Affiliation:
1. French National Centre for Scientific Research
2. University of Paris-Saclay
3. University of Paris-Sud
4. King's College London
5. École Normale Supérieure
Abstract
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.
Funder
Engineering and Physical Sciences Research Council
National Science Foundation
Subject
General Physics and Astronomy
Cited by
11 articles.
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