复流行上的随机全纯场

Random polynomials, or more generally, linear combinations of functions with random coefficients serve as a model for eigenfunctions of chaotic quantum systems. The principal problem concerns the asymptotic behavior of typical geometric structures as the degree of the polynomial goes to infinity. A natural generalization of random polynomials on complex manifolds are holomorphic sections of complex line bundles. In this proposal, following established developments in semiclassical analysis, random geometry and stochastic process, the PI plans to study the following three topics systematically:.1. Extremes and the point process of critical values of random sections. Following some recent results developed by the PI, this project concerns bounds and asymptotic expansions of the excursion probability, the limiting behavior of the rare events of the critical values above the high levels..2. The minimal and maximal spacing distribution of zeros of Gaussian random holomorphic sections (polynomials)..3. Zeros and critical points: the local behavior between zeros and critical points, the rescaling limit of the conditional expectation of critical points and conditional expectation of zeros.

随机多项式或者更广泛一点的随机函数可以用来模拟不同的量子系统。对于这样随机函数，最核心的问题是当随机函数个数趋近于无穷的时候，随机函数的几何性态的变化。线丛的随机截面是随机多项式在紧复流行上的自然推广。本项目的目标是利用在经典分析，随机几何以及随机过程里的方法来系统地研究线丛的随机截面的以下三个问题： .1.随机全纯场的极值点分布问题：利用已知的方法和结论，主要研究随机全纯场的游离概率以及高值极值的小概率事件的渐进性态。.2.高斯随机截面（多项式）零点的空间距离分布。.3.随机全纯场零点和临界点的分布问题：零点和临界点的局部关系，零点和临界点的条件期望。

本项目主要研究对象为随机几何领域，共有三篇文章发表，分别为随机复几何中随机截面的零点问题，和随机球谐函数的最大值分布问题。关于随机函数的最大值问题，已经有近半个世纪的研究，在本项目中，申请人和以色列理工的Robert Adler教授合作，首次给出了具体的事例，可以利用Weyl管道公式具体给出随机球谐函数的最大值分布。