Metastability in the dilute Ising model

TL;DR

This paper investigates how even minimal dilution in the dilute Ising model significantly accelerates the metastable to stable phase transition, due to rare high-dilution regions acting as catalysts.

Contribution

It demonstrates that small dilution levels can drastically reduce relaxation times in the dilute Ising model, revealing a catalyst effect not present in the regular model.

Findings

Small dilution dramatically reduces relaxation time.

Rare high-dilution regions act as catalysts.

Metastability is highly sensitive to dilution.

Abstract

Consider Glauber dynamics for the Ising model on the hypercubic lattice with a positive magnetic field. Starting from the minus configuration, the system initially settles into a metastable state with negative magnetization. Slowly the system relaxes to a stable state with positive magnetization. Schonmann and Shlosman showed that in the two dimensional case the relaxation time is a simple function of the energy required to create a critical Wulff droplet. The dilute Ising model is obtained from the regular Ising model by deleting a fraction of the edges of the underlying graph. In this paper we show that even an arbitrarily small dilution can dramatically reduce the relaxation time. This is because of a catalyst effect---rare regions of high dilution speed up the transition from minus phase to plus phase.