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      Large random correlations in individual mean field spin glass samples

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          Abstract

          We argue that complex systems must possess long range correlations and illustrate this idea on the example of the mean field spin glass model. Defined on the complete graph, this model has no genuine concept of distance, but the long range character of correlations is translated into a broad distribution of the spin-spin correlation coefficients for almost all realizations of the random couplings. When we sample the whole phase space we find that this distribution is so broad indeed that at low temperatures it essentially becomes uniform, with all possible correlation values appearing with the same probability. The distribution of correlations inside a single phase space valley is also studied and found to be much narrower.

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          Solvable Model of a Spin-Glass

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            Eigenvalues of the stability matrix for Parisi solution of the long-range spin-glass

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              Extensive eigenvalues in spin-spin correlations: a tool for counting pure states in Ising spin glasses

              We study the nature of the broken ergodicity in the low temperature phase of Ising spin glass systems, using as a diagnostic tool the spectrum of eigenvalues of the spin-spin correlation function. We show that multiple extensive eigenvalues of the correlation matrix \(C_{ij}\equiv \) occur if and only if there is replica symmetry breaking. We support our arguments with Exchange Monte-Carlo results for the infinite-range problem. Here we find multiple extensive eigenvalues in the RSB phase for \(N \agt 200\), but only a single extensive eigenvalue for phases with long-range order but no RSB. Numerical results for the short range model in four spatial dimensions, for \(N\le 1296\), are consistent with the presence of a single extensive eigenvalue, with the subdominant eigenvalue behaving in agreement with expectations derived from the droplet model. Because of the small system sizes we cannot exclude the possibility of replica symmetry breaking with finite size corrections in this regime.
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                Author and article information

                Journal
                15 October 2010
                2010-11-02
                Article
                10.1088/1742-5468/2011/02/P02009
                1010.3237
                fcead934-c60a-444c-88f2-b8d24f1741cb

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                J. Stat. Mech. (2011) P02009
                Added a few references and a comment phrase
                cond-mat.dis-nn

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