# Erdös Group

## Mathematics of Disordered Quantum Systems and Matrices

What do the eigenvalues of large matrices look like? How do energy levels of large quantum systems behave? According to Eugene Wigner's pioneering vision from 1955, these two questions have the same answer and it is universal in a sense that it depends only on the basic symmetry type of the system, but is otherwise independent of the details.

Quantum physics has been a permanent inspiration for mathematical concepts, problems and even methods. Erdos' group studies physical systems where two well-established branches of mathematics meet. Functional analysis and theory of operators are well suited to, in fact essentially developed for, studying quantum mechanics. On the other hand, probability theory becomes essential when the complete physical system is too complex for a deterministic description. In this case, additional degrees of freedom are described only by their statistical properties.

Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, large sums of independent random variables tend to be Gaussian by the central limit theorem. Since quantum systems are described by operators or by large matrices, the natural analogous question is to find universality behind large random matrices. The right concept was found by Eugene Wigner in 1955 who predicted that the local statistics of eigenvalues tend to follow a universal pattern. In fact, he envisioned that the very same statistics describe energy levels of any sufficient chaotic quantum system. This bold conjecture has been verified numerically and experimentally in many situations, but it still defies a rigorous proof for any realistic physical model.

A central example is impurities in metal, described by random Schrödinger operators. Metallic lattices are typically good conductors since their regular structure allows the forming of coherent electronic states that extend throughout the sample. If the regular arrangement of ions is slightly disturbed, one still expects conductance, perhaps a weaker one. However, this fundamental fact has not been established with full mathematical rigor. In contrast, it is proven that the sample becomes an insulator for large disorder (Anderson transition). Local spectral statistics can distinguish between the two regimes; the general belief is that random matrix statistics prevails in the conductance regime.

Historically, the primary motivation for studying random matrices comes from quantum systems. However, random matrix statistics appear ubiquitous in Nature, with applications ranging from statistics to network theory. Our research focuses on developing mathematical tools to understand this fascinating phenomenon and to prove rigorously that it indeed occurs in real-world models.

**Contact**

Laszlo Erdös

Institute of Science and Technology Austria (IST Austria)

Am Campus 1

A – 3400 Klosterneuburg

Phone: +43 (0)2243 9000-5601

Fax: +43 (0)2243 9000 2000

Email: laszlo.erdos@
ist.ac.at

**Assistant**

Alexandra Mally

Phone: +43 (0)2243 9000-1105

E-mail: alexandra.mally@
ist.ac.at

**Team**

- Heikki Oskari Ajanki, Postdoc (supported by ERC)
- Torben Krüger, PhD Student (supported by SFB TR 12 of DFG)
- Christian Sadel, Postdoc (ISTFellow)
- Kevin Schnelli, Postdoc (supported by ERC)
- Dirk Zeindler, Postdoc (ISTFellow)

**Selected Publications**

- Erdös L, Yau H-T. 2012. Universality of local spectral statistics of random matrices. Bull. Amer. Math. Soc. 49, no.3, 377–414.
- Erdös L, Yau H-T, Yin J. 2012. Rigidity of Eigenvalues of Generalized Wigner Matrices. Adv. Math. 229, no. 3, 1435–1515.
- Erdös L, Schlein B, Yau H-T. 2011. Universality of Random Matrices and Local Relaxation Flow. Invent. Math. 185, no.1, 75–119.

**Career**

Since 2013 Professor, IST Austria

2003–2013 Chair of Applied Mathematics (C4/W3), Ludwig-Maximilians University, Munich

1998–2003 Assistant, Associate, Full Professor, Georgia Institute of Technology, Atlanta

1995–1998 Courant Instructor/Assistant Professor, Courant Institute, New York University, New York

1994–1995 Postdoctoral researcher, ETH Zürich

1990–1994 PhD in Mathematics, Princeton University

1985–1990 Diploma in Mathematics, Eotvos Lorand University, Budapest

#### Selected Distinctions

2013 ERC Advanced Grant