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Delocalization and Diffusion Profile for Random Band Matrices
Authors:László Erdős  Antti Knowles  Horng-Tzer Yau  Jun Yin
Affiliation:1. Institute of Mathematics, University of Munich, Theresienstrasse 39, 80333, Munich, Germany
2. Courant Institute, New York University, 251 Mercer Street, New York, NY, 10012, USA
3. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA
4. Department of Mathematics, University of Wisconsin, Madison, WI, 53706, USA
Abstract:We consider Hermitian and symmetric random band matrices H = (h xy ) in ${d,geqslant,1}$ d ? 1 dimensions. The matrix entries h xy , indexed by ${x,y in (mathbb{Z}/Lmathbb{Z})^d}$ x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances ${s_{xy} = mathbb{E} |h_{xy}|^2}$ s x y = E | h x y | 2 . We assume that s xy is negligible if |x ? y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if ${Wgg L^{4/5}}$ W ? L 4 / 5 . We also show that the magnitude of the matrix entries ${|{G_{xy}}|^2}$ | G x y | 2 of the resolvent ${G=G(z)=(H-z)^{-1}}$ G = G ( z ) = ( H - z ) - 1 is self-averaging and we compute ${mathbb{E} |{G_{xy}}|^2}$ E | G x y | 2 . We show that, as ${Ltoinfty}$ L → ∞ and ${Wgg L^{4/5}}$ W ? L 4 / 5 , the behaviour of ${mathbb{E} |G_{xy}|^2}$ E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.
Keywords:
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