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 \(n\rightarrow + \infty \) with \(\alpha \) fixed, the associated moment generating function enjoys asymptotics of the form

$$\begin{aligned} \exp \big ( C_{1} n + C_{2} + o(1) \big ), \end{aligned}$$

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.