# 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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Article dans une revue
Annals of Probability, Institute of Mathematical Statistics, 2007, 35 (3), pp.833-866. 〈10.1214/009117906000000881〉
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https://hal.archives-ouvertes.fr/hal-00002875
Contributeur : Olivier Garet <>
Soumis le : mardi 31 juillet 2007 - 12:50:35
Dernière modification le : jeudi 3 mai 2018 - 15:32:06
Document(s) archivé(s) le : jeudi 23 septembre 2010 - 15:55:04

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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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