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# 5: A sketch of the phase diagram of the disordered XY model. The solid curves (red and blue), corresponding to η = 0 in eq. (2.75), separates the low temperature phase (the bounded region below it) and the high temperature phase. The red dashed curve is the wrong prediction that misses the freezing transition, and continues analytically the red solid curve beyond its validity. In particular, it would imply a re-entrance transition indicated by the dashed arrow. The correct phase separation for T < T * is instead given by the blue straight line, making the re-entrance disappear.

Source publication

This thesis presents original results in two domains of disordered statistical physics: logarithmic correlated Random Energy Models (logREMs), and localization transitions in long-range random matrices. In the first part devoted to logREMs, we show how to characterise their common properties and model--specific data. Then we develop their replica s...

## Citations

... Such a transition is known to be associated with additional log-correction factors and we extend the results of Refs. [28,35] for these corrections to any temperature in the Appendix A not only for log-REMs but also for the standard REM, filling a gap in the literature. ...

... , M be an "ordinary" logREM discrete potential sequence with zero mean and logarithmically decaying correlations. We refer to Ref. [35] (section 2.2.1) for a more precise definition. Here, we will concentrate on a few principle examples, which are all one-dimensional: ...

... A simple argument to understand the above result is the following ( [35], sect. 2.1.4). ...

We address systematically an apparent nonphysical 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 one-dimensional model appearing in decaying Burgers turbulence with log-correlated initial conditions and, finally, the two-dimensional log-correlated random-energy model (logREM) introduced in Cao et al. [Phys. Rev. Lett. 118, 090601 (2017)] based on the two-dimensional Gaussian free field 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 nonphysical vanishing of the moment generating function for some values of parameters is related to the termination point transition (i.e., prefreezing). We study the associated universal log corrections in the frozen phase, both for logREMs and for the standard REM, filling a gap in the literature. For the above mentioned integrable instances of logREMs, we predict the nontrivial free-energy cumulants describing non-Gaussian fluctuations on the top of the Gaussian with extensive variance. Some of the predictions are tested numerically.

... [12,19], and is closely related to the probabilistic construction of LFT using the 2d Gaussian Free Field [20][21][22]. Our previous work [24,25] revisited the connection and made it concrete. In particular we showed that LFT correlation functions provide the Gibbs measure statistics of a ther-mal particle in a 2d Gaussian Free Field plus an attractive deterministic logarithmic potential, a prototypical representative of the logREM class. ...

... This saturation, induced by the presence of atoms dominating the value of large moments, corresponds, in the LFT approach, to a competition between discrete and continuum terms in the operator product expansion (OPE) [30][31][32]. This remarkable link has had multiple consequences, including: the prediction of universal log-corrections associated with the transition [24,25], the extension to arbitrary temperature of recent results [3,4] on the overlap distribution of directed polymers on a Cayley tree, and the resolution of some standing puzzles concerning models such as the log-fractional Brownian motion [27,33]. ...

... -Setting n = −a/β, we obtain a fractional moment of the Gibbs probability weight p a/β β,1 , which undergoes a termination point transition when a = Q/2; this transition has been studied by the LFT mapping in Refs. [24,25]. Via the Girsanov transform, the corresponding moment Z n a = Z −a/β a = e aFa describes positive large deviations of the free energy F a . ...

We study transitions in log-correlated Random Energy Models (logREMs) that are related to the violation of a Seiberg bound in Liouville field theory (LFT): the binding transition and the termination point transition (a.k.a. pre-freezing). By means of LFT-logREM mapping, replica symmetry breaking and traveling-wave equation techniques, we unify both transitions in a two-parameter diagram, which describes the free energy large deviations of logREMs with a deterministic background log potential, or equivalently, the joint moments of the free energy and Gibbs measure in logREMs without background potential. Under the LFT-logREM mapping, the transitions correspond to the competition of discrete and continuous terms in a four-point correlation function. Our results provide a statistical interpretation of a peculiar non-locality of the operator product expansion in LFT. The results are re-derived by a traveling-wave equation calculation, which shows that the features of LFT responsible for the transitions are reproduced in a simple model of diffusion with absorption. We examine also the problem by a replica symmetry breaking analysis. It complements the previous methods and reveals a rich large deviation structure of the free energy of logREMs with a deterministic background log potential. Many results are verified in the integrable circular logREM, by a replica-Coulomb gas integral approach. The related problem of common length (overlap) distribution is also considered. We provide a traveling-wave equation derivation of the LFT predictions announced in a precedent work.

... Such a transition is known to be associated with additional log-correction factors and we extend the results of Refs. [28,35] for these corrections to any temperature in the Appendix A not only for log-REMs but also for the standard REM, filling a gap in the literature. ...

... , M be an "ordinary" logREM discrete potential sequence with zero mean and logarithmically decaying correlations. We refer to Ref. [35] (section 2.2.1) for a more precise definition. Here, we will concentrate on a few principle examples, which are all one-dimensional: ...

... A simple argument to understand the above result is the following ( [35], sect. 2.1.4). ...

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.