  We measure the Voronoi density probability distribution function (PDF) for
both dark matter and halos in N-body simulations. For the dark matter, Voronoi
densities represent the matter density field smoothed on a uniform mass scale,
which approximates the Lagrangian density field. For halos, the Voronoi
densities contain information about the local environment of each halo. We
measure the halo virial masses, the total amount of dark matter within each
halo Voronoi cell, and the halo Voronoi cell volumes, and we show how halo
abundances depend on these three quantities. We then study the
position-dependent Voronoi density PDF, measured within finite subregions of
the Universe, using separate universe simulations. We demonstrate that the
spatial variation of the position-dependent PDF is due to large-scale density
fluctuations, indicating that the position-dependent PDF is a biased tracer of
large-scale structure. We measure this bias for the dark matter, and interpret
it as the bias of regions of the Lagrangian density field that are selected
based on density. For the halos, this bias can be interpreted as a form of
assembly bias. We present the mapping from late-time to early-time Voronoi
density for each simulation dark matter particle, which is highly stochastic.
We compare the median of this stochastic map with spherical collapse
calculations and discuss challenges involved in modeling the evolution of the
density field on these scales.
