My question is motivated by the following recent paper:

http://arxiv.org/abs/1110.6333

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 $1\leq i,j\leq n$.

I have the following questions:

1. What is know about the structure of $M_{n}$? For instance, what is the expectation of $\mathbb{E}(M_{n})$ at least in a few examples?

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?