Freezing transitions and extreme values: random matrix theory, ζ (1/2 + it) and disordered landscapes

Moments of moments of the characteristic polynomials of random orthogonal and symplectic matrices

By using asymptotics of Toeplitz+Hankel determinants, we establish formulae for the asymptotics of the moments of the moments of the characteristic polynomials of random orthogonal and symplectic matrices, as the matrix size tends to infinity. Our results are analogous to those that Fahs obtained for random unitary matrices in (Fahs B. 2021 Communications in Mathematical Physics 383 , 685–730. (doi: 10.1007/s00220-021-03943-0 )). A key feature of the formulae we derive is that the phase transitions in the moments of moments are seen to depend on the symmetry group in question in a significant way.

On the Critical-Subcritical Moments of Moments of Random Characteristic Polynomials: A GMC Perspective

We study the 'critical moments' of subcritical Gaussian multiplicative chaos (GMCs) in dimensions d ≤ 2 . In particular, we establish a fully explicit formula for the leading order asymptotics, which is closely related to large deviation results for GMCs and demonstrates a similar universality feature. We conjecture that our result correctly describes the behaviour of analogous moments of moments of random matrices, or more generally structures which are asymptotically Gaussian and log-correlated in the entire mesoscopic scale. This is verified for an integer case in the setting of circular unitary ensemble, extending and strengthening the results of Claeys et al. and Fahs to higher-order moments.

Moments of Moments and Branching Random Walks

We calculate, for a branching random walk X n ( l ) to a leaf l at depth n on a binary tree, the positive integer moments of the random variable 1 2 n ∑ l = 1 2 n e 2 β X n ( l ) , for β ∈ R . We obtain explicit formulae for the first few moments for finite n. In the limit n → ∞ , our expression coincides with recent conjectures and results concerning the moments of moments of characteristic polynomials of random unitary matrices, supporting the idea that these two problems, which both fall into the class of logarithmically correlated Gaussian random fields, are related to each other.

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.

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)PRLTAO0031-900710.1103/PhysRevLett.118.090601] 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.

Localized growth and branching random walks with time correlations

We generalize a model of growth over a disordered environment, to a large class of Itō processes. In particular, we study how the microscopic properties of the noise influence the macroscopic growth rate. The present model can account for growth processes in large dimensions and provides a bed to understand better the tradeoff between exploration and exploitation. An additional mapping to the Schrödinger equation readily provides a set of disorders for which this model can be solved exactly. This mean-field approach exhibits interesting features, such as a freezing transition and an optimal point of growth, which can be studied in detail, and gives yet another explanation for the occurrence of the Zipf law in complex, well-connected systems.

Liouville Field Theory and Log-Correlated Random Energy Models

An exact mapping is established between the c≥25 Liouville field theory (LFT) and the Gibbs measure statistics of a thermal particle in a 2D Gaussian free field plus a logarithmic confining potential. The probability distribution of the position of the minimum of the energy landscape is obtained exactly by combining the conformal bootstrap and one-step replica symmetry-breaking methods. Operator product expansions in the LFT allow us to unveil novel universal behaviors of the log-correlated random energy class. High-precision numerical tests are given.

Quantum graphs whose spectra mimic the zeros of the Riemann zeta function

One of the most famous problems in mathematics is the Riemann hypothesis: that the nontrivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs that have the same oscillating part of the density of states as the Riemann zeros, offering an explanation of the overall minus sign. The smooth part is completely different, and hence also the spectrum, but the graphs pick out the low-lying zeros.