  The empirical HOD model of Wang et al. 2006 fits, by construction, both the
stellar mass function and correlation function of galaxies in the local
Universe. In contrast, the semi-analytical models of De Lucia & Blazoit 2007
(DLB07) and Guo et al. 2011 (Guo11), built on the same dark matter halo merger
trees than the empirical model, still have difficulties in reproducing these
observational data simultaneously. We compare the relations between the stellar
mass of galaxies and their host halo mass in the three models, and find that
they are different. When the relations are rescaled to have the same median
values and the same scatter as in Wang et al., the rescaled DLB07 model can fit
both the measured galaxy stellar mass function and the correlation function
measured in different galaxy stellar mass bins. In contrast, the rescaled Guo11
model still over-predicts the clustering of low-mass galaxies. This indicates
that the detail of how galaxies populate the scatter in the stellar mass --
halo mass relation does play an important role in determining the correlation
functions of galaxies. While the stellar mass of galaxies in the Wang et al.
model depends only on halo mass and is randomly distributed within the scatter,
galaxy stellar mass depends also on the halo formation time in semi-analytical
models. At fixed value of infall mass, galaxies that lie above the median
stellar mass -- halo mass relation reside in haloes that formed earlier, while
galaxies that lie below the median relation reside in haloes that formed later.
This effect is much stronger in Guo11 than in DLB07, which explains the
over-clustering of low mass galaxies in Guo11. Our results illustrate that the
assumption of random scatter in the relation between stellar and halo mass as
employed by current HOD and abundance matching models may be problematic in
case a significant assembly bias exists in the real Universe.
