  Consider the partition function of a directed polymer in an IID field. We
assume that both tails of the negative and the positive part of the field are
at least as light as exponential. It is a well-known fact that the free energy
of the polymer is equal to a deterministic constant for almost every
realization of the field and that the upper tail of the large deviations is
exponential. The lower tail of the large deviations is typically lighter than
exponential. In this paper we provide a method to obtain estimates on the rate
of decay of the lower tail of the large deviations, which are sharp up to
multiplicative constants. As a consequence, we show that the lower tail of the
large deviations exhibits three regimes, determined according to the tail of
the negative part of the field. Our method is simple to apply and can be used
to cover other oriented and non-oriented models including first/last-passage
percolation and the parabolic Anderson model
