, Volume 12, Issue 7, pp 1227-1319

Quantum Diffusion and Delocalization for Band Matrices with General Distribution

Abstract

We consider Hermitian and symmetric random band matrices H in ${d \geqslant 1}$ dimensions. The matrix elements H xy , indexed by ${x,y \in \Lambda \subset \mathbb{Z}^d}$ , are independent and their variances satisfy ${\sigma_{xy}^2:=\mathbb{E} |{H_{xy}}|^2 = W^{-d} f((x - y)/W)}$ for some probability density f. We assume that the law of each matrix element H xy is symmetric and exhibits subexponential decay. We prove that the time evolution of a quantum particle subject to the Hamiltonian H is diffusive on time scales ${t\ll W^{d/3}}$ . We also show that the localization length of the eigenvectors of H is larger than a factor ${W^{d/6}}$ times the band width W. All results are uniform in the size |Λ| of the matrix. This extends our recent result (Erdős and Knowles in Commun. Math. Phys., 2011) to general band matrices. As another consequence of our proof we show that, for a larger class of random matrices satisfying ${\sum_x\sigma_{xy}^2=1}$ for all y, the largest eigenvalue of H is bounded with high probability by ${2 + M^{-2/3 + \varepsilon}}$ for any ${\varepsilon > 0}$ , where ${M := 1 / (\max_{x,y}\sigma_{xy}^2)}$ .

Communicated by Abdelmalek Abdesselam.
L. Erdős’s research was partially supported by SFB-TR 12 Grant of the German Research Council.
A. Knowles’s research was partially supported by NSF Grant DMS-0757425.