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

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 nonlocality of the operator product expansion in LFT. The results are rederived 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.

Figure 1

Diagram for the logREM observable $\overline{p_{\beta ,1}^{a/\beta }{Z}_0^{n+a/\beta }}=\overline{e^{-a{\varphi }_1}{Z}_0^n}$, Eq. (10), obtained by the LFT approach (or by the traveling-wave equation approach, see Sec. 5). The parameter space is drawn with axes $n\beta +a$ and $a$. We indicate the three regimes (U, B, C), which are characterized by different dominating terms in the LFT correlation function, see Sec. 3c. The locus of binding and termination point transition follow from the discussion below Eq. (10). The LFT-logREM mapping is only valid in a half of the diagram with $n<0$, i.e., above the dash-dotted diagonal. The rest of the diagram, drawn with pale colors, is a formal extrapolation, see Eq. (29), and will be corrected by the RSB approach; see Fig. 3.

Figure 2

Illustration of a dominating configuration according to the one-step RSB Ansatz. The horizontal axis represents the space on which is defined the logREM potential. The circles represent the position $j_{\mu }$ of the $n$ replicas. The $n_0$ replicas in the circle are attached to the charge at $j=1$. The other $(n-n_0)$ replicas form groups of size $m$, whose positions are not fixed. Section 4a2 discusses the RSB of the $(n-n_0)$ replicas away from the charge (optimization of $m$); Sec. 4a3 discusses the optimization of $n_0$.

Figure 3

Complete large deviation diagram of the logREM with a charge for any fixed temperature $1/\beta$, obtained by computing the observable $\overline{e^{-a{\varphi }_1}{Z}_0^n}=M^{a^2}\overline{Z_a^n}$ [see Eqs. (9) and (7)] by the RSB approach. The axes are the same as in Fig. 1. The four regimes: unbound (U), critical (C), and bound (B), and log-normal (LN), drawn in different colours, are separated by first- (solid) and second-order (dashed) transitions. The diagonal $n=0$ describes the typical fluctuation of the free energy $F_a$ of the logREM with one charge. The binding transition happens when crossing the tricritical point $(Q/2,Q/2)$ along the diagonal (the full large deviation theory of $F_a$ is discussed in Sec. 4b). The termination point transition of the Gibbs probability weight happens when crossing U-C boundary along the $a$ axis. The region $n>0$ is not accessible in the LFT approach, whose formal prediction misses the log-normal regime; see Fig. 1. We denoted $b=min(1,\beta )$ and $Q=b+b^{-1}$ in agreement with Eq. (4). The triple (T) point at which join the U, LN, and B regime boundaries has coordinates $(n\beta +a=1/b+b/2,a=b/2)$.

Figure 4

Illustration of the large deviation function of the free energy $F_a$ of logREMs with one charge. The three panels correspond to Eqs. (39a), (39d), and (39e), from top to down. The large deviation function $\mathcal{L}(\widehat{y}=F_a/lnM)$ Eq. (39) is drawn in solid curve. Dots on the curve indicate a nonanalyticity.

Figure 5

(a) A sample of the BBM model, with the common length between a pair particles $i$ and $j$ illustrated. The two families of particles after the first branching are drawn in different colours/gray scale. (b) A sample of the drifted BBM. The drifting particle (in black, leftmost, beside the arrow indicating the drifting velocity) happens to be the left-most one; note that this is not generally the case.

Figure 6

Illustration of dominant contributions to the observable $\mathcal{O}$, Eq. (57), in terms of a single diffusive particle with an absorbing wall. The initial and final positions correspond to the log of the Gibbs probability weight at site 1 and the free energy of logREMs without charges, respectively; see Eq. (59) [compare also to Eq. (43)].

Figure 7

Illustration of the observable Eq. (69) and the time variables $\widehat{\mathsf{q}}$ and $\tau =t-\widehat{\mathsf{q}}$. The trajectory of two marked particles are drawn as red bold curves. The particles indicated by pink filled bullets have common length $\ge \widehat{\mathsf{q}}$ with $j,k$, and form the “subtree.” The behavior of its Gibbs measure characterizes the different regimes; see Eq. (81).

Figure 8

The regimes of the generalized multifractal spectrum $f(\widehat{x},\widehat{y})$ of logREM without charge, defined in Eq. (40). The expressions for U, C, and B regimes are given in Eqs. (42). The other first-order interpolation regimes are given in Eqs. (D2). The whole triangular region corresponds via Legendre transform to the triple point $\mathrm{T}$ described in Fig. 3. Note that the unbound and bound regimes occupy the whole semiaxes $\widehat{y}=-Q$ and $\widehat{x}=0$, respectively. The noted points on them are not critical. When $\beta \ge 1$, both the T and U/LN interpolation regions become degenerate.