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One step replica symmetry breaking and extreme order statistics of logarithmic REMs

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X. Cao, Y. V. Fyodorov, P. Le Doussal

Original languageEnglish
Article number011
JournalSciPost Physics
Issue number2
PublishedDec 2016

Bibliographical note

Funding Information: Acknowledgements. We thank P.-L. Arguin for helpful explanations on the decorated Poisson point process. XC is grateful to S. Franz and A. Rosso for stimulating discussions at early stages of this work. XC and PLD thank the hospitality of KITP (under Grant No. NSF PHY11-25915), where part of this work was done. The research at King’s College London was financially supported by EPSRC grant EP/N009436/1 “The many faces of random characteristic polynomials”. Publisher Copyright: Copyright X. Cao et al. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

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Building upon the one-step replica symmetry breaking formalism, we show that the extreme values of a general class of Euclidean-space logarithmic correlated random energy models behave as a randomly shifted decorated exponential Poisson point process in the thermodynamic limit. The distribution of the random shift is determined solely by the large-distance ( "infra-red", IR) limit of the model, and is equal to the free energy distribution at the critical temperature up to a translation. the decoration process is determined solely by the small-distance ("ultraviolet", UV) limit, in terms of the biased minimal process. We discuss the relations of our approach with that based on the freezing/duality conjecture, and connections to results in the probability literature. Our approach allowed us to derive the general and explicit formulae for the joint probability density of depths of the first and second minima (as well its higher-order generalizations) in terms of model-specific contributions from UV as well as IR limits. In particular, we show that the distribution of the second minimum is independent of UV data, and depends on IR behaviour via a single parameter, the mean value of the gap. For a given log-correlated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of 1/f-noise.

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