Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes
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- 作者：Leonid Petrov (1) (2)
- 关键词：60C05 ; 60G55 ; 82C22
- 刊名：Probability Theory and Related Fields
- 出版年：2014
- 出版时间：December 2014
- 年：2014
- 卷：160
- 期：3-4
- 页码：429-487
- 全文大小：3,257 KB
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1. Department of Mathematics, Northeastern University, 360 Huntington ave., Boston, MA, 02115, USA
2. Dobrushin Mathematics Laboratory, Kharkevich Institute for Information Transmission Problems, Moscow, Russia - ISSN：1432-2064
文摘
A Gelfand–Tsetlin scheme of depth \(N\) is a triangular array with \(m\) integers at level \(m\) , \(m=1,\ldots ,N\) , 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):263-02, 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).