Large deviations for the chemical distance in supercritical Bernoulli percolation
Résumé
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 $$\Big{ 0 \communique x, \frac{D(0,x)}{\mu(x)}\notin (1-\epsilon, 1+\epsilon) \Big\}$$ 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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