May 2024 Bulk universality and quantum unique ergodicity for random band matrices in high dimensions
Changji Xu, Fan Yang, Horng-Tzer Yau, Jun Yin
Author Affiliations +
Ann. Probab. 52(3): 765-837 (May 2024). DOI: 10.1214/23-AOP1670


We consider Hermitian random band matrices H=(hxy) on the d-dimensional lattice (Z/LZ)d, where the entries hxy=hyx are independent centered complex Gaussian random variables with variances sxy=E|hxy|2. The variance matrix S=(sxy) has a banded profile so that sxy is negligible if |xy| exceeds the band width W. For dimensions d7, we prove the bulk eigenvalue universality of H under the condition WL95/(d+95). Assuming that WLε 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)=(Hz)1 up to ImzW5L5d. The local law implies that the bulk eigenvector entries of H are of order O(W5/2Ld/2+5/2) with high probability.

Funding Statement

The work of Fan Yang is partially supported by the National Key R&D Program of China (No. 2023YFA1010400).
The work of Horng-Tzer Yau is partially supported by the NSF Grants DMS-1855509 and DMS-2153335, and a Simons Investigator award.
The work of Jun Yin is partially supported by the NSF Grant DMS-1802861 and Simons Fellows in Mathematics award 85515.


We would like to thank Paul Bourgade for the helpful discussions. We are very grateful to an anonymous referee for the helpful comments, which has resulted in a significant improvement of the paper.

Fan Yang is also affiliated with the Beijing Institute of Mathematical Sciences and Applications.


Download Citation

Changji Xu. Fan Yang. Horng-Tzer Yau. Jun Yin. "Bulk universality and quantum unique ergodicity for random band matrices in high dimensions." Ann. Probab. 52 (3) 765 - 837, May 2024.


Received: 1 December 2022; Revised: 1 October 2023; Published: May 2024
First available in Project Euclid: 23 April 2024

Digital Object Identifier: 10.1214/23-AOP1670

Primary: 60B20
Secondary: 15B52 , 82B44

Keywords: Bulk universality , delocalization , Quantum unique ergodicity , random band matrices

Rights: Copyright © 2024 Institute of Mathematical Statistics


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Vol.52 • No. 3 • May 2024
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