Local elliptic law

  title={Local elliptic law},
  author={Johannes Alt and Torben Kruger},
The empirical eigenvalue distribution of the elliptic random matrix ensemble tends to the uniform measure on an ellipse in the complex plane as its dimension tends to infinity. We show this convergence on all mesoscopic scales slightly above the typical eigenvalue spacing in the bulk spectrum with an optimal convergence rate. As a corollary we obtain complete delocalisation for the corresponding eigenvectors in any basis. 
Real eigenvalues of elliptic random matrices
We consider the real eigenvalues of an (N ×N) real elliptic Ginibre matrix whose entries are correlated through a non-Hermiticity parameter τN ∈ [0, 1]. In the almost-Hermitian regime where 1 − τN =
Randomly coupled differential equations with elliptic correlations
We consider the long time asymptotic behavior of a large system of $N$ linear differential equations with random coefficients. We allow for general elliptic correlation structures among the


Stability of the matrix Dyson equation and random matrices with correlations
We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the
Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices
We consider a class of elliptic random matrices which generalize two classical ensembles from random matrix theory: Wigner matrices and random matrices with iid entries. In particular, we establish a
The Elliptic Law
We show that, under some general assumptions on the entries of a random complex $n \times n$ matrix $X_n$, the empirical spectral distribution of $\frac{1}{\sqrt{n}} X_n$ converges to the uniform law
Edge universality for non-Hermitian random matrices
It is proved that their local eigenvalue statistics near the spectral edge, the unit circle, coincide with those of the Ginibre ensemble, i.e. when the matrix elements of $$X$$ X are Gaussian.
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 away
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 in
Edge scaling limits for a family of non-Hermitian random matrix ensembles
A family of random matrix ensembles interpolating between the Ginibre ensemble of n × n matrices with iid centered complex Gaussian entries and the Gaussian unitary ensemble (GUE) is considered. The
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 obtained
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 the
Low rank perturbations of large elliptic random matrices
A number of interesting results concerning elliptic random matrices whose entries have finite fourth moment are proved; these results include a bound on the least singular value and the asymptotic behavior of the spectral radius.