Typical versus averaged overlap distribution in spin glasses: Evidence for droplet scaling theory

We consider the statistical properties over disordered samples (J) of the overlap distribution P-J(q) which plays the role of an order parameter in spin glasses. We show that near zero temperature (i) the typical overlap distribution is exponentially small in the central region of -1 < q < 1: P-typ(q) = e(<(lnPJ(q))over bar>) = e(-beta N theta phi(q)), where theta is the droplet exponent defined here with respect to the total number N of spins (in order to consider also fully connected models in which the notion of length does not exist); (ii) the rescaled variable v = -[ln P-J(q)]/N-theta remains an O(1) random positive variable describing sample-to-sample fluctuations; (iii) the averaged distribution <(P-J(q))over bar> is nontypical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution I-J (q(0)) = integral(q0)(0) 0 dq P-J (q). These results are derived explicitly for the spherical mean-field model with theta = 1/3, phi(q) = 1 - q(2), and the random variable v corresponds to the rescaled difference between the two largest eigenvalues of Gaussian orthogonal ensemble random matrices. Then we compare numerically the typical and averaged overlap distributions for the long-ranged one-dimensional Ising spin glass with random couplings decaying as J (r) proportional to r(-sigma) for various values of the exponent sigma, corresponding to various droplet exponents theta(sigma), and for the mean-field Sherrington-Kirkpatrick model (corresponding formally to the s = 0 limit of the previous model). Our conclusion is that future studies on spin glasses should measure the typical values of the overlap distribution P-typ(q) or of the cumulative overlap distribution I-typ(q(0)) = e(<(ln IJ(q0))over bar>) to obtain clearer conclusions on the nature of the spin-glass phase.