Abstract
We consider a general class of random matrices whose entries are centred random variables, independent up to a symmetry constraint. We establish precise high-probability bounds on the averages of arbitrary monomials in the resolvent matrix entries. Our results generalize the previous results of Erdős et al. (Ann Probab, arXiv:1103.1919, 2013; Commun Math Phys, arXiv:1103.3869, 2013; J Combin 1(2):15–85, 2011) which constituted a key step in the proof of the local semicircle law with optimal error bound in mean-field random matrix models. Our bounds apply to random band matrices and improve previous estimates from order 2 to order 4 in the cases relevant to applications. In particular, they lead to a proof of the diffusion approximation for the magnitude of the resolvent of random band matrices. This, in turn, implies new delocalization bounds on the eigenvectors. The applications are presented in a separate paper (Erdős et al., arXiv:1205.5669, 2013).
Article PDF
Similar content being viewed by others
References
Erdős L., Knowles A.: Quantum diffusion and delocalization for band matrices with general distribution. Ann. H. Poincaré 12, 1227–1319 (2011)
Erdős L., Knowles A.: Quantum diffusion and eigenfunction delocalization in a random band matrix model. Commun. Math. Phys. 303, 509–554 (2011)
Erdős, L., Knowles, A., Yau, H.T., Yin, J.: Delocalization and diffusion profile for random band matrices. Commun. Math. Phys. arXiv:1205.5669 (2013, preprint)
Erdős, L., Knowles, A., Yau, H.T., Yin, J.: The local semicircle law for a general class of random matrices. arXiv:1212.0164 (preprint)
Erdős, L., Knowles, A., Yau, H.T., Yin, J.: Spectral statistics of Erdős–Rényi graphs I: local semicircle law. Ann. Probab. arXiv:1103.1919 (2013, preprint)
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. arXiv:1103.3869 (2013, preprint)
Erdős L., Péché S., Ramirez J.A., Schlein B., Yau H.T.: Bulk universality for Wigner matrices. Commun. Pure Appl. Math. 63, 895–925 (2010)
Erdős L., Ramirez J., Schlein B., Tao T., Vu V., Yau H.T.: Bulk universality for Wigner Hermitian matrices with subexponential decay. Math. Res. Lett. 17, 667–674 (2010)
Erdős L., Ramirez J., Schlein B., Yau H.T.: Universality of sine-kernel for Wigner matrices with a small Gaussian perturbation. Electron. J. Probab. 15, 526–604 (2010)
Erdős L., Schlein B., Yau H.T.: Local semicircle law and complete delocalization for Wigner random matrices. Commun. Math. Phys. 287, 641–655 (2009)
Erdős L., Schlein B., Yau H.T.: Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices. Ann. Probab. 37, 815–852 (2009)
Erdős L., Schlein B., Yau H.T.: Wegner estimate and level repulsion for Wigner random matrices. Int. Math. Res. Not. 2010, 436–479 (2009)
Erdős L., Schlein B., Yau H.T.: Universality of random matrices and local relaxation flow. Invent. Math. 185(1), 75–119 (2011)
Erdős L., Schlein B., Yau H.T., Yin J.: The local relaxation flow approach to universality of the local statistics of random matrices. Ann. Inst. Henri Poincaré (B) 48, 1–46 (2012)
Erdős, L., Yau, H.T., Yin, J.: Bulk universality for generalized Wigner matrices. arXiv:1001.3453 (preprint)
Erdős, L., Yau, H.T., Yin, J.: Rigidity of eigenvalues of generalized Wigner matrices. Adv. Math. arXiv:1007.4652 (to appear, preprint)
Erdős L., Yau H.T., Yin J.: Universality for generalized Wigner matrices with Bernoulli distribution. J. Combin. 1(2), 15–85 (2011)
Pillai, N.S., Yin, J.: Universality of covariance matrices. arXiv:1110.2501 (preprint)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Anton Bovier.
L. Erdős was partially supported by SFB-TR 12 Grant of the German Research Council.
A. Knowles was partially supported by NSF grant DMS-0757425.
H.-T. Yau was partially supported by NSF grants DMS-0757425, 0804279.
Rights and permissions
About this article
Cite this article
Erdős, L., Knowles, A. & Yau, HT. Averaging Fluctuations in Resolvents of Random Band Matrices. Ann. Henri Poincaré 14, 1837–1926 (2013). https://doi.org/10.1007/s00023-013-0235-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-013-0235-y