Random matrices beyond Wigner-Dyson-Mehta

Project Information

RMTBEYOND

Grant agreement ID: 101020331

DOI

10.3030/101020331

EC signature date 18 July 2021

Start date 1 October 2021

End date 30 September 2026

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 912 439,00

EU contribution

€ 1 912 439,00

Coordinated by

INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA
Austria

Periodic Reporting for period 3 - RMTBEYOND (Random matrices beyond Wigner-Dyson-Mehta)

Reporting period: 2024-10-01 to 2026-03-31

Summary of the context and overall objectives of the project

Random matrices, originally invented by E. Wigner in the 1950’s to model energy level statistics of heavy nuclei, play a fundamental role in a broad range of modelling quantum mechanical processes, as well as in statistics, evolutionary biology and neuroscience. Random matrices are known to exhibit spectral universality, meaning that their eigenvalue statistics are largely independent of the details of the ensemble. Our project extends this vision of beyond the traditional eigenvalue gap statistics to cover eigenvectors and expectation values of observables measured in experiments. This requires developing new tools, most importantly the theory of multi-resolvent
local laws which serve as a work-horse for many of our results. The second main focus is to extend the well established hermitian theory to the more general and complicated non-hermitian situations, especially because several applications in neuroscience naturally go beyond the hermitian world. As a theoretical research in mathematics, its direct impact on society is not realistically expected, but a deeper understanding of non-hermitian random matrices plays an important role how and to what extent random matrix models can be used in applied sciences.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

The most important work performed so far is the development of the theory of multi-resolvent local laws for large random matrices. This has been done on two different levels with increasing complexity and precision, the third level is currently in progress. Within the study of hermitian random matrices, so far we used this theory to establish functional central limit theorem. More importantly, using Girko’s hermitisation trick, we could transfer information on hermitian matrices to non-hermitian ones. On one hand we obtained very accurate bounds on the extremal eigenvalues and we proved sharp lower and upper bounds on the eigenvector overlaps. As an unexpected byproduct, this gave rise to new bounds on the eigenvalue condition number which are used in numerical linear algebra to estimate to smoothing (regularising) effect of the noise in ill-conditioned problems.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

The extremal eigenvalue statistics and the eigenvector overlap bounds within the non-hermitian theory of random matrices go well beyond the state of the art. By the end of the project period, on one hand we expect
to prove the full universality of the extremal non-hermitian eigenvalues. On the other hand we expect to extend the complete theory on eigenvector overlaps, on eigenstate thermalisation phenomenon and on functional central limit theorems from Wigner matrices to very general classes of mean field random matrices, including matrix elements with variable variances and even correlations.