Random Eigenvalues

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Random eigenvalues How does the shape of an object influence the sound that it gives? This question must be as old as music, but its relevance extends far beyond. In modern telecommunications, data analysis, physics and even manufacturing, the study of spectral properties of objects plays a central role. The goal of this project is to study the connection between the geometric structure and spectra in many classical and modern probabilistic models. The simplest model to study spectra is random matrices with independent entries. The principle of universality due to Wigner states that several models should have local behavior that is similar to this setup. Indeed, there are at least twenty fundamentally different models in that physicists believe should exhibit such behavior, but essentially none has been mathematically shown to do so. This is surprising given that this should be the analogue of the central limit theorem in the context of spectral questions. The most famous random matrix ensemble is Dyson beta ensemble, which has a parameter beta, representing symmetry class. In previous work, we have shown that for arbitrary beta, the point process of eigenvalues (both at the edge and in the bulk) have a limit as the dimension tends to infinity. In both cases, the limiting point process are eigenvalues of a random operator. The method of proof is to take scaling limits of the matrices themselves. In later work, for the edge case, we have carried out this program at the edge for all uniformly convex polynomials V, showing that the limit does not depend on V. A related area is random Schrodinger operetors. These are simple models for alloys, with a parameter s representing impurity. When s is large, it is believed that the eigenvalues of these matrices operator follow a Poisson statistics, and that for small s, in high dimensions, they exhibit random matrix-like behavior. In previous work, we have proved that random matrix-like behavior does exist in for such operators, but our proof is limited to very special situations. We would like to extend the proof for other cases. Another model for random Schrodinger operators is the eigenvalues of percolation clusters. Here, very little is known, not even the regularity of the limiting global eigenvalue distribution. It is clear that it always has a dense set of atoms. We have shown that on trees and planar lattices, there is also a continuous part. This question is still not understood for higher dimensions.