Connecting Faint End Slopes of the Lyman-$\alpha$ emitter and
  Lyman-break Galaxy Luminosity Functions
  We predict Lyman-$\alpha$ (Ly$\alpha$) luminosity functions (LFs) of
Ly$\alpha$-selected galaxies (Ly$\alpha$ emitters, or LAEs) at $z=3-6$ using
the phenomenological model of Dijkstra & Wyithe (2012). This model combines
observed UV-LFs of Lyman-break galaxies (LBGs, or drop out galaxies), with
constraints on their distribution of Ly$\alpha$ line strengths as a function of
UV-luminosity and redshift. Our analysis shows that while Ly$\alpha$ LFs of
LAEs are generally not Schechter functions, these provide a good description
over the luminosity range of $\log_{10}( L_{\alpha}/{\rm erg}\,{\rm
s}^{-1})=41-44$. Motivated by this result, we predict Schechter function
parameters at $z=3-6$. Our analysis further shows that (i) the faint end slope
of the Ly$\alpha$ LF is steeper than that of the UV-LF of Lyman-break galaxies,
(with a median $\alpha_{Ly\alpha} < -2.0$ at $z\gtrsim 4$), and (ii) a
turn-over in the Ly$\alpha$ LF of LAEs at Ly$\alpha$ luminosities $10^{40}$ erg
s$^{-1}<L_{\alpha}\lesssim 10^{41}$ erg s$^{-1}$ may signal a flattening of
UV-LF of Lyman-break galaxies at $-12>M_{\rm UV}>-14$. We discuss the
implications of these results - which can be tested directly with upcoming
surveys - for the Epoch of Reionization.
