# Laplacian on manifolds and random matrix theory

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.

• "Quantum chaos" is a term you might want to look up. Here's a reference that seems like it might be relevant: ncbi.nlm.nih.gov/pmc/articles/PMC58499 Dec 29 '20 at 5:38
• @QiaochuYuan, Thanks for the reference. I will check it out. I am familiar with the buzzword you mentioned but was hoping an expert might point me to something more specific related to the Laplacian. Dec 29 '20 at 6:15
• If $g$ has a very strong disorder, I think one should not be too surprise to see Anderson localisation phenomenom there. (But this is a completely behaviour.) Dec 29 '20 at 16:39
• I was just looking at a related problem in "Quantum chaos, random matrix theory, and the Riemann $\zeta$ function" by Bourgade and Keating (bourbaphy.fr/keating.pdf), in particular pp.119-121, so I'm keenly interested in these general relationships and any nice monographs sketching them. There is much literature on the topics. Any recommendations would be appreciated. Dec 29 '20 at 18:54

Much of the literature on random metrics constructs these as follows. Start from a reference metric $$g_0$$ on the compact Riemannian manifold $$M$$. The corresponding Laplacian $$\Delta_0$$ has eigenfunctions $$\phi_j$$ with eigenvalues $$\lambda_j$$. Draw a set $$\{a\}$$ of random numbers $$a_j$$, independently from a normal distribution, then the random metric is $$g_{\{a\}}=\exp\left(\sum_j e^{-\lambda_j}a_j\phi_j\right)g_0.$$ Each random metric is conformal to the reference $$g_0$$. The weight $$e^{-\lambda}$$ may be replaced by another function $$F(\lambda)$$ which decreases to zero when $$\lambda\rightarrow\infty$$.

These lecture notes discuss the spectral statistics of random metrics constructed in this way. (See also the course web site.)

There are quite a few connections. I will mention a result of mine where the connection is explicit and essential. Fix the metric $$g$$. Set $$m=\dim M$$ and assume that $${\rm vol}_g(M)=1$$.

Denote by $$\DeclareMathOperator{\spec}{spec}$$ $$\newcommand{\bR}{\mathbb{R}}$$ $$\spec(\Delta)$$ the spectrum of $$\Delta$$ $$\spec(\Delta)= \big\{\, 0=\lambda_0<\lambda_1\leq \lambda_2\leq \cdots \,\big\},$$ where each eigenvalue is repeated according to its multiplicity.

Fix an orthonormal eigenbasis $$(\Psi_k)_{k\geq0}$$ of $$L^2(M,g)$$ $$\Delta \Psi_k = \lambda_k\Psi_k,\;\;\forall k.$$ Pick a nonnegative even Schwartz function $$w:\bR\to[0,\infty)$$. (E.g. $$w(x)=e^{-x^2}$$) Fix $$\newcommand{\ve}{{\varepsilon}}$$ $$\ve>0$$ and set $$w_\ve(x)=w(\ve x)$$. Consider the random Fourier series $$U^\ve=\sum_{k\geq 0} X_k w_\ve\big(\sqrt{\lambda_k}\big)\Psi_k,$$ where $$X_k$$ are independent standard normal random variables.

Random matrices appear when you study the distribution of critical points and critical values of the random function $$U^\ve$$ as $$\ve\searrow 0$$. $$\DeclareMathOperator{\Hess}{Hess}$$ $$\newcommand{\bp}{\boldsymbol{p}}$$

The Hessian of $$U^\ve$$ at a point $$\bp\in M$$ is a random matrix $$\Hess_\bp^\ve$$. Suitably rescaled it converges in distribution to a classical random matrix of the form $$\newcommand{\one}{\boldsymbol{1}}$$

$$X_m\one +A_m$$

where $$X_m$$ is a normal random variable with mean zero and variance depending only on $$m$$ and $$w$$ and $$A_m$$ belongs to GOE, the Gaussian Orthogonal Ensemble.

The critical values of $$U^\ve$$ are with high confidence distributed in an interval of the form $$[\ve^{-m},\ve^{-m}]$$.

If you denote by $$\spec(U^\ve)$$ the set of critical values of $$U^\ve$$ then we obtain a random measure on $$\bR$$ $$\mu^\ve=\sum_{c\in\spec(U^\ve)}\delta_{\ve^mc}.$$ Its expectation $$\newcommand{\bE}{\mathbb{E}}$$ $$\bar{\mu}^\ve:=\bE[\mu^\ve]$$ is a deterministic measure on $$\bR$$. Its mass $$N^\ve:=\bar{\mu}^\ve[\bR]$$ is the expected number of critical points.

Suppose that $$A_{m+1}$$ is belongs to the GOE of symmetric matrices of dimension $$(m+1)\times (m+1)$$. We obtain similarly a random\probability measure $$\sigma_{m+1}=\frac{1}{m+1}\sum_{\lambda\in \spec{A_{m+1}}\delta_\lambda.$$ Its expectation $$\bar{\sigma}_{m+1}=\bE[\sigma_{m+1}]$$ bis a deterministic probability measure on $$\bR$$ that has a rather explicit, albeit complicated, description.

In the paper I mentioned I show that $$\bar{\mu}^\ve\to Z_m e^{-b_mx^2}\bar{\sigma}_{m+1}$$ weakly in the sense of measures as $$\ve\searrow 0$$.

I am not very precise here because this is true up to certain very explicit rescalings depending only on $$m$$ and $$w$$. The constants $$Z_m$$ and $$b_m$$ are also explicit and depend only on $$m$$ and $$w$$.

The normalizing constant $$Z_m$$ ensures that the right-hand-side above is indeed a probability measure.