Random band matrices in the delocalized phase, III: averaging fluctuations

• 成果类型: Article
• 署名作者: Yang, Fan; Yin, Jun
• 署名单位: University of Pennsylvania; University of California System; University of California Los Angeles
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-01013-5
• 发表日期: 2021
• 页码: 451-540
• 关键词: spectral statistics; characteristic-polynomials; scaling properties; UNIVERSALITY; localization; diffusion

摘要

We consider a general class of symmetric or Hermitian random band matrices H = (h(xy))(x, y.)is an element of([1,N]d) in any dimension d >= 1, where the entries are independent, centered random variables with variances s(xy) = E-vertical bar hxy vertical bar 2. We assume that sxy vanishes if vertical bar x - y vertical bar exceeds the band width W, and we are interested in the mesoscopic scale with 1 << W << N. Define the generalized resolvent of H as G(H, Z) := (H - Z)(-1), where Z is a deterministic diagonal matrix with entries Z(xx) is an element of C+ for all x. Then we establish a precise high-probability bound on certain averages of polynomials of the resolvent entries. As an application of this fluctuation averaging result, we give a selfcontained proof for the delocalization of random band matrices in dimensions d >= 2. More precisely, for any fixed d >= 2, we prove that the bulk eigenvectors of H are delocalized in certain averaged sense if N <= W1+ d/2. This improves the corresponding results in He and Marcozzi (Diffusion profile for random band matrices: a short proof, 2018. arXiv:1804.09446) that imposed the assumption N << W1+d/d+1, and the results in Erd os and Knowles (Ann Henri Poincare12(7):1227-1319, 2011; Commun Math Phys 303(2): 509-554, 2011) that imposed the assumption N << W1+d/6. For 1D random band matrices, our fluctuation averaging result was used in Bourgade et al. (J Stat Phys 174:1189-1221, 2019; Random band matrices in the delocalized phase, I: quantum unique ergodicity and universality, 2018. arXiv:1807.01559) to prove the delocalization conjecture and bulk universality for random band matrices with N << W-4/3.