@article{6240,
abstract = {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 (Probab. Theory Related Fields (2018)) offers a unified treatment of many structured matrix ensembles.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H and Nemish, Yuriy},
issn = {02460203},
journal = {Annales de l'institut Henri Poincare},
number = {2},
pages = {661--696},
title = {{Location of the spectrum of Kronecker random matrices}},
doi = {10.1214/18-AIHP894},
volume = {55},
year = {2019},
}
@article{566,
abstract = {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 a disk and in case of identical variances, the limiting density is uniform. In this special case, the local circular law by Bourgade et. al. [11,12] shows that the empirical density converges even locally on scales slightly above the typical eigenvalue spacing. In the general case, the limiting density is typically inhomogeneous and it is obtained via solving a system of deterministic equations. Our main result is the local inhomogeneous circular law in the bulk spectrum on the optimal scale for a general variance profile of the entries of X.
},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
journal = {Annals Applied Probability },
number = {1},
pages = {148--203},
publisher = {Institute of Mathematical Statistics},
title = {{Local inhomogeneous circular law}},
doi = {10.1214/17-AAP1302},
volume = {28},
year = {2018},
}
@phdthesis{149,
abstract = {The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.},
author = {Alt, Johannes},
pages = {456},
publisher = {IST Austria},
title = {{Dyson equation and eigenvalue statistics of random matrices}},
doi = {10.15479/AT:ISTA:TH_1040},
year = {2018},
}
@unpublished{6183,
abstract = {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 the complex upper half-plane, $a$ is
a self-adjoint element of $\mathcal{A}$ and $S$ is a positivity-preserving
linear operator on $\mathcal{A}$. We show that $m$ is the Stieltjes transform
of a compactly supported $\mathcal{A}$-valued measure on $\mathbb{R}$. Under
suitable assumptions, we establish that this measure has a uniformly
$1/3$-H\"{o}lder continuous density with respect to the Lebesgue measure, which
is supported on finitely many intervals, called bands. In fact, the density is
analytic inside the bands with a square-root growth at the edges and internal
cubic root cusps whenever the gap between two bands vanishes. The shape of
these singularities is universal and no other singularity may occur. We give a
precise asymptotic description of $m$ near the singular points. These
asymptotics generalize the analysis at the regular edges given in the companion
paper on the Tracy-Widom universality for the edge eigenvalue statistics for
correlated random matrices [arXiv:1804.07744] and they play a key role in the
proof of the Pearcey universality at the cusp for Wigner-type matrices
[arXiv:1809.03971,arXiv:1811.04055]. We also extend the finite dimensional band
mass formula from [arXiv:1804.07744] to the von Neumann algebra setting by
showing that the spectral mass of the bands is topologically rigid under
deformations and we conclude that these masses are quantized in some important
cases.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
booktitle = {arXiv:1804.07752},
pages = {72},
title = {{The Dyson equation with linear self-energy: Spectral bands, edges and cusps}},
year = {2018},
}
@unpublished{6184,
abstract = {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 the self-consistent density of states. In
particular, we establish a strong form of band rigidity which excludes
mismatches between location and label of eigenvalues close to internal edges in
these general models.},
author = {Alt, Johannes and Erdös, László and Krüger, Torben H and Schröder, Dominik J},
booktitle = {arXiv:1804.07744},
pages = {26},
title = {{Correlated random matrices: Band rigidity and edge universality}},
year = {2018},
}
@article{1010,
abstract = {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 with arbitrary variances. The limiting eigenvalue density that generalizes the Marchenko-Pastur law is determined by solving a system of nonlinear equations. Our entrywise and averaged local laws are on the optimal scale with the optimal error bounds. They hold both in the square case (hard edge) and in the properly rectangular case (soft edge). In the latter case we also establish a macroscopic gap away from zero in the spectrum of XX∗. },
author = {Alt, Johannes and Erdös, László and Krüger, Torben H},
issn = {10836489},
journal = {Electronic Journal of Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Local law for random Gram matrices}},
doi = {10.1214/17-EJP42},
volume = {22},
year = {2017},
}
@article{550,
abstract = {For large random matrices X with independent, centered entries but not necessarily identical variances, the eigenvalue density of XX* is well-approximated by a deterministic measure on ℝ. We show that the density of this measure has only square and cubic-root singularities away from zero. We also extend the bulk local law in [5] to the vicinity of these singularities.},
author = {Alt, Johannes},
issn = {1083589X},
journal = {Electronic Communications in Probability},
publisher = {Institute of Mathematical Statistics},
title = {{Singularities of the density of states of random Gram matrices}},
doi = {10.1214/17-ECP97},
volume = {22},
year = {2017},
}
@article{1677,
abstract = {We consider real symmetric and complex Hermitian random matrices with the additional symmetry hxy = hN-y,N-x. The matrix elements are independent (up to the fourfold symmetry) and not necessarily identically distributed. This ensemble naturally arises as the Fourier transform of a Gaussian orthogonal ensemble. Italso occurs as the flip matrix model - an approximation of the two-dimensional Anderson model at small disorder. We show that the density of states converges to the Wigner semicircle law despite the new symmetry type. We also prove the local version of the semicircle law on the optimal scale.},
author = {Alt, Johannes},
journal = {Journal of Mathematical Physics},
number = {10},
publisher = {American Institute of Physics},
title = {{The local semicircle law for random matrices with a fourfold symmetry}},
doi = {10.1063/1.4932606},
volume = {56},
year = {2015},
}