Bulk universality and quantum unique ergodicity for random band matrices in high dimensions

2022/07/29 by Xu, Changji, Yang, Fan, Yau, Horng-Tzer +1 · 1 citation
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2207.14533

Abstract

We consider Hermitian random band matrices H=(h_xy) on the d-dimensional lattice (\mathbb Z/L \mathbb Z)^d, where the entries h_xy= h_yx are independent centered complex Gaussian random variables with variances s_xy=\mathbb E|h_xy|². The variance matrix S=(s_xy) has a banded profile so that s_xy is negligible if |x-y| exceeds the band width W. For dimensions d≥ 7, we prove the bulk eigenvalue universality of H under the condition W ≫ L^95/(d+95). Assuming that W≥ L^ε for a small constant ε>0, we also prove the quantum unique ergodicity for the bulk eigenvectors of H and a sharp local law for the Green's function G(z)=(H-z)^-1 up to Im z ≫ W^-5L^5-d. The local law implies that the bulk eigenvector entries of H are of order O(W^-5/2L^-d/2+5/2) with high probability.

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