Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

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

 
 

 

Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram


Authors: Andrei Okounkov and Nikolai Reshetikhin
Journal: J. Amer. Math. Soc. 16 (2003), 581-603
MSC (2000): Primary 05E05, 60G55
DOI: https://doi.org/10.1090/S0894-0347-03-00425-9
Published electronically: March 3, 2003
MathSciNet review: 1969205
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Schur process is a time-dependent analog of the Schur measure on partitions studied by A. Okounkov in Infinite wedge and random partitions, Selecta Math., New Ser. 7 (2001), 57-81. Our first result is that the correlation functions of the Schur process are determinants with a kernel that has a nice contour integral representation in terms of the parameters of the process. This general result is then applied to a particular specialization of the Schur process, namely to random 3-dimensional Young diagrams. The local geometry of a large random 3-dimensional diagram is described in terms of a determinantal point process on a 2-dimensional lattice with the incomplete beta function kernel (which generalizes the discrete sine kernel). A brief discussion of the universality of this answer concludes the paper.


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

  • 1. A. Borodin, A. Okounkov, and G. Olshanski, On asymptotics of the Plancherel measures for symmetric groups, J. Amer. Math. Soc. 13 (2000), no. 3, 481-515. MR 2001g:05103
  • 2. R. Burton and R. Pemantle, Local characteristics, entropy and limit theorems for spanning trees and domino tilings via transfer-impendances, Ann. Prob. 21 (1993), 1329-1371. MR 94m:60019
  • 3. R. Cerf and R. Kenyon, The low-temperature expansion of the Wulff crystal in the 3D Ising model, Comm. Math. Phys. 222 (2001), 147-179. MR 2002i:82046
  • 4. H. Cohn, N. Elkies, and J. Propp, Local statistics for random domino tilings of the Aztec diamond, Duke Math. J. 85 (1996), no. 1, 117-166. MR 97k:52026
  • 5. H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings, math.CO/0008220, J. Amer. Math. Soc. 14 (2001), 297-346. MR 2002k:82038
  • 6. L. Faddeev and R. Kashaev, Quantum dilogarithm, hep-th/9310070, Modern Phys. Lett. A 9 (1994), 427-434. MR 95i:11150
  • 7. P. Ferrari and H. Spohn, Step fluctuations for a faceted crystal, cond-mat/0212456.
  • 8. K. Johansson, Discrete orthogonal polynomials and the Plancherel measure, math.CO/ 9906120, Ann. of Math (2) 153 (2001), 259-296. MR 2002g:05188
  • 9. K. Johansson, Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices, math.ph/0006020, Comm. Math. Phys. 215 (2001), 683-705. MR 2002j:15024
  • 10. K. Johansson, Non-intersecting paths, random tilings, and random matrices, math.PR/ 0011250, Probab. Theory Related Fields 123 (2002), 225-280.
  • 11. V. Kac, Infinite dimensional Lie algebras, Cambridge University Press. MR 87c:17023
  • 12. S. Karlin and G. McGregor, Coincidence probabilities, Pacific J. Math 9 (1959), 1141-1164. MR 22:5072
  • 13. R. Kenyon, Local statistics of lattice dimers, Ann. Inst. H. Poincaré, Prob. et Stat. 33 (1997), 591-618. MR 99b:82039
  • 14. R. Kenyon, The planar dimer model with a boundary: a survey, CRM Monogr. Ser., Vol. 13, Amer. Math. Soc., Providence, RI, 2000, pp. 307-328. MR 2002e:82011
  • 15. I. G. Macdonald, Symmetric functions and Hall polynomials, Clarendon Press, 1995. MR 96h:05207
  • 16. A. Okounkov, Infinite wedge and random partitions, Selecta Math., New Ser., 7 (2001), 57-81, math.RT/9907127. MR 2002f:60019
  • 17. A. Okounkov, Symmetric functions and random partitions, Symmetric functions 2001: Surveys of Developments and Perspectives, edited by S. Fomin, Kluwer Academic Publishers, 2002.
  • 18. M. Praehofer and H. Spohn, Scale Invariance of the PNG Droplet and the Airy Process, math.PR/0105240, J. Statist. Phys. 108 (2002), 1071-1106.
  • 19. A. Vershik, talk at the 1997 conference on Formal Power Series and Algebraic Combinatorics, Vienna.

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 05E05, 60G55

Retrieve articles in all journals with MSC (2000): 05E05, 60G55


Additional Information

Andrei Okounkov
Affiliation: Department of Mathematics, University of California at Berkeley, Evans Hall #3840, Berkeley, California 94720-3840
Email: okounkov@math.berkeley.edu

Nikolai Reshetikhin
Affiliation: Department of Mathematics, University of California at Berkeley, Evans Hall #3840, Berkeley, California 94720-3840
Email: reshetik@math.berkeley.edu

DOI: https://doi.org/10.1090/S0894-0347-03-00425-9
Received by editor(s): December 8, 2001
Published electronically: March 3, 2003
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society