Distributions of eigenvalues of large Euclidean matrices generated from [formula omitted] balls and spheres

Abstract

Let x1,…,xn be points randomly chosen from a set G⊂RN and f(x) be a function. The Euclidean random matrix is given by Mn=(f(‖xi−xj‖2))n×n where ‖⋅‖ is the Euclidean distance. When N is fixed and n→∞ we prove that μˆ(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 μˆ(Mn) when G is the lp unit ball or sphere with p⩾1. As corollaries, we obtain the limit of μˆ(An) with An=(d(xi,xj))n×n and d being the geodesic distance on the ordinary unit sphere in RN. We also obtain the limit of μˆ(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‖γ))n×n and (exp⁡(−d(xi,xj)γ))n×n. Our results partially confirm a conjecture by Do and Vu [14].