Delocalization and Diffusion Profile for Random Band Matrices

By L. Erdős, Antti Knowles, Horng-Tzer Yau, Jun Yin. 2012

140 citations recorded by the source corpus.

We consider Hermitian and symmetric random band matrices H = (hxy) in $${d\,\geqslant\,1}$$d⩾1 dimensions. The matrix entries hxy, indexed by $${x,y \in (\mathbb{Z}/L\mathbb{Z})^d}$$x,y∈(Z/LZ)d, are independent, centred random variables with variances $${s_{xy} = \mathbb{E} |h_{xy}|^2}$$sxy=E|hxy|2. We assume that sxy is negligible if |x − y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if $${W\gg L^{4/5}}$$W≫L4/5. We also show that the magnitude of the matrix entries $${|{G_{xy}}|^2}$$|Gxy|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|Gxy|2. We show that, as $${L\to\infty}$$L→∞ and $${W\gg L^{4/5}}$$W≫L4/5, the behaviour of $${\mathbb{E} |G_{xy}|^2}$$E|Gxy|2 is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.

Subfields: Applied Mathematics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/85aa428e1d1884e7f2d03938b8fe4069b71b736d

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Citations in the corpus

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No-gaps delocalization for general random matrices (2015, background): “Proof of Theorem 6.1 on invertibility for general distributions 41 References 44”
Delocalization of eigenvectors of random matrices with independent entries (2013, background): “Delocalization properties with varying degrees of strength and generality were then established for several other symmetric and Hermitian ensembles – band matrices [5, 6, 9], sparse matrices (adjacency matrices of Erdös-Renyi graphs) [7, 8], heavy-tailed matrices [2, 1], and sample covariance matric”
Lectures on the local semicircle law for Wigner matrices (2016, background): “So far an optimal local law and eigenvector delocalization are open problems, although partial progress has been made in [31, 32, 37, 36, 5].”
Spiked separable covariance matrices and principal components (2019, methodology): “We will use the following notion of stochastic domination, which was first introduced in [19] and subsequently used in many works on random matrix theory, such as [7, 8, 9, 20, 21, 30].”
Extremal eigenvalues and eigenvectors of deformed Wigner matrices (2013, background): “We refer to [42, 13, 16] for more details on the localization/delocalization for random band matrices.”
A Dynamical Approach to Random Matrix Theory (2017, background): “Finally, the self-consistent matrix equation for E|Gxy|(2) first appeared in [54] and it yielded the diffusion profile for the resolvent.”