Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

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...

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Bibliographic Details
Published in:Probability theory and related fields Vol. 160; no. 3-4; pp. 429 - 487
Main Author: Petrov, Leonid
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2014
Springer Nature B.V
Subjects:
Arrays
Asymptotic methods
Asymptotic properties
Economics
Ergodic processes
Finance
Growth models
Insurance
Invariants
Management
Mathematical analysis
Mathematical and Computational Biology
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operations Research/Decision Theory
Planes
Polygons
Polynomials
Probability
Probability Theory and Stochastic Processes
Quantitative Finance
Slopes
Statistics for Business
Studies
Theoretical
Tiling
Topological manifolds
60C05
82C22
60G55
ISSN:0178-8051, 1432-2064
Online Access:Get full text
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Summary: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. We work with a class of polygons which allows arbitrarily large number of sides. We show that the local limit behavior of random tilings (as all dimensions of the polygon grow) is directed by ergodic translation invariant Gibbs measures. The slopes of these measures coincide with the ones of tangent planes to the corresponding limit shapes described by Kenyon and Okounkov (Acta Math 199(2):263–302, 2007 ). We also prove that at the edge of the limit shape, the asymptotic behavior of random tilings is given by the Airy process. In particular, our results cover the most investigated case of random boxed plane partitions (when the polygon is a hexagon).
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ISSN:0178-8051
1432-2064
DOI:10.1007/s00440-013-0532-x