Random Eigenvalues

一个物体的形状如何影响它发出的声音？这个问题肯定和音乐一样古老，但它的相关性远不止于此。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。本项目的目的是研究许多经典和现代概率模型中几何结构与光谱之间的联系。研究光谱最简单的模型是具有独立条目的随机矩阵。Wigner提出的普适性原则指出，几个模型应该具有与此设置相似的局部行为。事实上，物理学家认为至少有20种根本不同的模型应该表现出这样的行为，但基本上没有一个在数学上被证明是这样的。这是令人惊讶的，因为这应该是光谱问题中的中心极限定理的类似物。最著名的随机矩阵系综是Dyson系综，它有一个参数β，表示对称类。在以前的工作中，我们已经表明，对于任意β，特征值的点过程（在边缘和大块）有一个极限，因为维度趋于无穷。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标度极限。在以后的工作中，对于边缘情况，我们在所有一致凸多项式V的边缘上执行了这个程序，表明极限不依赖于V，相关区域是随机薛定谔算子。这些是合金的简单模型，参数s表示杂质。当s较大时，可以认为这些矩阵算子的特征值遵循泊松统计量，当s较小时，在高维中，它们表现出随机的类矩阵行为。在以前的工作中，我们已经证明了这些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。我们想把这个证明推广到其他情况。随机薛定谔算子的另一个模型是渗透簇的特征值。在这里，我们知道的很少，甚至不知道极限全局特征值分布的规律性。很明显，它总是有一组密集的原子。我们已经证明，在树和面格上，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.