Small ball probability for multiple singular values of symmetric random matrices

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.