Large deviations for the macroscopic motion of an interface

TL;DR

This paper analyzes the most probable macroscopic interface motion in a stochastic Ising model, deriving bounds on deviation costs and showing that rapid transitions tend to concentrate on nucleation events.

Contribution

It provides explicit quantitative estimates for deviation costs and links the probability of interface motion to nucleation phenomena in a stochastic lattice system.

Findings

Derived bounds for deviation cost functional

Explicit error terms valid at macroscopic scale

Probability concentrates on nucleation for fast transitions

Abstract

We study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of Ising spins with Kac interaction evolving in time according to Glauber (non-conservative) dynamics. Such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions. We derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit error terms which are valid also in the macroscopic scale. Furthermore, using the result of a companion paper about the minimizers of this cost functional for the macroscopic motion of the interface in a fixed time, we prove that the probability of such events can concentrate on nucleations should the transition happen fast enough.