Limiting distributions of the rightmost eigenvalue of the truncated circular Unitary ensemble across all truncation regimes
Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Authors: Ma, Yutao, Meng, Xujia
Language: English
Subfield: Statistics and Probability
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221814264