  The recent discovery of a 125 GeV Higgs, as well as the lack of any positive
findings in searches for supersymmetry, has renewed interest in both the
supersymmetric Higgs sector and fine-tuning. Here, we continue our study of the
phenomenological MSSM (pMSSM), discussing the light Higgs and fine-tuning
within the context of two sets of previously generated pMSSM models. We find an
abundance of models with experimentally-favored Higgs masses and couplings. We
investigate the decay modes of the light Higgs in these models, finding strong
correlations between many final states. We then examine the degree of
fine-tuning, considering contributions from each of the pMSSM parameters at up
to next-to-leading-log order. In particular, we examine the fine-tuning
implications for our model sets that arise from the discovery of a 125 GeV
Higgs. Finally, we investigate a small subset of models with low fine-tuning
and a light Higgs near 125 GeV, describing the common features of such models.
We generically find a light stop and bottom with complex decay patterns into a
set of light electroweak gauginos, which will make their discovery more
challenging and may require novel search techniques.
