Open Access
February 2015 Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field
Leonid Petrov
Ann. Probab. 43(1): 1-43 (February 2015). DOI: 10.1214/12-AOP823


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].


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Leonid Petrov. "Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field." Ann. Probab. 43 (1) 1 - 43, February 2015.


Published: February 2015
First available in Project Euclid: 12 November 2014

zbMATH: 1315.60062
MathSciNet: MR3298467
Digital Object Identifier: 10.1214/12-AOP823

Primary: 60G55
Secondary: 60C05 , 60G15 , 82C22

Keywords: Determinantal point processes , Dimer model , Gaussian free field , height function , Random lozenge tilings

Rights: Copyright © 2015 Institute of Mathematical Statistics

Vol.43 • No. 1 • February 2015
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