### Abstract

The paper continues the development of the rigorous supersymmetric transfer matrix approach to the random band matrices started in (J Stat Phys 164:1233–1260, 2016; Commun Math Phys 351:1009–1044, 2017). We consider random Hermitian block band matrices consisting of W× W random Gaussian blocks (parametrized by j, k∈ Λ = [1 , n] ^{d}∩ Z^{d}) with a fixed entry’s variance J_{jk}= δ_{j}_{,}_{k}W^{- 1}+ βΔ _{j}_{,}_{k}W^{- 2}, β> 0 in each block. Taking the limit W→ ∞ with fixed n and β, we derive the sigma-model approximation of the second correlation function similar to Efetov’s one. Then, considering the limit β, n→ ∞, we prove that in the dimension d= 1 the behaviour of the sigma-model approximation in the bulk of the spectrum, as β≫ n, is determined by the classical Wigner–Dyson statistics.

Original language | English (US) |
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Pages (from-to) | 627-664 |

Number of pages | 38 |

Journal | Journal of Statistical Physics |

Volume | 172 |

Issue number | 2 |

DOIs | |

State | Published - Jul 1 2018 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Keywords

- Random band matrices
- Sigma-model approximation
- Transfer matrix approach
- Universality

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## Cite this

*Journal of Statistical Physics*,

*172*(2), 627-664. https://doi.org/10.1007/s10955-018-1969-1