can be regarded as the stellar mass function (SMF) for halos of mass $M$. We apply this formalism at low redshift and up to the highest redshifts where reliable observational stellar mass estimates are available ($0.1 \lesssim z \lesssim 4$). In this way, we derive a parameterized relationship between dark matter halo mass and galaxy mass as a function of redshift.

Using a parameterized relationship has several advantages. First, it provides a convenient way for other researchers to make use of our results and obtain an expression for stellar mass as a function of halo mass. Second, it is straightforward to include scatter in the relation, which is physically more realistic: one just has to choose a number drawn from an assumed random distribution and add that to the average relation. Finally, it is straightforward to treat central and satellite galaxies separately and assume different relations between stellar and halo mass for those populations. However, here we make the assumption that both populations follow the same relation, which has consequences for the clustering predictions of our model.

Using the CMF derived *only* from constraints from the observed SMF, we compute the predicted (projected) galaxy CF at $z\sim 0$ as a 