article
The Dyson equation with linear self-energy: Spectral bands, edges and cusps
published
yes
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 \( -m(z)^{-1} = z\1 - a + S[m(z)] \) on a von Neumann algebra A with the constraint Imm≥0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of A and S is a positivity-preserving linear operator on A. We show that m is the Stieltjes transform of a compactly supported A-valued measure on R. Under suitable assumptions, we establish that this measure has a uniformly 1/3-Hölder 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 [the first author et al., Ann. Probab. 48, No. 2, 963--1001 (2020; Zbl 1434.60017)] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner-type matrices [G. Cipolloni et al., Pure Appl. Anal. 1, No. 4, 615--707 (2019; Zbl 07142203); the second author et al., Commun. Math. Phys. 378, No. 2, 1203--1278 (2020; Zbl 07236118)]. We also extend the finite dimensional band mass formula from [the first author et al., loc. cit.] 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.
https://research-explorer.ista.ac.at/download/14694/14695/2020_DocumentaMathematica_Alt.pdf
application/pdfno
EMS Press2020
eng
General Mathematics
Documenta Mathematica
1431-0635
1431-0643
1804.0775210.4171/dm/780
251421-1539
https://research-explorer.ista.ac.at/record/6183
J. Alt, L. Erdös, T.H. Krüger, Documenta Mathematica 25 (2020) 1421–1539.
Alt J, Erdös L, Krüger TH. 2020. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. Documenta Mathematica. 25, 1421–1539.
Alt, Johannes, László Erdös, and Torben H Krüger. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>Documenta Mathematica</i>. EMS Press, 2020. <a href="https://doi.org/10.4171/dm/780">https://doi.org/10.4171/dm/780</a>.
Alt, J., Erdös, L., & Krüger, T. H. (2020). The Dyson equation with linear self-energy: Spectral bands, edges and cusps. <i>Documenta Mathematica</i>. EMS Press. <a href="https://doi.org/10.4171/dm/780">https://doi.org/10.4171/dm/780</a>
Alt, Johannes, et al. “The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps.” <i>Documenta Mathematica</i>, vol. 25, EMS Press, 2020, pp. 1421–539, doi:<a href="https://doi.org/10.4171/dm/780">10.4171/dm/780</a>.
J. Alt, L. Erdös, and T. H. Krüger, “The Dyson equation with linear self-energy: Spectral bands, edges and cusps,” <i>Documenta Mathematica</i>, vol. 25. EMS Press, pp. 1421–1539, 2020.
Alt J, Erdös L, Krüger TH. The Dyson equation with linear self-energy: Spectral bands, edges and cusps. <i>Documenta Mathematica</i>. 2020;25:1421-1539. doi:<a href="https://doi.org/10.4171/dm/780">10.4171/dm/780</a>
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