  The polarization of the cosmic microwave background radiation will have a
distribution of singularities and anti-singularities, points where the
polarization vanishes for topological reasons. The statistics of polarization
singularities provides a non-trivial scheme to analyze the polarization maps
that is distinct from the usual two-point correlation functions. Here we
characterize the statistics of the singularity distribution in simulated
polarization maps, and make predictions that can be compared with ongoing and
upcoming observations. We use three different characterizations: the nearest
neighbor distance between singularities, the critical exponent \nu that
captures the scaling of total charge q within a closed curve of length L (q
\propto L^\nu), and the angular two point angular correlation functions for
singularities of similar and opposite charge. In general, we find that the
distribution of singularities is random except on scales less than about 10
deg, where singularities of the same charge repel and those of opposite charge
attract. These conclusions appear to be extremely robust with respect to
variations in the underlying cosmological model and the presence of
non-Gaussianity; the only exception we found are cases where statistical
isotropy is grossly violated. This suggests that, within the assumption of
statistical isotropy, the distribution is a robust feature of the last
scattering surface and potentially may be used as a tool to discriminate
effects that occur during photon propagation from the last scattering surface
to the present epoch.
