Periodic Reporting for period 4 - RandMat (Spectral Statistics of Structured Random Matrices)
Berichtszeitraum: 2021-07-01 bis 2022-09-30
The purpose of this project is a better mathematical understanding of certain classes of large random matrices. Up to very recently, random matrix theory has been mainly focused on mean-field models with independent entries. In this project I instead consider random matrices that incorporate some nontrivial structure. I focus on two types of structured random matrices that arise naturally in important applications and lead to a rich mathematical behaviour: (1) random graphs with fixed degrees, such as random regular graphs, and (2) random band matrices, which constitute a good model of disordered quantum Hamiltonians.
The goals are strongly motivated by the applications to spectral graph theory and quantum chaos for (1) and to the physics of conductance in disordered media for (2). Specifically, I work in the following directions. First, derive precise bounds on the locations of the extremal eigenvalues and the spectral gap, ultimately obtaining their limiting distributions. Second, characterize the spectral statistics in the bulk of the spectrum, using both eigenvalue correlation functions on small scales and linear eigenvalue statistics on intermediate mesoscopic scales. Third, prove the delocalization of eigenvectors and derive the distribution of their components. These results will address several of the most important questions about the structured random matrices (1) and (2), such as expansion properties of random graphs, hallmarks of quantum chaos in random regular graphs, crossovers in the eigenvalue statistics of disordered conductors, and quantum diffusion.
To achieve these goals I combine tools introduced in my previous work, such as local resampling of graphs and subdiagram resummation techniques, and in addition develop novel, robust techniques to address the more challenging goals. I expect the output of this project to contribute significantly to the understanding of structured random matrices.
The goals are strongly motivated by the applications to spectral graph theory and quantum chaos for (1) and to the physics of conductance in disordered media for (2). Specifically, I work in the following directions. First, derive precise bounds on the locations of the extremal eigenvalues and the spectral gap, ultimately obtaining their limiting distributions. Second, characterize the spectral statistics in the bulk of the spectrum, using both eigenvalue correlation functions on small scales and linear eigenvalue statistics on intermediate mesoscopic scales. Third, prove the delocalization of eigenvectors and derive the distribution of their components. These results will address several of the most important questions about the structured random matrices (1) and (2), such as expansion properties of random graphs, hallmarks of quantum chaos in random regular graphs, crossovers in the eigenvalue statistics of disordered conductors, and quantum diffusion.
To achieve these goals I combine tools introduced in my previous work, such as local resampling of graphs and subdiagram resummation techniques, and in addition develop novel, robust techniques to address the more challenging goals. I expect the output of this project to contribute significantly to the understanding of structured random matrices.
The first main area of research is the statistics of eigenvalues and eigenvectors of sparse random matrices, such as the adjacency matrix of the Erdös-Rényi graph. We pursued a major programme of analysing the spectral phase diagram of the Erdös-Rényi graph. We established the coexistence of a fully delocalized and a semilocalized phase. Moreover, we proved that a subregion of the semilocalized phase is in fact fully localized. As a consequence, we established the existence of a mobility edge for the Erdös-Rényi graph, separating the fully delocalized phase from a fully localized phase. Moreover we established the complete delocalization of the eigenvectors of the Laplacian matrix of the Erdös-Rényi graph and uncovered the fluctuations of its largest and smallest eigenvalues, in particular obtaining the law of its spectral gap. Finally, we give a precise expansion of the eigenvalue density correlations for mean-field random matrices on mesoscopic scales, which uncovers new non-universal phenomena and provides a continuous bridge between known results on macroscopic and microscopic spectral scales.
The second area of research is on the limiting behaviour of the interacting quantum Bose at positive temperature. We showed that such an interacting Bose gas is asymptotically described by a Euclidean field theory in dimensions up to 3. We also analysed the corresponding time-dependent problem in dimension one.
The second area of research is on the limiting behaviour of the interacting quantum Bose at positive temperature. We showed that such an interacting Bose gas is asymptotically described by a Euclidean field theory in dimensions up to 3. We also analysed the corresponding time-dependent problem in dimension one.
The project is expected to yield new insights into the distribution of eigenvalues and eigenvectors of random graphs and random band matrices. In particular, we expect a significant advance in the understanding of eigenvalues and eigenvectors of Erdos-Renyi graphs, stochastic block models, and random regular graphs. In addition, we expect new results on the mesoscopic eigenvalue correlations of mean-field and band matrices.