The Annals of Probability

Characteristic Exponents for Two-Dimensional Bootstrap Percolation

Enrique D. Andjel

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Abstract

Bootstrap percolation is a model in which an element of $\mathbf{Z}^2$ becomes occupied in one time unit if two appropriately chosen neighbors are occupied. Schonmann [4] proved that starting from a Bernoulli product measure of positive density, the distribution of the time needed to occupy the origin decays exponentially. We show that for $\alpha > 1$, the exponent can be taken as $\delta p^{2\alpha}$ for some $\delta > 0$, thus showing that the associated characteristic exponent is at most two. Another characteristic exponent associated to this model is shown to be equal to one.

Article information

Source
Ann. Probab., Volume 21, Number 2 (1993), 926-935.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989275

Digital Object Identifier
doi:10.1214/aop/1176989275

Mathematical Reviews number (MathSciNet)
MR1217573

Zentralblatt MATH identifier
0787.60120

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Bootstrap percolation exponential rates characteristic exponents

Citation

Andjel, Enrique D. Characteristic Exponents for Two-Dimensional Bootstrap Percolation. Ann. Probab. 21 (1993), no. 2, 926--935. doi:10.1214/aop/1176989275. https://projecteuclid.org/euclid.aop/1176989275


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