Nucleation and growth in two dimensions

Bollobás, Béla; Griffiths, Simon; Morris, Robert; Rolla, Leonardo; Smith, Paul
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 {\$}\backslashmathbb{\{}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.
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