Petrov, Leonid Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes. (English) Zbl 1315.60013 Probab. Theory Relat. Fields 160, No. 3-4, 429-487 (2014). Summary: A Gelfand-Tsetlin scheme of depth \(N\) is a triangular array with \(m\) integers at level \(m\), \(m=1,\dots,N\), 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 R. Kenyon and A. Okounkov [Acta Math. 199, No. 2, 263–302 (2007; Zbl 1156.14029)]. 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). Cited in 49 Documents MSC: 60C05 Combinatorial probability 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 82C22 Interacting particle systems in time-dependent statistical mechanics Citations:Zbl 1156.14029 PDFBibTeX XMLCite \textit{L. Petrov}, Probab. Theory Relat. 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