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Dimers on Riemann surfaces I: Temperleyan forests

Nathanaël Berestycki, Benoit Laslier, Gourab Ray

Abstract

This is the first article in a series of two papers in which we study the Temperleyan dimer model on an arbitrary bounded Riemann surface of finite topolgical type. The end goal of both papers is to prove the convergence of height fluctuations to a universal and conformally invariant scaling limit. In this part we show that the dimer model on the Temperleyan superposition of a graph embedded on the surface and its dual is well posed, provided that we remove an appropriate number of punctures. We further show that the resulting dimer configuration is in bijection with an object which we call Temperleyan forest, whose law is characterised in terms of a certain topological condition. Finally we discuss the relation between height differences and Temperleyan forest, and give a criterion guaranteeing the convergence of the height fluctuations in terms of the Temperleyan forest.

Dimers on Riemann surfaces I: Temperleyan forests

Abstract

This is the first article in a series of two papers in which we study the Temperleyan dimer model on an arbitrary bounded Riemann surface of finite topolgical type. The end goal of both papers is to prove the convergence of height fluctuations to a universal and conformally invariant scaling limit. In this part we show that the dimer model on the Temperleyan superposition of a graph embedded on the surface and its dual is well posed, provided that we remove an appropriate number of punctures. We further show that the resulting dimer configuration is in bijection with an object which we call Temperleyan forest, whose law is characterised in terms of a certain topological condition. Finally we discuss the relation between height differences and Temperleyan forest, and give a criterion guaranteeing the convergence of the height fluctuations in terms of the Temperleyan forest.

Paper Structure

This paper contains 32 sections, 46 theorems, 116 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Background and main result
  3. Relation with previous work
  4. Overview of the proof and organisation of the paper
  5. Background and setup
  6. Riemann surfaces and embedding
  7. Classification of surfaces.
  8. Universal cover
  9. Intrinsic and topological winding
  10. Discretization setup
  11. Height function and forms
  12. Temperley's bijection on Riemann surfaces
  13. Notion of Temperleyan cycle rooted spanning forest; bijection
  14. Criterion for a wired CRSF to be Temperleyan
  15. Wilson's algorithm to generate wired oriented CRSF
  16. ...and 17 more sections

Key Result

Theorem 1.1

Let $M$ denote a bounded Riemann surface, possibly with a boundary, which is not the sphere or a simply connected domain. Let $G^{\# \delta}$ be a sequence of Temperleyan discretisations of $M$ satisfying the Invariance principle and a crossing estimate for random walk described in sec:setup. Let $h converges as $\delta \to 0$ in law and the limit is conformally invariant and independent of the se

Figures (12)

  • Figure 1: The graphs $\Gamma, \Gamma^\dagger, \hat{G}$, and $G$. The graph $\Gamma$ is pictured in red and $\Gamma^\dagger$ in black.
  • Figure 2: Removing a white vertex (puncture) to create the dimer graph $G'$.
  • Figure 5: A non-Temperleyan CRSF (in blue). The surface $M$ is the "pair of pants": a domain of the plane with two holes (in grey in the picture). In this example, $t$ does not contain any cycle, and hence any connected component flows to the boundary of $M$. Its dual $t^\dagger$ must contain a component with two cycles which overlap. The cycles go around each of the two holes, and must be connected, as otherwise there would have to be a path in $t$ separating them; however this is impossible as such a path would have to connect two distinct boundary points. So $t$ is not Temperleyan. Note that $\chi = 2-2g - b = -1$ here. See Lemma \ref{['lem:annulus']} for a more general argument.
  • Figure 6: Illustration of the local transformation in Temperley's bijection. Left: a dimer configuration on a superposition graph. Right: a collection of oriented edges forming dual oriented CRSF.
  • Figure 7: The definition of the height function in terms of winding.
  • ...and 7 more figures

Theorems & Definitions (95)

  • Theorem 1.1
  • Lemma 2.1: Lemma 2.1 in BLR16
  • Lemma 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.5: Conformal invariance of assumptions
  • proof
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8: Unique path lifting property
  • ...and 85 more