Volume estimates of metric balls in manifolds find diverse applications in communications and information theory. In this paper, we derive some new results for the volume of a metric ball in unitary group under Frobenius norm topological metric. Our first result is an integral representation of the exact volume, which involves a Toeplitz determinant of Bessel functions. The connection to matrix-variate hypergeometric functions leads from the exact finite size formula to an asymptotic one. The convergence of the obtained limiting formula is exceptionally fast due to the underlying mock-Gaussian behavior.
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Wei et al. (2015) studied this question.