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.

**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.

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