  We investigate the large deviation behavior of the overlap probability
density in the Sherrington--Kirkpatrick model from several analytical
perspectives. First we analyze the spin glass phase using the coupled replica
scheme. Here generically $\frac1N \log P_N(q)$ $\approx$ $- {\cal A}$
$((|q|-q_{EA})^3$, and we compute the first correction to the expansion of $\A$
in powers of $T_c-T$. We study also the $q=1$ case, where $P(q)$ is know
exactly. Finally we study the paramagnetic phase, where exact results valid for
all $q$'s are obtained. The overall agreement between the various points of
view is very satisfactory. Data from large scale numerical simulations show
that the predicted behavior can be detected already on moderate lattice sizes.
