Directed polymer in random environment and last passage percolation

The sequence of random probability measures ${\nu }_n$ that gives a path of length $n$, $\text{unsur}n$ times the sum of the random weights collected along the paths, is shown to satisfy a large deviations principle with good rate function the Legendre transform of the free energy of the associated directed polymer in a random environment. Consequences on the asymptotics of the typical number of paths whose collected weight is above a fixed proportion are then drawn.