Probability Theory and Related Fields

, Volume 166, Issue 3–4, pp 971–1023 | Cite as

On the asymptotics of dimers on tori

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Abstract

We study asymptotics of the dimer model on large toric graphs. Let \({\mathbb {L}}\) be a weighted \(\mathbb {Z}^2\)-periodic planar graph, and let \(\mathbb {Z}^2 E\) be a large-index sublattice of \(\mathbb {Z}^2\). For \({\mathbb {L}}\) bipartite we show that the dimer partition function \({{\textsf {\textit{Z}}}}_E\) on the quotient \({\mathbb {L}}/(\mathbb {Z}^2 E)\) has the asymptotic expansion
$$\begin{aligned} {{\textsf {\textit{Z}}}}=\exp \{A\,\mathbf {f}_0+{\textsf {fsc}}+o(1)\} \end{aligned}$$
where A is the area of \({\mathbb {L}}/(\mathbb {Z}^2 E)\), \(\mathbf {f}_0\) is the free energy density in the bulk, and \({\textsf {fsc}}\) is a finite-size correction term depending only on the conformal shape of the domain together with some parity-type information. Assuming a conjectural condition on the zero locus of the dimer characteristic polynomial, we show that an analogous expansion holds for \({\mathbb {L}}\) non-bipartite. The functional form of the finite-size correction differs between the two classes, but is universal within each class. Our calculations yield new information concerning the distribution of the number of loops winding around the torus in the associated double-dimer models.

Mathematics Subject Classification

82B20 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Richard W. Kenyon
    • 1
  • Nike Sun
    • 2
  • David B. Wilson
    • 3
  1. 1.Department of MathematicsBrown UniversityProvidenceUSA
  2. 2.Department of StatisticsStanford UniversityStanfordUSA
  3. 3.Microsoft ResearchOne Microsoft WayWashingtonUSA

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