Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

Abstract

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from $${\mathbb{N}}$$ to $${\mathbb{R}}$$ such that f(N)/ln N→+∞ and f(N)/N→0 as N goes to ∞ the laws of {x∈ $${\mathbb{R}}$$ 2 : d(x, C)≤f(N)}/N satisfy a large deviations principle with respect to the L 1 metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.