Quantum Diffusion and Delocalization for Band Matrices with General Distribution |
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Authors: | L��szl�� Erd?s Antti Knowles |
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Institution: | 1. Institute of Mathematics, University of Munich, Theresienstr. 39, 80333, Munich, Germany 2. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA
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Abstract: | We consider Hermitian and symmetric random band matrices H in
d \geqslant 1{d \geqslant 1} dimensions. The matrix elements H
xy
, indexed by
x,y ? L ì \mathbbZd{x,y \in \Lambda \subset \mathbb{Z}^d}, are independent and their variances satisfy
sxy2:=\mathbbE |Hxy|2 = W-d f((x - y)/W){\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 << Wd/3{t\ll W^{d/3}} . We also show that the localization length of the eigenvectors of H is larger than a factor Wd/6{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
?xsxy2=1{\sum_x\sigma_{xy}^2=1} for all y, the largest eigenvalue of H is bounded with high probability by 2 + M-2/3 + e{2 + M^{-2/3 + \varepsilon}} for any ${\varepsilon > 0}${\varepsilon > 0}, where M : = 1 / (maxx,ysxy2){M := 1 / (\max_{x,y}\sigma_{xy}^2)} . |
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