Distributions of Eigenvalues of Large Euclidean Matrices Generated from Three Manifolds

Let x1, · · · ,xn be points randomly chosen from a set G ⊂ R and f(x) be a function. A special Euclidean random matrix is given by Mn = (f(∥xi − xj∥))n×n. When p 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 p and n go to infinity with n/p → y ∈ (0,∞), we obtain the explicit limit of μ̂(Mn) when G is the unit sphere Sp−1 or the unit ball Bp(0, 1) and the explicit limit of μ̂((Mn − apIn)/bp) for G = [0, 1], where ap and bp are constants. As corollaries, we obtain the limit of μ̂(An) with An = (d(xi,xj))n×n and d being the geodesic distance on S p−1. We also obtain the limit of μ̂(An) for the Euclidean distance matrix An = (∥xi−xj∥)n×n as G is Sp−1 or Bp(0, 1). The limits are the law of a+bV where a and b are explicit constants and V follows the Marčenko-Pastur law. The same are also obtained for other examples including (exp(−λ∥xi − xj∥))n×n and (exp(−λd(xi,xj)))n×n.