The MUSE-Wide survey: Three-dimensional clustering analysis of
  Lyman-$\alpha$ emitters at $3.3<z<6$
  We present an analysis of the spatial clustering of 695 Ly$\alpha$-emitting
galaxies (LAE) in the MUSE-Wide survey. All objects have spectroscopically
confirmed redshifts in the range $3.3<z<6$. We employ the K-estimator of
Adelberger et al. (2005), adapted and optimized for our sample. We also explore
the standard two-point correlation function approach, which is however less
suited for a pencil-beam survey such as ours. The results from both approaches
are consistent. We parametrize the clustering properties by, (i) modelling the
clustering signal with a power law (PL), and (ii) adopting a Halo Occupation
Distribution (HOD) model. Applying HOD modeling, we infer a large-scale bias of
$b_{\rm{HOD}}=2.80^{+0.38}_{-0.38}$ at a median redshift of the number of
galaxy pairs $\langle z_{\rm pair}\rangle\simeq3.82$, while the PL analysis
results in $b_{\rm{PL}}=3.03^{+1.51}_{-0.52}$
($r_0=3.60^{+3.10}_{-0.90}\;h^{-1}$Mpc and $\gamma=1.30^{+0.36}_{-0.45}$). The
implied typical dark matter halo (DMH) mass is
$\log(M_{\rm{DMH}}/[h^{-1}\rm{M}_\odot])=11.34^{+0.23}_{-0.27}$. We study
possible dependencies of the clustering signal on object properties by
bisecting the sample into disjoint subsets, considering Ly$\alpha$ luminosity,
UV absolute magnitude, Ly$\alpha$ equivalent width, and redshift as variables.
We find a suggestive trend of more luminous Ly$\alpha$ emitters residing in
more massive DMHs than their lower Ly$\alpha$ luminosity counterparts. We also
compare our results to mock LAE catalogs based on a semi-analytic model of
galaxy formation and find a stronger clustering signal than in our observed
sample. By adopting a galaxy-conserving model we estimate that the LAEs in the
MUSE-Wide survey will typically evolve into galaxies hosted by halos of
$\log(M_{\rm{DMH}}/[h^{-1}\rm{M}_\odot])\approx13.5$ at redshift zero,
suggesting that we observe the ancestors of present-day galaxy groups.
