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.
Highlights
With respect to the Gumbel Poisson point process which describes the minima of the uncorrelated REM, the novel ingredient of shifted decorated Gumbel Poisson point process (SDPPP), i.e., the decoration process, describes the internal structure of blocks of extreme values which share a near ancestor, and are highly correlated
This equation is equivalent to the saying that the all the minima Vmin, Vmin,1, . . . are generated by a randomly shifted decorated Gumbel Poisson point process (SDPPP), such that the random shift has the same distribution as the free energy at critical temperature β = 1, and the decoration process is given by the biased minima vmin, vmin,1, . . . . Their statistics are characterized in terms of minima of the local logREM umin, umin,1, . . . by (44), which is a straightforward generalization of (35)
In Appendix A, we provide a full replica symmetry breaking analysis of logREMs, which show 1RSB results from the general definition (1)
Summary
Extreme value statistics of logarithmically correlated random energy model (logREMs) are the subject of active recent research by both physicists and mathematicians. With respect to the Gumbel Poisson point process which describes the minima of the uncorrelated REM, the novel ingredient of SDPPP, i.e., the decoration process, describes the internal structure of blocks of extreme values which share a near ancestor, and are highly correlated Such a picture is expected to apply in the context of non-hierarchical logREMs, in particular those related to 2d GFF and 1/ f -noise, in which the blocks are those of extreme values given by nearby points.
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