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Local Law and Eigenvector Delocalization for Random Lifts

  • Jiaoyang Huang
  • Ruizhe Xu
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Abstract

We establish a quantitative local law for random N-lifts of a fixed finite connected simple graph with minimum degree at least three. Away from the singular energies of the universal-cover spectrum, we obtain high-probability entrywise Green's function estimates down to spectral scales (log N)C/N, uniformly in the bulk and near regular edges and cusps. As a consequence, we establish complete eigenvector delocalization: every normalized eigenvector u whose eigenvalue stays a small but fixed positive distance away from the singular energies satisfies |u|∞≤(log N)C/√N with high probability.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026local,
  title = {{Local Law and Eigenvector Delocalization for Random Lifts}},
  author = {Jiaoyang Huang and Ruizhe Xu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.37463},
  eprint = {2610.37463},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.37463},
}
RIS
TY  - UNPB
TI  - Local Law and Eigenvector Delocalization for Random Lifts
AU  - Jiaoyang Huang
AU  - Ruizhe Xu
PY  - 2026
DA  - 2026/10/27
PB  - arXiv
DO  - 10.48550/arxiv.2610.37463
UR  - https://arxiv.org/abs/2610.37463
AB  - We establish a quantitative local law for random N-lifts of a fixed finite connected simple graph with minimum degree at least three. Away from the singular energies of the universal-cover spectrum, we obtain high-probability entrywise Green's function estimates down to spectral scales (log N)^C/N, uniformly in the bulk and near regular edges and cusps. As a consequence, we establish complete eigenvector delocalization: every normalized eigenvector u whose eigenvalue stays a small but fixed positive distance away from the singular energies satisfies |u|_∞≤(log N)^C/√N with high probability.
ER  -