Macroscopic loops in the 3d double-dimer model

Abstract

The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in ${\mathbb{Z}}^d$, $d>2$, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from [2, 19] that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is ‘long’, i.e., the distance between its end-points is proportional to the diameter of the box. Our paper shows that, when $d>2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is long even in the presence of a positive density of monomers.