Diffusion Profile for Random Band Matrices : A Short Proof

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Author(s)

Detail(s)

Original languageEnglish
Pages (from-to)666-716
Journal / PublicationJournal of Statistical Physics
Volume177
Issue number4
Online published13 Sept 2019
Publication statusPublished - Nov 2019
Externally publishedYes

Abstract

Let H be a Hermitian random matrix whose entries Hxy are independent, centred random variables with variances Sxy = E|Hxy|2, where x, y ∈ (Z/LZ)and ⩾ 1. The variance Sxy is negligible if |x − y| is bigger than the band width W. For =1 we prove that if W1+2/7 then the eigenvectors of H are delocalized and that an averaged version of |Gxy (z)|2 exhibits a diffusive behaviour, where (z) = (z)−1 is the resolvent of H. This improves the previous assumption W1+1/4 of Erdős et al. (Commun Math Phys 323:367–416, 2013). In higher dimensions ⩾ 2, we obtain similar results that improve the corresponding ones from Erdős et al. (Commun Math Phys 323:367–416, 2013). Our results hold for general variance profiles Sxy and distributions of the entries Hxy. The proof is considerably simpler and shorter than that of Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013), Erdős et al. (Commun Math Phys 323:367–416, 2013). It relies on a detailed Fourier space analysis combined with isotropic estimates for the fluctuating error terms. It is completely self-contained and avoids the intricate fluctuation averaging machinery from Erdős et al. (Ann Henri Poincaré 14:1837–1925, 2013).

Research Area(s)

  • math-ph, math.MP, math.PR, 15B52, 82B44, 82C44

Bibliographic Note

Information for this record is supplemented by the author(s) concerned.

Citation Format(s)

Diffusion Profile for Random Band Matrices: A Short Proof. / He, Yukun; Marcozzi, Matteo.
In: Journal of Statistical Physics, Vol. 177, No. 4, 11.2019, p. 666-716.

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review