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    The Scaling Behavior of Classical Wave Transport in Mesoscopic Media at the Localization Transition

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    The propagation of classical wave in disordered media at the Anderson localization transition is studied. Our results show that the classical waves may follow a different scaling behavior from that for electrons. For electrons, the effect of weak localization due to interference of recurrent scattering paths is limited within a spherical volume because of electron-electron or electron-phonon scattering, while for classical waves, it is the sample geometry that determine the amount of recurrent scattering paths that contribute. It is found that the weak localization effect is weaker in both cubic and slab geometry than in spherical geometry. As a result, the averaged static diffusion constant D(L) scales like ln(L)/L in cubic or slab geometry and the corresponding transmission follows ~ln L/L^2. This is in contrast to the behavior of D(L)~1/L and ~1/L^2 obtained previously for electrons or spherical samples. For wave dynamics, we solve the Bethe-Salpeter equation in a disordered slab with the recurrent scattering incorporated in a self-consistent manner. All of the static and dynamic transport quantities studied are found to follow the scaling behavior of D(L). We have also considered position-dependent weak localization effects by using a plausible form of position-dependent diffusion constant D(z). The same scaling behavior is found, i.e., ~ln L/L^2.Comment: 11 pages, 12 figures. Submitted to Phys. Rev. B on 3 May 200

    Delocalization and Diffusion Profile for Random Band Matrices

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    We consider Hermitian and symmetric random band matrices H=(hxy)H = (h_{xy}) in d1d \geq 1 dimensions. The matrix entries hxyh_{xy}, indexed by x,y \in (\bZ/L\bZ)^d, are independent, centred random variables with variances s_{xy} = \E |h_{xy}|^2. We assume that sxys_{xy} is negligible if xy|x-y| exceeds the band width WW. In one dimension we prove that the eigenvectors of HH are delocalized if WL4/5W\gg L^{4/5}. We also show that the magnitude of the matrix entries \abs{G_{xy}}^2 of the resolvent G=G(z)=(Hz)1G=G(z)=(H-z)^{-1} is self-averaging and we compute \E \abs{G_{xy}}^2. We show that, as LL\to\infty and WL4/5W\gg L^{4/5}, the behaviour of \E |G_{xy}|^2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions
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