  We study Lagrangian Perturbation Theory (LPT) and its regularization in the
Effective Field Theory (EFT) approach. We evaluate the LPT displacement with
the same phases as a corresponding $N$-body simulation, which allows us to
compare perturbation theory to the non-linear simulation with significantly
reduced cosmic variance, and provides a more stringent test than simply
comparing power spectra. We reliably detect a non-vanishing leading order EFT
coefficient and a stochastic displacement term, uncorrelated with the LPT
terms. This stochastic term is expected in the EFT framework, and, to the best
of our understanding, is not an artifact of numerical errors or transients in
our simulations. This term constitutes a limit to the accuracy of perturbative
descriptions of the displacement field and its phases, corresponding to a $1\%$
error on the non-linear power spectrum at $k=0.2 h$/Mpc at $z=0$. Predicting
the displacement power spectrum to higher accuracy or larger wavenumbers thus
requires a model for the stochastic displacement.
