  We consider the standard first passage percolation on $\mathbb{Z}^{d}$: with
each edge of the lattice we associate a random capacity. We are interested in
the maximal flow through a cylinder in this graph. Under some assumptions
Kesten proved in 1987 a law of large numbers for the rescaled flow. Chayes and
Chayes established that the large deviations far away below its typical value
are of surface order, at least for the Bernoulli percolation and cylinders of
certain height. Thanks to another approach we extend here their result to
higher cylinders, and we transport this result to the model of first passage
percolation.
