Nucleation and growth in two dimensions

TL;DR

This paper analyzes a two-dimensional infection process on a grid, providing detailed descriptions of its evolution, infection times, and sharp thresholds, thereby extending and refining previous models like bootstrap and first-passage percolation.

Contribution

It offers a precise characterization of the infection dynamics on , sharpening prior results and determining typical infection times and thresholds for a broad parameter range.

Findings

Determines typical infection times up to a constant factor.

Establishes a sharp threshold for the infection process.

Provides a detailed evolution description of the process.

Abstract

We consider a dynamical process on a graph $G$, in which vertices are infected (randomly) at a rate which depends on the number of their neighbours that are already infected. This model includes bootstrap percolation and first-passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph $\mathbb{Z}^2$, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.