  The joint probability distribution of matter overdensity and galaxy counts in
cells is a powerful probe of cosmology, and the extent to which variance in
galaxy counts at fixed matter density deviates from Poisson shot noise is not
fully understood. The lack of informed bounds on this stochasticity is
currently the limiting factor in constraining cosmology with the galaxy-matter
PDF. We investigate stochasticity in the conditional distribution of galaxy
counts at fixed matter density and present a halo occupation distribution
(HOD)-based approach for obtaining plausible ranges for stochasticity
parameters. To probe the high-dimensional space of possible galaxy-matter
connections, we derive HODs which conserve linear galaxy bias and number
density to produce redMaGiC-like galaxy catalogs within the AbacusSummit suite
of N-body simulations. We study the impact of individual HOD parameters and
cosmology on stochasticity and perform a Monte Carlo search in HOD parameter
space, subject to the constraints on bias and density. In mock catalogs
generated by the selected HODs, shot noise in galaxy counts spans both
sub-Poisson and super-Poisson values, ranging from 80% to 133% of Poisson
variance at mean matter density. Nearly all derived HODs show a positive
relationship between local matter density and stochasticity. For galaxy
catalogs with higher stochasticity, quadratic galaxy bias is required for an
accurate description of the conditional PDF of galaxy counts at fixed matter
density. The presence of galaxy assembly bias also substantially extends the
range of stochasticity in the super-Poisson direction. This HOD-based approach
leverages degrees of freedom in the galaxy-halo connection to obtain informed
bounds on model nuisance parameters and can be adapted to other
parametrizations of stochasticity, in particular to motivate prior ranges for
cosmological analyses.
