  We measure the clustering of low-redshift SDSS galaxies as a function of
stellar mass ($10.0<\log(M_*/M_\odot)<11.5$) and specific star formation rate
(sSFR) and compare the results to models of the galaxy--halo connection. We
find that the auto-correlation functions of central galaxies exhibit little
dependence on sSFR, with the well-known stronger clustering of quiescent
galaxies mainly attributable to satellites. Because halo assembly history is
known to affect distinct halo clustering, this result implies that there is
little net correlation between halo assembly history and central galaxy sSFR.
However, cross-correlations with satellites are stronger for quiescent centrals
than star-forming centrals, consistent with quiescent centrals having more
satellites in their haloes at fixed $M_*$, as found in SDSS group catalogues.
We model the galaxy--halo connection in an $N$-body simulation by assigning
sSFRs to central galaxies in three different ways. Two of the models depend on
halo assembly history (being based on halo accretion rate or concentration),
while the third is independent of halo assembly history (being based on peak
halo circular velocity, $V_\text{peak}$, a proxy for halo mass). All three
models replicate the observed auto-correlations of central galaxies, while only
the $V_\text{peak}$ model reproduces the observed cross-correlations with
satellites. This further suggests that the effects of halo assembly history may
not be easily seen in auto-correlations of centrals and implies that a more
complete understanding of central galaxy clustering may require more than
auto-correlations of centrals alone. Additionally, the good agreement with the
$V_\text{peak}$ model supports the idea that quiescent galaxies reside in more
massive haloes than star-forming galaxies at fixed $M_*$.
