2018
DOI: 10.1007/s00440-018-0835-z
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Stability of the matrix Dyson equation and random matrices with correlations

Abstract: 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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Cited by 58 publications
(150 citation statements)
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“…Remark 3.2. The operator F first appeared in [4]. In the current context we use it only in the limit ℑ(z) → 0.…”
Section: Exact Solution Atmentioning
confidence: 99%
See 1 more Smart Citation
“…Remark 3.2. The operator F first appeared in [4]. In the current context we use it only in the limit ℑ(z) → 0.…”
Section: Exact Solution Atmentioning
confidence: 99%
“…By differentiating in this additional parameter the product of resolvents is expressed in terms of the entries of the resolvent of the linearized matrix. The resolvent of the linearized matrix is then studied with the matrix Dyson equation [4]. Our formulas for functions of X and X * are then applied to compute the long time asymptotics of the norm and autocorrelation of solutions to the differential equations introduced at the beginning of this paper.…”
Section: Introductionmentioning
confidence: 99%
“…Note that (2.7) is very similar to the matrix Dyson equations (MDE) extensively studied in the literature in connection with large random matrices (see e.g. [40] and [2]). Their solutions typically give the deterministic part of the resolvent of a random matrix.…”
Section: Definition 24 (Nilpotent Linearization) a Linearization Ofmentioning
confidence: 73%
“…Only recently have people proved results on models with general correlation structure. In [6,4,2], bulk universality is proved for matrices where the correlation decays fast enough. In a recent paper [9], Erdös et al consider a model where the correlation between matrix entries has a power law decay of order d ≥ 12 in the long range and d ≥ 2 in the short range.…”
Section: Introductionmentioning
confidence: 99%
“…where R has sufficiently fast decay on off-diagonal entries. A similar strategy based on the smoothing effect of F is also used in [2]. Then we can embed and apply stability of the continuous solution.…”
Section: Introductionmentioning
confidence: 99%