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

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  • Raphaël Cerf

    (Université Paris Sud)

Abstract

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 $${\mathbb{N}}$$ to $${\mathbb{R}}$$ such that f(N)/ln N→+∞ and f(N)/N→0 as N goes to ∞ the laws of {x∈ $${\mathbb{R}}$$ 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.

Suggested Citation

  • Raphaël Cerf, 2000. "Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric," Journal of Theoretical Probability, Springer, vol. 13(2), pages 491-517, April.
    • Handle: RePEc:spr:jotpro:v:13:y:2000:i:2:d:10.1023_a:1007841407417
    DOI: 10.1023/A:1007841407417
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