Abstract
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.
Similar content being viewed by others
References
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)
Chang C.C., Yau H.-T.: Fluctuations of one dimensional Ginzburg–Landau models in nonequilibrium. Commun. Math. Phys. 145, 209–239 (1992)
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)
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)
Fritz J.: On the hydrodynamic limit of a Ginzburg Landau lattice model. Probab. Theory Relat. Fields 81, 291–318 (1989)
Funaki T.: Stochastic Interface Models. Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, Vol. 1869, pp. 103–274. Springer, Berlin (2005)
Funaki T., Spohn H.: Motion by mean curvature from the Ginzburg–Landau \({\nabla\phi}\) interface model. Commun. Math. Phys. 85, 1–36 (1997)
Georgii H.-O.: Gibbs Measures and Phase Transitions. Walter de Gruyter, Berlin (2011)
Kenyon R.: Lectures on Dimers. Statistical Mechanics, IAS/Park City Mathematics Series, Vol. 16, pp. 191–230. American Mathematical Society, Providence, RI (2009)
Kenyon R., Okounkov A., Sheffield S.: Dimers and amoebae. Ann. Math. 163, 1019–1056 (2006)
Kipnis C., Landim C.: Scaling Limits of Interacting Particle Systems. Springer, Berlin (1999)
Laslier B., Toninelli F.L.: Lozenge tilings, Glauber dynamics and macroscopic shape. Commun. Math. Phys. 338, 1287–1326 (2015)
Laslier B., Toninelli F.L.: Hydrodynamic limit for a lozenge tiling Glauber dynamics. Ann. Henri Poincaré Theor. Math. Phys. 18, 2007–2043 (2017)
Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific, Singapore (1996)
Luby M., Randall D., Sinclair A.: Markov Chain Algorithms for Planar Lattice Structures. SIAM J. Comput. 31, 167–192 (2001)
Nishikawa T.: Hydrodynamic limit for the Ginzburg–Landau \({\nabla\phi}\) interface model with boundary conditions. Commun. Math. Phys. 127, 205–227 (2003)
Sheffield, S.: Random Surfaces. Astésque, (2005)
Spohn H.: Large Scale Dynamics of Interacting Particles. Springer, Berlin (1991)
Spohn H.: Interface motion in models with stochastic dynamics. J. Stat. Phys. 71, 1081–1132 (1993)
Toninelli F.L.: A (2 + 1)-dimensional growth process with explicit stationary measure. Ann. Probab. 45, 2899–2940 (2017)
Wilson D.B.: Mixing times of lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14, 274–325 (2004)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by H. Duminil-Copin
Rights and permissions
About this article
Cite this article
Laslier, B., Toninelli, F.L. Lozenge Tiling Dynamics and Convergence to the Hydrodynamic Equation. Commun. Math. Phys. 358, 1117–1149 (2018). https://doi.org/10.1007/s00220-018-3095-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-018-3095-y