## The Annals of Probability

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

### Characteristic Exponents for Two-Dimensional Bootstrap Percolation

#### 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