Random Matrices and Disordered Systems

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In the recent work, the universality conjecture for the local spectral statistics was solved for both Winger ensembles and beta ensembles. The goal of this proposal is to extend this understanding to large classes of random matrices. The objective is to understand the grand vision of Wigner, which asserts that local spectral statistics of large correlated quantum systems are universal. Specifically, the following projects are proposed: 1. The spectral properties of Wigner matrices with random potentials. These are toy models for random Schrodinger operators on lattice. 2. The statistics of both eigenvalues and eigenvectors in band and sparse matrices. In addition to these two projects, local statistics at the level of individual eigenvalues will also be studied. More specifically, the intention is to look into the following problems: 3. The single gap universality, i.e., identify the probability distribution of a single gap in random matrices. 4. The universality of local statistics at a fixed energy. 5. Identify the Gaussian fluctuations of a single eigenvalue in Wigner matrices, i.e., extending Gustavsson's result for GUE to all Wigner matrices. 6. Establish the edge universality via Dyson's Brownian motion. These problems will be approached using the Helffer and Sjostrand representation of correlation functions and the parabolic regularity in the work of Caffarelli, Chan and Vasseur. Eugene Wigner's grand vision asserts that local spectral statistics of large correlated quantum systems are universal. In our recent work, we solved the universality conjecture for the local spectral statistics for both Wigner ensembles and beta ensembles. The goal of this proposal is to extend these findings to large classes of random matrices and expand our understanding of Wigner's vision.