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Eigenvalues of Euclidean random matrices
Author(s) -
Bordenave Charles
Publication year - 2008
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20228
Subject(s) - mathematics , adjacency matrix , random matrix , eigenvalues and eigenvectors , combinatorics , measure (data warehouse) , euclidean geometry , limit (mathematics) , limiting , euclidean distance , circular law , infinity , discrete mathematics , graph , mathematical analysis , convergence of random variables , random variable , geometry , statistics , physics , mechanical engineering , quantum mechanics , database , sum of normally distributed random variables , computer science , engineering
Abstract We study the spectral measure of large Euclidean random matrices. The entries of these matrices are determined by the relative position of n random points in a compact set Ω n of ℝ d . Under various assumptions, we establish the almost sure convergence of the limiting spectral measure as the number of points goes to infinity. The moments of the limiting distribution are computed, and we prove that the limit of this limiting distribution as the density of points goes to infinity has a nice expression. We apply our results to the adjacency matrix of the geometric graph. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008