Eigenvalues of truncated unitary matrices: disk counting statistics

Abstract

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