article published László Erdös author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603 Antti Knowles author Horng Yau author Jun Yin author We consider Hermitian and symmetric random band matrices H = (h xy ) in d⩾1 d ⩾ 1 dimensions. The matrix entries h xy , indexed by x,y∈(Z/LZ)d x , y ∈ ( Z / L Z ) d , are independent, centred random variables with variances sxy=E|hxy|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 W≫L4/5 W ≫ L 4 / 5 . We also show that the magnitude of the matrix entries |Gxy|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 E|Gxy|2 E | G x y | 2 . We show that, as L→∞ L → ∞ and W≫L4/5 W ≫ L 4 / 5 , the behaviour of E|Gxy|2 E | G x y | 2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions. Springer2013 10.1007/s00220-013-1773-3 3231367 - 416 yes Erdös L, Knowles A, Yau H, Yin J. 2013. Delocalization and diffusion profile for random band matrices. Communications in Mathematical Physics. 323(1), 367–416. Erdös, L., Knowles, A., Yau, H., &#38; Yin, J. (2013). Delocalization and diffusion profile for random band matrices. <i>Communications in Mathematical Physics</i>. Springer. <a href="https://doi.org/10.1007/s00220-013-1773-3">https://doi.org/10.1007/s00220-013-1773-3</a> Erdös, László, Antti Knowles, Horng Yau, and Jun Yin. “Delocalization and Diffusion Profile for Random Band Matrices.” <i>Communications in Mathematical Physics</i>. Springer, 2013. <a href="https://doi.org/10.1007/s00220-013-1773-3">https://doi.org/10.1007/s00220-013-1773-3</a>. L. Erdös, A. Knowles, H. Yau, and J. Yin, “Delocalization and diffusion profile for random band matrices,” <i>Communications in Mathematical Physics</i>, vol. 323, no. 1. Springer, pp. 367–416, 2013. Erdös L, Knowles A, Yau H, Yin J. Delocalization and diffusion profile for random band matrices. <i>Communications in Mathematical Physics</i>. 2013;323(1):367-416. doi:<a href="https://doi.org/10.1007/s00220-013-1773-3">10.1007/s00220-013-1773-3</a> L. Erdös, A. Knowles, H. Yau, J. Yin, Communications in Mathematical Physics 323 (2013) 367–416. Erdös, László, et al. “Delocalization and Diffusion Profile for Random Band Matrices.” <i>Communications in Mathematical Physics</i>, vol. 323, no. 1, Springer, 2013, pp. 367–416, doi:<a href="https://doi.org/10.1007/s00220-013-1773-3">10.1007/s00220-013-1773-3</a>. 26972018-12-11T11:59:07Z2021-01-12T06:59:07Z