  Provided a random realization of the cosmological model, observations of our
cosmic neighborhood now allow us to build simulations of the latter down to the
non-linear threshold. The resulting local Universe models are thus accurate up
to a given residual cosmic variance. Namely some regions and scales are
apparently not constrained by the data and seem purely random. Drawing
conclusions together with their uncertainties involves then statistics implying
a considerable amount of computing time. By applying the constraining algorithm
to paired fixed fields, this paper diverts the original techniques from their
first use to efficiently disentangle and estimate uncertainties on local
Universe simulations obtained with random fields. Paired fixed fields differ
from random realizations in the sense that their Fourier mode amplitudes are
fixed and they are exactly out of phase. Constrained paired fixed fields show
that only 20% of the power spectrum on large scales (> tens of megaparsecs) is
purely random. Namely 80% of it is partly constrained by the large scale /
small scale data correlations. Additionally, two realizations of our local
environment obtained with paired fixed fields of the same pair constitute an
excellent non-biased average or quasi-linear realization of the latter, namely
the equivalent of hundreds of constrained simulations. The variance between
these two realizations gives the uncertainty on the achievable local Universe
simulations. These two simulations will permit enhancing faster our local
cosmic web understanding thanks to a drastically reduced required computational
time to appreciate its modeling limits and uncertainties.
