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Linear speed large deviations for percolation clusters. (English) Zbl 1060.60097
Summary: 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 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 \(n\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F10 Large deviations
82B43 Percolation
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