On the decreasing fraction of Strong Ly$\alpha$ Emitters around $z$
  $\sim$ $6$-$7$
  The fraction of galaxies with strong Ly$\alpha$ emission has been observed to
decrease rapidly with redshift at $z \ge 6$, after a gradual increase at $z<
6$. This has been interpreted as a hint of the reionization of the
intergalactic medium (IGM): the emitted Ly$\alpha$ photons would be scattered
by an increasingly neutral IGM at $z>6$. We study this effect by modeling the
ionization and Ly$\alpha$ radiative transfer in the infall region and the IGM
around a Ly$\alpha$ emitting galaxy (LAE), for a spherical halo model with the
mean density and radial velocity profiles in the standard $\Lambda$CDM
cosmological scenario. We find that the expected fast increase of the ionizing
background intensity toward the end of the reionization epoch implies a rapid
evolution of halo infall regions from being self-shielded against the external
ionizing background to being mostly ionized. Whereas self-shielded infall
regions can scatter the Ly$\alpha$ photons over a much larger area than the
commonly used apertures for observing LAEs, the same infalling gas is no longer
optically thick to the Ly$\alpha$ emission line after it is ionized by the
external background, making the Ly$\alpha$ emission more compact and brighter
within the observed apertures. Based on this simple model, we show that the
observed drop in the abundance of LAEs at $z>6$ does not imply a rapid increase
with redshift of the fraction of the whole IGM volume that is atomic, but is
accounted for by a rapid increase of the neutral fraction in the infall regions
around galaxy host halos.
