Large deviations and continuity estimates for the derivative of a random model of log⁡∣ζ∣\log |\zeta| on the critical line

In this paper, we study the random field \begin{equation*} X(h) \circeq \sum_{p \leq T} \frac{\text{Re}(U_p \, p^{-i h})}{p^{1/2}}, \quad h\in [0,1], \end{equation*} where $(U_p, \, p ~\text{primes})$ is an i.i.d. sequence of uniform random variables on the unit circle in $\mathbb{C}$. Harper (2013) showed that $(X(h), \, h\in (0,1))$ is a good model for the large values of $(\log |\zeta(\frac{1}{2} + i (T + h))|, \, h\in [0,1])$ when $T$ is large, if we assume the Riemann hypothesis. The asymptotics of the maximum were found in Arguin, Belius & Harper (2017) up to the second order, but the tightness of the recentered maximum is still an open problem. As a first step, we provide large deviation estimates and continuity estimates for the field's derivative $X'(h)$. The main result shows that, with probability arbitrarily close to $1$, \begin{equation*} \max_{h\in [0,1]} X(h) - \max_{h\in \mathcal{S}} X(h) = O(1), \end{equation*} where $\mathcal{S}$ a discrete set containing $O(\log T \sqrt{\log \log T})$ points.Comment: 7 pages, 0 figur

Extremes of the two-dimensional Gaussian free field with scale-dependent variance

In this paper, we study a random field constructed from the two-dimensional Gaussian free field (GFF) by modifying the variance along the scales in the neighborhood of each point. The construction can be seen as a local martingale transform and is akin to the time-inhomogeneous branching random walk. In the case where the variance takes finitely many values, we compute the first order of the maximum and the log-number of high points. These quantities were obtained by Bolthausen, Deuschel and Giacomin (2001) and Daviaud (2006) when the variance is constant on all scales. The proof relies on a truncated second moment method proposed by Kistler (2015), which streamlines the proof of the previous results. We also discuss possible extensions of the construction to the continuous GFF.Comment: 30 pages, 4 figures. The argument in Lemma 3.1 and 3.4 was revised. Lemma A.4, A.5 and A.6 were added for this reason. Other typos were corrected throughout the article. The proof of Lemma A.1 and A.3 was simplifie

Poisson-Dirichlet statistics for the extremes of a log-correlated Gaussian field

We study the statistics of the extremes of a discrete Gaussian field with logarithmic correlations at the level of the Gibbs measure. The model is defined on the periodic interval $[0,1]$, and its correlation structure is nonhierarchical. It is based on a model introduced by Bacry and Muzy [Comm. Math. Phys. 236 (2003) 449-475] (see also Barral and Mandelbrot [Probab. Theory Related Fields 124 (2002) 409-430]), and is similar to the logarithmic Random Energy Model studied by Carpentier and Le Doussal [Phys. Rev. E (3) 63 (2001) 026110] and more recently by Fyodorov and Bouchaud [J. Phys. A 41 (2008) 372001]. At low temperature, it is shown that the normalized covariance of two points sampled from the Gibbs measure is either $0$ or $1$. This is used to prove that the joint distribution of the Gibbs weights converges in a suitable sense to that of a Poisson-Dirichlet variable. In particular, this proves a conjecture of Carpentier and Le Doussal that the statistics of the extremes of the log-correlated field behave as those of i.i.d. Gaussian variables and of branching Brownian motion at the level of the Gibbs measure. The method of proof is robust and is adaptable to other log-correlated Gaussian fields.Comment: Published in at http://dx.doi.org/10.1214/13-AAP952 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org