Random matrices, universality and disordered quantum systems

Project Period: 2014-03-01 – 2019-08-31
Externally Funded
Acronym
RANMAT
Principal Investigator
László Erdös
Department(s)
Erdős Group
Description
"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."
Grant Number
338804
Funding Organisation
FP7_ERC

34 Publications

2018 | Journal Article | IST-REx-ID: 181   OA
Power law decay for systems of randomly coupled differential equations
L. Erdös, T.H. Krüger, D.T. Renfrew, SIAM Journal on Mathematical Analysis 50 (2018) 3271–3290.
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2017 | Journal Article | IST-REx-ID: 1337   OA
Universality for general Wigner-type matrices
O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields 169 (2017) 667–727.
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2017 | Journal Article | IST-REx-ID: 1010
Local law for random Gram matrices
J. Alt, L. Erdös, T.H. Krüger, Electronic Journal of Probability 22 (2017) 25.
View | Files available | DOI | arXiv
 
2016 | Journal Article | IST-REx-ID: 1157   OA
Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population
J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.
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2018 | Journal Article | IST-REx-ID: 5971   OA
Bounds on the norm of Wigner-type random matrices
L. Erdös, P. Mühlbacher, Random Matrices: Theory and Applications (2018).
View | DOI | Download (ext.) | arXiv
 
2017 | Journal Article | IST-REx-ID: 615   OA
Universality for random matrix flows with time dependent density
L. Erdös, K. Schnelli, Annales de l’institut Henri Poincare (B) Probability and Statistics 53 (2017) 1606–1656.
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2014 | Journal Article | IST-REx-ID: 2225   OA
Isotropic local laws for sample covariance and generalized Wigner matrices
A. Bloemendal, L. Erdös, A. Knowles, H. Yau, J. Yin, Electronic Journal of Probability 19 (2014) 33.
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2016 | Journal Article | IST-REx-ID: 1280   OA
Fixed energy universality for generalized wigner matrices
P. Bourgade, L. Erdös, H. Yau, J. Yin, Communications on Pure and Applied Mathematics 69 (2016) 1815–1881.
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2014 | Conference Paper | IST-REx-ID: 1507   OA
Random matrices, log-gases and Hölder regularity
L. Erdös, in:, Kyung Moon SA Co. Ltd., 2014, pp. 214–236.
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2016 | Journal Article | IST-REx-ID: 1881   OA
Extremal eigenvalues and eigenvectors of deformed Wigner matrices
J. Lee, K. Schnelli, Probability Theory and Related Fields 164 (2016) 165–241.
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2017 | Journal Article | IST-REx-ID: 483   OA
Universality for a class of random band matrices
P. Bourgade, L. Erdös, H. Yau, J. Yin, Advances in Theoretical and Mathematical Physics 21 (2017) 739–800.
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2015 | Journal Article | IST-REx-ID: 1677
The local semicircle law for random matrices with a fourfold symmetry
J. Alt, Journal of Mathematical Physics 56 (2015).
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2017 | Journal Article | IST-REx-ID: 1207   OA
Local law of addition of random matrices on optimal scale
Z. Bao, L. Erdös, K. Schnelli, Communications in Mathematical Physics 349 (2017) 947–990.
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2016 | Journal Article | IST-REx-ID: 1219   OA
Bulk universality for deformed wigner matrices
J. Lee, K. Schnelli, B. Stetler, H. Yau, Annals of Probability 44 (2016) 2349–2425.
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2018 | Journal Article | IST-REx-ID: 429   OA
Stability of the matrix Dyson equation and random matrices with correlations
O.H. Ajanki, L. Erdös, T.H. Krüger, Probability Theory and Related Fields (2018).
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2018 | Journal Article | IST-REx-ID: 690   OA
Local law and Tracy–Widom limit for sparse random matrices
J. Lee, K. Schnelli, Probability Theory and Related Fields 171 (2018).
View | DOI | Download (ext.) | arXiv
 
2018 | Journal Article | IST-REx-ID: 70   OA
Transition to shocks in TASEP and decoupling of last passage times
P. Nejjar, Latin American Journal of Probability and Mathematical Statistics 15 (2018) 1311–1334.
View | Files available | DOI | arXiv
 
2017 | Journal Article | IST-REx-ID: 1144
Fluctuations of functions of Wigner matrices
L. Erdös, D.J. Schröder, Electronic Communications in Probability 21 (2017).
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2018 | Journal Article | IST-REx-ID: 556   OA
The free boundary Schur process and applications
D. Betea, J. Bouttier, P. Nejjar, M. Vuletic, Annales Henri Poincare 19 (2018) 3663–3742.
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2018 | Journal Article | IST-REx-ID: 1012   OA
Fluctuations of rectangular young diagrams of interlacing wigner eigenvalues
L. Erdös, D.J. Schröder, International Mathematics Research Notices 2018 (2018) 3255–3298.
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2019 | Journal Article | IST-REx-ID: 6240
Location of the spectrum of Kronecker random matrices
J. Alt, L. Erdös, T.H. Krüger, Y. Nemish, Annales de l’institut Henri Poincare 55 (2019) 661–696.
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2016 | Journal Article | IST-REx-ID: 1434   OA
Local stability of the free additive convolution
Z. Bao, L. Erdös, K. Schnelli, Journal of Functional Analysis 271 (2016) 672–719.
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2016 | Journal Article | IST-REx-ID: 1489   OA
Local spectral statistics of Gaussian matrices with correlated entries
O.H. Ajanki, L. Erdös, T.H. Krüger, Journal of Statistical Physics 163 (2016) 280–302.
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2017 | Journal Article | IST-REx-ID: 1528   OA
Delocalization for a class of random block band matrices
Z. Bao, L. Erdös, Probability Theory and Related Fields 167 (2017) 673–776.
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2015 | Journal Article | IST-REx-ID: 1864   OA
The Altshuler–Shklovskii formulas for random band matrices II: The general case
L. Erdös, A. Knowles, Annales Henri Poincare 16 (2015) 709–799.
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2014 | Journal Article | IST-REx-ID: 2019   OA
Phase transition in the density of states of quantum spin glasses
L. Erdös, D.J. Schröder, Mathematical Physics, Analysis and Geometry 17 (2014) 441–464.
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2017 | Journal Article | IST-REx-ID: 447   OA
Fluctuations of the competition interface in presence of shocks
P. Ferrari, P. Nejjar, Revista Latino-Americana de Probabilidade e Estatística 9 (2017) 299–325.
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2017 | Book | IST-REx-ID: 567
A dynamical approach to random matrix theory
L. Erdös, H. Yau, A Dynamical Approach to Random Matrix Theory, American Mathematical Society, 2017.
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2017 | Journal Article | IST-REx-ID: 721   OA
Singularities of solutions to quadratic vector equations on the complex upper half plane
O.H. Ajanki, T.H. Krüger, L. Erdös, Communications on Pure and Applied Mathematics 70 (2017) 1672–1705.
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2017 | Journal Article | IST-REx-ID: 733   OA
Convergence rate for spectral distribution of addition of random matrices
Z. Bao, L. Erdös, K. Schnelli, Advances in Mathematics 319 (2017) 251–291.
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2019 | Journal Article | IST-REx-ID: 6511   OA
Local single ring theorem on optimal scale
Z. Bao, L. Erdös, K. Schnelli, Annals of Probability 47 (2019) 1270–1334.
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2017 | Journal Article | IST-REx-ID: 550
Singularities of the density of states of random Gram matrices
J. Alt, Electronic Communications in Probability 22 (2017).
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2018 | Thesis | IST-REx-ID: 149
Dyson equation and eigenvalue statistics of random matrices
J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, IST Austria, 2018.
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2019 | Thesis | IST-REx-ID: 6179
From Dyson to Pearcey: Universal statistics in random matrix theory
D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix Theory, IST Austria, 2019.
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