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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References

  • Baik, J., Kriecherbauer, T., McLaughlin, K. T. R. and Miller, P. D. (2007). Discrete Orthogonal Polynomials: Asymptotics and Applications. Annals of Mathematics Studies 164. Princeton Univ. Press, Princeton, NJ.
    Mathematical Reviews (MathSciNet): MR2283089
  • Borodin, A. (2011). Determinantal point processes. In Oxford Handbook of Random Matrix Theory (G. Akemann, J. Baik and P. Di Francesco, eds.). Oxford Univ. Press, Oxford.
    Mathematical Reviews (MathSciNet): MR2920518
  • Borodin, A. and Ferrari, P. L. (2008). Anisotropic growth of random surfaces in $2+1$ dimensions. Available at arXiv:0804.3035 [math-ph].
    arXiv: 0804.3035
    Mathematical Reviews (MathSciNet): MR3148098
    Digital Object Identifier: doi:10.1007/s00220-013-1823-x
  • Borodin, A., Ferrari, P. L. and Sasamoto, T. (2008). Large time asymptotics of growth models on space-like paths. II. PNG and parallel TASEP. Comm. Math. Phys. 283 417–449.
    Mathematical Reviews (MathSciNet): MR2430639
    Digital Object Identifier: doi:10.1007/s00220-008-0515-4
  • Borodin, A. and Gorin, V. (2009). Shuffling algorithm for boxed plane partitions. Adv. Math. 220 1739–1770.
    Mathematical Reviews (MathSciNet): MR2493180
    Digital Object Identifier: doi:10.1016/j.aim.2008.11.008
  • Borodin, A., Gorin, V. and Rains, E. M. (2010). $q$-distributions on boxed plane partitions. Selecta Math. (N.S.) 16 731–789.
    Mathematical Reviews (MathSciNet): MR2734330
    Digital Object Identifier: doi:10.1007/s00029-010-0034-y
  • Borodin, A., Okounkov, A. and Olshanski, G. (2000). Asymptotics of Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 481–515 (electronic).
    Mathematical Reviews (MathSciNet): MR1758751
    Digital Object Identifier: doi:10.1090/S0894-0347-00-00337-4
  • Cohn, H., Kenyon, R. and Propp, J. (2001). A variational principle for domino tilings. J. Amer. Math. Soc. 14 297–346 (electronic).
    Mathematical Reviews (MathSciNet): MR1815214
    Digital Object Identifier: doi:10.1090/S0894-0347-00-00355-6
  • Cohn, H., Larsen, M. and Propp, J. (1998). The shape of a typical boxed plane partition. New York J. Math. 4 137–165 (electronic).
    Mathematical Reviews (MathSciNet): MR1641839
  • Destainville, N. (1998). Entropy and boundary conditions in random rhombus tilings. J. Phys. A 31 6123–6139.
    Mathematical Reviews (MathSciNet): MR1637731
    Digital Object Identifier: doi:10.1088/0305-4470/31/29/005
  • Destainville, N., Mosseri, R. and Bailly, F. (1997). Configurational entropy of codimension-one tilings and directed membranes. J. Stat. Phys. 87 697–754.
    Mathematical Reviews (MathSciNet): MR1459040
    Digital Object Identifier: doi:10.1007/BF02181243
  • Duits, M. (2011). The Gaussian free field in an interlacing particle system with two jump rates. Available at arXiv:1105.4656 [math-ph].
    arXiv: 1105.4656
    Mathematical Reviews (MathSciNet): MR3020314
    Digital Object Identifier: doi:10.1002/cpa.21419
  • Gorin, V. E. (2008). Nonintersecting paths and the Hahn orthogonal polynomial ensemble. Funct. Anal. Appl. 42 180–197.
    Mathematical Reviews (MathSciNet): MR2454474
    Digital Object Identifier: doi:10.4213/faa2910
  • Hough, J. B., Krishnapur, M., Peres, Y. and Virág, B. (2006). Determinantal processes and independence. Probab. Surv. 3 206–229.
    Mathematical Reviews (MathSciNet): MR2216966
    Digital Object Identifier: doi:10.1214/154957806000000078
    Project Euclid: euclid.ps/1146832696
  • Kenyon, R. (2008). Height fluctuations in the honeycomb dimer model. Comm. Math. Phys. 281 675–709.
    Mathematical Reviews (MathSciNet): MR2415464
    Digital Object Identifier: doi:10.1007/s00220-008-0511-8
  • Kenyon, R. (2009). Lectures on dimers. In Statistical Mechanics. IAS/Park City Math. Ser. 16 191–230. Amer. Math. Soc., Providence, RI.
    Mathematical Reviews (MathSciNet): MR2523460
  • Kenyon, R. and Okounkov, A. (2007). Limit shapes and the complex Burgers equation. Acta Math. 199 263–302.
    Mathematical Reviews (MathSciNet): MR2358053
    Digital Object Identifier: doi:10.1007/s11511-007-0021-0
  • Kenyon, R., Okounkov, A. and Sheffield, S. (2006). Dimers and amoebae. Ann. of Math. (2) 163 1019–1056.
    Mathematical Reviews (MathSciNet): MR2215138
    Digital Object Identifier: doi:10.4007/annals.2006.163.1019
  • Kuan, J. (2011). The Gaussian free field in interlacing particle systems. Available at arXiv:1109.4444 [math-ph].
    arXiv: 1109.4444
  • Okounkov, A. (2002). Symmetric functions and random partitions. In Symmetric Functions 2001: Surveys of Developments and Perspectives (S. Fomin, ed.). Kluwer Academic, Dordrecht.
    Mathematical Reviews (MathSciNet): MR2059364
    Digital Object Identifier: doi:10.1007/978-94-010-0524-1_6
  • Okounkov, A. and Reshetikhin, N. (2003). Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram. J. Amer. Math. Soc. 16 581–603 (electronic).
    Mathematical Reviews (MathSciNet): MR1969205
    Digital Object Identifier: doi:10.1090/S0894-0347-03-00425-9
  • Petrov, L. (2012). Asymptotics of random lozenge tilings via Gelfand–Tsetlin schemes. Available at arXiv:1202.3901 [math.PR].
    arXiv: 1202.3901
  • Sheffield, S. (2005). Random surfaces. Astérisque 304 vi+175.
    Mathematical Reviews (MathSciNet): MR2251117
  • Sheffield, S. (2007). Gaussian free fields for mathematicians. Probab. Theory Related Fields 139 521–541.
    Mathematical Reviews (MathSciNet): MR2322706
    Digital Object Identifier: doi:10.1007/s00440-006-0050-1
  • Soshnikov, A. (2000). Determinantal random point fields. Russian Math. Surveys 55 923–975.
    Mathematical Reviews (MathSciNet): MR1799012
    Digital Object Identifier: doi:10.4213/rm321

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