Nodal domains of Random plane waves

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