tokens: mode i is given by where P i is the discrete version of the power spectrum P ( k ). In the fixed-power approach, the PDF for mode i is instead given by where δ D indicates the Dirac delta-function. One can sample from Pr f straightforwardly by setting with θ i drawn with uniform probability between 0 and 2π, and θ − i = −θ i . The second of the pair of simulations is then generated by transforming θ i → π + θ i (Pontzen et al. 2016 ). Sampling from Pr f results in an ensemble that is not equivalent to sampling from Pr g . However, Pr f can stand in place of Pr g for many practical calculations (the analytic justification is discussed in Section 3 ). We verified that, despite the fixed amplitudes, the one-point input overdensity PDF in real space, , is still a Gaussian deviate owing to the central limit theorem. Furthermore, Fig. 1 shows the distribution of overdensities in the initial conditions at z = 9, δ IC , averaged over spheres of 8 h −1 Mpc radius for a subset of traditional simulations (grey lines) and the two paired-and-fixed (orange lines; these overlap almost perfectly). The corresponding PDF for the combined volume of the two paired-and-fixed simulations is shown by the red dots. There is excellent agreement between this characterization of the density fields of traditional and fixed simulations, with both following a near-Gaussian distribution. The mild skewness (which also agrees between the cases) arises from the 2LPT particle displacements. Figure 1. The probability density function of non-linear overdensities in spheres of radius 8 h −1 Mpc, as measured in the fixed simulations (orange lines) and in an ensemble of Gaussian simulations (grey lines), at the starting redshift, z = 9. The blue line and red circles show the mean of the respective cases, as indicated by the legend. 2.2 Results Fig. 2 shows the dark matter power spectrum measured from the z = 1 outputs. In the top panel, the results of the fixed pair (red circles) are indistinguishable from the traditional ensemble mean (blue line) over all the scales plotted, confirming that the approach correctly predicts the ensemble average power spectrum in linear and in non-linear regimes. Figure 2. The power spectrum of the dark matter at z = 1. In the top panel, measurements from the ensemble of 300 traditional simulations are shown as grey lines, with the mean shown by a blue line. The solid red circles show the average of the two simulations in the paired-and-fixed set. Finally, the horizontal dotted line marks the shot noise limit. In the bottom panel, we show the differences with respect to the average ensemble measurement, in units of the standard deviation in the ensemble. As in the top panel, red symbols show the final estimate from the pair of fixed simulations. We additionally show residuals in each of the two individual fixed simulations by the orange lines. The envelopes bounded by dashed lines mark a 1 per cent (left) and 0.1 per cent (right) uncertainty in
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