  We consider travelling-wave parametric down-conversion in the high-gain
regime and present the experimental demonstration of the quantum character of
the spatial fluctuations in the system. In addition to showing the presence of
sub-shot noise fluctuations in the intensity difference, we demonstrate that
the peak value of the normalized spatial correlations between signal and idler
lies well above the line marking the boundary between the classical and the
quantum domain. This effect is equivalent to the apparent violation of the
Cauchy-Schwartz inequality, predicted by some of us years ago, which represents
a spatial analogue of photon antibunching in time. Finally, we analyse
numerically the transition from the quantum to the classical regime when the
gain is increased and we emphasize the role of the inaccuracy in the
determination of the symmetry center of the signal/idler pattern in the
far-field plane.
