Haagerup’s phase transition at polydisc slicing

We establish a sharp comparison inequality between the negative moments and the second moment of the magnitude of sums of independent random vectors uniform on three-dimensional Euclidean spheres. This provides a probabilistic extension of the Oleszkiewicz–Pełczyński polydisc slicing result. The Haagerup-type phase transition occurs exactly when the $p$-norm recovers volume, in contrast to the real case. We also obtain partial results in higher dimensions.