  Using a large N-body cosmological simulation combined with a subgrid
treatment of galaxy formation, we study the formation and evolution of the
galaxy and cluster population in a comoving volume (100 Mpc)^3 in a LCDM
universe. At z = 0, our computational volume contains 1788 clusters with mass
M_cl > 1.1x10^12 Msun, including 18 massive clusters with M_cl > 10^14 Msun. It
also contains 1 088 797 galaxies with mass M_gal > 2x10^9 Msun and luminosity L
> 9.5x10^5 Lsun. For each cluster, we identified the brightest cluster galaxy
(BCG). We then computed the fraction f_BNC of clusters in which the BCG is not
the closest galaxy to the center of the cluster in projection, and the ratio
Dv/s, where Dv is the difference in radial velocity between the BCG and the
whole cluster, and s is the radial velocity dispersion of the cluster. f_BNC
increases from 0.05 for low-mass clusters (M_cl ~ 10^12 Msun) to 0.5 for
high-mass ones (M_cl > 10^14 Msun), with no dependence on cluster redshift. The
values of Dv/s vary from 0 to 1.8. These results are consistent with previous
observational studies, and indicate that the central galaxy paradigm, which
states that the BCG should be at rest at the center of the cluster, is usually
valid, but exceptions are too common to be ignored. Analysis of the merger
trees for the 18 most massive clusters in the simulation reveals that 16 of
these clusters have experienced major mergers in the past. These mergers leave
each cluster in a non-equilibrium state, but eventually the cluster settles
into an equilibrium configuration, unless it is disturbed by another major
merger. We found evidence that these mergers are responsible for the off-center
positions and peculiar velocities of some BCGs. Our results thus support the
merging-group scenario, in which some clusters form by the merger of smaller
groups in which the galaxies have already formed.
