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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^{- βN^θ ϕ(q)} $, where $θ$ 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 $v = - (\ln P_{\cal J}(q))/N^θ$ 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 $θ=1/3$, $ϕ(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 $J(r) \propto r^{-σ}$ for various values of the exponent $σ$, corresponding to various droplet exponents $θ(σ)$, and for the mean-field SK-model (corresponding formally to the $σ=0$ 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.