Xiangyu Cao, Yan V. Fyodorov, Pierre Le Doussal
SciPost Phys. 1, 011 (2016) ·
published 19 December 2016

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Building upon the onestep replica symmetry breaking formalism, duly
understood and ramified, we show that the sequence of ordered extreme values of a general class of Euclideanspace 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 largedistance ("infrared", 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 smalldistance ("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 logREMs. 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 higherorder generalizations) in terms of modelspecific 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 logcorrelated field this parameter can be evaluated numerically, and we provide several numerical tests of our theory using the circular model of $1/f$noise.