  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.
