Overlaps, eigenvalue gaps, and pseudospectrum under real Ginibre and absolutely continuous perturbations

Let G_n be an n × n matrix with real i.i.d. N(0, 1/n) entries, let A be a real n × n matrix with ||A|| ≤ 1, and let γ ∈ (0, 1). We show that with probability 0.99, A + γ G_n has all of its eigenvalue condition numbers bounded by O(n^5/2/γ ^3/2) and eigenvector condition number bounded by O(n³/γ ^3/2). Furthermore, we show that for any s > 0, the probability that A+ γ G_n has two eigenvalues within distance at most s of each other is O(n⁴s^2/7/γ ^5/2). In fact, we show the above statements hold in the more general setting of non-Gaussian perturbations with real, independent, absolutely continuous entries with a finite moment assumption and appropriate normalization. This extends the previous work (Banks et al. (2019)) which proved an eigenvector condition number bound of O(n^3/2/γ) for the simpler case of complex i.i.d. Gaussian matrix perturbations. The case of real perturbations introduces several challenges stemming from the weaker anticoncentration properties of real vs. complex random variables. A key ingredient in our proof is new lower tail bounds on the small singular values of the complex shifts z − (A + γ G_n) which recover the tail behavior of the complex Ginibre ensemble when ℑz ≠ 0. This yields sharp control on the area of the pseudospectrum Λ_ε(A + γ G_n) in terms of the pseudospectral parameter ε > 0, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.