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Transfer Matrix Approach to 1d Random Band Matrices: Density of States

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Abstract

We study the special case of \(n\times n\) 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by the matrix \(J=(-W^2\triangle +1)^{-1}\). Assuming that \(n\ge CW\log W\gg 1\), we prove that the averaged density of states coincides with the Wigner semicircle law up to the correction of order \(W^{-1}\).

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Acknowledgments

We are very grateful to Sasha Sodin, who drew our attention to the transfer matrix approach in application to 1d random band matrices, for many fruitful discussions. T. Shcherbina was supported by NSF Grant DMS-1128155.

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Authors and Affiliations

  1. Institute for Low Temperature Physics, Kharkiv, Ukraine

    Mariya Shcherbina

  2. Mathematical Department, Karazin Kharkiv National University, Kharkiv, Ukraine

    Mariya Shcherbina

  3. School of Mathematics, Institute for Advanced Study, Princeton, USA

    Tatyana Shcherbina

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

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Shcherbina, M., Shcherbina, T. Transfer Matrix Approach to 1d Random Band Matrices: Density of States. J Stat Phys 164, 1233–1260 (2016). https://doi.org/10.1007/s10955-016-1593-x

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  • DOI: https://doi.org/10.1007/s10955-016-1593-x

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