Overlaps, Eigenvalue Gaps, and Pseudospectrum under real Ginibre and Absolutely Continuous Perturbations
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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(n^3 /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(n^4 s^1/3/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 z≠ 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.
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