Nodal domains of Random plane waves

薛定谔/亥姆霍兹方程可以用来模拟量子力学中二维动力学系统的行为。这是一个长期存在的猜想，这个方程的解（是确定性的）是由一种特殊类型的随机高斯函数称为随机平面波很好地模拟的。这个猜想还不严密。这样的系统可能是可积的或混沌的，这取决于系统的域的形状，并且它是rectified的，这些情况下可以区分的节点域的解决方案的分布。这激发了对随机平面波节点域的研究，更一般地说，也激发了对随机高斯函数节点域的研究。最近，人们提出了随机平面波和渗流之间的联系。具体地说，随机平面波的节点域被证明具有临界渗流模型的近似分布，而随机平面波的水平集具有非临界模型的水平集。这些联系已经在物理学文献中得到了非严格的推导，并进行了数值测试。这些测试的结果表明，已经提出的精确渗流模型是不正确的，因为它预测的参数值没有被模拟匹配。然而，进一步的工作已经测试了在渗流模型中通用的参数，并发现结果与预测相匹配。这表明节点域的分布可能与另一种渗流模型的分布相匹配。这是一个数学领域，有许多悬而未决的问题和未经证明的假设。在我的研究中，我的目标是探索随机平面波和渗流之间的联系，特别是在节点域。我可以通过一些可能的途径进行工作; 1）提出并检验了随机平面波节点域分布的另一种特殊渗流模型。2）研究了随机平面波与Schramm-Loewner演化之间的可能联系。Schramm-Loewner演化（SLE）被证明是许多渗流过程的标度极限，因此，如果随机平面波可以很好地用渗流过程来模拟，那么很自然地可以预期它与SLE的联系。一些基本的数值试验结果与这种联系是一致的。3）考虑特定区域上的随机平面波。在2维单位球的情况下，节点域的数量的渐近性已被严格推导出，并显示出指数集中在一个固定的值。考虑到其他特定领域或领域类别可能会为某些命题的严格证明开辟道路。本项目属于EPSRC流体动力学研究领域的福尔斯

The Schrodinger/Helmholtz equation can be used to model the behaviour of 2 dimensional dynamic systems in quantum mechanics. It is a long-standing conjecture that the solutions to this equation (which are deterministic) are well modelled by a special type of random Gaussian function known as the random plane wave. This conjecture is not yet rigorous. Such systems may be integrable or chaotic depending on the shape of the domain of the system, and it is conjectured that these cases may be distinguished by the distribution of nodal domains of the solutions. This has motivated the study of the nodal domains of random plane waves, and, more generally, those of random Gaussian functions.More recently, a link has been proposed between random plane waves and percolation. Specifically, the nodal domains of random plane waves are conjectured to have the approximate distribution of a critical percolation model, and the level sets of random plane waves to have those of non-critical models. These links have been derived non-rigorously in the Physics literature and tested numerically. The results of these tests suggest that the precise percolation model which has been proposed is incorrect, as the parameter values it predicts have not been matched by the simulations. However further work has tested parameters which are universal across percolation models, and found the results to match the predictions. This suggests that the distribution of nodal domains may match that of an alternative percolation model.This is an area of mathematics with many open questions and unproven conjectures. In my research, I aim to explore the connections between random plane waves and percolation, specifically in relation to nodal domains. There are a number of possible avenues through which I could proceed;1) Proposing and testing an alternative specific percolation model for the distribution of nodal domains of random plane waves.2) Investigating possible links between random plane waves and Schramm-Loewner evolutions.Schramm-Loewner evolutions (SLEs) are conjectured to be the scaling limits of many percolation processes, so if random plane waves are well modelled by a percolation process, then itt is natural to expect a connection with SLEs. Some basic numerical tests have been performed which are consistent with such a connection.3) Considering random plane waves on particular domains. In the case of the 2-dimensional unit sphere, the asymptotics of the number of nodal domains has been rigorously derived and shown to concentrate exponentially around a fixed value. Considering other specific domains or classes of domains may open the way to rigorous proofs of some conjectures.This project falls within the EPSRC Fluid Dynamics research area