Large deviations for the chemical distance in supercritical Bernoulli percolation

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 behavior of this random metric, and we prove that, for an appropriate norm μ depending on the dimension and the percolation parameter, the probability of the event \\ 0↔ x,D(0,x)/μ(x)∉ (1-ε ,1+ε )\ \ exponentially decreases when ‖x‖1 tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.