Mermin-Wagner theorem for dimers, monomer double-dimers, and spatial random permutations

TL;DR

The paper develops a unified framework for dimer, monomer double-dimer, and spatial permutation models on two-dimensional-like graphs, proving a $Mermin ext{-}Wagner$ theorem in this broad setting. It introduces a novel complex spin representation that yields reflection positivity and a Bogoliubov-type inequality, enabling rigorous decay bounds for two-point functions without requiring positivity of the Gibbs measure. The main results include a bound on monomer–monomer correlations in the dimer model on 2D slabs, absence of macroscopic loops and decay of correlations in the monomer double-dimer model for all monomer activities, and absence of long-range order and macroscopic cycles in spatial random permutations at all temperatures. These insights extend beyond planar/integrable cases to non-planar graphs and general colorings (multi-occupancy and $2N$-color variants), with connections to loop $O(N)$ structures and potential implications for related Bose-gas-type representations.

Abstract

We study a generalisation of the double-dimer model that encompasses several models of interest, including the monomer double-dimer model, spatial random permutations, the dimer model, and the spin $O(N)$ model, and which is also related to the loop $O(N)$ model. We show that on two-dimensional-like graphs (such as slabs), both the correlation function and the probability that a loop visits two vertices decay to zero as the distance between the vertices diverges. Our approach is based on the introduction of a new complex spin representation for all models in this class, together with a new proof of the Mermin-Wagner theorem that does not require positivity of the Gibbs measure. Even for the well-studied dimer and double-dimer models our results are new: since they do not rely on exact solvability or Kasteleyn's theorem, they apply beyond the planar-graph setting.

Figures (1)

• Figure 1.1: Left: A configuration of the monomer double-dimer model in a box of $\mathbb{Z}^2$. Center: A configuration of the spatial permutation model in a box of $\mathbb{Z}^2$, where fixed points of the permutation are represented by isolated vertices. This configuration corresponds to the one on the left. Right: A configuration of the monomer double-dimer model in $\Omega_{o,x}$ when $x$ and $o$ have odd distance. Since $o$ and $x$ are the only vertices incident to exactly one dimer, while all other vertices are incident to either none or two dimers, the existence of a self-avoiding path with $o$ and $x$ as endpoints follows.

Theorems & Definitions (58)

• Theorem 1.1
• Theorem 1.2
• Remark 2.1
• Theorem 2.2
• Definition 3.1: Enlarged graph
• Definition 3.2: Dimer cardinalities
• Definition 3.3: Match functions
• Definition 3.4: Configurations
• Definition 3.5: Local time
• Definition 3.6: Measure