Haagerup’s phase transition at polydiscslicing

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-Pe lczyński 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.