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

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 $Z_1,\dots ,Z_n$ be the eigenvalues of its upper-left $n\times n$ principal submatrix. We determine the limiting distribution of ${\max }_{1\le i\le n}\mathrm{\Re }{Z}_i$ under a unified normalization: the limiting law is Gumbel when $q:=m-n\to \mathrm{\infty }$ and is of Weibull type when $q$ is fixed. The fixed-$q$ limiting laws converge to the Gumbel law as $q\to \mathrm{\infty }$. We also give a unified formulation of the normalizing constants for the spectral radius that recovers the previously known limiting laws.