Random Eigenvalues

随机本征值物体的形状如何影响其发出的声音？这个问题肯定和音乐一样古老，但它的相关性远远不止于此。在现代通信、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。这个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光谱之间的联系。研究光谱最简单的模型是具有独立条目的随机矩阵。维格纳的普适性原理指出，几个模型应该具有类似于这种设置的局部行为。事实上，物理学家认为至少有20个根本不同的模型应该表现出这样的行为，但基本上没有一个模型在数学上被证明是这样的。考虑到这应该是光谱问题背景下的中心极限定理的类比，这是令人惊讶的。最著名的随机矩阵系综是Dyson Beta系综，它有一个参数β，代表对称类。在以前的工作中，我们已经证明了对于任意的β，当维度趋于无穷大时，特征值的点过程(无论是在边缘还是在整体上)都是有极限的。在这两种情况下，极限点过程都是随机算子的特征值。证明的方法是取矩阵本身的比例极限。在后面的工作中，对于边缘情况，我们对所有一致凸多项式V在边缘处进行了这一程序，证明了极限不依赖于V。这些是合金的简单模型，参数S代表杂质。当S较大时，认为这些矩阵算子的本征值服从泊松统计，而对于较小的S，在高维时，它们表现出类似随机矩阵的行为。在以前的工作中，我们已经证明了对于这类算子确实存在随机类矩阵行为，但我们的证明仅限于非常特殊的情况。我们想将证据扩大到其他情况。随机薛定谔算符的另一种模型是渗流团簇的本征值。在这里，人们对此知之甚少，甚至连极限全局特征值分布的正则性都不知道。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.