Random distance matrices

Question

My question is motivated by the following recent paper:

Gadgil, Siddhartha; Krishnapur, Manjunath, Lipschitz correspondence between metric measure spaces and random distance matrices, Int. Math. Res. Not. 2013, No. 24, 5623-5644 (2013), arXiv:1110.6333, MR3144175, Zbl 1292.60005.

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?

Answer 1

I think that 1) is false.
Imagine 2 manifolds defined as:

$$(x_1,x_2) \in (0,1) \times (0,1)$$

and

$$(x_1,x_2,x_3) \in (0,1) \times (0,1) \times (0,1)$$

and you choose $x$ randomly.

Define the distance between points $(x,y)$ as as: $$ d(x,y) = \begin{cases} 1 & \text{if }x_1 \neq y_1,\\ \| x - y \|_2 &\text{if }x_1 = y_1. \end{cases} $$ They are different manifolds because 1 is locally $\Bbb R$ and 1 is locally $\Bbb R^2$, but almost always $M_n = 1 - I^{n \times n}$, so the random matrix (and therefore it's eigenvalues) are the same.