GOLDRUSH. III. A Systematic Search of Protoclusters at $z\sim4$ Based on
  the $>100\,\mathrm{deg^2}$ Area
  We conduct a systematic search for galaxy protoclusters at $z\sim3.8$ based
on the latest internal data release (S16A) of the Hyper SuprimeCam Subaru
strategic program (HSC-SSP). In the Wide layer of the HSC-SSP, we investigate
the large-scale projected sky distribution of $g$-dropout galaxies over an area
of $121\,\mathrm{deg^2}$, and identify 216 large-scale overdense regions
($>4\sigma$ overdensity significance) that are good protocluster candidates. Of
these, 37 are located within $8\,\mathrm{arcmin}$ ($3.4\,\mathrm{physicalMpc}$)
from other protocluster candidates of higher overdensity, and are expected to
merge into a single massive structure by $z=0$. Therefore, we find 179 unique
protocluster candidates in our survey. A cosmological simulation that includes
projection effects predicts that more than 76\% of these candidates will evolve
into galaxy clusters with halo masses of at least $10^{14}\,M_{\odot}$ by
$z=0$. The unprecedented size of our protocluster candidate catalog allowed us
to perform, for the first time, an angular clustering analysis of the
systematic sample of protocluster candidates. We find a correlation length of
$35.0\,h^{-1}\,\mathrm{Mpc}$. The relation between correlation length and
number density of $z\sim3.8$ protocluster candidates is consistent with the
prediction of the $\Lambda$CDM model, and the correlation length is similar to
that of rich clusters in the local universe. This result suggests that our
protocluster candidates are tracing similar spatial structures as those
expected of the progenitors of rich clusters and enhances the confidence that
our method to identify protoclusters at high redshifts is robust. In the coming
years, our protocluster search will be extended to the entire HSC-SSP Wide sky
coverage of $\sim1400\,\mathrm{deg^2}$ to probe cluster formation over a wide
redshift range of $z\sim2\mathrm{-}6$.
