Faint Quasars Live in the Same Number Density Environments as Lyman
  Break Galaxies at z ~ 4
  Characterizing high-z quasar environments is key to understanding the
co-evolution of quasars and the surrounding galaxies. To restrict their global
picture, we statistically examine the g-dropout galaxy overdensity distribution
around 570 faint quasar candidates at z ~ 4, based on the Hyper Suprime-Cam
Subaru Strategic Program survey. We compare the overdensity significances of
g-dropout galaxies around the quasars with those around g-dropout galaxies, and
find no significant difference between their distributions. A total of 4 (22)
out of the 570 faint quasars, 0.7_{-0.4}^{+0.4} (3.9_{-0.8}^{+0.8}) %, are
found to be associated with the > 4 sigma overdense regions within an angular
separation of 1.8 (3.0) arcmin, which is the typical size of protoclusters at
this epoch. This is similar to the fraction of g-dropout galaxies associated
with the > 4 sigma overdense regions. This result is consistent with our
previous work that 1.3_{-0.9}^{+0.9} % and 2.0_{-1.1}^{+1.1} % of luminous
quasars detected in the Sloan Digital Sky Survey exist in the > 4 sigma
overdense regions within 1.8 and 3.0 arcmin separations, respectively.
Therefore, we suggest that the galaxy number densities around quasars are
independent of their luminosity, and most quasars do not preferentially appear
in the richest protocluster regions at z ~ 4. The lack of an apparent positive
correlation between the quasars and the protoclusters implies that: i) the
gas-rich major merger rate is relatively low in the protocluster regions, ii)
most high-z quasars may appear through secular processes, or iii) some
dust-obscured quasars exist in the protocluster regions.
