Asymptotics of Uniformly Random Lozenge Tilings of Polygons. Gaussian Free Field

Abstract. We study large-scale height fluctuations of random stepped sur-faces corresponding to uniformly random lozenge tilings of polygons on the triangular lattice. For a class of polygons (which allows arbitrarily large num-ber of sides), we show that these fluctuations are asymptotically governed by a Gaussian free (massless) field. This complements the similar result obtained by Kenyon [Ken08] about tilings of regions without frozen facets of the limit shape. In our asymptotic analysis we use the explicit double contour integral for-mula for the determinantal correlation kernel of the model obtained previously in [Pet12]. 1. Introduction an