Constraining Reionization in Progress at z=5.7 with Lyman-$\alpha$
  Emitters: Voids, Peaks, and Cosmic Variance
  A number of independent observations suggest that the intergalactic medium
was significantly neutral at $z=7$ and that reionization was, perhaps, still in
progress at $z=5.7$. The narrowband survey, SILVERRUSH, has mapped over $2,000$
Lyman-$\alpha$ emitters (LAEs) at these redshifts. Previous analyses have
assumed that reionization was over by $z=5.7$, but this data may actually
sample the final stages of reionization when the last neutral islands were
relegated to the cosmic voids. Motivated by these developments, we reexamine
LAE void and peak statistics and their ability to constrain reionization. We
construct models of the LAE distribution in (1 Gpc$/h$)$^3$ volumes, spanning a
range of neutral fractions at $z=5.7$ and 6.6. Models with a higher neutral
fraction show an enhanced probability of finding holes in the LAE distribution.
When comparing models at fixed mean surface density, however, LAEs obscured by
neutral gas in the voids must be compensated by visible LAEs elsewhere. Hence,
in these models the likelihood of finding an over-dense peak is also enhanced
in the latter half of reionization. Compared to the widely used angular
two-point correlation function (2PCF), we find that the void probability
function (VPF) provides a more sensitive test of models during the latter half
of reionization. By comparison, at neutral fractions $\sim 50\%$, the VPF and a
simple peak thresholding statistic are both similar to the 2PCF in constraining
power. Lastly, we find that the cosmic variance and large-scale asymmetries
observed in the SILVERRUSH fields are consistent with large-scale structure in
a $\Lambda$CDM universe.
