On Euclidean random matrices in high dimension

TL;DR

This paper investigates the spectral properties of high-dimensional Euclidean random matrices with entries defined by a function of the Euclidean distance between i.i.d. vectors, providing conditions for eigenvalue distribution convergence.

Contribution

It offers new sufficient conditions for the empirical eigenvalue distribution to converge in high-dimensional regimes, extending understanding of Euclidean random matrices.

Findings

Eigenvalue distribution converges under certain conditions

Application to log-concave random vectors

Provides theoretical framework for high-dimensional Euclidean matrices

Abstract

In this note, we study the n x n random Euclidean matrix whose entry (i,j) is equal to f (|| Xi - Xj ||) for some function f and the Xi's are i.i.d. isotropic vectors in Rp. In the regime where n and p both grow to infinity and are proportional, we give some sufficient conditions for the empirical distribution of the eigenvalues to converge weakly. We illustrate our result on log-concave random vectors.