Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes

• 成果类型: Article
• 署名作者: Petrov, Leonid
• 署名单位: Northeastern University; Kharkevich Institute for Information Transmission Problems of the RAS
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-013-0532-x
• 发表日期: 2014
• 页码: 429-487
• 关键词: nonintersecting paths; plancherel measures; fluctuations; BOUNDARY; entropy

摘要

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 Nth 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).