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

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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 μ(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 μ(Mn) when G is the lp unit ball or sphere with p ≥ 1. As corollaries, we obtain the limit of μ(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 μ(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].

Original languageEnglish (US)
Pages (from-to)14-36
Number of pages23
JournalLinear Algebra and Its Applications
Volume473
DOIs
StatePublished - May 15 2015

Keywords

  • Empirical distributions of eigenvalues
  • Euclidean matrix
  • Geodesic distance
  • L-norm uniform distribution
  • Marčenko-Pastur law
  • Random matrix
  • l ball
  • l sphere

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