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Journal of the American Mathematical Society

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ISSN 1088-6834 (online) ISSN 0894-0347 (print)

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Asymptotics of Plancherel measures for symmetric groups
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by Alexei Borodin, Andrei Okounkov and Grigori Olshanski HTML | PDF
J. Amer. Math. Soc. 13 (2000), 481-515 Request permission

Abstract:

We consider the asymptotics of the Plancherel measures on partitions of $n$ as $n$ goes to infinity. We prove that the local structure of a Plancherel typical partition in the middle of the limit shape converges to a determinantal point process with the discrete sine kernel. On the edges of the limit shape, we prove that the joint distribution of suitably scaled 1st, 2nd, and so on rows of a Plancherel typical diagram converges to the corresponding distribution for eigenvalues of random Hermitian matrices (given by the Airy kernel). This proves a conjecture due to Baik, Deift, and Johansson by methods different from the Riemann-Hilbert techniques used in their original papers and from the combinatorial proof given by the second author. Our approach is based on an exact determinantal formula for the correlation functions of the poissonized Plancherel measures in terms of a new kernel involving Bessel functions. Our asymptotic analysis relies on the classical asymptotic formulas for the Bessel functions and depoissonization techniques.
References
  • D. Aldous and P. Diaconis, Hammersley’s interacting particle process and longest increasing subsequences, Probab. Theory Related Fields 103 (1995), no. 2, 199–213. MR 1355056, DOI 10.1007/BF01204214
  • AD2 —, Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc., 36, 1999, 413–432. BDJ1 J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations, Journal of AMS, 12, 1999, 1119–1178. BDJ2 —, On the distribution of the length of the second row of a Young diagram under Plancherel measure, math.CO/9901118.
  • Philippe Biane, Permutation model for semi-circular systems and quantum random walks, Pacific J. Math. 171 (1995), no. 2, 373–387. MR 1372234
  • B2 —, Representations of symmetric groups and free probability, Adv. Math. 138, 1998, no. 1, 126–181. Bo A. Borodin, Riemann-Hilbert problem and the discrete Bessel kernel, math.CO/9912093, to appear in Intern. Math. Res. Notices. BO A. Borodin and G. Olshanski, Distribution on partitions, point processes, and the hypergeometric kernel, math.RT/9904010, to appear in Comm. Math. Phys. BO3 —, Z-measures on partitions, Robinson-Schensted-Knuth correspondence, and $\beta =2$ random matrix ensembles, math.CO/9905189, to appear in Proceedings of the 1999 MSRI Workshop on Random Matrices and their Applications.
  • Peter A. Clarkson and J. Bryce McLeod, A connection formula for the second Painlevé transcendent, Arch. Rational Mech. Anal. 103 (1988), no. 2, 97–138. MR 946971, DOI 10.1007/BF00251504
  • D. J. Daley and D. Vere-Jones, An introduction to the theory of point processes, Springer Series in Statistics, Springer-Verlag, New York, 1988. MR 950166
  • P. J. Forrester, The spectrum edge of random matrix ensembles, Nuclear Phys. B 402 (1993), no. 3, 709–728. MR 1236195, DOI 10.1016/0550-3213(93)90126-A
  • F2 —, Random walks and random permutations, math.CO/9907037.
  • Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
  • Philippe Jacquet and Wojciech Szpankowski, Analytical de-Poissonization and its applications, Theoret. Comput. Sci. 201 (1998), no. 1-2, 1–62. MR 1625392, DOI 10.1016/S0304-3975(97)00167-9
  • Kurt Johansson, The longest increasing subsequence in a random permutation and a unitary random matrix model, Math. Res. Lett. 5 (1998), no. 1-2, 63–82. MR 1618351, DOI 10.4310/MRL.1998.v5.n1.a6
  • J2 —, Discrete orthogonal polynomials and the Plancherel measure, math.CO/9906120.
  • Serguei Kerov, Gaussian limit for the Plancherel measure of the symmetric group, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), no. 4, 303–308 (English, with English and French summaries). MR 1204294
  • 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 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 27 (1993), no. 2, 104–117. MR 1251166, DOI 10.1007/BF01085981
  • S. V. Kerov, Asymptotics of the separation of roots of orthogonal polynomials, Algebra i Analiz 5 (1993), no. 5, 68–86 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 5 (1994), no. 5, 925–941. MR 1263315
  • Ke4 —, A differential model of growth of Young diagrams, Proceedings of the St. Petersburg Math. Soc., 4, 1996, 167–194.
  • Sergei Kerov, Interlacing measures, Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, vol. 181, Amer. Math. Soc., Providence, RI, 1998, pp. 35–83. MR 1618739, DOI 10.1090/trans2/181/02
  • B. F. Logan and L. A. Shepp, A variational problem for random Young tableaux, Advances in Math. 26 (1977), no. 2, 206–222. MR 1417317, DOI 10.1016/0001-8708(77)90030-5
  • I. G. Macdonald, Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. With contributions by A. Zelevinsky; Oxford Science Publications. MR 1354144
  • O A. Okounkov, Random matrices and random permutations, math.CO/9903176 O2 —, Infinite wedge and measures on partitions, math.RT/9907127 O3 —, $SL(2)$ and $z$-measures, math.RT/0002135, to appear in Proceedings of the 1999 MSRI Workshop on Random Matrices and their Applications.
  • E. M. Rains, Increasing subsequences and the classical groups, Electron. J. Combin. 5 (1998), Research Paper 12, 9. MR 1600095
  • Amitai Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), no. 2, 115–136. MR 625890, DOI 10.1016/0001-8708(81)90012-8
  • Erhard Seiler and Barry Simon, On finite mass renormalizations in the two-dimensional Yukawa model, J. Mathematical Phys. 16 (1975), no. 11, 2289–2293. MR 403484, DOI 10.1063/1.522458
  • C. Schensted, Longest increasing and decreasing subsequences, Canadian J. Math. 13 (1961), 179–191. MR 121305, DOI 10.4153/CJM-1961-015-3
  • Timo Seppäläinen, A microscopic model for the Burgers equation and longest increasing subsequences, Electron. J. Probab. 1 (1996), no. 5, approx. 51 pp.}, issn=1083-6489, review= MR 1386297, doi=10.1214/EJP.v1-5,
  • Barry Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math. 24 (1977), no. 3, 244–273. MR 482328, DOI 10.1016/0001-8708(77)90057-3
  • Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149
  • So A. Soshnikov, Determinantal random point fields, math.PR/0002099.
  • Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174. MR 1257246
  • Craig A. Tracy and Harold Widom, Introduction to random matrices, Geometric and quantum aspects of integrable systems (Scheveningen, 1992) Lecture Notes in Phys., vol. 424, Springer, Berlin, 1993, pp. 103–130. MR 1253763, DOI 10.1007/BFb0021444
  • TW3 —, On the distribution of the lengths of the longest monotone subsequences in random words, math.CO/9904042.
  • A. M. Vershik, Statistical mechanics of combinatorial partitions, and their limit configurations, Funktsional. Anal. i Prilozhen. 30 (1996), no. 2, 19–39, 96 (Russian, with Russian summary); English transl., Funct. Anal. Appl. 30 (1996), no. 2, 90–105. MR 1402079, DOI 10.1007/BF02509449
  • VK1 A. Vershik and S. Kerov, Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux, Soviet Math. Dokl., 18, 1977, 527–531.
  • A. M. Vershik and S. V. Kerov, Asymptotic theory of the characters of a symmetric group, Funktsional. Anal. i Prilozhen. 15 (1981), no. 4, 15–27, 96 (Russian). MR 639197
  • 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 (Russian). MR 783703
  • Albert Eagle, Series for all the roots of the equation $(z-a)^m=k(z-b)^n$, Amer. Math. Monthly 46 (1939), 425–428. MR 6, DOI 10.2307/2303037
  • Harold Widom, Random Hermitian matrices and (nonrandom) Toeplitz matrices, Toeplitz operators and related topics (Santa Cruz, CA, 1992) Oper. Theory Adv. Appl., vol. 71, Birkhäuser, Basel, 1994, pp. 9–15. MR 1300210
  • Harold Widom, The strong Szegő limit theorem for circular arcs, Indiana Univ. Math. J. 21 (1971/72), 277–283. MR 288495, DOI 10.1512/iumj.1971.21.21022
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Additional Information
  • Alexei Borodin
  • Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104–6395 and Dobrushin Mathematics Laboratory, Institute for Problems of Information Transmission, Bolshoy Karetny 19, 101447, Moscow, Russia
  • Email: borodine@math.upenn.edu
  • Andrei Okounkov
  • Affiliation: University of Chicago, Department of Mathematics, 5734 University Ave., Chicago, Illinois 60637
  • Address at time of publication: Department of Mathematics, University of California at Berkeley, Evans Hall, Berkeley, California 94720-3840
  • MR Author ID: 351622
  • ORCID: 0000-0001-8956-1792
  • Email: okounkov@math.berkeley.edu
  • Grigori Olshanski
  • Affiliation: Dobrushin Mathematics Laboratory, Institute for Problems of Information Transmission, Bolshoy Karetny 19, 101447, Moscow, Russia
  • MR Author ID: 189402
  • Email: olsh@glasnet.ru
  • Received by editor(s): September 15, 1999
  • Published electronically: April 13, 2000
  • Additional Notes: The second author is supported by NSF grant DMS-9801466, and the third author is supported by the Russian Foundation for Basic Research under grant 98-01-00303.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 481-515
  • MSC (1991): Primary 05E10, 60C05
  • DOI: https://doi.org/10.1090/S0894-0347-00-00337-4
  • MathSciNet review: 1758751