Objectif 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 provedfor 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 therandom 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 theproject is a conceptual understanding of spectral universality and the development of robust analytical tools to study strongly correlated systems. Champ scientifique natural sciencesphysical sciencesquantum physicsnatural sciencesphysical sciencesclassical mechanicsstatistical mechanicsnatural sciencesmathematicsapplied mathematicsstatistics and probability Programme(s) FP7-IDEAS-ERC - Specific programme: "Ideas" implementing the Seventh Framework Programme of the European Community for research, technological development and demonstration activities (2007 to 2013) Thème(s) ERC-AG-PE1 - ERC Advanced Grant - Mathematical foundations Appel à propositions ERC-2013-ADG Voir d’autres projets de cet appel Régime de financement ERC-AG - ERC Advanced Grant Institution d’accueil INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA Contribution de l’UE € 1 754 717,00 Adresse Am Campus 1 3400 Klosterneuburg Autriche Voir sur la carte Région Ostösterreich Niederösterreich Wiener Umland/Nordteil Type d’activité Higher or Secondary Education Establishments Contact administratif Barbara Abraham (Ms.) Chercheur principal Laszlo Erdös (Prof.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée Bénéficiaires (1) Trier par ordre alphabétique Trier par contribution de l’UE Tout développer Tout réduire INSTITUTE OF SCIENCE AND TECHNOLOGY AUSTRIA Autriche Contribution de l’UE € 1 754 717,00 Adresse Am Campus 1 3400 Klosterneuburg Voir sur la carte Région Ostösterreich Niederösterreich Wiener Umland/Nordteil Type d’activité Higher or Secondary Education Establishments Contact administratif Barbara Abraham (Ms.) Chercheur principal Laszlo Erdös (Prof.) Liens Contacter l’organisation Opens in new window Site web Opens in new window Coût total Aucune donnée