Large deviations for the chemical distance in supercritical Bernoulli percolation

Details Large deviations for the chemical distance in supercritical Bernoulli percolation Large deviations for the chemical distance in supercritical Bernoulli percolation

2004-09-18

arxiv.org/abs/math/0409317v3

Annals of Probability 35, 3 (2007) 833-866

Mathematics

Probability

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

Scientific paper

10.1214/009117906000000881

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 \[\biggl\{0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\epsilon, 1+\epsilon) \biggr\}\] 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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