Glauber Dynamics

Abstract

The goal of this chapter is to extend the analysis of Chap. 17 to volumes that grow moderately fast as the temperature decreases. We run the Glauber dynamics on a large torus starting from a random initial configuration where all the droplets (= clusters of plus-spins) are small. For low temperature, and in the parameter range corresponding to the metastable regime, the transition from the metastable state (with only subcritical droplets) to the stable state (with one or more supercritical droplets) is triggered by the appearance of a single critical droplet somewhere in the torus. We show that the average time until this happens is inversely proportional to the volume and is driven by the same quantities as for small volumes, a property we refer to as homogeneous nucleation. This scaling is valid as long as the average nucleation time tends to infinity. Sections 19.1–19.4 develop the key steps of the proof. Sections 19.5–19.6 provide some key ingredients that are needed along the way.

La complexion qui fait le talent pour les petites choses est contraire à celle qu’il faut pour le talent des grandes.   (François de La Rochefoucauld, Réflexions)