Dyson equation and eigenvalue statistics of random matrices Dissertation Thesis

Author(s): Alt, Johannes
Advisor(s): Erdős, László
Committee Chair(s): Jösch, Maximilian
Committee Member(s): Maas, Jan
Title: Dyson equation and eigenvalue statistics of random matrices
Affiliation IST Austria
Abstract: 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.
Publication Title: IST Dissertation
Degree Granting Institution: IST Austria  
Degree: PhD
Degree Date: 2018-07-12
Start Page: 1
Total Pages: 456
Sponsor: European Research Council under ERC Advanced Grant, RANMAT, No. 338804.
DOI: 10.15479/AT:ISTA:TH_1040
Notes: I am grateful to all current and former members of the research group of my supervisor, László Erdős, at IST Austria during my PhD studies. I have benefited a lot from many inspiring discussions with them and the motivating atmosphere. Moreover, I very much appreciate the support of my thesis committee with the internal members, László Erdős and Jan Maas, from IST Austria and Jiří Černý as external member. Some people have directly contributed to some research projects of this thesis. They are László Erdős, Torben Krüger, Yuriy Nemish and Dominik Schröder (in alphabetical order). I am very grateful to them for our fruitful collaborations which have been a constant source of inspiration and motivation to me. Overall, my PhD studies have been a very good and rewarding experience for me. To a large extend, this is due to my supervisor, László Erdős. He has always made a lot of time to listen to my concerns and questions, to answer them, to introduce me to new mathematical ideas, to discuss about my research projects and to teach me how to efficiently communicate mathematics. László has been the best supervisor I could have imagined and, for that, I want to express my deep gratitude to him.
Open access: yes (repository)
IST Austria Authors
  1. Johannes Alt
    5 Alt
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