  Our first main result is that correlations between monomers in the dimer
model in $\mathbb{Z}^d$ do not decay to zero when $d > 2$. This is the first
rigorous result about correlations in the dimer model in dimensions greater
than two and shows that the model behaves drastically differently than in two
dimensions, in which case it is integrable and correlations are known to decay
to zero polynomially. Such a result is implied by our more general, second main
result, which states the occurrence of a phase transition in the model of
lattice permutations, which is related to the quantum Bose gas. More precisely,
we consider a self-avoiding walk interacting with lattice permutations and we
prove that, in the regime of fully-packed loops, such a walk is `long' and the
distance between its end-points grows linearly with the diameter of the box.
These results follow from the derivation of a version of the infrared bound
from a new general probabilistic settings, with coloured loops and walks
interacting at sites and walks entering into the system from some `virtual'
vertices.
