On loops in the complement to dimers

TL;DR

This paper investigates the structure of ergodic translation-invariant Gibbs measures for the dimer model on the hexagonal lattice, revealing conditions under which the complement forms either infinitely many loops or a single bi-infinite path.

Contribution

It demonstrates that non-frozen measures must have either infinitely many loops or a unique bi-infinite path, using flip operations and percolation theory techniques.

Findings

Non-frozen measures have either infinitely many loops or a single bi-infinite path.

Classical dimer results imply positive density of certain hexagons, enabling percolation methods.

Excludes the coexistence of multiple bi-infinite paths in such measures.

Abstract

We consider ergodic translation-invariant Gibbs measures for the dimer model (i.e. perfect matchings) on the hexagonal lattice. The complement to a dimer configuration is a fully-packed loop configuration: each vertex has degree two. This is also known as the loop $O(1)$ model at $x=\infty$. We show that, if the measure is non-frozen, then it exhibits either infinitely many loops around every face or a unique bi-infinite path. Our main tool is the flip (or XOR) operation: if a hexagon contains exactly three dimers, one can replace them by the other three edges. Classical results in the dimer theory imply that such hexagons appear with a positive density. Up to some extent, this replaces the finite-energy property and allows to make use of tools from the percolation theory, in particular the Burton--Keane argument, to exclude existence of more than one bi-infinite path.