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Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes

  • Yutao Ma
  • Xujia Meng
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Abstract

We study the limiting distribution of the largest real part of the eigenvalues of truncated unitary matrices. Let Um be an m× m Haar-distributed unitary matrix, and let Z1,…,Zn be the eigenvalues of its upper-left n× n principal submatrix. We determine the limiting distribution of max1≤ i≤ nℜ Zi under a unified normalization: the limiting law is Gumbel when q:=m−n→∞ and is of Weibull type when q is fixed. The fixed-q limiting laws converge to the Gumbel law as q→∞. We also give a unified formulation of the normalizing constants for the spectral radius that recovers the previously known limiting laws.

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Cite

BibTeXRIS
BibTeX
@misc{ma2026limiting,
  title = {{Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes}},
  author = {Yutao Ma and Xujia Meng},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16216},
  eprint = {2610.16216},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16216},
}
RIS
TY  - UNPB
TI  - Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes
AU  - Yutao Ma
AU  - Xujia Meng
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16216
UR  - https://arxiv.org/abs/2610.16216
AB  - We study the limiting distribution of the largest real part of the eigenvalues of truncated unitary matrices. Let U_m be an m× m Haar-distributed unitary matrix, and let Z_1,…,Z_n be the eigenvalues of its upper-left n× n principal submatrix. We determine the limiting distribution of max_1≤ i≤ nℜ Z_i under a unified normalization: the limiting law is Gumbel when q:=m−n→∞ and is of Weibull type when q is fixed. The fixed-q limiting laws converge to the Gumbel law as q→∞. We also give a unified formulation of the normalizing constants for the spectral radius that recovers the previously known limiting laws.
ER  -