---
OA_place: publisher
_id: '19540'
abstract:
- lang: eng
text: "This thesis deals with several different models for complex quantum mechanical
systems and is structured in three main parts. \r\n\t\r\nIn Part I, we study mean
field random matrices as models for quantum Hamiltonians. Our focus lies on proving
concentration estimates for resolvents of random matrices, so-called local laws,
mostly in the setting of multiple resolvents. These estimates have profound consequences
for eigenvector overlaps and thermalization problems. More concretely, we obtain,
e.g., the optimal eigenstate thermalization hypothesis (ETH) uniformly in the
spectrum for Wigner matrices, an optimal lower bound on non-Hermitian eigenvector
overlaps, and prethermalization for deformed Wigner matrices.\tIn order to prove
our novel multi-resolvent local laws, we develop and devise two main methods,
the static Psi-method and the dynamical Zigzag strategy. \r\n\t\r\nIn Part II,
we study Bardeen-Cooper-Schrieffer (BCS) theory, the standard mean field microscopic
theory of superconductivity. We focus on asymptotic formulas for the characteristic
critical temperature and energy gap of a superconductor and prove universality
of their ratio in various physical regimes. Additionally, we investigate multi-band
superconductors and show that inter-band coupling effects can only enhance the
critical temperature. \r\n\t\r\nIn Part III, we study quantum lattice systems.
On the one hand, we show a strong version of the local-perturbations-perturb-locally
(LPPL) principle for the ground state of weakly interacting quantum spin systems
with a uniform on-site gap. On the other hand, we introduce a notion of a local
gap and rigorously justify response theory and the Kubo formula under the weakened
assumption of a local gap. \r\n\t\r\nAdditionally, we discuss two classes of problems
which do not fit into the three main parts of the thesis. These are deformational
rigidity of Liouville metrics on the torus and relativistic toy models of particle
creation via interior-boundary-conditions (IBCs). "
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Sven Joscha
full_name: Henheik, Sven Joscha
id: 31d731d7-d235-11ea-ad11-b50331c8d7fb
last_name: Henheik
orcid: 0000-0003-1106-327X
citation:
ama: 'Henheik SJ. Modeling complex quantum systems : Random matrices, BCS theory,
and quantum lattice systems. 2025. doi:10.15479/AT-ISTA-19540'
apa: 'Henheik, S. J. (2025). Modeling complex quantum systems : Random matrices,
BCS theory, and quantum lattice systems. Institute of Science and Technology
Austria. https://doi.org/10.15479/AT-ISTA-19540'
chicago: 'Henheik, Sven Joscha. “Modeling Complex Quantum Systems : Random Matrices,
BCS Theory, and Quantum Lattice Systems.” Institute of Science and Technology
Austria, 2025. https://doi.org/10.15479/AT-ISTA-19540.'
ieee: 'S. J. Henheik, “Modeling complex quantum systems : Random matrices, BCS theory,
and quantum lattice systems,” Institute of Science and Technology Austria, 2025.'
ista: 'Henheik SJ. 2025. Modeling complex quantum systems : Random matrices, BCS
theory, and quantum lattice systems. Institute of Science and Technology Austria.'
mla: 'Henheik, Sven Joscha. Modeling Complex Quantum Systems : Random Matrices,
BCS Theory, and Quantum Lattice Systems. Institute of Science and Technology
Austria, 2025, doi:10.15479/AT-ISTA-19540.'
short: 'S.J. Henheik, Modeling Complex Quantum Systems : Random Matrices, BCS Theory,
and Quantum Lattice Systems, Institute of Science and Technology Austria, 2025.'
corr_author: '1'
date_created: 2025-04-10T21:21:18Z
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id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
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orcid: 0000-0001-5366-9603
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lattice systems'
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OA_place: publisher
_id: '20575'
abstract:
- lang: eng
text: "This thesis deals with eigenvalue and eigenvector universality results for
random matrix ensembles equipped with non-trivial spatial structure. We consider
both mean-field models with a general variance profile (Wigner-type matrices)
and correlation structure (correlated matrices) among the entries, as well as
non-mean-field random band matrices with bandwidth W >> N^(1/2).\r\n\r\nTo extract
the universal properties of random matrix spectra and eigenvectors, we obtain
concentration estimates for their resolvent, the local laws, which generalize
the celebrated Wigner semicircle law for a broad class of random matrices to much
finer spectral scales. The local laws hold for both a single resolvent as well
as for products of multiple resolvents, known as resolvent chains, and express
the remarkable approximately-deterministic behavior of these objects down to the
microscopic scale.\r\n\r\nOur primary tool for establishing the local laws is
the dynamical Zigzag strategy, which we develop in the setting of spatially-inhomogeneous
random matrices. Our proof method systematically addresses the challenges arising
from non-trivial spatial structures and is robust to all types of singularities
in the spectrum, as we demonstrate in the correlated setting. Furthermore, we
incorporate the analysis of the deterministic resolvent chain approximations into
the dynamical framework of the Zigzag strategy, synthesizing a unified toolkit
for establishing multi-resolvent local laws.\r\n\r\nUsing these methods, we prove
complete eigenvector delocalization, the Eigenstate Thermalization Hypothesis,
and Wigner-Dyson universality in the bulk for random band matrices down to the
optimal bandwidth W >> N^(1/2). For mean-field ensembles, we establish universality
of local eigenvalue statistics at the cups for random matrices with correlated
entries, and the Eigenstate Thermalization Hypothesis for Wigner-type matrices
in the bulk of the spectrum.\r\n\r\nFinally, this thesis also contains other applications
of the multi-resolvent local laws to spatially-inhomogeneous random matrices,
obtained prior to the development of the Zigzag strategy. In particular, we provide
a complete analysis of mesoscopic linear-eigenvalue statistics of Wigner-type
matrices in all spectral regimes, including the novel cusps, and rigorously establish
the prethermalization phenomenon for deformed Wigner matrices.\r\n\r\nThe main
body of this thesis consists of seven research papers (listed on page xi), each
presented in a separate chapter with its own introduction and all relevant context,
suitable to be read independently. We ask the reader’s indulgence for the repetitions
in the historical overviews and other minor redundancies that remain among the
chapters as a result. The overall Introduction, preceding the chapters, provides
a condensed, informal summary of the main ideas and concepts at the core of these
works.\r\n"
acknowledgement: "The work comprising this thesis was supported by the ERC Advanced
Grant \"RMTBeyond\"\r\nNo.101020331 awarded to my advisor."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Volodymyr
full_name: Riabov, Volodymyr
id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
last_name: Riabov
citation:
ama: Riabov V. Universality in random matrices with spatial structure. 2025. doi:10.15479/AT-ISTA-20575
apa: Riabov, V. (2025). Universality in random matrices with spatial structure.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT-ISTA-20575
chicago: Riabov, Volodymyr. “Universality in Random Matrices with Spatial Structure.”
Institute of Science and Technology Austria, 2025. https://doi.org/10.15479/AT-ISTA-20575.
ieee: V. Riabov, “Universality in random matrices with spatial structure,” Institute
of Science and Technology Austria, 2025.
ista: Riabov V. 2025. Universality in random matrices with spatial structure. Institute
of Science and Technology Austria.
mla: Riabov, Volodymyr. Universality in Random Matrices with Spatial Structure.
Institute of Science and Technology Austria, 2025, doi:10.15479/AT-ISTA-20575.
short: V. Riabov, Universality in Random Matrices with Spatial Structure, Institute
of Science and Technology Austria, 2025.
corr_author: '1'
date_created: 2025-10-29T19:12:24Z
date_published: 2025-11-03T00:00:00Z
date_updated: 2026-04-07T12:32:20Z
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---
OA_place: publisher
_id: '17164'
abstract:
- lang: eng
text: "This thesis is structured into two parts. In the first part, we consider
the random\r\nvariable X := Tr(f1(W)A1 . . . fk(W)Ak) where W is an N × N Hermitian
Wigner matrix, k ∈ N, and we choose (possibly N-dependent) regular functions f1,
. . . , fk as well as\r\nbounded deterministic matrices A1, . . . , Ak. In this
context, we prove a functional central\r\nlimit theorem on macroscopic and mesoscopic
scales, showing that the fluctuations of X\r\naround its expectation are Gaussian
and that the limiting covariance structure is given\r\nby a deterministic recursion.
We further give explicit error bounds in terms of the scaling\r\nof f1, . . .
, fk and the number of traceless matrices among A1, . . . , Ak, thus extending\r\nthe
results of Cipolloni, Erdős and Schröder [40] to products of arbitrary length
k ≥ 2.\r\nAnalyzing the underlying combinatorics leads to a non-recursive formula
for the variance\r\nof X as well as the covariance of X and Y := Tr(fk+1(W)Ak+1
. . . fk+ℓ(W)Ak+ℓ) of similar\r\nbuild. When restricted to polynomials, these
formulas reproduce recent results of Male,\r\nMingo, Peché, and Speicher [107],
showing that the underlying combinatorics of noncrossing partitions and annular
non-crossing permutations continue to stay valid beyond\r\nthe setting of second-order
free probability theory. As an application, we consider the\r\nfluctuation of
Tr(eitW A1e\r\n−itW A2)/N around its thermal value Tr(A1) Tr(A2)/N2 when t\r\nis
large and give an explicit formula for the variance.\r\nThe second part of the
thesis collects three smaller projects focusing on different random\r\nmatrix
models. In the first project, we show that a class of weakly perturbed Hamiltonians\r\nof
the form Hλ = H0 + λW, where W is a Wigner matrix, exhibits prethermalization.\r\nThat
is, the time evolution generated by Hλ relaxes to its ultimate thermal state via
an\r\nintermediate prethermal state with a lifetime of order λ\r\n−2\r\n. As the
main result, we obtain\r\na general relaxation formula, expressing the perturbed
dynamics via the unperturbed\r\ndynamics and the ultimate thermal state. The proof
relies on a two-resolvent global law\r\nfor the deformed Wigner matrix Hλ.\r\nThe
second project focuses on correlated random matrices, more precisely on a correlated
N × N Hermitian random matrix with a polynomially decaying metric correlation\r\nstructure.
A trivial a priori bound shows that the operator norm of this model is stochastically
dominated by √\r\nN. However, by calculating the trace of the moments of the matrix\r\nand
using the summable decay of the cumulants, the norm estimate can be improved to
a\r\nbound of order one.\r\nIn the third project, we consider a multiplicative
perturbation of the form UA(t) where U\r\nis a unitary random matrix and A = diag(t,
1, ..., 1). This so-called UA model was\r\nfirst introduced by Fyodorov [73] for
its applications in scattering theory. We give a\r\ngeneral description of the
eigenvalue trajectories obtained by varying the parameter t and\r\nintroduce a
flow of deterministic domains that separates the outlier resulting from the\r\nrank-one
perturbation from the typical eigenvalues for all sub-critical timescales. The\r\nresults
are obtained under generic assumptions on U that hold for various unitary random\r\nmatrices,
including the circular unitary ensemble (CUE) in the original formulation of\r\nthe
model."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Jana
full_name: Reker, Jana
id: e796e4f9-dc8d-11ea-abe3-97e26a0323e9
last_name: Reker
citation:
ama: 'Reker J. Central limit theorems for random matrices: From resolvents to free
probability. 2024. doi:10.15479/at:ista:17164'
apa: 'Reker, J. (2024). Central limit theorems for random matrices: From resolvents
to free probability. Institute of Science and Technology Austria. https://doi.org/10.15479/at:ista:17164'
chicago: 'Reker, Jana. “Central Limit Theorems for Random Matrices: From Resolvents
to Free Probability.” Institute of Science and Technology Austria, 2024. https://doi.org/10.15479/at:ista:17164.'
ieee: 'J. Reker, “Central limit theorems for random matrices: From resolvents to
free probability,” Institute of Science and Technology Austria, 2024.'
ista: 'Reker J. 2024. Central limit theorems for random matrices: From resolvents
to free probability. Institute of Science and Technology Austria.'
mla: 'Reker, Jana. Central Limit Theorems for Random Matrices: From Resolvents
to Free Probability. Institute of Science and Technology Austria, 2024, doi:10.15479/at:ista:17164.'
short: 'J. Reker, Central Limit Theorems for Random Matrices: From Resolvents to
Free Probability, Institute of Science and Technology Austria, 2024.'
corr_author: '1'
date_created: 2024-06-24T11:23:29Z
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keyword:
- Random Matrices
- Spectrum
- Central Limit Theorem
- Resolvent
- Free Probability
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id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
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OA_place: publisher
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abstract:
- lang: eng
text: "In the first part of the thesis we consider Hermitian random matrices. Firstly,
we consider sample covariance matrices XX∗ with X having independent identically
distributed (i.i.d.) centred entries. We prove a Central Limit Theorem for differences
of linear statistics of XX∗ and its minor after removing the first column of X.
Secondly, we consider Wigner-type matrices and prove that the eigenvalue statistics
near cusp singularities of the limiting density of states are universal and that
they form a Pearcey process. Since the limiting eigenvalue distribution admits
only square root (edge) and cubic root (cusp) singularities, this concludes the
third and last remaining case of the Wigner-Dyson-Mehta universality conjecture.
The main technical ingredients are an optimal local law at the cusp, and the proof
of the fast relaxation to equilibrium of the Dyson Brownian motion in the cusp
regime.\r\nIn the second part we consider non-Hermitian matrices X with centred
i.i.d. entries. We normalise the entries of X to have variance N −1. It is well
known that the empirical eigenvalue density converges to the uniform distribution
on the unit disk (circular law). In the first project, we prove universality of
the local eigenvalue statistics close to the edge of the spectrum. This is the
non-Hermitian analogue of the TracyWidom universality at the Hermitian edge. Technically
we analyse the evolution of the spectral distribution of X along the Ornstein-Uhlenbeck
flow for very long time\r\n(up to t = +∞). In the second project, we consider
linear statistics of eigenvalues for macroscopic test functions f in the Sobolev
space H2+ϵ and prove their convergence to the projection of the Gaussian Free
Field on the unit disk. We prove this result for non-Hermitian matrices with real
or complex entries. The main technical ingredients are: (i) local law for products
of two resolvents at different spectral parameters, (ii) analysis of correlated
Dyson Brownian motions.\r\nIn the third and final part we discuss the mathematically
rigorous application of supersymmetric techniques (SUSY ) to give a lower tail
estimate of the lowest singular value of X − z, with z ∈ C. More precisely, we
use superbosonisation formula to give an integral representation of the resolvent
of (X − z)(X − z)∗ which reduces to two and three contour integrals in the complex
and real case, respectively. The rigorous analysis of these integrals is quite
challenging since simple saddle point analysis cannot be applied (the main contribution
comes from a non-trivial manifold). Our result\r\nimproves classical smoothing
inequalities in the regime |z| ≈ 1; this result is essential to prove edge universality
for i.i.d. non-Hermitian matrices."
acknowledgement: I gratefully acknowledge the financial support from the European
Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie
Grant Agreement No. 665385 and my advisor’s ERC Advanced Grant No. 338804.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Giorgio
full_name: Cipolloni, Giorgio
id: 42198EFA-F248-11E8-B48F-1D18A9856A87
last_name: Cipolloni
orcid: 0000-0002-4901-7992
citation:
ama: Cipolloni G. Fluctuations in the spectrum of random matrices. 2021. doi:10.15479/AT:ISTA:9022
apa: Cipolloni, G. (2021). Fluctuations in the spectrum of random matrices.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:9022
chicago: Cipolloni, Giorgio. “Fluctuations in the Spectrum of Random Matrices.”
Institute of Science and Technology Austria, 2021. https://doi.org/10.15479/AT:ISTA:9022.
ieee: G. Cipolloni, “Fluctuations in the spectrum of random matrices,” Institute
of Science and Technology Austria, 2021.
ista: Cipolloni G. 2021. Fluctuations in the spectrum of random matrices. Institute
of Science and Technology Austria.
mla: Cipolloni, Giorgio. Fluctuations in the Spectrum of Random Matrices.
Institute of Science and Technology Austria, 2021, doi:10.15479/AT:ISTA:9022.
short: G. Cipolloni, Fluctuations in the Spectrum of Random Matrices, Institute
of Science and Technology Austria, 2021.
corr_author: '1'
date_created: 2021-01-21T18:16:54Z
date_published: 2021-01-25T00:00:00Z
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name: Random matrices, universality and disordered quantum systems
publication_identifier:
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publication_status: published
publisher: Institute of Science and Technology Austria
status: public
supervisor:
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full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
title: Fluctuations in the spectrum of random matrices
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2021'
...
---
OA_place: publisher
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abstract:
- lang: eng
text: "In the first part of this thesis we consider large random matrices with arbitrary
expectation and a general slowly decaying correlation among its entries. We prove
universality of the local eigenvalue statistics and optimal local laws for the
resolvent in the bulk and edge regime. The main novel tool is a systematic diagrammatic
control of a multivariate cumulant expansion.\r\nIn the second part we consider
Wigner-type matrices and show that at any cusp singularity of the limiting eigenvalue
distribution the local eigenvalue statistics are uni- versal and form a Pearcey
process. Since the density of states typically exhibits only square root or cubic
root cusp singularities, our work complements previous results on the bulk and
edge universality and it thus completes the resolution of the Wigner- Dyson-Mehta
universality conjecture for the last remaining universality type. Our analysis
holds not only for exact cusps, but approximate cusps as well, where an ex- tended
Pearcey process emerges. As a main technical ingredient we prove an optimal local
law at the cusp, and extend the fast relaxation to equilibrium of the Dyson Brow-
nian motion to the cusp regime.\r\nIn the third and final part we explore the
entrywise linear statistics of Wigner ma- trices and identify the fluctuations
for a large class of test functions with little regularity. This enables us to
study the rectangular Young diagram obtained from the interlacing eigenvalues
of the random matrix and its minor, and we find that, despite having the same
limit, the fluctuations differ from those of the algebraic Young tableaux equipped
with the Plancharel measure."
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
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full_name: Schröder, Dominik J
id: 408ED176-F248-11E8-B48F-1D18A9856A87
last_name: Schröder
orcid: 0000-0002-2904-1856
citation:
ama: 'Schröder DJ. From Dyson to Pearcey: Universal statistics in random matrix
theory. 2019. doi:10.15479/AT:ISTA:th6179'
apa: 'Schröder, D. J. (2019). From Dyson to Pearcey: Universal statistics in
random matrix theory. Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:th6179'
chicago: 'Schröder, Dominik J. “From Dyson to Pearcey: Universal Statistics in Random
Matrix Theory.” Institute of Science and Technology Austria, 2019. https://doi.org/10.15479/AT:ISTA:th6179.'
ieee: 'D. J. Schröder, “From Dyson to Pearcey: Universal statistics in random matrix
theory,” Institute of Science and Technology Austria, 2019.'
ista: 'Schröder DJ. 2019. From Dyson to Pearcey: Universal statistics in random
matrix theory. Institute of Science and Technology Austria.'
mla: 'Schröder, Dominik J. From Dyson to Pearcey: Universal Statistics in Random
Matrix Theory. Institute of Science and Technology Austria, 2019, doi:10.15479/AT:ISTA:th6179.'
short: 'D.J. Schröder, From Dyson to Pearcey: Universal Statistics in Random Matrix
Theory, Institute of Science and Technology Austria, 2019.'
corr_author: '1'
date_created: 2019-03-28T08:58:59Z
date_published: 2019-03-18T00:00:00Z
date_updated: 2026-04-08T13:55:03Z
day: '18'
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- '519'
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call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
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supervisor:
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full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
title: 'From Dyson to Pearcey: Universal statistics in random matrix theory'
type: dissertation
user_id: ba8df636-2132-11f1-aed0-ed93e2281fdd
year: '2019'
...
---
OA_place: publisher
_id: '149'
abstract:
- lang: eng
text: 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.
alternative_title:
- ISTA Thesis
article_processing_charge: No
author:
- first_name: Johannes
full_name: Alt, Johannes
id: 36D3D8B6-F248-11E8-B48F-1D18A9856A87
last_name: Alt
citation:
ama: Alt J. Dyson equation and eigenvalue statistics of random matrices. 2018. doi:10.15479/AT:ISTA:TH_1040
apa: Alt, J. (2018). Dyson equation and eigenvalue statistics of random matrices.
Institute of Science and Technology Austria. https://doi.org/10.15479/AT:ISTA:TH_1040
chicago: Alt, Johannes. “Dyson Equation and Eigenvalue Statistics of Random Matrices.”
Institute of Science and Technology Austria, 2018. https://doi.org/10.15479/AT:ISTA:TH_1040.
ieee: J. Alt, “Dyson equation and eigenvalue statistics of random matrices,” Institute
of Science and Technology Austria, 2018.
ista: Alt J. 2018. Dyson equation and eigenvalue statistics of random matrices.
Institute of Science and Technology Austria.
mla: Alt, Johannes. Dyson Equation and Eigenvalue Statistics of Random Matrices.
Institute of Science and Technology Austria, 2018, doi:10.15479/AT:ISTA:TH_1040.
short: J. Alt, Dyson Equation and Eigenvalue Statistics of Random Matrices, Institute
of Science and Technology Austria, 2018.
corr_author: '1'
date_created: 2018-12-11T11:44:53Z
date_published: 2018-07-12T00:00:00Z
date_updated: 2026-04-08T14:11:37Z
day: '12'
ddc:
- '515'
- '519'
degree_awarded: PhD
department:
- _id: LaEr
doi: 10.15479/AT:ISTA:TH_1040
ec_funded: 1
file:
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oa_version: Published Version
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project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
call_identifier: FP7
grant_number: '338804'
name: Random matrices, universality and disordered quantum systems
publication_identifier:
issn:
- 2663-337X
publication_status: published
publisher: Institute of Science and Technology Austria
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pubrep_id: '1040'
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relation: part_of_dissertation
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- id: '1677'
relation: part_of_dissertation
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status: public
supervisor:
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full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
title: Dyson equation and eigenvalue statistics of random matrices
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
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