Quantifying scatter in galaxy formation at the lowest masses

We predict the stellar mass -- halo mass (SMHM) relationship for dwarf galaxies, using simulated galaxies with peak halo masses of M$_{\rm peak} = 10^{11}$ M$_{\odot}$ down into the ultra-faint dwarf range to M$_{\rm peak} =$ 10$^7$ M$_{\odot}$. Our simulated dwarfs have stellar masses of M$_{\rm star} = $ 790 M$_{\odot}$ to $8.2 \times 10^8$ M$_{\odot}$, with corresponding $V$-band magnitudes from $-2$ to $-18.5$. For M$_{\rm peak} > 10^{10}$ M$_{\odot}$, the simulated SMHM relationship agrees with literature determinations, including exhibiting a small scatter of 0.3 dex. However, the scatter in the SMHM relation increases for lower-mass halos. We first present results for well-resolved halos that contain a simulated stellar population, but recognize that whether a halo hosts a galaxy is inherently mass resolution dependent. We thus adopt a probabilistic model to populate "dark" halos below our resolution limit to predict an "intrinsic" slope and scatter for the SMHM relation. We fit linearly growing log-normal scatter in stellar mass, which grows to more than 1 dex at M$_{\rm peak}$ $=$ 10$^8$ M$_{\odot}$. At the faintest end of the SMHM relation probed by our simulations, a galaxy cannot be assigned a unique halo mass based solely on its luminosity. Instead, we provide a formula to stochastically populate low-mass halos following our results. Finally, we show that our growing log-normal scatter steepens the faint-end slope of the predicted stellar mass function.