Alt, Johannes; Erdős, László; Krüger, Torben

The Dyson Equation with Linear Self-Energy: Spectral Bands, Edges and Cusps

Doc. Math. 25, 1421-1539 (2020)
DOI: 10.25537/dm.2020v25.1421-1539
Communicated by Friedrich Götze


We study the unique solution \(m\) of the Dyson equation \[ -m(z)^{-1} = z\1 - a + S[m(z)] \] on a von Neumann algebra \(\mathcal{A}\) with the constraint \(\text{Im}\, m\ge 0\). 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 \(1/3\)-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 [\textit{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.

Mathematics Subject Classification

60B20, 46L54, 60F05


Dyson equation, positive operator-valued measure, Stieltjes transform, band rigidity, eigenvalue distribution


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Alt, Johannes
Section of Mathematics, University of Geneva, 2-4 Rue du Lièvre, CH-1211 Genève 4, Switzerland
Erdős, László
IST Austria, Am Campus 1, A-3400 Klosterneuburg, Austria
Krüger, Torben
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 København, Denmark