Limit theorems for zeroes of Gaussian processes.

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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.