Journal of the American Mathematical Society

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



Asymptotics of Plancherel measures for symmetric groups

Authors: Alexei Borodin, Andrei Okounkov and Grigori Olshanski
Journal: J. Amer. Math. Soc. 13 (2000), 481-515
MSC (1991): Primary 05E10, 60C05
Published electronically: April 13, 2000
MathSciNet review: 1758751
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information


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 [Enhancements On Off] (What's this?)

  • 1. 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, 10.1007/BF01204214
  • 2. -, Longest increasing subsequences: From patience sorting to the Baik-Deift-Johansson theorem, Bull. Amer. Math. Soc., 36, 1999, 413-432. CMP 99:17
  • 3. 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. CMP 99:15
  • 4. -, On the distribution of the length of the second row of a Young diagram under Plancherel measure, math.CO/9901118.
  • 5. Philippe Biane, Permutation model for semi-circular systems and quantum random walks, Pacific J. Math. 171 (1995), no. 2, 373–387. MR 1372234
  • 6. -, Representations of symmetric groups and free probability, Adv. Math. 138, 1998, no. 1, 126-181. CMP 99:01
  • 7. A. Borodin, Riemann-Hilbert problem and the discrete Bessel kernel, math.CO/9912093, to appear in Intern. Math. Res. Notices.
  • 8. A. Borodin and G. Olshanski, Distribution on partitions, point processes, and the hypergeometric kernel, math.RT/9904010, to appear in Comm. Math. Phys.
  • 9. -, 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.
  • 10. 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, 10.1007/BF00251504
  • 11. 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
  • 12. P. J. Forrester, The spectrum edge of random matrix ensembles, Nuclear Phys. B 402 (1993), no. 3, 709–728. MR 1236195, 10.1016/0550-3213(93)90126-A
  • 13. -, Random walks and random permutations, math.CO/9907037.
  • 14. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
  • 15. Philippe Jacquet and Wojciech Szpankowski, Analytical de-Poissonization and its applications, Theoret. Comput. Sci. 201 (1998), no. 1-2, 1–62. MR 1625392, 10.1016/S0304-3975(97)00167-9
  • 16. 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, 10.4310/MRL.1998.v5.n1.a6
  • 17. -, Discrete orthogonal polynomials and the Plancherel measure, math.CO/9906120.
  • 18. 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
  • 19. 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, 10.1007/BF01085981
  • 20. 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
  • 21. -, A differential model of growth of Young diagrams, Proceedings of the St. Petersburg Math. Soc., 4, 1996, 167-194. CMP 2000:07
  • 22. 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
  • 23. 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, 10.1016/0001-8708(77)90030-5
  • 24. 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
  • 25. A. Okounkov, Random matrices and random permutations, math.CO/9903176
  • 26. -, Infinite wedge and measures on partitions, math.RT/9907127
  • 27. -, $SL(2)$ and $z$-measures, math.RT/0002135, to appear in Proceedings of the 1999 MSRI Workshop on Random Matrices and their Applications.
  • 28. E. M. Rains, Increasing subsequences and the classical groups, Electron. J. Combin. 5 (1998), Research Paper 12, 9 pp. (electronic). MR 1600095
  • 29. Amitai Regev, Asymptotic values for degrees associated with strips of Young diagrams, Adv. in Math. 41 (1981), no. 2, 115–136. MR 625890, 10.1016/0001-8708(81)90012-8
  • 30. 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 0403484
  • 31. C. Schensted, Longest increasing and decreasing subsequences, Canad. J. Math. 13 (1961), 179–191. MR 0121305
  • 32. Timo Seppäläinen, A microscopic model for the Burgers equation and longest increasing subsequences, Electron. J. Probab. 1 (1996), no. 5, approx. 51 pp. (electronic). MR 1386297, 10.1214/EJP.v1-5
  • 33. Barry Simon, Notes on infinite determinants of Hilbert space operators, Advances in Math. 24 (1977), no. 3, 244–273. MR 0482328
  • 34. Barry Simon, Trace ideals and their applications, London Mathematical Society Lecture Note Series, vol. 35, Cambridge University Press, Cambridge-New York, 1979. MR 541149
  • 35. A. Soshnikov, Determinantal random point fields, math.PR/0002099.
  • 36. Craig A. Tracy and Harold Widom, Level-spacing distributions and the Airy kernel, Comm. Math. Phys. 159 (1994), no. 1, 151–174. MR 1257246
  • 37. 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, 10.1007/BFb0021444
  • 38. -, On the distribution of the lengths of the longest monotone subsequences in random words, math.CO/9904042.
  • 39. 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, 10.1007/BF02509449
  • 40. 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.
  • 41. 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
  • 42. 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
  • 43. G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England; The Macmillan Company, New York, 1944. MR 0010746
  • 44. 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
  • 45. Harold Widom, The strong Szegő limit theorem for circular arcs, Indiana Univ. Math. J. 21 (1971/1972), 277–283. MR 0288495

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 05E10, 60C05

Retrieve articles in all journals with MSC (1991): 05E10, 60C05

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

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

Grigori Olshanski
Affiliation: Dobrushin Mathematics Laboratory, Institute for Problems of Information Transmission, Bolshoy Karetny 19, 101447, Moscow, Russia

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.
Article copyright: © Copyright 2000 American Mathematical Society