Random Eigenvalues

物体的形状如何影响它发出的声音？这个问题一定和音乐一样古老，但它的相关性远远超出了音乐。在现代电信、数据分析、物理甚至制造业中，对物体光谱特性的研究起着核心作用。本项目的目标是研究许多经典和现代概率模型中几何结构和谱之间的联系。研究光谱的最简单模型是具有独立元素的随机矩阵。维格纳的普适性原理指出，几个模型应该有类似于这种设置的局部行为。事实上，至少有20个物理学家认为应该表现出这种行为的根本不同的模型，但基本上没有一个被数学证明是这样的。这是令人惊讶的，因为这应该是类似的中心极限定理的背景下，谱问题。最著名的随机矩阵系综是Dyson beta系综，它有一个参数beta，代表对称类。在以前的工作中，我们已经表明，对于任意的β，点过程的特征值（在边缘和在散装）有一个限制的维数趋于无穷大。在这两种情况下，极限点过程是随机算子的特征值。证明的方法是取矩阵本身的标度极限。在以后的工作中，对于边缘的情况下，我们已经进行了这个程序在边缘的所有一致凸多项式V，表明限制不依赖于V。一个相关的领域是随机薛定谔operetors。这些是合金的简单模型，参数s代表杂质。当s较大时，这些矩阵算子的特征值遵循泊松统计，并且对于较小的s，在高维中，它们表现出随机的矩阵状行为。在以前的工作中，我们已经证明了随机矩阵的行为确实存在这样的运营商，但我们的证明是有限的非常特殊的情况。我们想把证明扩展到其他情况。随机薛定谔算子的另一个模型是渗流集团的本征值。在这里，我们所知甚少，甚至连极限全局特征值分布的规律性都不知道。很明显，它总是有一组密集的原子。我们已经证明，在树和平面格上，也有一个连续的部分。这个问题在更高的维度上仍然没有被理解。

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.