  We present the analysis of the diffuse, low column density HI environment of
18 MHONGOOSE galaxies. We obtained deep observations with the Robert C. Byrd
Green Bank Telescope, and reached down to a 3sigma column density detection
limit of NHI=6.3x10^{17} cm^{-2} over a 20 km/s linewidth. We analyze the
environment around these galaxies, with a focus on HI gas that reaches column
densities below NHI=10^{19} cm^{-2}. We calculate the total amount of HI gas in
and around the galaxies revealing that nearly all of these galaxies contained
excess HI outside of their disks. We quantify the amount of diffuse gas in the
maps of each galaxy, defined by HI gas with column densities below 10^{19}
cm^{-2}, and find a large spread in percentages of diffuse gas. However, by
binning the percentage of diffuse HI into quarters, we find that the bin with
the largest number of galaxies is the lowest quartile (0-25\% diffuse HI). We
identified several galaxies which may be undergoing gas accretion onto the
galaxy disk using multiple methods of analysis, including azimuthally averaging
column densities beyond the disk, and identifying structure within our
integrated intensity (Moment 0) maps. We measured HI mass outside the disks of
most of our galaxies, with rising cumulative flux even at large radii. We also
find a strong correlation between the fraction of diffuse gas in a galaxy and
its baryonic mass, and test this correlation using both Spearman and Pearson
correlation coefficients. We see evidence of a dark matter halo mass threshold
of M_{halo}~10^{11.1} \msun{} in which galaxies with high fractions of diffuse
HI all reside below. It is in this regime in which cold-mode accretion should
dominate. Finally, we suggest a rotation velocity of v_{rot}~80 km\s as an
upper threshold to find diffuse gas-dominated galaxies.
