"Zero" temperature stochastic 3D ising model and dimer covering fluctuations: A first step towards interface mean curvature motion

• Published in: Communications on pure and applied mathematics Vol. 64; no. 6; pp. 778 - 831
• Main Authors: Caputo, Pietro, Martinelli, Fabio, Simenhaus, François, Toninelli, Fabio Lucio
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.06.2011; Wiley; John Wiley and Sons, Limited
• Subjects: Combinatorics; Combinatorics. Ordered structures; Comparative analysis; Designs and configurations; Differential geometry; Exact sciences and technology; General mathematics; General, history and biography; Geometry; Heuristic; Lattice theory; Mathematics; Order, lattices, ordered algebraic structures; Probability; Sciences and techniques of general use; Temperature; Lower bound; Stochastic model; Upper bound; Applied mathematics; Covering problem; Glauber dynamic; Boundary condition; Dynamic model; Ising model; Dimer coverings; Glauber dynamics; Mixing time
• ISSN: 0010-3640; 1097-0312
• DOI: 10.1002/cpa.20359

We consider the Glauber dynamics for the Ising model with “+” boundary conditions, at zero temperature or at a temperature that goes to zero with the system size (hence the quotation marks in the title). In dimension d = 3 we prove that an initial domain of linear size L of “−” spins disappears within a time τ+, which is at most L2(log L)c and at least L2/(c log L) for some c > 0. The proof of the upper bound proceeds via comparison with an auxiliary dynamics which mimics the motion by mean curvature that is expected to describe, on large time scales, the evolution of the interface between “+” and “−” domains. The analysis of the auxiliary dynamics requires recent results on the fluctuations of the height function associated to dimmer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factors, is the first rigorous confirmation of the Lifshitz law τ+ ≃ const × L2, conjectured on heuristic grounds [8, 13]. In dimension d = 2, τ+ can be shown to be of order L2 without logarithmic corrections: the upper bound was proven in [6], and here we provide the lower bound. For d = 2, we also prove that the spectral gap of the generator behaves like ${ c \over L}$ for L large, as conjectured in [2]. © 2011 Wiley Periodicals, Inc.