  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.
