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。长期以来的猜测是，这个方程的解(具有确定性)是由一种称为随机平面波的特殊类型的随机高斯函数很好地模拟的。这一猜想还不够严谨。这种系统可以是可积的，也可以是混沌的，这取决于系统的域的形状，并推测这些情况可以通过解的节点域的分布来区分。这推动了对随机平面波的节点域的研究，以及更广泛地研究随机高斯函数的节点域。最近，人们提出了随机平面波与渗流之间的联系。具体地，随机平面波的节点域具有临界渗流模型的近似分布，随机平面波的水平集具有非临界模型的近似分布。这些联系已经在物理学文献中非严格地推导出来，并进行了数值测试。这些试验的结果表明，所提出的精确渗流模型是不正确的，因为它预测的参数值与模拟结果不匹配。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符。这表明节点区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实的猜想。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，特别是与节域有关的联系。有许多可能的途径可供我继续；1)提出并测试用于随机平面波节点域分布的替代特定渗流模型；2)研究随机平面波与Schramm-Loewner演化之间的可能联系。Schramm-Loewner演化(SLEs)被推测为许多渗流过程的尺度极限，因此，如果随机平面波由渗流过程很好地模拟，那么ITT自然期望与SLEs建立联系。已经进行了一些基本的数值试验，这与这种联系是一致的。3)考虑特定区域上的随机平面波。在二维单位球的情况下，节点区域数目的渐近性已经被严格地推导出来，并且被证明是指数地集中在一个固定值附近。考虑其他特定的域或域的类别可能会打开对某些猜想的严格证明的道路。该项目属于EPSRC流体动力学研究领域

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