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Nesting of double-dimer loops: local fluctuations and convergence to the nesting field of CLE(4)

Mikhail Basok, Konstantin Izyurov

TL;DR

This work proves two central results about the double-dimer model in the upper half-plane: first, at a fixed bulk point $v$, the number of encircling double-dimer loops $N_{\delta}(v)$ satisfies a central limit theorem with $\mu_{\delta}=-\frac{1}{\pi^{2}}\log\delta+O(1)$ and $\sigma_{\delta}^{2}= -\frac{2}{3\pi^{2}}\log\delta+O(1)$, so $(N_{\delta}(v)-\mu_{\delta})/\sigma_{\delta} \Rightarrow \mathcal{N}(0,1)$; second, the nesting field $\varphi_{\delta}=N_{\delta}-\mathbb{E}N_{\delta}$ converges in distribution to the CLE${}_4$ nesting field $\varphi$ in the local Sobolev space $H^{-1-\nu}_{\mathrm{loc}}(\mathbb{C}^{+})$ for every $\nu>0$. The authors achieve this via a monodromy-enhanced Kasteleyn framework to control near-diagonal asymptotics of inverse operators, coupled with approximate Wick's rules for height fluctuations and meticulous regularization of nested loop observables. They then embed these discrete results into a Sobolev-field setting, develop a two-point approximation for CLE nesting fields, and leverage cylindrical-event convergence results for CLE${}_4$ to conclude the full nesting-field convergence of the double-dimer model. The work thus solidifies a path toward the full convergence of double-dimer loop laminations to CLE${}_4$ by establishing the Gaussian fluctuations and the Sobolev-continuity of nesting fields. Overall, it provides a rigorous link between discrete dimer observables and the conformal loop ensemble nesting fields, with explicit constants and a robust multiscale analytic framework.

Abstract

We consider the double-dimer model in the upper-half plane discretized by the square lattice with mesh size $δ$. For each point $x$ in the upper half-plane, we consider the random variable $N_δ(x)$ given by the number of the double-dimer loops surrounding this point. We prove that the normalized fluctuations of $N_δ(x)$ for a fixed $x$ are asymptotically Gaussian as $δ\to 0+$. Further, we prove that the double-dimer nesting field $N_δ(\cdot) - \mathbb{E}\, N_δ(\cdot)$, viewed as a random distribution in the upper half-plane, converges as $δ\to 0+$ to the nesting field of CLE(4) constructed by Miller, Watson and Wilson.

Nesting of double-dimer loops: local fluctuations and convergence to the nesting field of CLE(4)

TL;DR

This work proves two central results about the double-dimer model in the upper half-plane: first, at a fixed bulk point , the number of encircling double-dimer loops satisfies a central limit theorem with and , so ; second, the nesting field converges in distribution to the CLE nesting field in the local Sobolev space for every . The authors achieve this via a monodromy-enhanced Kasteleyn framework to control near-diagonal asymptotics of inverse operators, coupled with approximate Wick's rules for height fluctuations and meticulous regularization of nested loop observables. They then embed these discrete results into a Sobolev-field setting, develop a two-point approximation for CLE nesting fields, and leverage cylindrical-event convergence results for CLE to conclude the full nesting-field convergence of the double-dimer model. The work thus solidifies a path toward the full convergence of double-dimer loop laminations to CLE by establishing the Gaussian fluctuations and the Sobolev-continuity of nesting fields. Overall, it provides a rigorous link between discrete dimer observables and the conformal loop ensemble nesting fields, with explicit constants and a robust multiscale analytic framework.

Abstract

We consider the double-dimer model in the upper-half plane discretized by the square lattice with mesh size . For each point in the upper half-plane, we consider the random variable given by the number of the double-dimer loops surrounding this point. We prove that the normalized fluctuations of for a fixed are asymptotically Gaussian as . Further, we prove that the double-dimer nesting field , viewed as a random distribution in the upper half-plane, converges as to the nesting field of CLE(4) constructed by Miller, Watson and Wilson.
Paper Structure (16 sections, 46 theorems, 255 equations)

This paper contains 16 sections, 46 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Height function, loop statistics and monodromy
  3. Combinatorial relations between nesting and height function
  4. Kasteleyn matrix, dimers and the Gaussian free field
  5. Approximate Wick's rule and the asymptotic of mdelta and sigmadelta
  6. Loop statistics and Kasteleyn operator with an SL(2,C) monodromy
  7. Normalized fluctuations of N(x) and Kasteleyn operator with monodromy
  8. Full-plane operator with monodromy around one point
  9. Full-plane operator with monodromy around two points
  10. Proof of Theorem \ref{['thma:main1']}
  11. The double dimer nesting field and the CLE(4) nesting field
  12. Random fields belonging to a Sobolev class
  13. Two-point approximation of the CLE(kappa)-nesting field
  14. Cylindrical events and the convergence of mdelta to mdelta.
  15. Two-point approximation of the double-dimer nesting field
  16. ...and 1 more sections

Key Result

Theorem 1

Consider the double-dimer model in ${\mathbb{C}}^+_\delta$, let $v$ be a fixed point in the upper half-plane, and denote by $N_\delta(v)$ the number of double-dimer loops encircling $v$. Then we have

Theorems & Definitions (98)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • proof
  • Corollary 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 88 more