2012

Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes

Petrov, Leonid

Understand

A Gelfand-Tsetlin scheme of depth N is a triangular array with m integers at level m, m=1,...,N, subject to certain interlacing constraints.

We study the ensemble of uniformly random Gelfand-Tsetlin schemes with arbitrary fixed N-th row.
We obtain an explicit double contour integral expression for the determinantal correlation kernel of this ensemble (and also of its q-deformation).
This provides new tools for asymptotic analysis of uniformly random lozenge tilings of polygons on the triangular lattice; or, equivalently, of random stepped surfaces.

Built on

A. Erdélyi (ed.), Higher transcendental functions , McGraw–Hill, 1953
1953
Earlier work this paper cites.
H. W. Gould, The q q -Stirling numbers of first and second kinds , Duke Math. J. 28
1961
Earlier work this paper cites.
P. Kasteleyn, Graph theory and crystal physics , Graph Theory and Theoretical Physics, Academic Press, London, 1967, pp. 43–110
1967
Earlier work this paper cites.
I.G. Macdonald, Symmetric functions and Hall polynomials , 2nd ed., Oxford University Press, 1995
1995
Earlier work this paper cites.
N. Destainville, R. Mosseri, and F. Bailly, Configurational entropy of codimension-one tilings and directed membranes , J. Stat. Phys. 87
1997
Earlier work this paper cites.
R. Kenyon, Local statistics of lattice dimers , Annales de Inst. H. Poincaré, Probabilités et Statistiques 33
1997
Earlier work this paper cites.
H. Weyl, The Classical Groups. Their Invariants and Representations , Princeton University Press, 1997
Original
1997
Earlier work this paper cites.
H. Cohn, M. Larsen, and J. Propp, The shape of a typical boxed plane partition , New York J. Math 4
1998
Earlier work this paper cites.
N. Destainville, Entropy and boundary conditions in random rhombus tilings , J. Phys. A: Math. Gen. 31
1998
Earlier work this paper cites.
A. Borodin, A. Okounkov, and G. Olshanski, Asymptotics of Plancherel measures for symmetric groups , J. Amer. Math. Soc. 13
2000
Earlier work this paper cites.
A. Soshnikov, Determinantal random point fields , Russian Mathematical Surveys 55
2000
Earlier work this paper cites.
H. Cohn, R. Kenyon, and J. Propp, A variational principle for domino tilings , Journal of the AMS 14
2001
Earlier work this paper cites.
R. Stanley, Enumerative Combinatorics. Vol. 2 , Cambridge University Press, Cambridge, 2001, With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin
2001
Earlier work this paper cites.

Similar

K. Johansson, Non-intersecting paths, random tilings and random matrices , Probability theory and related fields 123
2002
Cited alongside, same era.
A. Okounkov, Symmetric functions and random partitions , Symmetric functions 2001: Surveys of Developments and Perspectives (S. Fomin, ed.), Kluwer Academic Publishers, 2002, arXiv:math/0309074 [math.CO]
2002
Cited alongside, same era.
M. Prähofer and H. Spohn, Scale invariance of the PNG droplet and the Airy process , J. Stat. Phys. 108
2002
Cited alongside, same era.
P.L. Ferrari and H. Spohn, Step fluctuations for a faceted crystal , Journal of Statistical Physics 113
2003
Cited alongside, same era.
A. Okounkov and N. Reshetikhin, Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram , Journal of the American Mathematical Society 16
2003
Cited alongside, same era.
by same author, Non-intersecting, simple, symmetric random walks and the extended Hahn kernel , Annales de l’Institut Fourier (Grenoble) 55
2005
Cited alongside, same era.
S. Sheffield, Random surfaces , Astérisque 304
2005
Cited alongside, same era.
J.B. Hough, M. Krishnapur, Y. Peres, and B. Virág, Determinantal processes and independence , Probability Surveys 3
2006
Cited alongside, same era.
K. Johansson and E. Nordenstam, Eigenvalues of GUE minors , Electron. J. Probab 11
2006
Cited alongside, same era.
R. Kenyon, A. Okounkov, and S. Sheffield, Dimers and amoebae , Ann. Math. 163
2006
Cited alongside, same era.
A. Okounkov and N.Y. Reshetikhin, The birth of a random matrix , Mosc. Math. J. 6
2006
Cited alongside, same era.
A. Borodin, P. Ferrari, M. Prähofer, and T. Sasamoto, Fluctuation properties of the TASEP with periodic initial configuration , J. Stat. Phys. 129
2007
Cited alongside, same era.

Then

R. Kenyon and A. Okounkov, Limit shapes and the complex Burgers equation , Acta Math. 199
2007
Later among the works it cites.
A. Okounkov and N. Reshetikhin, Random skew plane partitions and the Pearcey process , Communications in mathematical physics 269
2007
Later among the works it cites.
A. Borodin and P. Ferrari, Large time asymptotics of growth models on space-like paths I: PushASEP , Electron. J. Probab. 13
Original
2008
Later among the works it cites.
A. Borodin and J. Kuan, Asymptotics of Plancherel measures for the infinite-dimensional unitary group , Adv. Math. 219
Original
2008
Later among the works it cites.
V. Gorin, Nonintersecting paths and the Hahn orthogonal polynomial ensemble , Funct. Anal. Appl. 42
Original
2008
Later among the works it cites.
by same author, Height fluctuations in the honeycomb dimer model , Communications in Mathematical Physics 281
2008
Later among the works it cites.
A. Borodin and V. Gorin, Shuffling algorithm for boxed plane partitions , Advances in Mathematics 220
Original
2009
Later among the works it cites.
P.J. Forrester and E. Nordenstam, The anti-symmetric GUE minor process , Moscow Mathematical Journal 9
Original
2009
Later among the works it cites.
A. Borodin, V. Gorin, and E. Rains, q-Distributions on boxed plane partitions , Selecta Mathematica, New Series 16
Original
2010
Later among the works it cites.
C. Boutillier, S. Mkrtchyan, N. Reshetikhin, and P. Tingley, Random skew plane partitions with a piecewise periodic back wall , Annales Henri Poincare (2011), arXiv:0912.3968 [math-ph]
Original
2011
Later among the works it cites.
A. Borodin, Determinantal point processes , Oxford Handbook of Random Matrix Theory (G. Akemann, J. Baik, and P. Di Francesco, eds.), Oxford University Press, 2011, arXiv:0911.1153 [math.PR]
Original
2011
Later among the works it cites.
E. Nordenstam and B. Young, Domino shuffling on Novak half-hexagons and Aztec half-diamonds , The electronic journal of combinatorics 18
Original
2011
Later among the works it cites.
A. Borodin and G. Olshanski, The boundary of the Gelfand-Tsetlin graph: A new approach , Adv. Math. 230
Original
2012
Closest in time.

Beyond the bibliography

alphaXiv searches the wider corpus for related work and actual follow-ups.

Open on alphaXiv

alphaXiv is searching for related work…

Reading the bibliography…