Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes

• 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
• DOI: 10.1007/s00440-013-0532-x

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).