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 E0 in disorder models of two types: (i) for the directed polymer of length N in a two-dimensional medium, for which many exact results exist; (ii) for the Sherrington–Kirkpatrick spin-glass model of N spins, for which various possibilities have been proposed. Here we stress that it is very instructive to study, besides the behavior of the disorder average E0av(N) and that of the standard deviation ΔE0(N) ∼ Nωf that defines the fluctuation exponent ωf, also the full probability distribution Π(u) of the rescaled variable u = (E0(N) − E0av(N))/ΔE0(N): (a) numerically, the convergence towards Π(u) is usually very rapid, so data for rather small sizes but with high statistics allow one to measure the two tail exponents η± defined as ln Π (u → ±∞) ∼-|u|η|±; in the generic case 1<η±<+∞, this leads to explicit non-trivial terms in the asymptotic behaviors of the moments ZNn-bar of the partition function when the combination [|n|Nωf] becomes large; (b) simple rare events arguments can usually be found for providing explicit relations between η± and ωf; these rare events usually correspond to 'anomalous' large deviation properties of the generalized form