Making cosmological inferences from the observed galaxy clustering requires accurate predictions for the mean clustering statistics and their covariances. Those are affected by cosmic variance -- the statistical noise due to the finite number of harmonics. The cosmic variance can be suppressed by fixing the amplitudes of the harmonics instead of drawing them from a Gaussian distribution predicted by the inflation models. Initial realizations also can be generated in pairs with 180 degrees flipped phases to further reduce the variance. Here, we compare the consequences of using paired-and-fixed vs Gaussian initial conditions on the average dark matter clustering and covariance matrices predicted from N-body simulations. As in previous studies, we find no measurable differences between paired-and-fixed and Gaussian simulations for the average density distribution function, power spectrum and bispectrum. Yet, the covariances from paired-and-fixed simulations are suppressed in a complicated scale- and redshift-dependent way. The situation is particularly problematic on the scales of Baryon Acoustic Oscillations where the covariance matrix of the power spectrum is lower by only 20% compared to the Gaussian realizations, implying that there is not much of a reduction of the cosmic variance. The non-trivial suppression, combined with the fact that paired-and-fixed covariances are noisier than from Gaussian simulations, suggests that there is no path towards obtaining accurate covariance matrices from paired-and-fixed simulations. Because the covariances are crucial for the observational estimates of galaxy clustering statistics and cosmological parameters, paired-and-fixed simulations, though useful for some applications, cannot be used for the production of mock galaxy catalogs.
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