Research project is (co) funded by the Slovenian Research Agency.
UL Member: Faculty of Mathematics and Physics
Code: J1-70033
Project: Random matrices, graphs, and groups
Period: 1. 3. 2026 - 28. 2. 2029
Range per year: 1,89 FTE, category: B
Head: Roman Bessonov
Research activity: Natural sciences and mathematics
Project description:
The random matrix theory is used traditionally to describe complex systems with large number of parameters, when statistical properties of the system represent the main interest. The objectives of the project lie at the edge of spectral theory, quantum ergodicity, and group theory. One of key features of our approach will be the quantitative study of spectral properties of random matrices on the level of estimates of local eigenvalue statistics (universality problems) and spectral gaps for Schreier graphs generated by random groups (strong convergence problems). Here, the quantitative study assumes new accurate estimates of various statistics "in finite volume", i.e., when the size of n x n random matrices under consideration is large enough but fixed. Among the main analytic tools for getting these estimates we will use the entropy function of a measure - a new powerful instrument introduced in 2017 by project leader and S. Denisov. The entropy function is especially powerful in the study of problems of spectral theory of self-adjoint differential and finite difference operators such as Dirac systems, Krein strings, and Jacobi matrices. In the project, we will investigate the spectral properties on Laplacians on some special structured graphs - the antitrees, whose theory, in some sense, is close to the theory of orthogonal polynomials and Jacobi matrices (and includes the latter as a very special case). Our main aim in this part of the project is the proof of weak quantum ergodicity conjecture for antitrees in the spirit of N. Anantharaman and E. Le Masson, who recently proved it for regular graphs.
