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

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.

Figure 1

Left: large deviation rate function of the free energy ${\mathcal{F}}_B$ [Eq. (8)]. The variance ${\sigma }^2$ refers to that of ${\mathcal{F}}_B$. Right: the leading exponent of the moment generating function of ${\mathcal{F}}_B$ [Eq. (A1)]. The two functions are related by Legendre transform.

Figure 2

(a) Circular model: numerical calculation of the cumulants of the minimum of the fBm0 bridge. For the variance, $k=2$, we define $\overline{v_{min}^2}:=\overline{B_{min}^2}-{\overline{V_1^2}}^c$, i.e., the extensive Gaussian contribution is subtracted from the raw data. Higher cumulants are not affected by the Gaussian contribution, and are plotted as such: $\overline{v_{min}^k}:=\overline{B_{min}^k}$, $k>2$. A quadratic ansatz $a_1+a_1/lnM+a_2/{(lnM)}^2$ is used to account for the finite size scaling and extrapolate the $M\to \infty$ value from $M=2^8\to ,2^{23}$. The extrapolation is then compared to the predictions (22), plotted as markers at left. (b) Interval model: the same for the fBm0 on the interval $[0,1]$ (pinned interval model $M=2^{10},\cdots ,2^{23}$), compared with Eq. (31).

Figure 3

Numerical test of the predictions of the termination point transition, at the freezing transition $\beta =1$ (a), and in the frozen $\beta >1$ phase [(b), $\beta =2.5]$. The straight lines represent the log-correction predictions in Eqs. (A2) and (A3). Markers represent the numerical data obtained in the circular model (see [30] for simulation method), with sizes $M=2^8,\cdots ,2^{24}$ (translation invariance is used to enhance the statistics), with various values of $t$, above and below the critical value $t_c=Q/2$. The leading behavior $\mathrm{\Delta }lnM$ is extracted, according to Eq. (11). The analog test for $\beta <1$ can be found in Ref. [43], p. 114.

Figure 4

Low-order cumulant corrections to the minimum distribution of 2D logREM [Eq. (B6)]. (a) The variance correction $C_2$ in the whole region $\mathcal{R}$ [Eq. (B10)] (delimited by the yellow triangle). The behavior of the minimum variance in the short-distance bound phase (blue polygon on top right) and the large-distance escaping phase (red triangle on bottom left) the 2D logREM is also indicated [see Eqs. (B15) and (B14)]. (b) 3D version of (a) inside the region $\mathcal{R}$. (c) First cumulants $C_k$ along the line $a_1=a_2=a\in (1/2,1)$. $C_1$ is defined in the same way as Eq. (B6). $C_{1,3}$ diverge negatively when approaching any boundary of $\mathcal{R}$, while $C_{2,4}$ diverge positively (negatively) when undergoing a binding (unbinding) transition, respectively.