Study of the fluctuation of spectra of random operators

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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.