The Annals of Probability

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

Leonid Petrov

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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].

Article information

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

Dates
First available in Project Euclid: 12 November 2014

Permanent link to this document
https://projecteuclid.org/euclid.aop/1415801551

Digital Object Identifier
doi:10.1214/12-AOP823

Mathematical Reviews number (MathSciNet)
MR3298467

Zentralblatt MATH identifier
1315.60062

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

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

Citation

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. https://projecteuclid.org/euclid.aop/1415801551


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