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 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 $$\biggl\{0\leftrightarrow x,\frac{D(0,x)}{\mu(x)}\notin (1-\epsilon, 1+\epsilon) \biggr\}$$ exponentially decreases when $\|x\|_1$ tends to infinity. From this bound we also derive a large deviation inequality for the corresponding asymptotic shape result.