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

Cecile Monthus, Thomas Garel

Published 2009-12-15, updated 2010-02-25Version 2

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 $\kappa_{\pm}=1=\rho_{\pm}$).