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

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.regardless of the geometric nature of the model. In fact eq. (4) holds for both uncorrelated REM and logREM that satisfy (2), and a way to see this for uncorrelated REM is as follows [1]. By (2), one computes the microscopic ensemble entropy, defined as the log of the number of states up to a given energy: S(E = −2q ln M ) = ln M (1 − q) for q ∈ [0, 1] and S = 0 for q > 1, and notices the non-analyticity at q = 1 where β c = ∂S ∂E q=1−ǫ = 1. For correlated REMs, strictly speaking, the existence of one (and only one) transition is a result of RSB analysis, see Sect. IV A and A.

A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications

“…[31], [39], upon which GMC measures are built, from the complexity of mathematical problems that their stochastic dependence poses, and from the many applications in mathematical and theoretical physics and pure mathematics, in which GMC naturally appears. Without aiming for comprehension, we can mention applications to five areas: (1) conformal field theory and Liouville quantum gravity [3], [11], [29], [83], [85], (2) statistical mechanics of disordered energy landscapes [18], [19], [22], [37], [38], [39], [42], [43], [78], (3) random matrix theory [45], [46], [48], [57], [98], (4) statistical modeling of fully intermittent turbulence [23], [24], [25], [35], [68], [84], (5) conjectured [40], [44], [77] and some rigorous [90] applications to the behavior of the Riemann zeta function on the critical line.…”

Operator product expansion in Liouville field theory and Seiberg-type transitions in log-correlated random energy models

“…For logREMs without charge, the replica approach was developed in Refs. [19,39,45] (see Ref. [25], Section 2.3 for a more pedantic introduction).…”

“…They turn out to agree exactly with the linear parts of the large deviation theory prediction, eqs. (39) . This is a non-trivial test of the convexity assumption, which we will use again in Section IV C.…”

“…[39], when replicas form groups of size > 1, the internal structure of the group is non-trivial and contributes a factor depending on the short-distance details ("UV-data" ) of the model, in addition to the Coulomb gas integral, which depends only on the long-distance details ("IR-data"). The short-distance factor is known to have tangible effects, e.g., on the distribution of the second minimum of logREMs [39], but its analytical behavior is poorly understood. Now the right hand side eq.…”

“…(C15) does not contain such a shortdistance factor, C UV (α). However, we know [39] that this factor becomes trivial when the replica symmetry is unbroken, i.e., in the Unbound regime (of the β < 1 phase): C UV (α → a) → 1, so we expect that the asymptotic behaviors above are not affected by its presence, at least not too far away from the U/C boundary.…”