Uniformly positive correlations in the Dimer model and macroscopic interacting self-avoiding walk in ℤd, d≥3

Our first main result is that correlations between monomers in the dimer model in (Formula presented.) do not decay to 0 when (Formula presented.). This is the first rigorous result about correlations in the dimer model in dimensions greater than 2 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 endpoints 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. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.