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

TL;DR

This paper rigorously analyzes the zero-temperature Glauber dynamics of the 3D Ising model, confirming the expected quadratic time scale for domain disappearance and linking interface evolution to mean curvature motion, with insights from dimer covering fluctuations.

Contribution

It provides the first rigorous proof that the domain disappearance time in 3D is proportional to L^2, supporting the mean curvature motion heuristic, and establishes spectral gap behavior in 2D.

Findings

Disappearance time in 3D is between L^2/(c log L) and L^2 (log L)^c.

In 2D, the disappearance time is of order L^2 without logarithmic corrections.

Spectral gap in 2D behaves like c/L for large L.

Abstract

We consider the Glauber dynamics for the Ising model with "+" boundary conditions, at zero temperature or at temperature which 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 \tau_+ which is at most L^2(\log L)^c and at least L^2/(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 dimer coverings of the infinite honeycomb lattice. Our result, apart from the spurious logarithmic factor, is the first rigorous confirmation of the expected behavior \tau_+\simeq const\times L^2, conjectured on heuristic grounds. In dimension d=2, \tau_+ can be shown to be of order L^2 without logarithmic corrections: the upper bound was proven in [Fontes, Schonmann, Sidoravicius, 2002] and here we provide the lower bound. For d=2, we also prove that the spectral gap of the generator behaves like c/L for L large, as conjectured in [Bodineau-Martinelli, 2002].