Erdös, LászlóIST Austria ; Knowles, Antti; Yau, Horng-Tzer
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).
Annales Henri Poincare
1837 - 1926
Erdös L, Knowles A, Yau H. Averaging fluctuations in resolvents of random band matrices. Annales Henri Poincare. 2013;14(8):1837-1926. doi:10.1007/s00023-013-0235-y
Erdös, L., Knowles, A., & Yau, H. (2013). Averaging fluctuations in resolvents of random band matrices. Annales Henri Poincare. Birkhäuser. https://doi.org/10.1007/s00023-013-0235-y
Erdös, László, Antti Knowles, and Horng Yau. “Averaging Fluctuations in Resolvents of Random Band Matrices.” Annales Henri Poincare. Birkhäuser, 2013. https://doi.org/10.1007/s00023-013-0235-y.
L. Erdös, A. Knowles, and H. Yau, “Averaging fluctuations in resolvents of random band matrices,” Annales Henri Poincare, vol. 14, no. 8. Birkhäuser, pp. 1837–1926, 2013.
Erdös L, Knowles A, Yau H. 2013. Averaging fluctuations in resolvents of random band matrices. Annales Henri Poincare. 14(8), 1837–1926.
Erdös, László, et al. “Averaging Fluctuations in Resolvents of Random Band Matrices.” Annales Henri Poincare, vol. 14, no. 8, Birkhäuser, 2013, pp. 1837–926, doi:10.1007/s00023-013-0235-y.