For a general class of large non-Hermitian random block matrices X we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtained from the Dyson equation of the Hermitization of X as the self-consistent approximation of the pseudospectrum. We demonstrate that the analysis of the matrix Dyson equation from [3] offers a unified treatment of many structured matrix ensembles.

We consider products of independent square non-Hermitian random matrices. More precisely, let X1, . . . , Xn be independent N × N random matrices with independent entries (real or complex with independent real and imaginary parts) with zero mean and variance 1 N . Soshnikov-O'Rourke [15] and Götze-Tikhomirov [11] showed that the empirical spectral distribution of the product of n random matrices with iid entries converges toWe prove that if the entries of the matrices X1, . . . , Xn satisfy uniform subexponential decay condition, then in the bulk the convergence of the ESD of X1 · · · Xn to (1) holds up to the scale N −1/2+ε .

We consider products of independent square random non-Hermitian matrices. More precisely, let n ≥ 2 and let X1, . . . , Xn be independent N × N random matrices with independent centered entries (either real or complex with independent real and imaginary parts) with variance N −1 . In [10] and [15] it was shown that the limit of the empirical spectral distribution of the product X1 · · · Xn is supported in the unit disk. We prove that if the entries of the matrices X1, . . . , Xn satisfy uniform subexponential decay condition, then the spectral radius of X1 · · · Xn converges to 1 almost surely as N → ∞.

We consider general self-adjoint polynomials in several independent random matrices whose entries are centered and have the same variance. We show that under certain conditions the local law holds up to the optimal scale, i.e., the eigenvalue density on scales just above the eigenvalue spacing follows the global density of states which is determined by free probability theory. We prove that these conditions hold for general homogeneous polynomials of degree two and for symmetrized products of independent matrices with i.i.d. entries, thus establishing the optimal bulk local law for these classes of ensembles. In particular, we generalize a similar result of Anderson for anticommutator. For more general polynomials our conditions are effectively checkable numerically.

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