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 \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{μ(x)}\notin (1-ε, 1+ε) \biggr\}\] exponentially decreases when $\Vert x{\Vert }_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.

Published at http://dx.doi.org/10.1214/009117906000000881 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

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