Small ball probability for multiple singular values of symmetric random matrices

Let Aₙ be an n n random symmetric matrix with (A₈₉) ₈< ₉ 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 ₁ and ₂, where ₁ and 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 A₈₉ that involve all Gaussian convolutions (where ₌₈₍ () denotes the least singular value of a square matrix), P (₌₈₍ (Aₙ-₁ Iₙ) ₁n^-1/2, ₌₈₍ (Aₙ-₂ Iₙ) ₂n^-1/2) c₁₂+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.