Erdoes Group

Mathematics of Disordered Quantum Systems and Matrices

How do energy levels of large quantum systems behave? What do the eigenvalues of a typical large matrix look like? Surprisingly, these two very different questions have the same answer!

Large complex systems tend to develop universal patterns that often represent their essential characteristics. A pioneering vision of Eugene 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. Erdos’ group took up the challenge to verify Wigner’s vision with full mathematical rigor as well as to understand the underlying mechanism. Starting from the simplest model, a large random matrix with independent identically distributed entries, they are now able to deal with arbitrary distributions and even matrices with correlated entries. The mathematical ideas and tools developed along the way will extend the scope of random matrix theory and are likely to be used in its many applications beyond quantum physics such as wireless communications and statistics.

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Current Projects

Self-consistent resolvent equation and application in random matrices | Next order correction in the form factor for Wigner matrices | Local spectral universality for random band matrices | Spectral statistics of random matrices with correlated entries | Quantum spin glasses


Bao Z, Erdös L, Schnelli K. 2021. Equipartition principle for Wigner matrices. Forum of Mathematics, Sigma. 9, e44. View

Virosztek D. 2021. The metric property of the quantum Jensen-Shannon divergence. Advances in Mathematics. 380(3), 107595. View

Cipolloni G, Erdös L, Schröder DJ. 2021. Fluctuation around the circular law for random matrices with real entries. Electronic Journal of Probability. 26, 24. View

Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. IST Austria. View

Pitrik J, Virosztek D. 2021. A divergence center interpretation of general symmetric Kubo-Ando means, and related weighted multivariate operator means. Linear Algebra and its Applications. 609, 203–217. View

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since 2013 Professor, IST Austria
2003 – 2013 Chair of Applied Mathematics (C4/W3), Ludwig Maximilian University of Munich, Germany
1998 – 2003 Assistant, Associate, Full Professor, Georgia Institute of Technology, Atlanta, USA
1995 – 1998 Courant Instructor/Assistant Professor, Courant Institute, New York University, USA
1994 – 1995 Postdoc, ETH Zurich, Switzerland
1994 PhD, Princeton University, USA

Selected Distinctions

ISI Highly Cited Researcher
2017 Leonard Eisenbud Prize
2016 Foreign Member, Hungarian Academy of Sciences
2015 Corresponding Member, Austrian Academy of Sciences (ÖAW)
2015 Member, Academia Europaea
2014 Invited Speaker, ICM
2013 ERC Advanced Grant
2007 – 2016 Participant of SFB TR12, Symmetries and Universality
1999 – 2005 NSF Grants
1993 – 1994 Alfred P. Sloan Foundation Dissertation Fellowship

Additional Information

Download CV
View Laszlo Erdös’ website

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