Lozenge Tilings, Glauber Dynamics and Macroscopic Shape

Abstract

We study the Glauber dynamics on the set of tilings of a finite domain of the plane with lozenges of side 1/L. Under the invariant measure of the process (the uniform measure over all tilings), it is well known (Cohn et al. J Am Math Soc 14:297–346, 2001) that the random height function associated to the tiling converges in probability, in the scaling limit ${L {\to} {\infty}}$, to a non-trivial macroscopic shape minimizing a certain surface tension functional. According to the boundary conditions, the macroscopic shape can be either analytic or contain “frozen regions” (Arctic Circle phenomenon Cohn et al. N Y J Math 4:137–165, 1998; Jockusch et al. Random domino tilings and the arctic circle theorem, arXiv:math/9801068, 1998). It is widely conjectured, on the basis of theoretical considerations (Henley J Statist Phys 89:483–507, 1997; Spohn J Stat Phys 71:1081–1132, 1993), partial mathematical results (Caputo et al. Commun Math Phys 311:157–189, 2012; Wilson Ann Appl Probab 14:274–325, 2004) and numerical simulations for similar models (Destainville Phys Rev Lett 88:030601, 2002; cf. also the bibliography in Henley (J Statist Phys 89:483–507, 1997) and Wilson (Ann Appl Probab 14:274–325, 2004), that the Glauber dynamics approaches the equilibrium macroscopic shape in a time of order L ^2+o(1). In this work we prove this conjecture, under the assumption that the macroscopic equilibrium shape contains no “frozen region”.