Maxima of log-correlated fields: some recent developments*

Abstract

We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmically-correlated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov and Keating concerning the extreme value statistics, moments of moments, connections to Gaussian multiplicative chaos, and explicit formulae derived from the theory of symmetric functions.

Hierarchical Structure in the Trace Formula

“…It is worth remarking that the hierarchical structure described in Section 3 can be proved in the case of the explicit formula (5) for the Riemann zeta-function; see [10] for an overview. Moreover, it can be proved for the corresponding formula in Random Matrix Theory.…”

“…The timescale T H is often referred to as the Heisenberg time. For instance, the moments of log ζ (1/2 + iE) can be computed by analysing the sum on the right hand side of (5) restricted smoothly to r log p log (E/2π) [10,52]. And the restricted sum does approximate log ζ (1/2 + iE) point-wise for a set of values of E of measure close to one.…”

“…This is also the case for the Riemann zeta-function and for characteristic polynomials of random matrices. See [10] for a review. My purpose here is examine one particular aspect of the theory of logarithmically correlated Gaussian random fields in the context of the semiclassical theory of quantum chaotic systems.…”

“…where the convergence properties of the sum are subtle [37], but may be understood in a distributional sense. It can be shown that when n → ∞, the value distribution of log |P (A, θ)|/ 1 2 log n converges to a standard normal and that at different values of θ, log |P (A, θ)| is log-correlated [10]:…”

Large Deviation Estimates of Selberg's Central Limit Theorem and Applications

“…A hybrid regime interpolating between IID and log-correlated statistics was also proposed in [ADH21]. For more on recent developments in extreme values of log-correlated processes, see for example [BK22]. Theorem 1.1 can also be applied to improve current bounds for the moments of ζ in short intervals.…”