Small ball probability for multiple singular values of symmetric random matrices

Abstract

Let \(A_n\) be an \(n\times n\) random symmetric matrix with \((A_{ij})_{i< j}\) i.i.d. mean \(0\), variance 1, following a subGaussian distribution and diagonal elements i.i.d. following a subGaussian distribution with a fixed variance. We investigate the joint small ball probability that \(A_n\) has eigenvalues near two fixed locations \(\lambda_1\) and \(\lambda_2\), where \(\lambda_1\) and \(\lambda_2\) are sufficiently separated and in the bulk of the semicircle law. More precisely we prove that for a wide class of entry distributions of \(A_{ij}\) that involve all Gaussian convolutions (where \(\sigma_{min}(\cdot)\) denotes the least singular value of a square matrix), \[\mathbb{P}(\sigma_{min}(A_n-\lambda_1 I_n)\leq\delta_1n^{-1/2},\sigma_{min}(A_n-\lambda_2 I_n)\leq\delta_2n^{-1/2})\leq c\delta_1\delta_2+e^{-cn}.\] The given estimate approximately factorizes as the product of the estimates for the two individual events, which is an indication of quantitative independence. The estimate readily generalizes to \(d\) distinct locations. As an application, we upper bound the probability that there exist \(d\) eigenvalues of \(A_n\) asymptotically satisfying any fixed linear equation, which in particular gives a lower bound of the distance to this linear relation from any possible eigenvalue pair that holds with probability \(1-o(1)\), and rules out the existence of two equal singular values in generic regions of the spectrum.