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

Abstract

We consider the statistical properties over disordered samples $(\mathcal{J})$ of the overlap distribution ${P}_{\mathcal{J}}(q)$ 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 $\ensuremath{-}1<q<1$: ${P}^{\text{typ}}(q)={e}^{\overline{\mathrm{ln}{P}_{\mathcal{J}}(q)}}\ensuremath{\sim}{e}^{\ensuremath{-}\ensuremath{\beta}{N}^{\ensuremath{\theta}}\ensuremath{\phi}(q)}$, where $\ensuremath{\theta}$ 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 $v=\ensuremath{-}[\mathrm{ln}{P}_{\mathcal{J}}(q)]/{N}^{\ensuremath{\theta}}$ remains an $O(1)$ random positive variable describing sample-to-sample fluctuations; (iii) the averaged distribution $\overline{{P}_{\mathcal{J}}(q)}$ is nontypical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution ${I}_{\mathcal{J}}({q}_{0})\ensuremath{\equiv}{\ensuremath{\int}}_{0}^{{q}_{0}}dq\phantom{\rule{0.16em}{0ex}}{P}_{\mathcal{J}}(q)$. These results are derived explicitly for the spherical mean-field model with $\ensuremath{\theta}=1/3$, $\ensuremath{\phi}(q)=1\ensuremath{-}{q}^{2}$, 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 $J(r)\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\sigma}}$ for various values of the exponent $\ensuremath{\sigma}$, corresponding to various droplet exponents $\ensuremath{\theta}(\ensuremath{\sigma})$, and for the mean-field Sherrington-Kirkpatrick model (corresponding formally to the $\ensuremath{\sigma}=0$ limit of the previous model). Our conclusion is that future studies on spin glasses should measure the typical values of the overlap distribution ${P}^{\text{typ}}(q)$ or of the cumulative overlap distribution ${I}^{\text{typ}}({q}_{0})={e}^{\overline{\mathrm{ln}{I}_{\mathcal{J}}({q}_{0})}}$ to obtain clearer conclusions on the nature of the spin-glass phase.