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Quadratic vector equations on complex upper half-plane
We consider the nonlinear equation $-\frac{1}{m}=z+Sm$ with a parameter $z$ in the complex upper half plane $\mathbb{H} $, where $S$ is a positivity preserving symmetric linear operator acting onExpand
Local law for random Gram matrices
We prove a local law in the bulk of the spectrum for random Gram matrices XX∗, a generalization of sample covariance matrices, where X is a large matrix with independent, centered entries withExpand
Location of the spectrum of Kronecker random matrices
For a general class of large non-Hermitian random block matrices $\mathbf{X}$ we prove that there are no eigenvalues away from a deterministic set with very high probability. This set is obtainedExpand
Random Matrices with Slow Correlation Decay.
We consider large random matrices with a general slowly decaying correlation among its entries. We prove universality of the local eigenvalue statistics and optimal local laws for the resolvent awayExpand
Local inhomogeneous circular law
We consider large random matrices $X$ with centered, independent entries which have comparable but not necessarily identical variances. Girko's circular law asserts that the spectrum is supported inExpand
Spectral radius of random matrices with independent entries
We consider random $n\times n$ matrices $X$ with independent and centered entries and a general variance profile. We show that the spectral radius of $X$ converges with very high probability to theExpand
The Dyson equation with linear self-energy: spectral bands, edges and cusps
We study the unique solution $m$ of the Dyson equation \[ -m(z)^{-1} = z - a + S[m(z)] \] on a von Neumann algebra $\mathcal{A}$ with the constraint $\mathrm{Im} \, m\geq 0$. Here, $z$ lies in theExpand
Correlated Random Matrices: Band Rigidity and Edge Universality
We prove edge universality for a general class of correlated real symmetric or complex Hermitian Wigner matrices with arbitrary expectation. Our theorem also applies to internal edges of theExpand
Cusp Universality for Random Matrices II: The Real Symmetric Case
We prove that the local eigenvalue statistics of real symmetric Wigner-type matrices near the cusp points of the eigenvalue density are universal. Together with the companion paperExpand
Local semicircle law with imprimitive variance matrix
We extend the proof of the local semicircle law for generalized Wigner matrices given in MR3068390 to the case when the matrix of variances has an eigenvalue $-1$. In particular, this result providesExpand