Small ball probability for multiple singular values of symmetric random matrices

Let Aₙ be an n× n random symmetric matrix with (Aᵢⱼ)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 Aₙ has eigenvalues near two fixed locations λ₁ and λ₂, where λ₁ and λ₂ are sufficiently separated and in the bulk of the semicircle law. More precisely we prove that for a wide class of entry distributions of Aᵢⱼ that involve all Gaussian convolutions (where σₘᵢₙ(·) denotes the least singular value of a square matrix), P(σₘᵢₙ(Aₙ-λ₁ Iₙ)≤δ₁n-1/2,σₘᵢₙ(Aₙ-λ₂ Iₙ)≤δ₂n-1/2)≤ cδ₁δ₂+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 Aₙ 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.