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Let $G_n$ be an $n times n$ matrix with real i.i.d. $N(0,1/n)$ entries, let $A$ be a real $n times n$ matrix with $Vert A Vert le 1$, and let $gamma in (0,1)$. We show that with probability $0.99$, $A + gamma G_n$ has all of its eigenvalue condition numbers bounded by $Oleft(n^{5/2}/gamma^{3/2}right)$ and eigenvector condition number bounded by $Oleft(n^3 /gamma^{3/2}right)$. Furthermore, we show that for any $s > 0$, the probability that $A + gamma G_n$ has two eigenvalues within distance at most $s$ of each other is $Oleft(n^4 s^{1/3}/gamma^{5/2}right).$ 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 $Oleft(n^{3/2} / gammaright)$ for the simpler case of {em 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 $z-(A+gamma G_n)$ which recover the tail behavior of the complex Ginibre ensemble when $Im z eq 0$. This yields sharp control on the area of the pseudospectrum $Lambda_epsilon(A+gamma G_n)$ in terms of the pseudospectral parameter $epsilon>0$, which is sufficient to bound the overlaps and eigenvector condition number via a limiting argument.
For a fixed quadratic polynomial $mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $Ntimes N$ complex Ginibre matrices $X_1^N,dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =mathfrak{p}(X_1^N
Let $sqrt{N}+lambda_{max}$ be the largest real eigenvalue of a random $Ntimes N$ matrix with independent $N(0,1)$ entries (the `real Ginibre matrix). We study the large deviations behaviour of the limiting $Nrightarrow infty$ distribution $P[lambda_{
We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respe
We establish a quantitative criterion for an operator defined on a Galton-Watson random tree for having an absolutely continuous spectrum. For the adjacency operator, this criterion requires that the offspring distribution has a relative variance bel
In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogon