Eigenvalues of truncated unitary matrices: disk counting statistics

Let $T$ be an $n\times n$ truncation of an $(n+\alpha )\times (n+\alpha )$ Haar distributed unitary matrix. We consider the disk counting statistics of the eigenvalues of $T$. We prove that as $n\to +\mathrm{\infty }$ with $\alpha$ fixed, the associated moment generating function enjoys asymptotics of the form \begin{align*} \exp \big( C_{1} n + C_{2} + o(1) \big), \end{align*} where the constants $C_1$ and $C_2$ are given in terms of the incomplete Gamma function. Our proof uses the uniform asymptotics of the incomplete Beta function.