preprint
The Dyson equation with linear self-energy: Spectral bands, edges and cusps
submitted
Johannes
Alt
author 36D3D8B6-F248-11E8-B48F-1D18A9856A87
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Torben H
Krüger
author 3020C786-F248-11E8-B48F-1D18A9856A870000-0002-4821-3297
LaEr
department
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.
2018
eng
arXiv
1804.07752
https://research-explorer.ista.ac.at/record/149 https://research-explorer.ista.ac.at/record/14694
Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. arXiv, 1804.07752.
Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>ArXiv</i>, n.d.
J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and cusps,” <i>arXiv</i>. .
Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. <i>arXiv</i>.
Alt, J., Erdös, L., & Krüger, T. H. (n.d.). The Dyson equation with linear self-energy: Spectral bands, edges and cusps. <i>arXiv</i>.
J. Alt, L. Erdös, T.H. Krüger, ArXiv (n.d.).
Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>ArXiv</i>, 1804.07752.
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