Log-correlated random-energy models with extensive free-energy fluctuations: Pathologies caused by rare events as signatures of phase transitions

Abstract

We address systematically an apparent non-physical behavior of the free energy moment generating function for several instances of the logarithmically correlated models: the Fractional Brownian Motion with Hurst index H = 0 (fBm0) (and its bridge version), a 1D model appearing in decaying Burgers turbulence with log-correlated initial conditions, and finally, the two-dimensional logREM introduced in [Cao et al., Phys.Rev.Lett.,118,090601] based on the 2D Gaussian free field (GFF) with background charges and directly related to the Liouville field theory. All these models share anomalously large fluctuations of the associated free energy, with a variance proportional to the log of the system size. We argue that a seemingly non-physical vanishing of the moment generating function for some values of parameters is related to the termination point transition (a.k.a pre-freezing). We study the associated universal log corrections in the frozen phase, both for log-REMs and for the standard REM, filling a gap in the literature. For the above mentioned integrable instances of logREMs, we predict the non-trivial free energy cumulants describing non-Gaussian fluctuations on the top of the Gaussian with extensive variance. Some of the predictions are tested numerically.

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

“…Our goal in this section is to write down the analytic continuation and prove its involution invariance. We note that the physicists' approach to this problem can be found in [20], which leads to interesting conjectures about the maximum of the underlying gaussian field.…”

“…(12.60), (12.62), (12.63), and (12.64) are not Mellin transforms of probability distributions. We refer the interested reader to [20] and [21] for deep results on the subtle nature of the distribution of the maximum of the centered GFF fields. In addition, as shown in [20], the distribution of the maximum of the two-dimensional gaussian field with the covariance − log | r 1 − r 2 | can be similarly quantified in terms of the critical analytic continuation of the complex Selberg integral in Eq.…”

“…It can however be naturally interpreted as a rescaled limit of the moments of total mass of the two-dimensional GMC measure with the kernel − log | r 1 − r 2 | in the limit of the domain, over which it is defined, going to infinity, cf. [20] for details. Then, the analytic continuation of the complex Selberg integral becomes the mod-Gaussian limit of the REM (random energy model) that is associated with the GMC measure.…”