## Abstract

This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of N× N random band matrices H= (H _{ij} ) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances E| H _{ij} | ^{2} form a band matrix with typical band width 1 ≪ W≪ N. We consider the generalized resolvent of H defined as G(Z) : = (H- Z) ^{- 1} , where Z is a deterministic diagonal matrix such that Z _{ij} = (z1 _{1} _{⩽} _{i} _{⩽} _{W} + z~ 1 _{i} _{>} _{W} ) δ _{ij} , with two distinct spectral parameters z∈C+:={z∈C:Imz>0} and z~ ∈ C _{+} ∪ R. In this paper, we prove a sharp bound for the local law of the generalized resolvent G for W≫ N ^{3 / 4} . This bound is a key input for the proof of delocalization and bulk universality of random band matrices in Bourgade et al. (arXiv:1807.01559, 2018). Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in Yang and Yin (arXiv:1807.02447, 2018).

Original language | English (US) |
---|---|

Pages (from-to) | 1189-1221 |

Number of pages | 33 |

Journal | Journal of Statistical Physics |

Volume | 174 |

Issue number | 6 |

DOIs | |

State | Published - Mar 30 2019 |

## Keywords

- Band random matrix
- Delocalized phase
- Generalized resolvent

## ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics