Organized Chaos: Scatter in the relation between stellar mass and halo
  mass in small galaxies
  We use Local Group galaxy counts together with the ELVIS N-body simulations
to explore the relationship between the scatter and slope in the stellar mass
vs. halo mass relation at low masses, $M_\star \simeq 10^5 - 10^8 M_\odot$.
Assuming models with log-normal scatter about a median relation of the form
$M_\star \propto M_\mathrm{halo}^\alpha$, the preferred log-slope steepens from
$\alpha \simeq 1.8$ in the limit of zero scatter to $\alpha \simeq 2.6$ in the
case of $2$ dex of scatter in $M_\star$ at fixed halo mass. We provide fitting
functions for the best-fit relations as a function of scatter, including cases
where the relation becomes increasingly stochastic with decreasing mass. We
show that if the scatter at fixed halo mass is large enough ($\gtrsim 1$ dex)
and if the median relation is steep enough ($\alpha \gtrsim 2$), then the
"too-big-to-fail" problem seen in the Local Group can be self-consistently
eliminated in about $\sim 5-10\%$ of realizations. This scenario requires that
the most massive subhalos host unobservable ultra-faint dwarfs fairly often; we
discuss potentially observable signatures of these systems. Finally, we compare
our derived constraints to recent high-resolution simulations of dwarf galaxy
formation in the literature. Though simulation-to-simulation scatter in
$M_\star$ at fixed $M_\mathrm{halo}$ is large among separate authors ($\sim 2$
dex), individual codes produce relations with much less scatter and usually
give relations that would over-produce local galaxy counts.
