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October 1996 First passage times for threshold growth dynamics on ${\bf Z}\sp 2$
Janko Gravner, David Griffeath
Ann. Probab. 24(4): 1752-1778 (October 1996). DOI: 10.1214/aop/1041903205


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.


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Janko Gravner. David Griffeath. "First passage times for threshold growth dynamics on ${\bf Z}\sp 2$." Ann. Probab. 24 (4) 1752 - 1778, October 1996.


Published: October 1996
First available in Project Euclid: 6 January 2003

zbMATH: 0872.60077
MathSciNet: MR1415228
Digital Object Identifier: 10.1214/aop/1041903205

Primary: 60K35
Secondary: 52A10

Keywords: First passage time , metastability , nucleation , Poisson convergence , Shape theory

Rights: Copyright © 1996 Institute of Mathematical Statistics

Vol.24 • No. 4 • October 1996
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