Large deviations for the chemical distance in supercritical Bernoulli percolation

Abstract

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 $$\biggl\{\ 0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\varepsilon ,1+\varepsilon )\ \biggr\}$$ exponentially decreases when ‖x‖₁ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.