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

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 N to R such that f(N)/ln N (longrightarrow)+ (infinity) and f(N)/N (longrightarrow) 0 as N goes to (infinity) the laws of {x (element of) 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.