THE DYSON EQUATION WITH LINEAR SELF-ENERGY: SPECTRAL BANDS, EDGES AND CUSPS

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Standard

THE DYSON EQUATION WITH LINEAR SELF-ENERGY : SPECTRAL BANDS, EDGES AND CUSPS. / Alt, Johannes; Erdos, Laszlo; Kruger, Torben.

I: Documenta Mathematica, Bind 25, 2020, s. 1421-1539.

Publikation: Bidrag til tidsskriftTidsskriftartikelfagfællebedømt

Harvard

Alt, J, Erdos, L & Kruger, T 2020, 'THE DYSON EQUATION WITH LINEAR SELF-ENERGY: SPECTRAL BANDS, EDGES AND CUSPS', Documenta Mathematica, bind 25, s. 1421-1539. https://doi.org/10.25537/dm.2020v25.1421-1539

APA

Alt, J., Erdos, L., & Kruger, T. (2020). THE DYSON EQUATION WITH LINEAR SELF-ENERGY: SPECTRAL BANDS, EDGES AND CUSPS. Documenta Mathematica, 25, 1421-1539. https://doi.org/10.25537/dm.2020v25.1421-1539

Vancouver

Alt J, Erdos L, Kruger T. THE DYSON EQUATION WITH LINEAR SELF-ENERGY: SPECTRAL BANDS, EDGES AND CUSPS. Documenta Mathematica. 2020;25:1421-1539. https://doi.org/10.25537/dm.2020v25.1421-1539

Author

Alt, Johannes ; Erdos, Laszlo ; Kruger, Torben. / THE DYSON EQUATION WITH LINEAR SELF-ENERGY : SPECTRAL BANDS, EDGES AND CUSPS. I: Documenta Mathematica. 2020 ; Bind 25. s. 1421-1539.

Bibtex

@article{e643140142d24edcbb5c1a963fc669bf,
title = "THE DYSON EQUATION WITH LINEAR SELF-ENERGY: SPECTRAL BANDS, EDGES AND CUSPS",
abstract = "We study the unique solution m of the Dyson equation-m(z)(-1) = z1 - a + S[m(z)]on a von Neumann algebra A with the constraint Im m >= 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 I Under suitable assumptions, we establish that this measure has a uniformly 1/3-Holder 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 [8] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner type matrices [15, 19] We also extend the finite dimensional band mass formula from [8] 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.",
keywords = "Dyson equation, positive operator-valued measure, Stieltjes transform, band rigidity, INFORMATION MEASURE, RANDOM MATRICES, ENTROPY, ANALOGS",
author = "Johannes Alt and Laszlo Erdos and Torben Kruger",
year = "2020",
doi = "10.25537/dm.2020v25.1421-1539",
language = "English",
volume = "25",
pages = "1421--1539",
journal = "Documenta Mathematica",
issn = "1431-0635",
publisher = "Deutsche Mathematiker Vereinigung",

}

RIS

TY - JOUR

T1 - THE DYSON EQUATION WITH LINEAR SELF-ENERGY

T2 - SPECTRAL BANDS, EDGES AND CUSPS

AU - Alt, Johannes

AU - Erdos, Laszlo

AU - Kruger, Torben

PY - 2020

Y1 - 2020

N2 - We study the unique solution m of the Dyson equation-m(z)(-1) = z1 - a + S[m(z)]on a von Neumann algebra A with the constraint Im m >= 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 I Under suitable assumptions, we establish that this measure has a uniformly 1/3-Holder 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 [8] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner type matrices [15, 19] We also extend the finite dimensional band mass formula from [8] 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.

AB - We study the unique solution m of the Dyson equation-m(z)(-1) = z1 - a + S[m(z)]on a von Neumann algebra A with the constraint Im m >= 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 I Under suitable assumptions, we establish that this measure has a uniformly 1/3-Holder 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 [8] and they play a key role in the proof of the Pearcey universality at the cusp for Wigner type matrices [15, 19] We also extend the finite dimensional band mass formula from [8] 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.

KW - Dyson equation

KW - positive operator-valued measure

KW - Stieltjes transform

KW - band rigidity

KW - INFORMATION MEASURE

KW - RANDOM MATRICES

KW - ENTROPY

KW - ANALOGS

U2 - 10.25537/dm.2020v25.1421-1539

DO - 10.25537/dm.2020v25.1421-1539

M3 - Journal article

VL - 25

SP - 1421

EP - 1539

JO - Documenta Mathematica

JF - Documenta Mathematica

SN - 1431-0635

ER -

ID: 257706847