Moderate deviations for the chemical distance in Bernoulli percolation

19 pages, in english. A french version, entitled "Déviations modérées de la distance chimique" is also available (http://hal.archives-ouvertes.fr/hal-00401688/).International audienceIn this paper, we establish moderate deviations for the chemical distance in Bernoulli percolation. The chemical distance between two points is the length of the shortest open path between these two points. Thus, we study the size of random fluctuations around the mean value, and also the asymptotic behavior of this mean value. The estimates we obtain improve our knowledge of the convergence to the asymptotic shape. Our proofs rely on concentration inequalities proved by Boucheron, Lugosi and Massart, and also on the approximation theory of subadditive functions initiated by Alexander