  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.
