Study of the fluctuation of spectra of random operators

在本课题中，我们从点过程理论的角度对随机算子谱的统计涨落进行了一般的探讨，同时研究了具体随机薛定谔算子的水平统计量。除了这些研究之外，我们还对随机树进行了概率研究，这是一项由随机矩阵迹迹估计问题引发的研究。主要研究结果如下：“展开”是对随机算子的频谱进行归一化的过程，因此它可以被看作是一个典型的驻定点过程的实现。试图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不相交区间同时包含该算子的特征值的概率。然而，由于很难得到这个估计，整个证明仍然是不完整的。对于作为离散时间分支过程轨迹的一类随机树，在k= 0,1,2,0.4时，得到了结点个数为k的局部极限定理。得到了在树的空间上构造概率测度的一般扩展定理，这些树的顶点有“标记”，例如表示顶点的空间位置、年龄、类型等。

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.