Large deviations for the chemical distance in supercritical Bernoulli percolation

Olivier Garet (1) , Régine Marchand (2)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
2 IECN - Institut Élie Cartan de Nancy

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.

Mots clés: Percolation; first-passage percolation; chemical distance; shape theorem; large deviation inequalities

Domaines: Probabilités [math.PR]

Olivier Garet, Régine Marchand. Large deviations for the chemical distance in supercritical Bernoulli percolation. Annals of Probability, 2007, 35 (3), pp.833-866. ⟨10.1214/009117906000000881⟩. ⟨hal-00002875v3⟩