Extremes of log-correlated random fields and the Riemann zeta function, and some asymptotic results for various estimators in statistics

In this thesis, we study the extreme values of certain log-correlated random fields that are Gaussian (the scale-inhomogeneous Gaussian free field and the time-inhomogeneous branching random walk) or approximatively Gaussian (the log-modulus of the Riemann zeta function on the critical line and a randomized toy model of it), as well as asymptotic properties of various estimators in statistics. Apart from the introduction and conclusion, the thesis is divided in three parts, each containing three articles.  The first part contains three articles on log-correlated Gaussian fields. The first article shows the first order convergence of the maximum and the number of high points for the scale-inhomogeneous Gaussian free field on its full domain. The second article uses the results from the first article to show that the limiting law of the Gibbs measure is a Ruelle probability cascade with a certain number of effective scales (a tree of Poisson-Dirichlet processes). The third article shows the tightness of the recentered maximum for the time-inhomogeneous branching random walk.  The second part contains three articles on the Riemann zeta function. The first article shows that, at low temperature, the limiting law of the Gibbs measure for a randomized toy model of the log-modulus of zeta on the critical line is a Poisson-Dirichlet process. The second article deals with the open problem of the tightness of the recentered maximum for this toy model on an interval of length $O(1)$. We simplify the problem by showing that the continuous maximum is at the order of constant away from a discrete maximum over $O(\log T \sqrt{\log \log T})$ points. The third article shows the first order of convergence of the maximum and the free energy for the log-modulus of the Riemann zeta function on short intervals of length $O(\log^{\theta} T)$, $\theta > -1$, on the critical line.  The third part contains three articles treating various topics in asymptotic statistics. The first article shows the complete monotonicity of multinomial probabilities and opens the door to the study of the asymptotic properties of Bernstein estimators on the simplex. The second article shows a uniform law of large numbers for sums containing terms that ``blow up''.  The third article finds the limiting law of a modified score statistic when we test a given member of the exponential power distribution family against the family of asymmetric power distributions.  The thesis contains nine articles of which seven are already published in peer-reviewed journals. All the information is gathered on my personal website : https://sites.google.com/site/fouimet26/research.

Keyphrases: extreme value theory, Gaussian free field, branching random walk, inhomogeneous environment, Riemann zeta function, Gibbs measure, Ghirlanda-Guerra identities, ultrametricity, large deviations, asymptotic statistics, complete monotonicity, multinomial probabilities, Bernstein estimators, uniform law of large numbers, Laplace distribution, goodness-of-fit tests, probability, statistics, champs log-corrélés, log-correlated fields, mathematics, Gaussian fields