THE DYSON EQUATION WITH LINEAR SELF-ENERGY: SPECTRAL BANDS, EDGES AND CUSPS
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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 tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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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