Nucleation and growth in two dimensions

GRIFFITHS, SIMON; SMITH, PAUL; BOLLOBÁS, BÉLA; T ROLLA, LEONARDO; MORRIS, ROBERT

RANDOM STRUCTURES ALGORITHMS

JOHN WILEY & SONS INC

Año: 2020 vol. 56 p. 63 - 96

1042-9832

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 neighbors 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 Z2, 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.