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

Yutao Ma; Xujia Meng

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arXiv · 2610.16216

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 $U_m$ be an $m\times m$ Haar-distributed unitary matrix, and let $Z_1,\ldots,Z_n$ be the eigenvalues of its upper-left $n\times n$ principal submatrix. We determine the limiting distribution of $\max_{1\le i\le n}\Re Z_i$ under a unified normalization: the limiting law is Gumbel when $q:=m-n\to\infty$ and is of Weibull type when $q$ is fixed. The fixed-$q$ limiting laws converge to the Gumbel law as $q\to\infty$. 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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BibTeX RIS

Yutao Ma, Xujia Meng. 2026-10-12. Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes. https://arxiv.org/abs/2610.16216

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