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

TL;DR

This paper demonstrates that large-scale height fluctuations in random lozenge tilings of polygons are asymptotically described by a Gaussian free field, extending previous results to polygons with many sides.

Contribution

It establishes the Gaussian free field as the universal limit for height fluctuations in a broad class of polygonal tilings, using explicit integral formulas.

Findings

Height fluctuations converge to Gaussian free field

Results apply to polygons with arbitrarily many sides

Uses explicit double contour integral kernel formula

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].