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tokens: measure the target power spectrum , and ψ’s PDF, from the SB simulation. In using both from a single realization, the results may depend on peculiarities in that realization. The limitation is due to the lack of an available software to produce accurate fitting functions for the power spectrum and PDF of ψ. This introduced dependence is not a problem, as long as one has the aim of isolating the contribution from the PDF and the correlation of ψ to the specific realization examined, to assess how much information they convey with respect to the exact result. To emulate the final ψ field, we set the phases of the pre-translation Gaussian field to the phases of the initial density field of the realization. We do not smooth the measured ψ field, because both the power spectrum and the PDF are measured at the grid level, and one can check that the ψ field variance is already consistent with its power spectrum variance integrated up to the Nyquist frequency. In Fig. 12 , we can see that the PDF of the ψ realization from ITAM matches the one of the N -body simulation (much better than muscle , and particularly 2LPT), as it should by the virtue of the translation transform equation ( 2 ). Likewise, the Lagrangian-space ψ power spectrum of ITAM matches that of the N -body simulation better than the other prescriptions, as shown in Fig. 13 . The two standard simulation schemes have a mismatch, with 2LPT diverging the most at small scales and muscle departing the most on large scales. This confirms the detailed analysis of the power spectra of Lagrangian schemes conducted by Chan ( 2014 ), where it is also found that these techniques come short of matching the power spectrum measured in N -body simulations. Another benchmark of the goodness of a fast simulation scheme is the similarity between the resulting matter density and the N -body matter density. This is generally quantified by means of the cross-correlation spectrum, which for two fields A and B is Figure 12. PDFs of ψ for various schemes. The peak at ψ ∼ −3 is associated with collapsed halo particles. In muscle , halo particles give a Dirac-delta function at ψ = −3 (see equation 43 ). Figure 13. Transfer function of ψ, defined as the ratio of the power spectrum of ψ generated from the fast simulation scheme over the one measured from the N -body realization. The cross-correlations are plotted in the top panel of Fig. 14 . As we can see, the muscle implementation still performs better than ITAM, which in turn is better than 2LPT, at least on smaller scales. Examining this further, 2LPT is almost a local approach, in the sense that it considers the information of each voxel and its nearest neighbours (equation 37 ). ITAM by construction is non-local, as it involves the two-point correlation function. muscle is also non-local, but the non-locality is coming from a smoothing procedure on various scales centred around each voxel (equation 43 ). Considering that the one-point mapping of ITAM is very