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

Abstract

We consider the statistical properties over disordered samples 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 {\it typical} overlap distribution is exponentially small in the central region of $-1<q<1$: $P^{typ}(q)=e^{\overline{\ln P_{\mathcal{J}}(q)}}\sim e^{-\beta N^{\theta }\varphi (q)}$, 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 $v=-(\ln P_{\mathcal{J}}(q))/N^{\theta }$ remains an O(1) random positive variable describing sample-to sample fluctuations; (iii) the averaged distribution $\overline{P_{\mathcal{J}}(q)}$ is non-typical and dominated by rare anomalous samples. Similar statements hold for the cumulative overlap distribution $I_{\mathcal{J}}(q_0)\equiv {\int }_0^{q_0}dq{P}_{\mathcal{J}}(q)$. These results are derived explicitly for the spherical mean-field model with $\theta =1/3$, $\varphi (q)=1-q^2$, 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.

v2=final revised version (in particular new sections IIE,IIIC and Appendix B w.r.t. v1)

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Cited by in corpus (22)

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• Evidence against a mean field description of short-range spin glasses revealed through thermal boundary conditions
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• Number of thermodynamic states in the three-dimensional Edwards-Anderson spin glass
• The cumulative overlap distribution function in realistic spin glasses
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• Scaling of the largest dynamical barrier in the one-dimensional long-range Ising spin-glass
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• A singular-potential random matrix model arising in mean-field glassy systems
• Explicit generation of the branching tree of states in spin glasses
• Universality for the Toda algorithm to compute the largest eigenvalue of a random matrix
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• Ground state interface exponents of the diluted Sherrington-Kirkpatrick spin glass
• Tracy-Widom distributions for the Gaussian orthogonal and symplectic ensembles revisited: a skew-orthogonal polynomials approach
• Generalised Gibbs Ensemble for spherically constrained harmonic models
• Energy gap of quantum spin glasses: a projection quantum Monte Carlo study