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Study of the fluctuation of spectra of random operators

随机算子谱涨落的研究

基本信息

批准号：
17540100
负责人：
MINAMI Nariyuki
金额：
$ 2.24万
依托单位：
Keio University (2007)University of Tsukuba (2005-2006)
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

项目摘要

In this project, we made a general approach to the statistical fluctuation of spectra of random operators from the viewpoint of point process theory, and at the same time, studied the level statistics for a concrete random Schrodinger operator In addition to these researches, we made a probabilistic investigation of random trees, a study motivated by questions of estimating traces of random matrices. Main results are the followings:1. A mathematical formulation was made of the notion of "unfolding' which is a procedure of normalizing the spectrum of a random operator, so that it can be viewed as a typical realization of a stationary point process.2. An attempt was made to prove that the unfolded spectrum of the discrete random Schrodinger operator looks like a typical realization of the Poisson point process, and it was clarified that the proof boils down to estimating the probability that two disjoint intervals simultaneously contain eigenvalues of the operator. However, because of the difficulty in obtaining this estimate, the whole proof is still incomplete.3. Concerning a class of random trees obtained as trajectories of discrete time branching processes, a local limit theorem was obtained for the number of vertices having k children for k=0, 1, 2, .4. A general extension theorem was obtained for the construction of a probability measure on the space of trees whose vertices have "marks" representing for example spatial location, age, types etc. of vertices.

在这个项目中，我们从点过程理论的角度对随机算子谱的统计涨落提出了一般性的方法，同时研究了具体随机薛定谔算子的能级统计。除了这些研究之外，我们还对随机树进行了概率研究，这项研究的动机是估计随机矩阵迹的问题。主要结果如下： 1．对“展开”的概念进行了数学表述，“展开”是随机算子谱归一化的过程，因此可以将其视为驻点过程的典型实现。2.尝试证明离散随机薛定谔算子的展开谱看起来像泊松点过程的典型实现，并澄清了证明归结为估计两个概率不相交的区间同时包含算子的特征值。然而，由于获得这个估计的困难，整个证明仍然不完整。 3．关于作为离散时间分支过程的轨迹获得的一类随机树，对于具有 k 个子节点（k=0,1,2,.4）的顶点数量获得局部极限定理。获得了用于构造概率测度的一般可拓定理在其顶点具有代表例如顶点的空间位置、年龄、类型等的“标记”的树的空间上。