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The empirical HOD model of Wang et al. 2006 fits, by construction, both thestellar mass function and correlation function of galaxies in the localUniverse. 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 mergertrees than the empirical model, still have difficulties in reproducing theseobservational data simultaneously. We compare the relations between the stellarmass of galaxies and their host halo mass in the three models, and find thatthey are different. When the relations are rescaled to have the same medianvalues and the same scatter as in Wang et al., the rescaled DLB07 model can fitboth the measured galaxy stellar mass function and the correlation functionmeasured in different galaxy stellar mass bins. In contrast, the rescaled Guo11model still over-predicts the clustering of low-mass galaxies. This indicatesthat the detail of how galaxies populate the scatter in the stellar mass --halo mass relation does play an important role in determining the correlationfunctions 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-analyticalmodels. At fixed value of infall mass, galaxies that lie above the medianstellar mass -- halo mass relation reside in haloes that formed earlier, whilegalaxies that lie below the median relation reside in haloes that formed later.This effect is much stronger in Guo11 than in DLB07, which explains theover-clustering of low mass galaxies in Guo11. Our results illustrate that theassumption of random scatter in the relation between stellar and halo mass asemployed by current HOD and abundance matching models may be problematic incase a significant assembly bias exists in the real Universe.

On the scatter in the relation between stellar mass and halo mass: random or halo formation time dependent?