The environments of Ly$\alpha$ blobs I: Wide-field Ly$\alpha$ imaging of
  TN J1338-1942, a powerful radio galaxy at $z\simeq 4.1$ associated with a
  giant Ly$\alpha$ nebula
  We exploit wide-field Ly$alpha$ imaging with Subaru to probe the environment
around TN J1338-1942, a powerful radio galaxy with a >100 kpc Ly$\alpha$ halo
at z=4.11. We used a sample of Ly$\alpha$ emitters (LAEs) down to $\log(L_{\rm
Ly\alpha} [erg\, s^{-1}])\sim 42.8$ to measure the galaxy density around
TNJ1338, compared to a control sample from a blank field taken with the same
instrument. We found that TNJ1338 resides in a region with a peak overdensity
of $\delta_{\rm LAE}=2.8\pm 0.5$ on scales of $8\, h^{-1}\rm Mpc$ (on the sky)
and $112\, h^{-1}\rm Mpc$ (line of sight) in comoving coordinates. Adjacent to
this overdensity, we found a strong underdensity where virtually no LAEs are
detected. We used a semi-analytical model of LAEs derived from the Millennium
Simulation to compare our results with theoretical predictions. While the
theoretical density distribution is consistent with the blank field, overdense
regions such as that around TNJ1338 are very rare, with a number density of
$6.4\times 10^{-8}\rm Mpc^{-3}$ (comoving), corresponding to the densest < 0.4
percentile at $z\simeq 4.1$. We also found that the Ly$\alpha$ luminosity
function in the TNJ1338 field differs from that in the blank field: the number
of bright LAEs ($\log(L_{\rm Ly\alpha}[erg\,s^{-1}]) \gtrsim 43.3$) is
enhanced, while the number of fainter LAEs is relatively suppressed. These
results suggest that some powerful radio galaxies associated with Ly$\alpha$
nebulae reside in extreme overdensities on $\sim 3$--$6\, \rm Mpc$ scales,
where star-formation and AGN activity may be enhanced via frequent galaxy
mergers or high rates of gas accretion from the surroundings.
