My question is motivated by the following recent paper:

<cite authors="Gadgil, Siddhartha; Krishnapur, Manjunath">_Gadgil, Siddhartha; Krishnapur, Manjunath_, [**Lipschitz correspondence between metric measure spaces and random distance matrices**](https://doi.org/10.1093/imrn/rns208), Int. Math. Res. Not. 2013, No. 24, 5623-5644 (2013), arXiv:[1110.6333](http://arxiv.org/abs/1110.6333), [MR3144175](https://mathscinet.ams.org/mathscinet-getitem?mr=3144175), [Zbl 1292.60005](https://zbmath.org/?q=an:1292.60005).</cite>

Assume you have a metric space $(X,d)$ equipped with a Borel probability measure $\mu$. We can further assume that $X$ is a manifold, if necessary. Let $\{x_{k}\}$ be a sequence of point randomly chosen according to the measure $\mu$. Let us consider $M_{n}$ to be the corresponding distance matrix. More precisely, 
$$
M_{n}(i,j):=d(x_{i},x_{j})
$$
for $1\leq i,j\leq n$.

Let $\mu_n$ be the distribution of eigenvalues of the (empirical) random matrix $M_{n}$. This is a probability measure on the real line.

I have the following questions:

 1. To what extent can we distinguish two manifolds by looking at the measures $\mu_{n}$? Is it possible for two non-isomorphic manifolds to have the same measures $\mu_{n}$ for all values of $n$.
 
 2. Can we compute the limit eigenvalue distribution of $M_{n}$ as $n\to\infty$ with the appropriate normalization? For instance, let $X_{n}:=M_{n}/n$ can we compute
$$
m_{p}:=\lim_{n\to\infty}{\frac{1}{n}\mathbb{E}\Big[\mathrm{Tr}\big(X_{n}^{p}\big)\Big]}
$$
for the the $n$ dimensional torus $S^{1}\times\ldots\times S^{1}$ or the sphere with the uniform measure?