Annals of Probability

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

Leonid Petrov

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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].

Article information

Ann. Probab., Volume 43, Number 1 (2015), 1-43.

First available in Project Euclid: 12 November 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G55: Point processes
Secondary: 60G15: Gaussian processes 60C05: Combinatorial probability 82C22: Interacting particle systems [See also 60K35]

Random lozenge tilings dimer model height function Gaussian free field determinantal point processes


Petrov, Leonid. Asymptotics of uniformly random lozenge tilings of polygons. Gaussian free field. Ann. Probab. 43 (2015), no. 1, 1--43. doi:10.1214/12-AOP823.

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