Random Band Matrices in the Delocalized Phase, II: Generalized Resolvent Estimates

  • P. Bourgade
  • F. Yang
  • H.-T. Yau
  • J. YinEmail author


This is the second part of a three part series abut delocalization for band matrices. In this paper, we consider a general class of \(N\times N\) random band matrices \(H=(H_{ij})\) whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances \(\mathbb {E} |H_{ij}|^2\) form a band matrix with typical band width \(1\ll W\ll N\). We consider the generalized resolvent of H defined as \(G(Z):=(H - Z)^{-1}\), where Z is a deterministic diagonal matrix such that \(Z_{ij}=\left( z\mathbb {1}_{1\leqslant i \leqslant W}+\widetilde{z}\mathbb {1}_{ i > W} \right) \delta _{ij}\), with two distinct spectral parameters \(z\in \mathbb {C}_+:=\{z\in \mathbb {C}:{{\,\mathrm{Im}\,}}z>0\}\) and \(\widetilde{z}\in \mathbb {C}_+\cup \mathbb {R}\). In this paper, we prove a sharp bound for the local law of the generalized resolvent G for \(W\gg N^{3/4}\). 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).


Band random matrix Delocalized phase Generalized resolvent 


  1. 1.
    Bourgade, P., Erdős, L., Yau, H.-T., Yin, J.: Universality for a class of random band matrices. Adv. Theor. Math. Phys. 21(3), 739–800 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bourgade, P., Yau, H.-T., Yin, J.: Random band matrices in the delocalized phase, I: quantum unique ergodicity and universality (2018). arXiv:1807.01559
  3. 3.
    Erdős, L., Schlein, B., Yau, H.-T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287(2), 641–655 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Spectral statistics of Erdős-Rényi graphs II: Eigenvalue spacing and the extreme eigenvalues. Commun. Math. Phys. 314, 587–640 (2012)ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Erdős, L., Yau, H.-T., Yin, J.: Bulk universality for generalized Wigner matrices. Probab. Theory Relat. Fields 154(1–2), 341–407 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Erdős, L., Knowles, A., Yau, H.-T.: Averaging fluctuations in resolvents of random band matrices. Ann. Henri Poincaré 14, 1837–1926 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: Delocalization and diffusion profile for random band matrices. Commun. Math. Phys. 323(1), 367–416 (2013)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erdős, L., Knowles, A., Yau, H.-T., Yin, J.: The local semicircle law for a general class of random matrices. Electron. J. Prob. 18(59), 1–58 (2013)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Knowles, A., Yin, J.: The isotropic semicircle law and deformation of Wigner matrices. Commun. Pure Appl. Math. 66, 1663–1749 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Yang, F., Yin, J.: Random band matrices in the delocalized phase, III: averaging fluctuations (2018). arXiv:1807.02447

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Courant InstituteNew YorkUSA
  2. 2.University of California, Los AngelesLos AngelesUSA
  3. 3.Harvard UniversityCambridgeUSA

Personalised recommendations