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My question is motivated by the following recent paper:

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 $\mu_n$ the empirical eigenvalue distribution of the random matrix $M_{n}$. This is one 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?

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