Statistical properties of paired fixed fields
  The initial conditions of cosmological simulations are commonly drawn from a
Gaussian ensemble. The limited number of modes inside a simulation volume gives
rise to statistical fluctuations known as \textit{sample variance}, limiting
the accuracy of simulation predictions. Fixed fields offer an alternative
initialization strategy; they have the same power spectrum as standard Gaussian
fields but without intrinsic amplitude scatter at linear order. Paired fixed
fields consists of two fixed fields with opposite phases that cancel phase
correlations which otherwise induce second-order scatter in the non-linear
power spectrum. We study the statistical properties of those fields for 19
different quantities at different redshifts through a large set of 600 N-body
and 506 state-of-the-art magneto-hydrodynamic simulations covering a wide range
of scales, mass and spatial resolutions. We find that paired fixed simulations
do not introduce a bias on any of the examined quantities. We quantify the
statistical improvement brought by these simulations, over standard ones, on
different power spectra such as matter, halos, CDM, gas, stars, black-holes and
magnetic fields, finding that they can reduce their variance by factors as
large as $10^6$. We quantify the improvement achieved by fixing and by pairing,
showing that sample variance in some quantities can be highly suppressed by
pairing after fixing. Paired fixed simulations do not change the scatter in
quantities such as the probability distribution function of matter density, or
the halo, void or stellar mass functions. We argue that procedures aiming at
reducing the sample variance of those quantities are unlikely to work. Our
results show that paired fixed simulations do not affect either mean relations
or scatter of galaxy properties, and suggest that the information embedded in
1-pt statistics is highly complementary to that in clustering.
