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Universality for 1d Random Band Matrices

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Abstract

We consider 1d random Hermitian \(N\times N\) block band matrices consisting of \(W\times W\) random Gaussian blocks (parametrized by \(j,k \in \Lambda =[1,n]\cap \mathbb {Z}\), \(N=nW\)) with a fixed entry’s variance \(J_{jk}=W^{-1}(\delta _{j,k}+\beta \Delta _{j,k})\) in each block. Considering the limit \(W, n\rightarrow \infty \), we prove that the behaviour of the second correlation function of such matrices in the bulk of the spectrum, as \(W\gg \sqrt{N}\), is determined by the Wigner–Dyson statistics. The method of the proof is based on the rigorous application of supersymmetric transfer matrix approach developed in Shcherbina and Shcherbina (J Stat Phys 172:627–664, 2018).

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Authors and Affiliations

  1. Institute for Low Temperature Physics, Kharkiv, Ukraine

    Mariya Shcherbina

  2. Karazin Kharkiv National University, Kharkiv, Ukraine

    Mariya Shcherbina

  3. Department of Mathematics, Princeton University, Princeton, USA

    Tatyana Shcherbina

  4. Department of Mathematics, University of Wisconsin-Madison, Madison, USA

    Tatyana Shcherbina

Authors
  1. Mariya Shcherbina
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  2. Tatyana Shcherbina
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Correspondence to Tatyana Shcherbina.

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Communicated by L. Erdos

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Supported in part by NSF Grant DMS-1700009.

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Shcherbina, M., Shcherbina, T. Universality for 1d Random Band Matrices. Commun. Math. Phys. 385, 667–716 (2021). https://doi.org/10.1007/s00220-021-04135-6

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