Log-correlated Random Energy Models with extensive free energy fluctuations: pathologies caused by rare events as signatures of phase transitions

TL;DR

This paper investigates anomalous free energy fluctuations in log-correlated models, linking non-physical behaviors to phase transitions, and provides new predictions for non-Gaussian fluctuations with numerical validation.

Contribution

It systematically analyzes the non-physical behavior of free energy moment generating functions in log-correlated models and predicts non-Gaussian free energy fluctuations in integrable cases.

Findings

Identification of the termination point transition as cause of non-physical behavior.

Prediction of non-trivial free energy cumulants indicating non-Gaussian fluctuations.

Numerical tests supporting the theoretical predictions.

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.