Linear Speed Large Deviations for Percolation Clusters
Yevgeniy Kovchegov, UCLA Mathematics Department, USA
Scott Roger Sheffield, Microsoft Research
Abstract
Let $C_n$ be the origin-containing cluster in subcritical percolation on the lattice $frac{1}{n}
mathbb Z^d$, viewed as a random variable in the space $Omega$ of compact, connected,
origin-containing subsets of $mathbb R^d$, endowed with the Hausdorff metric $delta$. When $d
geq 2$, and $Gamma$ is any open subset of $Omega$, we prove that $$lim_{n rightarrow
infty}frac{1}{n} log P(C_n in Gamma) = -inf_{S in Gamma} lambda(S)$$ where $lambda(S)$ is
the one-dimensional Hausdorff measure of $S$ defined using the {em correlation norm}: $$||u|| :=
lim_{n rightarrow infty} - frac{1}{n} log P (u_n in C_n )$$ where $u_n$ is $u$ rounded to
the nearest element of $frac{1}{n}mathbb Z^d$. Given points $a^1, ldots, a^k in mathbb R^d$,
there are finitely many correlation-norm Steiner trees spanning these points and the origin. We
show that if the $C_n$ are each conditioned to contain the points $a^1_n, ldots, a^k_n$, then the
probability that $C_n$ fails to approximate one of these trees tends to zero exponentially in $n$.
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