### Abstract

We study the special case of n× n 1D Gaussian Hermitian random band matrices, when the covariance of the elements is determined by J=(-W2▵+1)-1. Assuming that the band width W≪n, we prove that the limit of the normalized second mixed moment of characteristic polynomials (as W, n→ ∞) is equal to one, and so it does not coincide with that for GUE. This complements the result of Shcherbina (J Stat Phys 155(3):466–499, 2014) and proves the existence of the expected crossover for 1D Hermitian random band matrices at W∼n on the level of characteristic polynomials.

Original language | English (US) |
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Pages (from-to) | 1009-1044 |

Number of pages | 36 |

Journal | Communications In Mathematical Physics |

Volume | 351 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2017 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

Shcherbina, M., & Shcherbina, T. (2017). Characteristic Polynomials for 1D Random Band Matrices from the Localization Side.

*Communications In Mathematical Physics*,*351*(3), 1009-1044. https://doi.org/10.1007/s00220-017-2849-2