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Large deviations for the chemical distance in supercritical Bernoulli percolation

2007
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Annals of Probability
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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.

doi:10.1214/009117906000000881
fatcat:i4hdpw7enbcdvbkg62cfof74cm