The density of states of 1D random band matrices via a supersymmetric transfer operator

Margherita Disertori, Martin Lohmann, Sasha Sodin

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

Recently,M. and T. Shcherbina proved a pointwise semicircle lawfor the density of states of one-dimensional Gaussian band matrices of large bandwidth. The main step of their proof is a new method to study the spectral properties of non-self-adjoint operators in the semiclassical regime. The method is applied to a transfer operator constructed from the supersymmetric integral representation for the density of states. We present a simpler proof of a slightly upgraded version of the semicircle law, which requires only standard semiclassical arguments and some peculiar elementary computations. The simplification is due to the use of supersymmetry, which manifests itself in the commutation between the transfer operator and a family of transformations of superspace, and was applied earlier in the context of bandmatrices by Constantinescu. Other versions of this supersymmetry have been a crucial ingredient in the study of the localization- delocalization transition by theoretical physicists.

Original languageEnglish
Pages (from-to)125-191
Number of pages67
JournalJournal of Spectral Theory
Volume11
Issue number1
DOIs
StatePublished - 2021

Funding

FundersFunder number
Newton Institute
Horizon 2020 Framework Programme639305
Natural Sciences and Engineering Research Council of Canada
Royal Society
European Research Council
Deutsche Forschungsgemeinschaft

    Keywords

    • Density of states
    • Random band matrices
    • Supersymmetry

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