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