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Random matrices, universality and disordered quantum systems

Final Report Summary - RANMAT (Random matrices, universality and disordered quantum systems)

The main goal of the project was to investigate the scope of E. Wigner's revolutionary vision on the universality of spectral statistics of large random matrices well beyond the original model of Wigner matrices.This vision, formalised mathematically by Dyson and Mehta in the 60's, predicts that the local eigenvalue statistics is insensitive to all details of the matrix ensemble, it depends only on the symmetry type. Wigner's prediction has been extensively verified by numerical simulations and even in a wide range of physical experiments, still the rigorous mathematical proof for the simplest i.i.d. case has only been achieved about ten years ago. The outcome of the completed project is to give rigorous proofs of Wigner's vision for very general matrix ensembles, including matrices with nontrivial correlation structure, band matrices and even non-Hermitian matrices in the edge regime. Along the way we have developed a systematic way to prove local laws for an even bigger class of ensembles that include polynomials and certain rational functions in i.i.d. random matrices, as well as random addition of Hermitian matrices. The methods developed in the project, especially the analysis of block matrices, have been tested in direct applications concerning (i) the capacity of MIMO channels in telecommunication; (ii) long time decay rate to equilibrium in a widely used model in neural networks and (iii) a revision of Beenakker's theory of scattering in quantum dots.The project has supported four PhD students and seven postdocs and it has made
IST Austria a major scientific center in random matrices.