84 research outputs found
Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
For complex Wigner-type matrices, i.e. Hermitian random matrices with
independent, not necessarily identically distributed entries above the
diagonal, we show that at any cusp singularity of the limiting eigenvalue
distribution the local eigenvalue statistics are universal and form a Pearcey
process. Since the density of states typically exhibits only square root or
cubic root cusp singularities, our work complements previous results on the
bulk and edge universality and it thus completes the resolution of the
Wigner-Dyson-Mehta universality conjecture for the last remaining universality
type in the complex Hermitian class. Our analysis holds not only for exact
cusps, but approximate cusps as well, where an extended Pearcey process
emerges. As a main technical ingredient we prove an optimal local law at the
cusp for both symmetry classes. This result is also used in the companion paper
[arXiv:1811.04055] where the cusp universality for real symmetric Wigner-type
matrices is proven.Comment: 58 pages, 2 figures. Updated introduction and reference
Universality for general Wigner-type matrices
We consider the local eigenvalue distribution of large self-adjoint random matrices with centered independent entries.
In contrast to previous works the matrix of variances is not assumed to be stochastic. Hence the density of states is
not the Wigner semicircle law. Its possible shapes are described in the
companion paper [1]. We show that as grows, the resolvent,
, converges to a diagonal matrix, , where
solves the vector equation that has
been analyzed in [1]. We prove a local law down to the smallest spectral
resolution scale, and bulk universality for both real symmetric and complex
hermitian symmetry classes.Comment: Changes in version 3: The format of pictures was changed to resolve a
conflict with certain pdf viewer
The Dyson equation with linear self-energy: spectral bands, edges and cusps
We study the unique solution of the Dyson equation on a von Neumann algebra with the constraint
. Here, lies in the complex upper half-plane, is
a self-adjoint element of and is a positivity-preserving
linear operator on . We show that is the Stieltjes transform
of a compactly supported -valued measure on . Under
suitable assumptions, we establish that this measure has a uniformly
-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 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.Comment: 72 pages, 4 figures. We added several equivalent characterisations
for a regular edge and an additional reference. Moreover, we included a proof
for Lemma B.2 and removed some typos and inconsistencie
Singularity degree of structured random matrices
We consider the density of states of structured Hermitian random matrices
with a variance profile. As the dimension tends to infinity the associated
eigenvalue density can develop a singularity at the origin. The severity of
this singularity depends on the relative positions of the zero submatrices. We
provide a classification of all possible singularities and determine the
exponent in the density blow-up, which we label the singularity degree
Spectral radius of random matrices with independent entries
We consider random matrices with independent and centered
entries and a general variance profile. We show that the spectral radius of
converges with very high probability to the square root of the spectral radius
of the variance matrix of when tends to infinity. We also establish the
optimal rate of convergence, that is a new result even for general i.i.d.
matrices beyond the explicitly solvable Gaussian cases. The main ingredient is
the proof of the local inhomogeneous circular law [arXiv:1612.07776] at the
spectral edge.Comment: 44 pages; Included additional explanations for one step of the proo
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