Large deviations for the contact process and two dimensional percolation (Q1090012)

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 \(Y_{\Lambda}\) of sites inside a square \(\Lambda\) which belong to the infinite cluster has the following large deviation properties. The probability that \(Y_{\Lambda}\) deviates from its expected value by a positive amount decays exponentially with the area of \(\Lambda\), while the probability that it deviates from its expected value by a negative amount decays exponentially with the perimeter of \(\Lambda\). These two problems are treated together in this paper because similar techniques (renormalization) are used for both.