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arXiv · 2305.08976

Eigenvalues of truncated unitary matrices: disk counting statistics

Yacin Ameur·Christophe Charlier·Philippe Moreillon
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Abstract

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 $n\to + \infty$ 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.

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Yacin Ameur, Christophe Charlier, Philippe Moreillon. 2023-05-15. Eigenvalues of truncated unitary matrices: disk counting statistics. https://arxiv.org/abs/2305.08976

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