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

Let x₁,...,x_n be points randomly chosen from a set G ⊂ ℝ^N and f(x) be a function. The Euclidean random matrix is given by M_n = (f(||x_i - x_j||²))_n×n where || · || is the Euclidean distance. When N is fixed and n → ∞ we prove that μ(M_n), the empirical distribution of the eigenvalues of M_n, 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 μ(M_n) when G is the l_p unit ball or sphere with p ≥ 1. As corollaries, we obtain the limit of μ(A_n) with A_n = (d(x_i, x_j))_n×n and d being the geodesic distance on the ordinary unit sphere in ℝ^N. We also obtain the limit of μ(A_n) for the Euclidean distance matrix A_n = (||x_i - x_j||)_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(-||x_i - x_j||^γ))_n×n and (exp(-d(x_i, x_j)^γ))_n×n. Our results partially confirm a conjecture by Do and Vu [14].