  Based on the Millennium Simulation we examine assembly bias for the halo
properties: shape, triaxiality, concentration, spin, shape of the velocity
ellipsoid and velocity anisotropy. For consistency we determine all these
properties using the same set of particles, namely all gravitationally
self-bound particles belonging to the most massive sub-structure of a given
friends-of-friends halo. We confirm that near-spherical and high-spin halos
show enhanced clustering. The opposite is true for strongly aspherical and
low-spin halos. Further, below the typical collapse mass, M*, more concentrated
halos show stronger clustering whereas less concentrated halos are less
clustered which is reversed for masses above M*. Going beyond earlier work we
show that: (1) oblate halos are more strongly clustered than prolate ones; (2)
the dependence of clustering on the shape of the velocity ellipsoid coincides
with that of the real-space shape, although the signal is stronger; (3) halos
with weak velocity anisotropy are more clustered, whereas radially anisotropic
halos are more weakly clustered; (4) for all highly clustered subsets we find
systematically less radially biased velocity anisotropy profiles. These
findings indicate that the velocity structure of halos is tightly correlated
with environment.
