Approximate Lifshitz law for the zero-temperature stochastic Ising model in any dimension

TL;DR

This paper rigorously proves an approximate Lifshitz law for the zero-temperature stochastic Ising model in four dimensions, showing the time for a hypercube to flip spins scales as L^2(log L)^c, extending understanding from lower dimensions.

Contribution

It provides the first rigorous proof of the Lifshitz law in four dimensions, leveraging three-dimensional interface results to establish the scaling behavior.

Findings

Proves T+ = O(L^2(log L)^c) for the four-dimensional Ising model.

Extends Lifshitz law validity from lower to higher dimensions.

Uses known three-dimensional interface fluctuation results to inform higher-dimensional analysis.

Abstract

We study the Glauber dynamics for the zero-temperature Ising model in dimension d=4 with "plus" boundary condition.Let T+ be the time needed for an hypercube of size L entirely filled with "minus" spins to become entirely "plus". We prove that T+ is O(L^2(log L)^c) for some constant c, not depending on the dimension. This brings further rigorous justification for the so-called "Lifshitz law" T+ = O(L^2) [5, 3] conjectured on heuristic grounds. The key point of our proof is to use the detail knowledge that we have on the three-dimensional problem: results for fluctuation of monotone interfaces at equilibrium and mixing time for monotone interfaces dynamics extracted from [2], to get the result in higher dimension.