Séminaire PMMH - Sergei Nechaev (Center Poncelet, CNRS UMI 2615)

Vendredi 3 avril 11:00-12:00 - Salle réunion PMMH 1

Anomalous statistics of extreme random processes

I am going to discuss two problems of extreme statistics in which unusual (but related to each other) features arise : i) statistics of two-dimensional "stretched" random walks over a semicircle with Kardar-Parisi-Zhang (KPZ) scaling, ii) spectral properties of symmetric three-diagonal random matrices (operators) whose off-diagonal elements can independently take values 0 and 1. The spectral density of the ensemble of such random matrices has a specific fractal (ultrametric) structure and the spectral statistics shares some number-theoretic properties related to the theory of modular forms. The edge of the spectral density of such matrices has a "Lifshitz tail", typical for the one-dimensional Anderson localization. I will show that the "Lifshitz tail" can be considered as the manifestation of KPZ scaling and statistics of large deviations. I expect also to highlight a relationship of the spectral properties of symmetric three-diagonal random matrices with the “phyllotaxis” (manifestation of Fibonacci sequences in nature).