Typical versus averaged overlap distribution in Spin-Glasses : Evidence for the droplet scaling theory

We consider the statistical properties over disordered samples of the overlap distribution $P_{\cal J}(q)$ which plays the role of an order parameter in spin-glasses. We show that near zero temperature (i) the {\it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $P^{typ}(q) = e^{\bar{\ln P_{\cal J}(q)}} \sim e^{- \beta N^{\theta} \phi(q)}$, 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 where the notion of length does not exist); (ii) the rescaled variable $v = - (\ln P_{\cal J}(q))/N^{\theta}$ remains an O(1) random positive variable describing sample-to sample fluctuations; (iii) the averaged distribution $\bar{P_{\cal J}(q)}$ is non-typical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution $I_{\cal J}(q_0) \equiv \int_{0}^{q_0} dq P_{\cal J}(q)$. 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 GOE 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 $J(r) \propto r^{-\sigma}$ for various values of the exponent $\sigma$, corresponding to various droplet exponents $\theta(\sigma)$, and for the mean-field SK-model (corresponding formally to the $\sigma=0$ limit of the previous model). Our conclusion is that future studies on spin-glasses should measure the {\it typical} values of the overlap distribution or of the cumulative overlap distribution to obtain clearer conclusions on the nature of the spin-glass phase.