The eigenvalue gap of inhomogeneous symmetric discrete random matrix

Abstract

Let A be an n x n symmetric random matrix whose upper-triangular entries are independent and follow possibly non-identical subgaussian distributions. This paper investigates the spectral properties of A, including its eigenvalues and eigenvectors. Firstly, we prove that for k <= n / log n, 1 <= i <= n - k and epsilon >= 0, P(the gap between the (i+k)-th and i-th eigenvalues is at most epsilon n^(-1/2)) <= (C epsilon)^((k^2 + k)/2) + exp(-c n), where the eigenvalues are ordered increasingly. Secondly, combining the recent result of Yi Han, we give a quantitative estimate of the singular values of A. For c log n <= k <= sqrt(n) and epsilon >= 0, we have P(the (n-k+1)-th smallest singular value of A is at most k epsilon n^(-1/2)) <= (C epsilon)^(c k^2) + exp(-c k n), where the singular values are ordered increasingly. Finally, based on the distance analytical framework developed for the eigenvalue gap, we further derive quantitative bounds for singular values and delocalization of eigenvectors. In particular, we establish a quantitative bound for the probability that some eigenvector of A exhibits no-gap delocalization, which improves the result of Rudelson and Vershynin.