Averaging Fluctuations in Resolvents of Random Band Matrices

作者：Erdos Laszlo^*; Knowles Antti; Yau Horng Tzer

来源：Annales Henri Poincare, 2013, 14(8): 1837-1926.

DOI：10.1007/s00023-013-0235-y

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 ErdAs 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 (ErdAs et al., arXiv:1205.5669, 2013).

• 出版日期2013-12