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Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices

  • Mohammed Osman
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Abstract

We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based on the inclusion-exclusion principle, the small probability comparison of Erdős--Xu, and partial diagonalisation of Gaussian divisible matrices.

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Cite

BibTeXRIS
BibTeX
@misc{osman2026precise,
  title = {{Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices}},
  author = {Mohammed Osman},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.11630},
  eprint = {2609.11630},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.11630},
}
RIS
TY  - UNPB
TI  - Precise Delocalisation and Gumbel Laws for Eigenvectors of Wigner Matrices
AU  - Mohammed Osman
PY  - 2026
DA  - 2026/09/10
PB  - arXiv
DO  - 10.48550/arxiv.2609.11630
UR  - https://arxiv.org/abs/2609.11630
AB  - We prove a delocalisation bound for eigenvectors of Wigner matrices with the precise relationship between the size of the largest entry and the decay exponent of the probability. We also prove that the largest entry of an individual eigenvector and the largest entry of all eigenvectors are both Gumbel distributed. The proof is based on the inclusion-exclusion principle, the small probability comparison of Erdős--Xu, and partial diagonalisation of Gaussian divisible matrices.
ER  -