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Eigenvalues of Euclidean random matrices. (English) Zbl 1158.15020
Summary: 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 \(\Omega _n\) of \(\mathbb R^d\). 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.

MSC:
15B52 Random matrices (algebraic aspects)
51K05 General theory of distance geometry
05C80 Random graphs (graph-theoretic aspects)
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
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