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 14:297346, 2001) that the random height function associated to the tiling converges in probability, in the scaling limit , to a nontrivial 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 4:137165, 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 89:483507, 1997; Spohn J Stat Phys 71:10811132, 1993), partial mathematical results (Caputo et al. Commun Math Phys 311:157189, 2012; Wilson Ann Appl Probab 14:274325, 2004) and numerical simulations for similar models (Destainville Phys Rev Lett 88:030601, 2002; cf. also the bibliography in Henley (J Statist Phys 89:483507, 1997) and Wilson (Ann Appl Probab 14:274325, 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".
 Publication:

Communications in Mathematical Physics
 Pub Date:
 September 2015
 DOI:
 10.1007/s0022001523967
 arXiv:
 arXiv:1310.5844
 Bibcode:
 2015CMaPh.338.1287L
 Keywords:

 Mathematics  Probability
 EPrint:
 38 pages, 5 figures