While we have presented results using just the paired MW/M31 halos in ELVIS, here we compare our main results—virial-infall times and group preprocessed fractions—for satellites in the isolated versus paired MW/M31 halos. This comparison is useful for a number of reasons. First, theoretically, it is interesting to understand the degree to which the larger-scale environment around a MW/M31 halo influences the infall histories of its satellite population. This comparison also informs whether theoretical models need to consider separately the satellite populations of paired versus isolated MW/M31 halos, for example, in order to understand satellites in the LG as opposed to in isolated MW/M31-like galaxies in the local volume.
Figure \ref{fig:infall.time_v_mass_isolated} shows \({t_{\rm first\,infall}^{\rm since}}\) (top) and \({t_{\rm MW/M31\,infall}^{\rm since}}\) (bottom) as a function of satellite \({M_{\rm star}}\), or subhalo \({M_{\rm peak}}\), similar to Figure \ref{fig:infall.time_v_mass}, but for the isolated MW/M31 halos. For comparison, dashed curves show the median values for the paired MW/M31 halos from Figure \ref{fig:infall.time_v_mass}. While satellites in paired MW/M31 halos first fell into any host halo slightly earlier, any such difference is small compared to the large scatter. Thus, we conclude that the virial-infall times of satellites do not depend significantly on whether their host halo is isolated or paired like the LG.
Similar to Figure \ref{fig:infall.fraction_v_mass} (top), Figure \ref{fig:infall.fraction_v_mass_isolated} shows the fraction of all satellites at \(z = 0\) that were a satellite in another host halo any time before falling into the MW/M31 halo, or at the time of falling into the MW/M31 halo, as a function of satellite mass. For comparison, the light dashed curves show the averages for the paired MW/M31 halos from Figure \ref{fig:infall.fraction_v_mass} (top). Here, differences between paired versus isolated MW/M31 halos are stronger, such that low-mass satellites are more likely to have been preprocessed if they are in paired MW/M31 halos, at a level comparable to the halo-to-halo standard deviation. This trend reverses slightly at higher mass, but here the difference is much smaller than the large scatter, so we do not consider it significant.
Most likely, the higher preprocessed fractions for satellites in the paired MW/M31 halos arises because, as \citet{GarrisonKimmel2014} noted, the paired MW/M31 halos have many more neighboring host halos within a few Mpc of them than the isolated MW/M31 halos, because the paired MW/M31 halos (almost by definition) reside in a preferentially higher-mass cosmic region. With more neighboring host halos around, the low-mass halos that end up as satellites in the paired MW/M31 halos are more likely first to have fallen into a neighboring host halo and be preprocessed. However, this difference in preprocessed fraction does not lead to a significant difference in infall times in Figure \ref{fig:infall.fraction_v_mass_isolated}, so while group preprocessing is more prevalent for the paired MW/M31 halos, the duration of this preprocessing is not longer.
Finally, the most significant difference that we find between the satellites in paired versus isolated MW/M31 halos was in Figure \ref{fig:nearest_distance_reionization}: during the epoch of reionization (\(z > 6\)), the progenitors of the satellites in the isolated MW/M31 halos were much (\(\sim 2 \times\)) closer to their nearest neighboring, more massive halo than those in the paired MW/M31 halos. This result may seem counterintuitive, given that the paired MW/M31 halos contain many more neighboring host halos at \(z = 0\). However, we find that these structures were diluted over a much larger volume at \(z > 6\) for the paired halos. Specifically, we randomly sub-sample all particles within \({R_{\rm vir}}\) of each MW/M31 halo at \(z = 0\) and trace their locations back to \(z > 6\), finding that the Lagrangian volume that contains all such particles was many (\(2 - 6\)) times larger for the paired MW/M31 halos. Thus, the satellite progenitors from the paired MW/M31 halos had fewer neighboring halos at a given distance at \(z > 6\) than those from the isolated MW/M31 halos.