Let $M$ be a compact Riemannian manifold with a metric $g$, and consider the spectrum of the Laplacian operator $\Delta$.

What is known about the relationship between this spectrum and random matrix theory?

In posing this question, I am imagining that the metric $g$ is drawn randomly from a suitable distribution. I am agnostic as to how this is done, but since the space of metrics on $M$ is somewhat unwieldy it may be simpler to consider special finite-dimensional spaces of $g$'s.

To be concrete, we could for instance imagine a two-dimensional Riemann surface of genus greater than one equipped with a uniformly negatively curved metric. Such metrics come in finite-dimensional families and it is natural to imagine drawing the metric from this set.

(Clearly in the genus one case the spectrum on a flat torus is not random, so I am also imagining that the topology of $M$ is suitably generic.)

In higher dimensions there are also sometimes natural finite-dimensional families of metrics, e.g. on Calabi-Yau manifolds and I am also interested to know what generic features of the spectra are known here as well.

As some physics motivation, if you consider a quantum particle moving on such a manifold, the energy levels are controlled by this spectrum. If the system is sufficiently generic, one expects chaotic behavior and hence some type of random matrix universality in properties of the spectrum.

Thanks for any answers or links to relevant literature!