Impact of Cosmic Variance on the Galaxy-Halo Connection for
  Lyman-$\alpha$ Emitters
  In this paper we study the impact of cosmic variance and observational
uncertainties in constraining the mass and occupation fraction, $f_{\rm occ}$,
of dark matter halos hosting Ly-$\alpha$ Emitting Galaxies (LAEs) at high
redshift. To this end, we construct mock catalogs from an N-body simulation to
match the typical size of observed fields at $z=3.1$ ($\sim 1 {\rm deg^2}$). In
our model a dark matter halo with mass in the range $M_{\rm min}<M_{\mathrm
h}<M_{\rm max}$ can only host one detectable LAE at most. We proceed to explore
the parameter space determined by $M_{\rm min}$,$M_{\rm max}$ and $f_{\rm occ}$
with a Markov Chain Monte-Carlo algorithm using the angular correlation
function (ACF) and the LAEs number density as observational constraints. We
find that the preferred minimum and maximum masses in our model span a wide
range $10^{10.0}h^{-1}{\rm{M_{\odot}}}\leq M_{\rm min} \leq
10^{11.1}h^{-1}{\rm{M_{\odot}}}$ , $10^{11.0}h^{-1}{\rm{M_{\odot}}}\leq M_{\rm
max} \leq 10^{13.0}h^{-1}{\rm{M_{\odot}}}$; followed by a wide range in the
occupation fraction $0.02\leq f_{\rm occ} \leq 0.30$. As a consequence the
median mass, $M_{50}$, of all the consistent models has a large uncertainty
$M_{50} = 3.16^{+9.34}_{-2.37}\times 10^{10}$$h^{-1}{\rm{M_{\odot}}}$. However,
we find that the same individual models have a relatively tight $1\sigma$
scatter around the median mass $\Delta M_{1\sigma} = 0.55^{+0.11}_{-0.31}$ dex.
We are also able to show that \focc\ is uniquely determined by $M_{\rm min}$,
regardless of $M_{\rm max}$. We argue that upcoming large surveys covering at
least $25$ deg$^{2}$ should be able to put tighter constraints on $M_{\rm min}$
and $f_{\rm occ}$ through the LAE number density distribution width constructed
over several fields of $\sim 1$ deg$^{2}$.
