The Annals of Probability

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

Janko Gravner and David Griffeath

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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.

Article information

Ann. Probab., Volume 24, Number 4 (1996), 1752-1778.

First available in Project Euclid: 6 January 2003

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 52A10: Convex sets in 2 dimensions (including convex curves) [See also 53A04]

Shape theory nucleation first passage time Poisson convergence metastability


Gravner, Janko; Griffeath, David. First passage times for threshold growth dynamics on ${\bf Z}\sp 2$. Ann. Probab. 24 (1996), no. 4, 1752--1778. doi:10.1214/aop/1041903205.

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