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Spectral Statistics of Structured Random Matrices

Periodic Reporting for period 2 - RandMat (Spectral Statistics of Structured Random Matrices)

Reporting period: 2018-07-01 to 2019-12-31

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 main results from this stage of the project are a precise understanding of the behaviour of the extreme eigenvalues of very sparse random graphs. We uncover a new crossover in the distribution of the extreme eigenvalues, which occurs when the mean expected degree is equal to the logarithm of the number of vertices, below which the extreme eigenvalues leave the bulk spectrum to form a cloud of outlier eigenvalues. Above the critical scale for the crossover, we prove that all corresponding eigenvectors are completely delocalized. This result is optimal in the sense that the conclusion is wrong at and below the critical scale. Moreover, we give a full analysis of the behaviour of the eigenvalues through the crossover by establishing rigidity bounds on the locations of all eigenvalues. This result uncovers a sharp transition in the location of the top eigenvalue, which detaches from the asymptotic bulk at a specific value of the expected degree of the vertices. The method used for this analysis relies on a novel combination of nonbacktracking matrices and a tridiagonal representation of the adjacency matrix, which will have several further applications in related problems. 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 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.
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