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Large deviations for the chemical distance in supercritical Bernoulli percolation

Abstract : The chemical distance $D(x,y)$ is the length of the shortest open path between two points $x$ and $y$ in an infinite Bernoulli percolation cluster. In this work, we study the asymptotic behaviour of this random metric, and we prove that, for an appropriate norm $\mu$ depending on the dimension and the percolation parameter, the probability of the event \[\biggl\{ 0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\varepsilon ,1+\varepsilon) \biggr\}\] exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.
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Submitted on : Tuesday, July 31, 2007 - 12:50:35 PM
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Olivier Garet, Régine Marchand. Large deviations for the chemical distance in supercritical Bernoulli percolation. Annals of Probability, Institute of Mathematical Statistics, 2007, 35 (3), pp.833-866. ⟨10.1214/009117906000000881⟩. ⟨hal-00002875v3⟩

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