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.
Download