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First passage times for threshold growth dynamics on \(\mathbb{Z}^ 2\). (English) Zbl 0872.60077

Summary: 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. 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.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
52A10 Convex sets in \(2\) dimensions (including convex curves)
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