thesis
Dyson equation and eigenvalue statistics of random matrices
ISTA Thesis
published
Johannes
Alt
author 36D3D8B6-F248-11E8-B48F-1D18A9856A87
László
Erdös
supervisor
LaEr
department
Random matrices, universality and disordered quantum systems
project
The eigenvalue density of many large random matrices is well approximated by a deterministic measure, the self-consistent density of states. In the present work, we show this behaviour for several classes of random matrices. In fact, we establish that, in each of these classes, the self-consistent density of states approximates the eigenvalue density of the random matrix on all scales slightly above the typical eigenvalue spacing. For large classes of random matrices, the self-consistent density of states exhibits several universal features. We prove that, under suitable assumptions, random Gram matrices and Hermitian random matrices with decaying correlations have a 1/3-Hölder continuous self-consistent density of states ρ on R, which is analytic, where it is positive, and has either a square root edge or a cubic root cusp, where it vanishes. We, thus, extend the validity of the corresponding result for Wigner-type matrices from [4, 5, 7]. We show that ρ is determined as the inverse Stieltjes transform of the normalized trace of the unique solution m(z) to the Dyson equation −m(z) −1 = z − a + S[m(z)] on C N×N with the constraint Im m(z) ≥ 0. Here, z lies in the complex upper half-plane, a is a self-adjoint element of C N×N and S is a positivity-preserving operator on C N×N encoding the first two moments of the random matrix. In order to analyze a possible limit of ρ for N → ∞ and address some applications in free probability theory, we also consider the Dyson equation on infinite dimensional von Neumann algebras. We present two applications to random matrices. We first establish that, under certain assumptions, large random matrices with independent entries have a rotationally symmetric self-consistent density of states which is supported on a centered disk in C. Moreover, it is infinitely often differentiable apart from a jump on the boundary of this disk. Second, we show edge universality at all regular (not necessarily extreme) spectral edges for Hermitian random matrices with decaying correlations.
https://research-explorer.ista.ac.at/download/149/6241/2018_thesis_Alt.pdf
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Institute of Science and Technology Austria2018
eng
2663-337X10.15479/AT:ISTA:TH_1040
456
https://research-explorer.ista.ac.at/record/1677 https://research-explorer.ista.ac.at/record/550 https://research-explorer.ista.ac.at/record/6183 https://research-explorer.ista.ac.at/record/566 https://research-explorer.ista.ac.at/record/1010 https://research-explorer.ista.ac.at/record/6240 https://research-explorer.ista.ac.at/record/6184
Alt, Johannes. <i>Dyson Equation and Eigenvalue Statistics of Random Matrices</i>. Institute of Science and Technology Austria, 2018, doi:<a href="https://doi.org/10.15479/AT:ISTA:TH_1040">10.15479/AT:ISTA:TH_1040</a>.
J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” Institute of Science and Technology Austria, 2018.
Alt, J. (2018). <i>Dyson equation and eigenvalue statistics of random matrices</i>. Institute of Science and Technology Austria. <a href="https://doi.org/10.15479/AT:ISTA:TH_1040">https://doi.org/10.15479/AT:ISTA:TH_1040</a>
Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices. Institute of Science and Technology Austria.
Alt J. Dyson equation and eigenvalue statistics of random matrices. 2018. doi:<a href="https://doi.org/10.15479/AT:ISTA:TH_1040">10.15479/AT:ISTA:TH_1040</a>
J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, Institute of Science and Technology Austria, 2018.
Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.” Institute of Science and Technology Austria, 2018. <a href="https://doi.org/10.15479/AT:ISTA:TH_1040">https://doi.org/10.15479/AT:ISTA:TH_1040</a>.
1492018-12-11T11:44:53Z2024-02-22T14:34:33Z