## Abstract

Let x_{1},...,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 δ_{0} 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].

Original language | English (US) |
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Pages (from-to) | 14-36 |

Number of pages | 23 |

Journal | Linear Algebra and Its Applications |

Volume | 473 |

DOIs | |

State | Published - 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