  It has been recently understood (arXiv:1212.2885, arXiv:1310.4764,
arXiv:1410.0605) that for a general class of percolation models on
$\mathbb{Z}^d$ satisfying suitable decoupling inequalities, which includes
i.a.\ Bernoulli percolation, random interlacements and level sets of the
Gaussian free field, large scale geometry of the unique infinite cluster in
strongly percolative regime is qualitatively the same; in particular, the
random walk on the infinite cluster satisfies the quenched invariance
principle, Gaussian heat-kernel bounds and local CLT.
  In this paper we consider the random walk loop soup on $\mathbb{Z}^d$ in
dimensions $d\geq 3$. An interesting aspect of this model is that despite its
similarity and connections to random interlacements and the Gaussian free
field, it does not fall into the above mentioned general class of percolation
models, since the required decoupling inequalities are not valid.
  We identify weaker (and more natural) decoupling inequalities and prove that
(a) they do hold for the random walk loop soup and (b) all the results about
the large scale geometry of the infinite percolation cluster proved for the
above mentioned class of models hold also for models that satisfy the weaker
decoupling inequalities. Particularly, all these results are new for the vacant
set of the random walk loop soup. (The range of the random walk loop soup has
been addressed by Chang arXiv:1504.07906 by a model specific approximation
method, which does not apply to the vacant set.)
  Finally, we prove that the strongly supercritical regime for the vacant set
of the random walk loop soup is non-trivial. It is expected, but open at the
moment, that the strongly supercritical regime coincides with the whole
supercritical regime.
