Linear Speed Large Deviations for Percolation Clusters

@article{Kovchegov2003LinearSL,
  title={Linear Speed Large Deviations for Percolation Clusters},
  author={Yevgeniy Kovchegov and Scott Sheffield},
  journal={Electronic Communications in Probability},
  year={2003},
  volume={8},
  pages={179-183}
}
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… Expand
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