Open Access
2023 Macroscopic loops in the 3d double-dimer model
Alexandra Quitmann, Lorenzo Taggi
Author Affiliations +
Electron. Commun. Probab. 28: 1-12 (2023). DOI: 10.1214/23-ECP536


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 Zd, 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.


The authors thank the German Research Foundation (project number 444084038, priority program SPP2265) for financial support. AQ additionally thanks the German Research Foundation through IRTG 2544 and the German Academic Exchange Service (grant number 57556281) for financial support. The authors also thank the editor and the two anonymous referees for carefully reviewing the paper.


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Alexandra Quitmann. Lorenzo Taggi. "Macroscopic loops in the 3d double-dimer model." Electron. Commun. Probab. 28 1 - 12, 2023.


Received: 27 June 2022; Accepted: 17 July 2023; Published: 2023
First available in Project Euclid: 4 August 2023

arXiv: 2206.08284
MathSciNet: MR4627413
zbMATH: 07734099
Digital Object Identifier: 10.1214/23-ECP536

Primary: 60K35 , 82B20 , 82B26

Keywords: Dimer model , double dimer model , random walk loop soups , Self-avoiding walk , statistical mechanics

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