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

TL;DR

This paper proves that correlations in the dimer model on high-dimensional lattices do not decay, revealing a phase transition in lattice permutations related to quantum Bose gases, with implications for understanding complex statistical mechanics models.

Contribution

It provides the first rigorous proof of non-decaying correlations in the dimer model for dimensions greater than two and establishes a phase transition in lattice permutation models.

Findings

Correlations in the dimer model do not decay in dimensions > 2.

A phase transition occurs in lattice permutation models.

Long-range order is demonstrated in high-dimensional lattice systems.

Abstract

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.