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

by Xiangyu Cao, Yan V. Fyodorov, Pierre Le Doussal

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Submission summary

Authors (as registered SciPost users): Xiangyu Cao
Submission information
Preprint Link:  (pdf)
Date accepted: 2016-12-05
Date submitted: 2016-10-10 02:00
Submitted by: Cao, Xiangyu
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Probability
  • Condensed Matter Physics - Theory
  • Mathematical Physics
  • Statistical and Soft Matter Physics
Approaches: Theoretical, Computational


Building upon the one-step replica symmetry breaking formalism, duly understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclidean-space logarithmically correlated random energy models (logREMs) behave in the thermodynamic limit as a randomly shifted decorated exponential Poisson point process. 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. Our approach provides connections of the replica framework to results in the probability literature and sheds further light on the freezing/duality conjecture which was the source of many previous results for log-REMs. In this way we 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 second min statistics is largely independent of details of UV data, whose influence is seen only through 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.

Published as SciPost Phys. 1, 011 (2016)

Submission & Refereeing History

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Submission 1610.02226v1 on 10 October 2016

Reports on this Submission

Anonymous Report 1 on 2016-12-1 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1610.02226v1, delivered 2016-12-01, doi: 10.21468/SciPost.Report.45


High quality, leading edge research


Perhaps very difficult to read for non specialists


This is a rather formidable paper on a topical subject, by well established researchers in that field. Although highly specialized and technical, the paper is full of new results and insights, and should certainly be published.

Requested changes


  • validity: top
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  • originality: top
  • clarity: high
  • formatting: good
  • grammar: good

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