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For the statistics of global observables in disordered systems, we discussthe matching between typical fluctuations and large deviations. We focus on thestatistics 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, wheremany exact results exist (ii) for the Sherrington-Kirkpatrick spin-glass modelof $N$ spins, where various possibilities have been proposed. Here we stressthat, besides the behavior of the disorder-average $E_0^{av}(N)$ and of thestandard deviation $ \Delta E_0(N) \sim N^{\omega_f}$ that defines thefluctuation exponent $\omega_f$, it is very instructive to study the fullprobability distribution $\Pi(u)$ of the rescaled variable $u=\frac{E_0(N)-E_0^{av}(N)}{\Delta E_0(N)}$ : (a) numerically, the convergencetowards $\Pi(u)$ is usually very rapid, so that data on rather small sizes butwith high statistics allow to measure the two tails exponents $\eta_{\pm}$defined as $\ln \Pi(u \to \pm \infty) \sim - | u |^{\eta_{\pm}}$. In thegeneric case $1< \eta_{\pm} < +\infty$, this leads to explicit non-trivialterms in the asymptotic behaviors of the moments $\bar{Z_N^n}$ of the partitionfunction when the combination $[| n | N^{\omega_f}]$ becomes large (b) simplerare 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 $R(w_{\pm} =\frac{E_0(N)-E_0^{av}(N)}{N^{\kappa_{\pm}}}) \sim e^{- N^{\rho_{\pm}} {\calR}_{\pm}(w_{\pm})}$ (the 'usual' large deviations formalism corresponds to$\kappa_{\pm}=1=\rho_{\pm}$).

journal of statistical mechanics

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