Periodic Reporting for period 4 - LogCorrelatedFields (Extremes in logarithmically correlated fields)
Okres sprawozdawczy: 2021-07-01 do 2022-06-30
The project is concerned with the study of extremes in a class of random processes known as ``log correlated processes". These processes involve long-range correlations, with the influence of each (dyadic) scale been roughly equal. Such fields occur in many physical models, of which we mention models of quantum gravity in two dimensions, branching random walks, cover times for two dimensional surfaces and graphs, and random matrices. Rapid progress has occurred during the first half of the current decade in the understanding of the Gaussian case, and this project builds on that and attempts to address the non-Gaussian setup, as well as deepen the understanding of the Gaussian setup, with the goal of obtaining a good understanding of the process of extrema. The results have implications in models of mathematical physics, as well as within probability theory.
Significant progress has been achieved in the understanding of the cover time of the sphere by (a thickening of) Brownian motion, and in the related (and easier) problem of cover time of the binary tree by random walk. In particular, the square-root of the cover time has been shown to have fluctuations of order 1 around its mean, and in the case of the tree, the square-root of the cover time, properly normalized and centered, possesses a limit law. Related results, at a lower degree of refinement, were also obtained for the extrema of determinants of random matrices, random surfaces of the Ginzburg-Landau type, and random polymers. In another direction, a study of the stochastic heat equation in dimension greater than or equal to 3 has been initiated, based in part of some of the tools developed in addressing log-correlated fields.
The main output of the project is the development of a set of techniques for handling extrema for processes with long range correlations, beyond the Gaussian case. We essentially completed the study of cover times for two dimensional manifolds and obtained limit laws for the maximum of determinants of random matrices. Maybe more important, the techniques we expect to develop tools (a combination of comparison, decoupling, and analytical) which will provide a method of analysis for a wide class of processes.