Communications in Mathematical Physics

, Volume 358, Issue 3, pp 1117–1149 | Cite as

Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation

  • Benoît Laslier
  • Fabio Lucio Toninelli


We study a reversible continuous-time Markov dynamics of a discrete (2 + 1)-dimensional interface. This can be alternatively viewed as a dynamics of lozenge tilings of the \({L\times L}\) torus, or as a conservative dynamics for a two-dimensional system of interlaced particles. The particle interlacement constraints imply that the equilibrium measures are far from being product Bernoulli: particle correlations decay like the inverse distance squared and interface height fluctuations behave on large scales like a massless Gaussian field. We consider a particular choice of the transition rates, originally proposed in Luby et al. (SIAM J Comput 31:167–192, 2001): in terms of interlaced particles, a particle jump of length n that preserves the interlacement constraints has rate 1/(2n). This dynamics presents special features: the average mutual volume between two interface configurations decreases with time (Luby et al. 2001) and a certain one-dimensional projection of the dynamics is described by the heat equation (Wilson in Ann Appl Probab 14:274–325, 2004). In this work we prove a hydrodynamic limit: after a diffusive rescaling of time and space, the height function evolution tends as \({L\to\infty}\) to the solution of a non-linear parabolic PDE. The initial profile is assumed to be C2 differentiable and to contain no “frozen region”. The explicit form of the PDE was recently conjectured (Laslier and Toninelli in Ann Henri Poincaré Theor Math Phys 18:2007–2043, 2017) on the basis of local equilibrium considerations. In contrast with the hydrodynamic equation for the Langevin dynamics of the Ginzburg–Landau model (Funaki and Spohn in Commun Math Phys 85:1–36, 1997; Nishikawa in Commun Math Phys 127:205–227, 2003), here the mobility coefficient turns out to be a non-trivial function of the interface slope.


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  1. 1.
    Caputo P., Martinelli F., Toninelli F.L.: Mixing times of monotone surfaces and SOS interfaces: a mean curvature approach. Commun. Math. Phys. 311, 157–189 (2012)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chang C.C., Yau H.-T.: Fluctuations of one dimensional Ginzburg–Landau models in nonequilibrium. Commun. Math. Phys. 145, 209–239 (1992)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chhita S., Ferrari P.L.: A combinatorial identity for the speed of growth in an anisotropic KPZ model. Ann. Inst. Henri Poincaré D 4(4), 453–477 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corwin I., Toninelli F.L.: Stationary measure of the driven two-dimensional q-Whittaker particle system on the torus. Electron. Commun. Probab. 21(44), 1–12 (2016)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fritz J.: On the hydrodynamic limit of a Ginzburg Landau lattice model. Probab. Theory Relat. Fields 81, 291–318 (1989)CrossRefzbMATHGoogle Scholar
  6. 6.
    Funaki T.: Stochastic Interface Models. Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, Vol. 1869, pp. 103–274. Springer, Berlin (2005)Google Scholar
  7. 7.
    Funaki T., Spohn H.: Motion by mean curvature from the Ginzburg–Landau \({\nabla\phi}\) interface model. Commun. Math. Phys. 85, 1–36 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Georgii H.-O.: Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin (2011)CrossRefzbMATHGoogle Scholar
  9. 9.
    Kenyon R.: Lectures on Dimers. Statistical Mechanics, IAS/Park City Mathematics Series, Vol. 16, pp. 191–230. American Mathematical Society, Providence, RI (2009)Google Scholar
  10. 10.
    Kenyon R., Okounkov A., Sheffield S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Laslier B., Toninelli F.L.: Lozenge tilings, Glauber dynamics and macroscopic shape. Commun. Math. Phys. 338, 1287–1326 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Laslier B., Toninelli F.L.: Hydrodynamic limit for a lozenge tiling Glauber dynamics. Ann. Henri Poincaré Theor. Math. Phys. 18, 2007–2043 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)CrossRefzbMATHGoogle Scholar
  15. 15.
    Luby M., Randall D., Sinclair A.: Markov Chain Algorithms for Planar Lattice Structures. SIAM J. Comput. 31, 167–192 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nishikawa T.: Hydrodynamic limit for the Ginzburg–Landau \({\nabla\phi}\) interface model with boundary conditions. Commun. Math. Phys. 127, 205–227 (2003)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Sheffield, S.: Random Surfaces. Astésque, (2005)Google Scholar
  18. 18.
    Spohn H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  19. 19.
    Spohn H.: Interface motion in models with stochastic dynamics. J. Stat. Phys. 71, 1081–1132 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Toninelli F.L.: A (2 + 1)-dimensional growth process with explicit stationary measure. Ann. Probab. 45, 2899–2940 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wilson D.B.: Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14, 274–325 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.LPMA - Univ. Paris DiderotParisFrance
  2. 2.Univ Lyon, CNRS, Université Claude Bernard Lyon 1, UMR 5208 Institut Camille JordanVilleurbanne CedexFrance

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