First passage times for threshold growth dynamics on ${\bf Z}\sp 2$

Abstract

In the threshold growth model on an integer lattice, the occupied set grows according to a simple local rule: a site becomes occupied iff it sees at least a threshold number of already occupied sites in its prescribed neighborhood. In this paper, we analyze the behavior of two-dimensional threshold growth dynamics started from a sparse Bernoulli density of occupied sites. We explain how nucleation of rare centers, invariant shapes and interaction between growing droplets influence the first passage time in the supercritical case. We also briefly address scaling laws for the critical case.