  Cosmological $N$-body simulations provide numerical predictions of the
structure of the Universe against which to compare data from ongoing and future
surveys, but the growing volume of the Universe mapped by surveys requires
correspondingly lower statistical uncertainties in simulations, usually
achieved by increasing simulation sizes at the expense of computational power.
It was recently proposed to reduce simulation variance without incurring
additional computational costs by adopting fixed-amplitude initial conditions.
This method has been demonstrated not to introduce bias in various statistics,
including the two-point statistics of galaxy samples typically used for
extracting cosmological parameters from galaxy redshift survey data, but
requires us to revisit current methods for estimating covariance matrices of
clustering statistics for simulations. In this work, we find that it is not
trivial to construct covariance matrices analytically for fixed-amplitude
simulations, but we demonstrate that EZmock (Effective Zel'dovich approximation
mock catalogue), the most efficient method for constructing mock catalogues
with accurate two- and three-point statistics, provides reasonable covariance
matrix estimates for such simulations. We further examine how the variance
suppression obtained by amplitude-fixing depends on three-point clustering,
small-scale clustering, and galaxy bias, and propose intuitive explanations for
the effects we observe based on the EZmock bias model.
