TY - JOUR
T1 - Random Band Matrices in the Delocalized Phase, II
T2 - Generalized Resolvent Estimates
AU - Bourgade, P.
AU - Yang, F.
AU - Yau, H. T.
AU - Yin, J.
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2019/3/30
Y1 - 2019/3/30
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
SN - 0022-4715
VL - 174
SP - 1189
EP - 1221
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 6
ER -