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

TL;DR

This paper extends the one-step replica symmetry breaking formalism to analyze the extreme value statistics of logarithmically correlated random energy models, revealing universal behaviors and explicit formulas for minima distributions.

Contribution

It introduces a novel connection between the replica framework and probability theory, providing explicit formulas for minima statistics in logREMs based on IR and UV limits.

Findings

The distribution of extreme values behaves as a randomly shifted decorated exponential Poisson process.

The second minimum statistics are largely independent of UV details, depending mainly on the mean gap.

Numerical tests confirm the theoretical predictions using the circular model of 1/f-noise.

Abstract

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.