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Large deviations for the contact process and two dimensional percolation
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  • Published: December 1988

Large deviations for the contact process and two dimensional percolation

  • R. Durrett1 &
  • R. H. Schonmann1,2 

Probability Theory and Related Fields volume 77, pages 583–603 (1988)Cite this article

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Summary

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 densityY Λ of sites inside a square Λ which belong to the infinite cluster has the following large deviation properties. The probability thatY Λ 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, Cornell University, White Hall, 14853, Ithaca, NY, USA

    R. Durrett & R. H. Schonmann

  2. Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 20570, 01000, São Paulo, SP, Brasil

    R. H. Schonmann

Authors
  1. R. Durrett
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  2. R. H. Schonmann
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Additional information

Partially supported by the National Science Foundation and the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell

Partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq (Brazil) and the U.S. Army Research Office through the Mathematical Sciences Institute at Cornell

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Durrett, R., Schonmann, R.H. Large deviations for the contact process and two dimensional percolation. Probab. Th. Rel. Fields 77, 583–603 (1988). https://doi.org/10.1007/BF00959619

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  • Received: 05 April 1987

  • Revised: 03 November 1987

  • Issue Date: December 1988

  • DOI: https://doi.org/10.1007/BF00959619

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Keywords

  • Stochastic Process
  • Probability Theory
  • Invariant Measure
  • Mathematical Biology
  • Similar Technique
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