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<p>We study the double–dimer model, the monomer double–dimer model, spatial random permutations, and the dimer model. For this entire class of models, we prove absence of long-range order on two-dimensional graphs, both planar and non-planar. More precisely, long-range order is excluded in the sense of vanishing spontaneous magnetization; moreover, correlations and the probability that a loop visits two vertices decay to zero in an averaged sense as the distance between the vertices diverges. As a further consequence, we show that giant loops are absent, in sharp contrast with the behavior in dimensions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(d&gt;2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>&gt;</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

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Mermin–Wagner Theorem for Dimers, Monomer Double-Dimers, and Spatial Random Permutations

  • 期刊论文

  • 发表日期：

    2026年7月9日

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  • Lorenzo Taggi,
  • Wei Wu

摘要

We study the double–dimer model, the monomer double–dimer model, spatial random permutations, and the dimer model. For this entire class of models, we prove absence of long-range order on two-dimensional graphs, both planar and non-planar. More precisely, long-range order is excluded in the sense of vanishing spontaneous magnetization; moreover, correlations and the probability that a loop visits two vertices decay to zero in an averaged sense as the distance between the vertices diverges. As a further consequence, we show that giant loops are absent, in sharp contrast with the behavior in dimensions \(d>2\) d > 2 . 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.