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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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.

Article information

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

Dates
First available in Project Euclid: 6 January 2003

Permanent link to this document
https://projecteuclid.org/euclid.aop/1041903205

Digital Object Identifier
doi:10.1214/aop/1041903205

Mathematical Reviews number (MathSciNet)
MR1415228

Zentralblatt MATH identifier
0872.60077

Subjects
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]

Keywords
Shape theory nucleation first passage time Poisson convergence metastability

Citation

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. https://projecteuclid.org/euclid.aop/1041903205


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