  Observations of colliding galaxy clusters with high relative velocity probe
the tail of the halo pairwise velocity distribution with the potential of
providing a powerful test of cosmology. As an example it has been argued that
the discovery of the Bullet Cluster challenges standard $\Lambda$CDM model
predictions. Halo catalogs from N-body simulations have been used to estimate
the probability of Bullet-like clusters. However, due to simulation volume
effects previous studies had to rely on a Gaussian extrapolation of the
pairwise velocity distribution to high velocities. Here, we perform a detail
analysis using the halo catalogs from the Dark Energy Universe Simulation Full
Universe Runs (DEUS-FUR), which enables us to resolve the high-velocity tail of
the distribution and study its dependence on the halo mass definition, redshift
and cosmology. Building upon these results we estimate the probability of
Bullet-like systems in the framework of Extreme Value Statistics. We show that
the tail of extreme pairwise velocities significantly deviates from that of a
Gaussian, moreover it carries an imprint of the underlying cosmology. We find
the Bullet Cluster probability to be two orders of magnitude larger than
previous estimates, thus easing the tension with the $\Lambda$CDM model.
Finally, the comparison of the inferred probabilities for the different
DEUS-FUR cosmologies suggests that observations of extreme interacting clusters
can provide constraints on dark energy models complementary to standard
cosmological tests.
