Random Eigenvalues

随机特征值

• 批准号: RGPIN-2014-06713
• 负责人: Virag, Balint
• 金额: $ 2.77万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635232
• 关键词: Random; Eigenvalues

Random eigenvaluesHow 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.

随机特征值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，但它的相关性远远超出了音乐。在现代电信、数据分析、物理学甚至制造领域，物体光谱特性的研究发挥着核心作用。该项目的目标是研究许多经典和现代概率模型中的几何结构和谱之间的联系。研究谱的最简单模型是具有独立条目的随机矩阵。维格纳提出的通用性原则指出，多个模型应该具有与此设置类似的局部行为。事实上，物理学家认为至少有二十个根本不同的模型应该表现出这种行为，但基本上没有一个模型在数学上被证明可以这样做。这是令人惊讶的，因为这应该是谱问题中中心极限定理的模拟。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。在之前的工作中，我们已经证明，对于任意 beta，特征值的点过程（无论是在边缘还是在整体）随着维数趋于无穷大而具有极限。在这两种情况下，极限点过程都是随机算子的特征值。证明的方法是采用矩阵本身的缩放极限。在后来的工作中，对于边缘情况，我们在边缘对所有一致凸多项式 V 执行了这个程序，表明极限不依赖于 V。A 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 代表杂质。当 s 较大时，人们认为这些矩阵算子的特征值遵循泊松统计，而对于较小的 s，在高维中，它们表现出类似矩阵的随机行为。在之前的工作中，我们已经证明了此类算子确实存在类似矩阵的随机行为，但我们的证明仅限于非常特殊的情况。我们希望将证明扩展到其他情况。随机薛定谔算子的另一个模型是渗滤簇的特征值。在这里，我们知之甚少，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，在树和平面晶格上，也存在连续的部分。对于更高的维度，这个问题仍然不被理解。