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

Uniformly positive correlations in the dimer model and phase transition in lattice permutations on $\mathbb{Z}^d$, $d > 2$, via reflection positivity