Distributions of eigenvalues of large Euclidean matrices generated from lp balls and spheres

Let x1, · · · ,xn be points randomly chosen from a set G ⊂ R and f(x) be a function. The Euclidean random matrix is given by Mn = (f(∥xi − xj∥))n×n where ∥ · ∥ is the Euclidean distance. When N is fixed and n → ∞ we prove that μ̂(Mn), the empirical distribution of the eigenvalues of Mn, converges to δ0 for a big class of functions of f(x). Assuming both N and n go to infinity proportionally, we obtain the explicit limit of μ̂(Mn) when G is the lp unit ball or sphere with p ≥ 1. As corollaries, we obtain the limit of μ̂(An) with An = (d(xi,xj))n×n and d being the geodesic distance on the ordinary unit sphere in R . We also obtain the limit of μ̂(An) for the Euclidean distance matrix An = (∥xi −xj∥)n×n. The limits are a+ bV where a and b are constants and V follows the Marčenko-Pastur law. The same are also obtained for other examples appeared in physics literature including (exp(−∥xi−xj∥))n×n and (exp(−d(xi,xj)))n×n. Our results partially confirm a conjecture by Do and Vu.