# Large deviations for the contact process and two dimensional percolation

Published

Journal Article

The following results are proved: 1) For the upper invariant measure of the basic one-dimensional supercritical contact process the density of 1's has the usual large deviation behavior: the probability of a large deviation decays exponentially with the number of sites considered. 2) For supercritical two-dimensional nearest neighbor site (or bond) percolation the density YΛ of sites inside a square Λ which belong to the infinite cluster has the following large deviation properties. The probability that YΛ deviates from its expected value by a positive amount decays exponentially with the area of Λ, while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of Λ. These two problems are treated together in this paper because similar techniques (renormalization) are used for both. © 1988 Springer-Verlag.

### Full Text

### Duke Authors

### Cited Authors

- Durrett, R; Schonmann, RH

### Published Date

- December 1, 1988

### Published In

### Volume / Issue

- 77 / 4

### Start / End Page

- 583 - 603

### Electronic International Standard Serial Number (EISSN)

- 1432-2064

### International Standard Serial Number (ISSN)

- 0178-8051

### Digital Object Identifier (DOI)

- 10.1007/BF00959619

### Citation Source

- Scopus