Linear Speed Large Deviations for Perco- Lation Clusters

Let Cn be the origin-containing cluster in subcritical percolation on the lattice 1 nZ , viewed as a random variable in the space Ω of compact, connected, origin-containing subsets of R, endowed with the Hausdorff metric δ. When d ≥ 2, and Γ is any open subset of Ω, we prove that lim n→∞ 1 n logP (Cn ∈ Γ) = − inf S∈Γ λ(S) where λ(S) is the one-dimensional Hausdorff measure of S defined using the correlation norm: ||u|| := lim n→∞ − 1 n logP (un ∈ Cn) where un is u rounded to the nearest element of 1 nZ . Given points a, . . . , a ∈ R, there are finitely many correlation-norm Steiner trees spanning these points and the origin. We show that if the Cn are each conditioned to contain the points a 1 n, . . . , a k n, then the probability that Cn fails to approximate one of these trees tends to zero exponentially in n.