  We consider nearest neighbour spatial random permutations on $\mathbb{Z}^d$.
In this case, the energy of the system is proportional the sum of all cycle
lengths, and the system can be interpreted as an ensemble of edge-weighted,
mutually self-avoiding loops. The constant of proportionality, $\alpha$, is the
order parameter of the model. Our first result is that in a parameter regime of
edge weights where it is known that a single self-avoiding loop is weakly space
filling, long cycles of spatial random permutations are still exponentially
unlikely. For our second result, we embed a self-avoiding walk into a
background of spatial random permutations, and condition it to cover a
macroscopic distance. For large values of $\alpha$ (where long cycles are very
unlikely) we show that this walk collapses to a straight line in the scaling
limit, and give bounds on the fluctuations that are almost sufficient for
diffusive scaling. For proving our results, we develop the concepts of spatial
strong Markov property and iterative sampling for spatial random permutations,
which may be of independent interest. Among other things, we use them to show
exponential decay of correlations for large values of $\alpha$ in great
generality.
