权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Limit theorems for zeroes of Gaussian processes.

高斯过程零点的极限定理。

基本信息

批准号：
EP/V002449/1
负责人：
Jeremiah Buckley
金额：
$ 34.78万
依托单位：
King's College London
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2021
资助国家：
英国
起止时间：
2021 至无数据
项目状态：
未结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FV002449%2F1
关键词：
Limit theorems zeroes Gaussian processes.

项目摘要

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.

随机点过程在数学和物理学中都是研究得很好的对象。许多物理现象可以用随机点过程来模拟，例如，排队的人到达的时间，星系中恒星的排列，以及重原子核的能级。经典的，也是最重要的随机点过程的例子是泊松点过程。泊松过程的定义特征是，当泊松过程被限制在不相交的集合中时，它是随机独立的。这意味着知道在给定位置有一个过程点并不影响附近有点的概率。在许多实际情况下，这种独立性是一种自然的假设，但在其他情况下显然是不可接受的。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的点上有一个粒子，那么附近就不太可能有粒子。相反，如果研究一种传染病的爆发，那么知道在一个给定的地点有一个病例，就更有可能在附近有病例。因此，研究不满足独立假设的随机点过程是一个很有意义的课题，本课题主要研究排斥过程。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，人们期望随机的零相互排斥。在这个建议中，我们考虑了空间中一个大区域中点的数目的行为。通常情况下，随机模型在某些渐近状态下接近平均值。要理解的有趣对象是这个平均值的（相对于平均值而言的）波动。这个项目试图在两种情况下理解这些波动。第一种是对于被限制在一条直线上的粒子，我们试图了解在一个很长的间隔内粒子的数量。第二种是局限于二维区域的粒子；这个区域是双曲空间，它是弯曲的而不是平坦的。从数学角度来看，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，我们感兴趣的是了解一个大区域内的粒子数量，但这次这个区域是一个大的（双曲）圆盘。这个建议还有第三条线索，它也处理一些随机场的零集。这个场是在球面上定义的，但是零集不再是点的集合，而是曲线的集合。这些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。我们提出建立一大类确定性（即非随机）函数，其中固定函数的曲线数渐近地接近于某些（自然定义的）随机场的平均曲线数。