Poisson statistics of one-dimensional random band matrices
Abstract
Consider an N× N random band matrix H with bandwidth W with centered real Gaussian entries. We prove that, under the (almost) sharp assumption W2≪ N/log N, the bulk local statistics of H are asymptotically a Poisson point process. Combined with the bulk universality results from arXiv:2501.01718 and arXiv:2506.06441, this establishes the Poisson-RMT transition of the local statistics of random band matrices, and complements the results about the localization-delocalization transition of the bulk eigenvectors previously established in arXiv:2501.01718 and arXiv:2506.06441, and arXiv:2508.05802. The key technical inputs are: (1) the fractional moment estimates from arXiv:2508.05802; (2) a novel estimate for the density of states, obtained from the local laws in arXiv:2501.01718 and arXiv:2506.06441.
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BibTeX
@misc{fan2026poisson,
title = {{Poisson statistics of one-dimensional random band matrices}},
author = {Jiaqi Fan and Guangyi Zou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23389},
eprint = {2609.23389},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23389},
}
RIS
TY - UNPB
TI - Poisson statistics of one-dimensional random band matrices
AU - Jiaqi Fan
AU - Guangyi Zou
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23389
UR - https://arxiv.org/abs/2609.23389
AB - Consider an N× N random band matrix H with bandwidth W with centered real Gaussian entries. We prove that, under the (almost) sharp assumption W^2≪ N/log N, the bulk local statistics of H are asymptotically a Poisson point process. Combined with the bulk universality results from arXiv:2501.01718 and arXiv:2506.06441, this establishes the Poisson-RMT transition of the local statistics of random band matrices, and complements the results about the localization-delocalization transition of the bulk eigenvectors previously established in arXiv:2501.01718 and arXiv:2506.06441, and arXiv:2508.05802. The key technical inputs are: (1) the fractional moment estimates from arXiv:2508.05802; (2) a novel estimate for the density of states, obtained from the local laws in arXiv:2501.01718 and arXiv:2506.06441.
ER -