Matching between typical fluctuations and large deviations in disordered
systems : application to the statistics of the ground state energy in the SK
spin-glass model

For the statistics of global observables in disordered systems, we discuss
the matching between typical fluctuations and large deviations. We focus on the
statistics of the ground state energy $E_0$ in two types of disordered models :
(i) for the directed polymer of length $N$ in a two-dimensional medium, where
many exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass model
of $N$ spins, where various possibilities have been proposed. Here we stress
that, besides the behavior of the disorder-average $E_0^{av}(N)$ and of the
standard deviation $\Delta E_0(N) \sim N^{\omega_f}$ that defines the
fluctuation exponent $\omega_f$, it is very instructive to study the full
probability distribution $\Pi(u)$ of the rescaled variable $u=
\frac{E_0(N)-E_0^{av}(N)}{\Delta E_0(N)}$ : (a) numerically, the convergence
towards $\Pi(u)$ is usually very rapid, so that data on rather small sizes but
with high statistics allow to measure the two tails exponents $\eta_{\pm}$
defined as $\ln \Pi(u \to \pm \infty) \sim - | u |^{\eta_{\pm}}$. In the
generic case $1< \eta_{\pm} < +\infty$, this leads to explicit non-trivial
terms in the asymptotic behaviors of the moments $\bar{Z_N^n}$ of the partition
function when the combination $[| n | N^{\omega_f}]$ becomes large (b) simple
rare events arguments can usually be found to obtain explicit relations between
$\eta_{\pm}$ and $\omega_f$. These rare events usually correspond to
'anomalous' large deviation properties of the generalized form $R(w_{\pm} =
\frac{E_0(N)-E_0^{av}(N)}{N^{\kappa_{\pm}}}) \sim e^{- N^{\rho_{\pm}} {\cal
R}_{\pm}(w_{\pm})}$ (the 'usual' large deviations formalism corresponds to
$\kappa_{\pm}=1=\rho_{\pm}$).Comment: 10 pages, 4 figure