Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field

Abstract

We study large-scale height fluctuations of random stepped surfaces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows arbitrarily large number of sides), we show that these fluctuations are asymptotically governed by a Gaussian free (massless) field. This complements the similar result obtained in Kenyon [Comm. Math. Phys. 281 (2008) 675-709] about tilings of regions without frozen facets of the limit shape. In our asymptotic analysis we use the explicit double contour integral formula for the determinantal correlation kernel of the model obtained previously in Petrov [Asymptotics of random lozenge tilings via Gelfand-Tsetlin schemes (2012) Preprint].

Published in at http://dx.doi.org/10.1214/12-AOP823 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (5)

• Uniform Asymptotics for Polynomials Orthogonal With Respect to a General Class of Discrete Weights and Universality Results for Associated Ensembles
• Gaussian free fields for mathematicians
• Asymptotics of Random Lozenge Tilings via Gelfand-Tsetlin Schemes
• The Gaussian free field in an interlacing particle system with two jump rates
• The Gaussian free field in interlacing particle systems

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