2021/04/25 by Yang, Fan, Yau, Horng-Tzer, Yin, Jun · 3 citations
#FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2104.12048
We consider Hermitian random band matrices H=(hxy) on the d-dimensional lattice (\mathbb Z/L\mathbb Z)d. The entries hxy are independent (up to Hermitian conditions) centered complex Gaussian random variables with variances sxy=\mathbb E|hxy|2. The variance matrix S=(sxy) has a banded structure so that sxy is negligible if |x-y| exceeds the band width W. In dimensions d≥ 8, we prove that, as long as W≥ Lε for a small constant ε>0, with high probability most bulk eigenvectors of H are delocalized in the sense that their localization lengths are comparable to L. Denote by G(z)=(H-z)-1 the Green's function of the band matrix. For \mathrm Im z≫ W2/L2, we also prove a widely used criterion in physics for quantum diffusion of this model, namely, the leading term in the Fourier transform of \mathbb E|Gxy(z)|2 with respect to x-y is of the form (\mathrm Im z + a(p))-1 for some a(p) quadratic in p, where p is the Fourier variable. Our method is based on an expansion of Txy=|m|2 ∑αsxα|Gαy|2 and it requires a self-energy renormalization up to error W-K for any large constant K independent of W and L. We expect that this method can be extended to non-Gaussian band matrices.