### 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) |
---|---|

Journal | Journal of Statistical Physics |

DOIs | |

State | Accepted/In press - Jan 1 2019 |

### Fingerprint

### Keywords

- Band random matrix
- Delocalized phase
- Generalized resolvent

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Journal of Statistical Physics*. https://doi.org/10.1007/s10955-019-02229-z

**Random Band Matrices in the Delocalized Phase, II : Generalized Resolvent Estimates.** / Bourgade, Paul; Yang, F.; Yau, H. T.; Yin, J.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Random Band Matrices in the Delocalized Phase, II

T2 - Generalized Resolvent Estimates

AU - Bourgade, Paul

AU - Yang, F.

AU - Yau, H. T.

AU - Yin, J.

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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= (Hij) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances E| Hij| 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 Zij= (z11 ⩽ i ⩽ W+ z~ 1i > W) δij, 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≫ N3 / 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).

AB - 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= (Hij) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances E| Hij| 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 Zij= (z11 ⩽ i ⩽ W+ z~ 1i > W) δij, 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≫ N3 / 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).

KW - Band random matrix

KW - Delocalized phase

KW - Generalized resolvent

UR - http://www.scopus.com/inward/record.url?scp=85060199121&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060199121&partnerID=8YFLogxK

U2 - 10.1007/s10955-019-02229-z

DO - 10.1007/s10955-019-02229-z

M3 - Article

AN - SCOPUS:85060199121

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

ER -