  Halo Models of large scale structure provide powerful and indispensable tools
for phenomenological understanding of the clustering of matter in the Universe.
While the halo model builds structures out of the superposition of haloes,
defining halo profiles in their outskirts - beyond their virial radii - becomes
increasingly ambiguous, as one cannot assign matter to individual haloes in a
clear way. In this paper, we tackle this question by using numerical N-body
simulations to find a systematic definition of mean halo profile that can be
extended to large radii. These profiles must be compensated and are the key
ingredients for the computation of cosmological correlation functions in the
Amended Halo Model. The latter, introduced in our earlier work
(arXiv:1912.04872), provides a more physically accurate phenomenological
description of nonlinear structure formation, which respects conservation laws
on large scales. Here, we extend this model from the matter auto-power spectrum
to the halo-matter cross-power spectra, using N-body simulations. We find that
the (dimensionless) compensated halo profile, $r^3 \times \rho(r)/M_{200c}$ has
a near-universal maximum in the small range $0.03-0.04$ around the virial
radius, $r \simeq r_{\rm 200c}$, independent of the halo mass. The profiles
cross zero into negative values in the halo outskirts, beyond 2-3$\times r_{\rm
200c}$, consistent with our previous results. We provide preliminary fitting
functions for compensated Navarro-Frenk-White (NFW) profiles, and this can be
used to compute more physical observables in the Amended Halo Model of large
scale structure.
