Macroscopic loops in the $3d$ double-dimer model

TL;DR

This paper demonstrates that in three or more dimensions, the double-dimer model exhibits macroscopic loops that differ from two-dimensional behavior, and that a critical monomer activity threshold exists where macroscopic self-avoiding walks persist.

Contribution

It proves the existence of macroscopic loops in the $3d$ double-dimer model and establishes a positive critical monomer activity threshold for macroscopic self-avoiding walks.

Findings

Loops are macroscopic in $ extbf{d} > 2$ dimensions.

Existence of a positive critical monomer activity threshold.

Self-avoiding walks remain macroscopic with monomers present.

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 Betz and Taggi (2019) and Taggi (2021) that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when $d >2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.