It was recently conjectured by Fyodorov, Hiary and Keating that the maximum of the characteristic polynomial on the unit circle of a N × N random unitary matrix sampled from the Haar measure grows like CN/(log N ) 3/4 for some random variable C. In this paper, we verify the leading order of this conjecture, that is, we prove that with high probability the maximum lies in the rangeThe method is based on identifying an approximate branching random walk in the Fourier decomposition of the characteristic polynomial, and uses techniques developed to describe the extremes of branching random walks and of other log-correlated random fields. A key technical input is the asymptotic analysis of Toeplitz determinants with dimension-dependent symbols.The original argument for these asymptotics followed the general idea that the statistical mechanics of 1/f -noise random energy models is governed by a freezing transition. We also prove the conjectured freezing of the free energy for random unitary matrices.

A recent conjecture of Fyodorov-Hiary-Keating states that the maximum of the absolute value of the Riemann zeta function on a typical bounded interval of the critical line is exp{log log T − 3 4 log log log T + O(1)}, for an interval at (large) height T . In this paper, we verify the first two terms in the exponential for a model of the zeta function, which is essentially a randomized Euler product. The critical element of the proof is the identification of an approximate tree structure, present also in the actual zeta function, which allows us to relate the maximum to that of a branching random walk.2000 Mathematics Subject Classification. 60G70, 11M06.

We prove the leading order of a conjecture by Fyodorov, Hiary, and Keating about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as T → ∞ for a set of t ∊ [T, 2T] of measure (1–o(1)) T, we have max|t−u|≤1log|ζ(12+iu)|=(1+o(1))loglogT. © 2018 Wiley Periodicals, Inc.

Abstract. The ε-cover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance ε > 0 from any point. Its leading order in the small ε-regime has been established by Dembo, Peres, Rosen and Zeitouni [Ann. of Math., 160 (2004)]. In this work, the second order correction is identified. The approach relies on a multi-scale refinement of the second moment method, and draws on ideas from the study of the extremes of branching Brownian motion.