BULK UNIVERSALITY AND QUANTUM UNIQUE ERGODICITY FOR RANDOM BAND MATRICES IN HIGH DIMENSIONS

• 成果类型: Article
• 署名作者: Xu, Changji; Yang, Fan; Yau, Horng-Tzer; Yin, Jun
• 署名单位: Harvard University; Tsinghua University; Harvard University; University of California System; University of California Los Angeles
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1670
• 发表日期: 2024
• 页码: 765-837
• 关键词: fixed-energy universality; local semicircle law; spectral statistics; characteristic-polynomials; scaling properties; delocalization; localization; eigenvectors; diffusion

摘要

We consider Hermitian random band matrices H = (h(xy)) on the d-dimensional lattice (Z/LZ)(d), where the entries h(xy) = (h) over bar (yx) are independent centered complex Gaussian random variables with variances s(xy) = E|h(xy)|(2). The variance matrix S = (s(xy)) has a banded profile so that sxy is negligible if |x - y| exceeds the band width W. For dimensions d >= 7, we prove the bulk eigenvalue universality of H under the condition W >> L95/(d+95). Assuming that W >= L-epsilon for a small constant epsilon > 0, we also prove the quantum unique ergodicity for the bulk eigenvectors of H and a sharp local law for the Green's function G(z) = (H - z)(-1) up to Im z >> W-5L5-d. The local law implies that the bulk eigenvector entries of H are of order O(W-5/2L-d/2+5/2) with high probability.