## "Random matrices, universality and disordered quantum systems"

**From** 2014-03-01 **to** 2019-02-28, ongoing project

## Project details

### Total cost:

EUR 1 754 717

### EU contribution:

EUR 1 754 717

### Coordinated in:

Austria
## Objective

"Large complex systems tend to develop universal patterns that often represent their essential characteristics. A pioneering vision of E. Wigner was that the distribution of the gaps between energy levels of complicated quantum systems depends only on the basic symmetry of the model and is otherwise independent of the physical details. This thesis has never been rigorously proved

for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Wigner also discovered that the statistics of gaps can be modelled by eigenvalues of large random matrices. Thus the natural questions, “How do energy levels behave?” and “What do eigenvalues of a typical large matrix look like?”, have surprisingly the same answer! This project will develop new tools to respond to the two main challenges that Wigner’s vision poses for mathematics.

First, prove that a large class of natural systems exhibits universality. The simplest model is the

random matrix itself, for which the original conjecture, posed almost fifty years ago, has recently been solved by the PI and coworkers. This breakthrough opens up the route to the universality for more realistic physical systems such as random band matrices, matrices with correlated entries and random Schrödinger operators. Second, eigenvalue statistics will be used to detect the basic dichotomy of disordered quantum systems, the Anderson metal-insulator transition. Third, describe the properties of the strongly correlated eigenvalues viewed as a point process.

Although this process appears as ubiquitous in Nature as the Poisson process or the Brownian motion, we still know only very little about it. Due to the very strong correlations, the standard toolboxes of probability theory and statistical mechanics are not applicable. The main impact of the

project is a conceptual understanding of spectral universality and the development of robust analytical tools to study strongly correlated systems."

## Related information

## Host Institution

Institute of Science and Technology Austria

Austria

**EU contribution: **EUR 1 754 717

Am Campus 1

3400 Klosterneuburg

Austria

## Beneficiaries

Institute of Science and Technology Austria

Austria

**EU contribution: **EUR 1 754 717

Am Campus 1

3400 Klosterneuburg

Austria

**Record Number**: 109988 / **Last updated on**: 2015-03-12
**Last updated on** 2015-03-12