probability, Probability and statistical physics in St. Petersburg, Proc. Sympos. Pure Math., vol. 91, Amer. Math. Soc., Providence, RI, 2016, pp. 155–214.
  • [7] Alexei Borodin, Andrei Okounkov, and Grigori Olshanski, Asymptotics of Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), no. 3, 481–515.
  • [8] Djalil Chafaï, Yan Doumerc, and Florent Malrieu, Processus des restaurants chinois et loi d’Ewens, Revue de Mathématiques Spéciales (RMS) 123 (2013), no. 3, 56–74.
  • [9] Ivan Corwin, The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), no. 1, 1130001, 76.
  • [10] Peter Donnelly and Geoffrey Grimmett, On the asymptotic distribution of large prime factors, J. London Math. Soc. (2) 47 (1993), no. 3, 395–404.
  • [11] Sergi Elizalde, Descent sets of cyclic permutations, Adv. in Appl. Math. 47 (2011), no. 4, 688–709.
  • [12] Nicholas M. Ercolani and Daniel Ueltschi, Cycle structure of random permutations with cycle weights, Random Structures Algorithms 44 (2014), no. 1, 109–133.
  • [13] W. J. Ewens, The sampling theory of selectively neutral alleles, Theoret. Population Biology 3 (1972), 87–112; erratum, ibid. 3 (1972), 240; erratum, ibid. 3 (1972), 376.
  • [14] Curtis Greene, An extension of Schensted’s theorem, Advances in Math. 14 (1974), 254–265.
  • [15] Lars Holst, The Poisson-Dirichlet distribution and its relatives revisited, 2001.
  • [16] Vladimir Ivanov and Grigori Olshanski, Kerov’s central limit theorem for the Plancherel measure on Young diagrams, Symmetric functions 2001: surveys of developments and perspectives, NATO Sci. Ser. II Math. Phys. Chem., vol. 74, Kluwer Acad. Publ., Dordrecht, 2002, pp. 93–151.
  • [17] Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), no. 1, 259–296.
  • [18] Kurt Johansson, Random matrices and determinantal processes, Mathematical statistical physics, Elsevier B. V., Amsterdam, 2006, pp. 1–55.
  • [19] S. Kerov, The asymptotics of interlacing sequences and the growth of continual Young diagrams, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 205 (1993), no. Differentsial prime naya Geom. Gruppy Li i Mekh. 13, 21–29, 179.
  • [20] S. Kerov, A differential model for the growth of Young diagrams, Proceedings of the St. Petersburg Mathematical Society, Vol. IV, Amer. Math. Soc. Transl. Ser. 2, vol. 188, Amer. Math. Soc., Providence, RI, 1999, pp. 111–130.
  • [21] S. V. Kerov, Transition probabilities of continual Young diagrams and the Markov moment problem, Funktsional. Anal. i Prilozhen. 27 (1993), no. 2, 32–49, 96.
  • [22] S. V. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis, Translations of Mathematical Monographs, vol. 219, American Mathematical Society, Providence, RI, 2003, Translated from the Russian manuscript by N. V. Tsilevich, With a foreword by A. Vershik and comments by G. Olshanski.
  • [23] Serguei Kerov, Grigori Olshanski, and Anatoli Vershik, Harmonic analysis on the infinite symmetric group. A deformation of the regular representation, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 8, 773–778.
  • [24] J. F. C. Kingman, Random partitions in population genetics, Proc. Roy. Soc. London Ser. A 361 (1978), no. 1704, 1–20.
  • [25] J. F. C. Kingman, Mathematics of genetic diversity, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 34, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1980.
  • [26] J. F. C. Kingman, S. J. Taylor, A. G. Hawkes, A. M. Walker, David Roxbee Cox, A. F. M. Smith, B. M. Hill, P. J. Burville, and T. Leonard, Random discrete distribution, J. Roy. Statist. Soc. Ser. B 37 (1975), 1–22, With a discussion by S. J. Taylor, A. G. Hawkes, A. M. Walker, D. R. Cox, A. F. M. Smith, B. M. Hill, P. J. Burville, T. Leonard and a reply by the author.
  • [27] Donald E. Knuth, Permutations, matrices, and generalized Young tableaux, Pacific J. Math. 34 (1970), 709–727.
  • [28] B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), no. 2, 206–222.
  • [29] Odile Macchi, The coincidence approach to stochastic point processes, Advances in Appl. Probability 7 (1975), 83–122.
  • [30] Peter McCullagh, Random permutations and partition models, (2011), 1170–1177.
  • [31] Carl Mueller and Shannon Starr, The length of the longest increasing subsequence of a random Mallows permutation, J. Theoret. Probab. 26 (2013), no. 2, 514–540.
  • [32] Jim Pitman and Marc Yor, The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator, Ann. Probab. 25 (1997), no. 2, 855–900.
  • [33] G. de B. Robinson, On the Representations of the Symmetric Group, Amer. J. Math. 60 (1938), no. 3, 745–760.
  • [34] Bruce E. Sagan, The symmetric group, second ed., Graduate Texts in Mathematics, vol. 203, Springer-Verlag, New York, 2001, Representations, combinatorial algorithms, and symmetric functions.
  • [35] C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179–191.
  • [36] Gérald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, second ed., Cours Spécialisés [Specialized Courses], vol. 1, Société Mathématique de France, Paris, 1995.
  • [37] Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174.
  • [38] N. V. Tsilevich, Stationary random partitions of a natural series, Teor. Veroyatnost. i Primenen. 44 (1999), no. 1, 55–73.
  • [39] Stanislaw M. Ulam, Monte Carlo calculations in problems of mathematical physics, Modern mathematics for the engineer: Second series, McGraw-Hill, New York, 1961, pp. 261–281.
  • [40] A. M. Vershik and S. V. Kerov, Asymptotic behavior of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Dokl. Akad. Nauk SSSR 233 (1977), no. 6, 1024–1027.
  • [41] A. M. Vershik and S. V. Kerov, Asymptotic behavior of the maximum and generic dimensions of irreducible representations of the symmetric group, Funktsional. Anal. i Prilozhen. 19 (1985), no. 1, 25–36, 96.