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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 $ ΔE_0(N) \sim N^{ω_f}$ that defines the fluctuation exponent $ω_f$, it is very instructive to study the full probability distribution $Π(u)$ of the rescaled variable $u= \frac{E_0(N)-E_0^{av}(N)}{ΔE_0(N)}$ : (a) numerically, the convergence towards $Π(u)$ is usually very rapid, so that data on rather small sizes but with high statistics allow to measure the two tails exponents $η_{\pm}$ defined as $\ln Π(u \to \pm \infty) \sim - | u |^{η_{\pm}}$. In the generic case $1< η_{\pm} < +\infty$, 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 $[| n | N^{ω_f}]$ becomes large (b) simple rare events arguments can usually be found to obtain explicit relations between $η_{\pm}$ and $ω_f$. These rare events usually correspond to 'anomalous' large deviation properties of the generalized form $R(w_{\pm} = \frac{E_0(N)-E_0^{av}(N)}{N^{κ_{\pm}}}) \sim e^{- N^{ρ_{\pm}} {\cal R}_{\pm}(w_{\pm})}$ (the 'usual' large deviations formalism corresponds to $κ_{\pm}=1=ρ_{\pm}$).