Random matrices with log-range correlations, and log-Sobolev inequalities

Kemp, Todd; Zimmermann, David

doi:10.5802/ambp.396

Cited by 3 publications

(1 citation statement)

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“…Further results in the local regime were achieved in [7] for finite range correlations, in [1] for correlated Gaussian entries and in [8] for more general correlations. For structured random matrices with correlations, see [17] (and references therein) for arbitrary correlations within log-size families of matrix entries, but with independence between these families, [19] for random band matrices with exchangeable entries and [12] for random band matrices with approximately uncorrelated entries.…”

Section: Introductionmentioning

confidence: 99%

Random Band and Block Matrices with Correlated Entries

Catalano¹,

Fleermann²,

Kirsch³

2022

Preprint

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In this paper, we derive limit laws for the empirical spectral distributions of random band and block matrices with correlated entries. In the first part of the paper, we study band matrices with approximately uncorrelated entries. We strengthen previously obtained results while requiring weaker assumptions, which is made possible by a refined application of the method of moments. In the second part of the paper, we introduce a new two-layered correlation structure we call SSB-HKW correlated, which enables the study of structured random matrices with correlated entries. Our results include semicircle laws in probability and almost surely, but we also obtain other limiting spectral distributions depending on the conditions. Simple necessary and sufficient conditions for the limit law to be the semicircle are provided. Our findings strengthen and extend many results already known.

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