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The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality

Ioan Manolescu, Maran Mohanarangan

Abstract

The study of the phase transition in planar FK-percolation on the square lattice has seen significant recent breakthroughs. The model undergoes a change in the nature of its phase transition at $q = 4$, transitioning from a continuous to a discontinuous regime. The aim of this article is to investigate the behaviour of the model in the discontinuous regime as $q > 4$ approaches the continuous transition point $4$ from above, while maintaining the critical parameter $p = p_c(q)$. We prove that in this limit, the correlation length becomes isotropic. The core of the proof builds upon the recently established rotational invariance of the large-scale features of the model at $q = 4$ (arXiv:2012.11672).

The Wulff crystal of self-dual FK-percolation becomes round when approaching criticality

Abstract

The study of the phase transition in planar FK-percolation on the square lattice has seen significant recent breakthroughs. The model undergoes a change in the nature of its phase transition at , transitioning from a continuous to a discontinuous regime. The aim of this article is to investigate the behaviour of the model in the discontinuous regime as approaches the continuous transition point from above, while maintaining the critical parameter . We prove that in this limit, the correlation length becomes isotropic. The core of the proof builds upon the recently established rotational invariance of the large-scale features of the model at (arXiv:2012.11672).
Paper Structure (30 sections, 17 theorems, 100 equations, 4 figures)

This paper contains 30 sections, 17 theorems, 100 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Definition of the model and main result
  3. Strategy of the proof
  4. Near-critical FK-percolation with q ≤ 4
  5. Preliminaries
  6. Isoradial graphs
  7. Definition and elementary properties of the isoradial random-cluster model
  8. Monotonic properties.
  9. Spatial Markov property.
  10. The track-exchange operator.
  11. Isoradial random-cluster model with q = 4
  12. Duality.
  13. RSW property.
  14. Mixing.
  15. Arm events and flower domains.
  16. ...and 15 more sections

Key Result

Theorem 1.1

For all $\varepsilon >0$, there exists $q_0 > 4$ such that for all $q \in (4,q_0]$ and any $\theta_1, \theta_2 \in [0,2\pi)$, we have

Figures (4)

  • Figure 1: Left: An isoradial rectangular lattice $\mathbb{L}(\alpha)$. The underlying diamond graph is drawn in thin black lines, while the actual lattice is drawn in red. The white points denote vertices of the dual lattice. Two tracks --- one horizontal and one vertical --- are shown as dashed grey lines. Note the different parameters $p_{\rm v}$ and $p_{\rm h}$ for the two edge orientations. Right: For $\alpha = \pi/2$, the diamond graph is $\mathbb{Z}^2$ and the lattice $\mathbb{L}(\pi/2)$ is the rotation of $\sqrt{2} \mathbb{Z}^2$ by $\frac{\pi}{4}$. In this case, the model is homogenous, as $p_{\rm v} = p_{\rm h} = \sqrt{q}/(1+\sqrt{q})$.
  • Figure 2: Left: the flower domain from $\Lambda_{2R}$ to $\Lambda_R$ is obtained by exploring all primal--dual interfaces starting on $\partial \Lambda_{2R}$ until they exit $\Lambda_{2R}$ or reach $\Lambda_R$. The explored region is grey and the flower domain is the white domain; its boundary is formed of one primal and one dual petal. Right: a good flower domain with the unique primal petal contained in the green region.
  • Figure 3: A cluster $\mathsf{C}$ with its extremum ${\rm Ext}_\theta(\mathsf{C})$ in direction $e_\theta$. In this example, ${\rm Ext}_\theta(\mathsf{C})$ does not belong to $\mathsf{C}$ itself. The cell associated to the extremum is shaded in light blue and the line $\langle \cdot, e_\theta \rangle = {\rm E}_\theta(\mathsf{C})$ is drawn in black.
  • Figure 4: The track-exchanges performed on a single period. From left to right: The initial lattice $\mathbb{L}_0$ with tracks of angle $\alpha = \pi/2$ at the bottom and $\beta < \pi/2$ at the top. The first two tracks to be exchanged are marked with dashed grey lines. Applying $\mathbf{S}_t \circ \dots \circ \mathbf{S}_0$ transforms $\mathbb{L}_0$ into $\mathbb{L}_{t+1}$ and a mixed block (colored in red) starts appearing in the middle. The cells in the mixed block at even timesteps are centred around rhombi with angle $\beta$. After more transformations, a $\beta$-block starts forming at the bottom and an $\alpha$-block at the top (both marked in blue). By time $2N$, the $\beta$-block and the $\alpha$-block have been exchanged completely. The interfaces of each lattice are marked with thick red lines.

Theorems & Definitions (29)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['thm:rotational_invariance']}
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3: GasManMohArmSeparation
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 3.1
  • ...and 19 more