  We calculate the void probability function (VPF) in simulations of
Lyman-$\alpha$ emitters (LAEs) across a wide redshift range ($z=3.1,\ 4.5,\
5.7,\ 6.6$). The VPF measures the zero-point correlation function (i.e. places
devoid of galaxies) and naturally connects to higher order correlation
functions while being computationally simple to calculate. We explore the
Poissonian and systematic errors on the VPF, specify its accuracy as a function
of average source density and the volume probed, and provide the appropriate
size scales to measure the VPF. At small radii the accuracy of the VPF is
limited by galaxy density, while at large radii the VPF is limited by the
number of independent volumes probed. We also offer guidelines for
understanding and quantifying the error in the VPF. We approximate the error in
the VPF by using independent sub-volumes of the catalogs, after finding that
jackknife statistics underestimate the uncertainty. We use the VPF to probe the
strength of higher order correlation functions by measuring and examining the
hierarchical scaling between the correlation functions using count-in-cells.
The negative binomial model (NBM) has been shown to best describe the scaling
between the two point correlation function and VPF for low-redshift galaxy
observations. We further test the fit of the NBM by directly deriving the
volume averaged two-point correlation function from the VPF and vice versa. We
find the NBM best describes the $z=3.1, 4.5, 5.7$ simulated LAEs, with a
1$\sigma$ deviation from the model in the $z=6.6$ catalog. This suggests that
LAEs show higher order clustering terms similar to those of normal low redshift
galaxies.
