Limit theorems for zeroes of Gaussian processes.

随机点过程在数学和物理中都是被广泛研究的对象。许多物理现象可以用随机点过程来模拟，例如，排队的人的到达时间，星系中恒星的排列，以及重原子核的能级。典型的，也是最重要的，随机点过程的例子是泊松过程。泊松过程的定义特征是当被限制到不相交的集合时，该过程是随机独立的。这意味着，知道过程在给定位置存在点并不影响附近有点的概率。在许多物理情况下，这种独立性是自然假设，但在其他情况下，这显然是不可接受的。例如，如果这些点代表电子(或其他带电粒子)，那么它们自然会相互排斥。如果我们知道在给定点有一个粒子，那么附近就不太可能有粒子。相比之下，如果一个人研究传染病的爆发，那么知道在给定的地点有病例的情况下，更有可能是附近有病例。出于这个原因，研究不满足独立性假设的随机点过程是很有意义的，而这个项目涉及的是排斥过程。建立这样一个过程的一种自然方法是考虑高斯过程的零集；在一些温和的条件下，人们预计随机的零点会相互排斥。在这个方案中，我们考虑了空间中大区域中点的数量的行为。通常情况下，随机模型在某种渐近状态下接近平均值。因此，需要理解的有趣对象是关于该平均值的(相对于平均值的小的)波动。这个项目试图在两个背景下理解这些波动。第一种是对于限制在一条线上的粒子，我们试图了解长时间间隔内的粒子数量。第二种是限制在二维区域内的粒子，这个区域是弯曲的而不是平坦的双曲空间。从数学的角度来看，双曲几何是一种非常自然的几何，也出现在许多物理环境中。同样，我们对了解一个大区域中的粒子数量很感兴趣，但这一次这个区域是一个大的(双曲)圆盘。这个提议还有第三条线，它也处理一些随机场的零集。该场是在球体上定义的，但零点集不再是一组点，而是一组曲线。这样的物体也被用来模拟一些物理现象，两个例子是量子混沌和宇宙背景微波辐射。在这里，我们感兴趣的是当“频率”很大时计算曲线的数量。我们建议建立一大类确定性(即非随机)函数，其中固定函数的曲线个数渐近于某些(自然定义的)随机场的平均曲线个数。

Random point processes are well-studied objects in both mathematics and physics. Many physical phenomena can be modelled by random point processes, for example, the arrival times of people in a queue, the arrangement of stars in a galaxy, and the energy levels of heavy nuclei of atoms. The classical, and most important, example of a random point process is the Poisson point process. The defining characteristic of the Poisson process is that the process is stochastically independent when restricted to disjoint sets. This means that knowing that there is a point of the process at a given location does not affect the probability that there are points nearby.In many physical situations this independence is a natural assumption, but it is obviously unacceptable in others. For example, if the points represent electrons (or other charged particles) then they naturally repel. If we know that there is a particle at a given point, then it is highly unlikely that there are particles nearby. In contrast, if one studies the outbreak of a contagious disease, then knowing that there is a case in a given location makes it much more likely that there are cases nearby. For this reason it is of interest to study random point processes that do not satisfy an independence assumption and this project is concerned with repulsive processes. One natural way to build such a process is to consider the zero set of a Gaussian process; under some mild conditions one expects the random zeroes to repel. In this proposal we consider the behaviour of the number of points in a large region in space.It is often the case that random models are close to the average value, in some asymptotic regime. The interesting object to understand is then the (small, relative to the average) fluctuations about this average value. This project seeks to understand these fluctuations in two contexts. The first is for particles confined to a line, we seek to understand the number of particles in a long interval. The second is for particles confined to a two-dimensional region; this region is the hyperbolic space which is curved and not flat. The hyperbolic geometry is a very natural one from a mathematical perspective, and appears in many physical contexts as well. Again, we are interested in understanding the number of particles in a large region but this time the region is a large (hyperbolic) disc.There is a third strand to this proposal which also treats the zero sets of some random field. This field is defined on the sphere, but the zero set is no longer a set of points but rather a collection of curves. Such objects are also used to model some physical phenomena, two examples are quantum chaos and cosmic background microwave radiation. Here we are interested in counting the number of curves when the "frequency" is large. We propose to establish a large class of deterministic (i.e., not random) functions where the number of curves of the fixed function is asymptotically close to the average number of curves of some (naturally defined) random field.