Isotropic self-consistent equations for mean-field random matrices

He, Yukun; Knowles, Antti; Rosenthal, Ron

doi:10.1007/s00440-017-0776-y

Cited by 40 publications

(42 citation statements)

References 38 publications

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“…Local laws have become a cornerstone in the analysis of spectral properties of large random matrices [4,8,20,23,29,35,37,51]. In its simplest form, a local law considers the normalized trace 1 N Tr G(ζ ) of the resolvent.…”

Section: A := E H S[r] := E (H − A)r(h − A)mentioning

confidence: 99%

Stability of the matrix Dyson equation and random matrices with correlations

Ajanki

Erdős

Krüger

2018

Probab. Theory Relat. Fields

145

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We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.

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Section: A := E H S[r] := E (H − A)r(h − A)mentioning

confidence: 99%

Stability of the matrix Dyson equation and random matrices with correlations

Ajanki

Erdős

Krüger

2018

Probab. Theory Relat. Fields

145

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“…for a centred random variable h. Unlike in previous works (e.g. [27], [32], [28], [9], [29], [26]), our object E G F * requires recursive cumulant expansions in order to control the error terms. The first expansion by (3.3) leads to the Schwinger-Dyson (or self-consistent) equation…”

Section: )mentioning

confidence: 99%

“…For a real symmetric Wigner matrix H, we shall use the generalized Stein Lemma [2,36], which can be viewed a more precise and quantitative version of Stein's method. It was developed in the context of random matrix theory in [6,7,27,28,31]. The proof of a slightly different version can be found in in [28].…”

Section: Preliminariesmentioning

confidence: 99%

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Mesoscopic eigenvalue density correlations of Wigner matrices

Knowles

2019

Probab. Theory Relat. Fields

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The celebrated Wigner-Gaudin-Mehta-Dyson (WGMD) (or sine kernel) statistics of random matrix theory describes the universal correlations of eigenvalues on the microscopic scale, i.e. correlations of eigenvalue densities at points whose separation is comparable to the typical eigenvalue spacing. We investigate to what extent these statistics remain valid on mesoscopic scales, where the densities are measured at points whose separation is much larger than the typical eigenvalue spacing. In the mesoscopic regime, density-density correlations are much weaker than in the microscopic regime. More precisely, we compute the connected two-point spectral correlation function of a Wigner matrix at two mesoscopically separated points. We obtain a precise and explicit formula for the two-point function. Among other things, this formula implies that the WGMD statistics are valid to leading order on all mesoscopic scales, that the real symmetric terms contain subleading corrections matching precisely the WGMD statistics, while in the complex Hermitian case these subleading corrections are absent. We also uncover non-universal subleading correlations, which dominate over the universal ones beyond a certain intermediate mesoscopic scale. Our formula reproduces, in the extreme macroscopic regime, the well-known non-universal correlations of linear statistics on the macroscopic scale. Thus, our results bridges, within one continuous unifying picture, two previously unrelated classes of results in random matrix theory: correlations of linear statistics and microscopic eigenvalue universality. The main technical achievement of the proof is the development of an algorithm that allows to compute asymptotic expansions up to arbitrary accuracy of arbitrary correlation functions of mesoscopic linear statistics for Wigner matrices. Its proof is based on a hierarchy of Schwinger-Dyson equations for a sufficiently large class of polynomials in the entries of the Green function. The hierarchy is indexed by a tree, whose depth is controlled using stopping rules. A key ingredient in the derivation of the stopping rules is a new estimate on the density of states, which we prove to have bounded derivatives of all order on all mesoscopic scales.

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“…Diagonal deformations (A is diagonal) are easier to handle, this class was considered even for a large diagonal in [60,61,64,66]. The general A was considered in [54]. Finally, a very challenging direction is to depart from the mean field condition, i.e.…”

Section: The Three Step Strategymentioning

confidence: 99%

The matrix Dyson equation and its applications for random matrices

Erdős¹

2019

Random Matrices

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These lecture notes are a concise introduction of recent techniques to prove local spectral universality for a large class of random matrices. The general strategy is presented following the recent book with H.T. Yau [45]. We extend the scope of this book by focusing on new techniques developed to deal with generalizations of Wigner matrices that allow for non-identically distributed entries and even for correlated entries. This requires to analyze a system of nonlinear equations, or more generally a nonlinear matrix equation called the Matrix Dyson Equation (MDE). We demonstrate that stability properties of the MDE play a central role in random matrix theory. The analysis of MDE is based upon joint works with J. Alt, O. Ajanki, D. Schröder and T.

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Isotropic self-consistent equations for mean-field random matrices

Cited by 40 publications

References 38 publications

Stability of the matrix Dyson equation and random matrices with correlations

Stability of the matrix Dyson equation and random matrices with correlations

Mesoscopic eigenvalue density correlations of Wigner matrices

The matrix Dyson equation and its applications for random matrices

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