Delocalization and Diffusion Profile for Random Band Matrices

Abstract

We consider Hermitian and symmetric random band matrices $H=(h_{xy})$ in $d\ge 1$ dimensions. The matrix entries $h_{xy}$, 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 $s_{xy}$ is negligible if $|x-y|$ exceeds the band width $W$. In one dimension we prove that the eigenvectors of $H$ are delocalized if $W\gg L^{4/5}$. We also show that the magnitude of the matrix entries $\abs{G_{xy}}^2$ of the resolvent $G=G(z)=(H-z{)}^{-1}$ is self-averaging and we compute $\E \abs{G_{xy}}^2$. We show that, as $L\to \infty$ and $W\gg L^{4/5}$, 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.

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Semicircle law on short scales and delocalization of eigenvectors for Wigner random matrices

Cited by in corpus (18)

• Averaging Fluctuations in Resolvents of Random Band Matrices
• Stability of the Matrix Dyson Equation and Random Matrices with Correlations
• Delocalization of eigenvectors of random matrices with independent entries
• Characteristic polynomials for 1D random band matrices from the localization side
• The Altshuler-Shklovskii Formulas for Random Band Matrices I: the Unimodular Case
• Local Marchenko-Pastur Law at the Hard Edge of Sample Covariance Matrices
• From Extreme Values of I.I.D. Random Fields to Extreme Eigenvalues of Finite-volume Anderson Hamiltonian
• Edge universality of separable covariance matrices
• Random band matrices in the delocalized phase, II: Generalized resolvent estimates
• Bounds for the Stieltjes Transform and the Density of States of Wigner Matrices
• Transfer matrix approach to 1d random band matrices: density of states
• Diffusion Profile for Random Band Matrices: a Short Proof
• Local circular law for the product of a deterministic matrix with a random matrix
• Sample canonical correlation coefficients of high-dimensional random vectors: local law and Tracy-Widom limit
• Delocalization and quantum diffusion of random band matrices in high dimensions II: $T$-expansion
• Delocalization and continuous spectrum for ultrametric random operators
• Random sparse generators of Markovian evolution and their spectral properties
• Circular Law for Random Block Band Matrices with Genuinely Sublinear Bandwidth