  We investigate the evolution in the dark matter halo circular velocity
function, measured at a fixed physical radius of 20 kpc (v_20), which is likely
to be a good proxy for galaxy circular velocity, in the Millennium-II
simulation. We find that the v_20 function evolves remarkably little since z ~
4. We analyze the histories of the main progenitors of halos, and we find that
the dark matter distribution within the central 20 kpc of massive halos has
been in place since early times. This provides evidence for the inside-out
growth of haloes. The constancy of the central circular velocity of halos may
offer a natural explanation for the observational finding that the galaxy
circular velocity is an excellent predictor of various galaxy properties. Our
results also indicate that we can expect a significant number of galaxies with
high circular velocities already at z=4 (more than one per 10^6 (Mpc/h)^3 with
circular velocities in excess of 450 km/s, and more than one per 10^4.5
(Mpc/h)^3 with circular velocities in excess of 350 km/s). Finally, adding
baryonic mass and using a simple model for halo adiabatic contraction, we find
remarkable agreement with the velocity dispersion functions inferred
observationally by Bezanson et al. (2011) up to z ~ 1 and down to about 220
km/s.
