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

Statistics of Extremes in Eigenvalue-Counting Staircases

We consider the number N_{θ_{A}}(θ) of eigenvalues e^{iθ_{j}} of a random unitary matrix, drawn from CUE_{β}(N), in the interval θ_{j}∈[θ_{A},θ]. The deviations from its mean, N_{θ_{A}}(θ)-E[N_{θ_{A}}(θ)], form a random process as function of θ. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any β>0. It exhibits combined features of standard counting statistics of fermions (free for β=2 and with Sutherland-type interaction for β≠2) in an interval and extremal statistics of the fractional Brownian motion with Hurst index H=0. The β=2 results are expected to apply to the statistics of zeroes of the Riemann Zeta function.

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.