Exploring the phase transition of planar FK-percolation

TL;DR

This work analyzes the phase transition of planar FK-percolation (random-cluster model) across all q ≥ 1 on Z^2, establishing a self-dual critical point p_sd = √q/(1+√q) and proving a dichotomy between continuous and discontinuous transitions through a renormalization–RSW framework. It connects FK-percolation to the six-vertex model via the Baxter–Kelland–Wu (BKW) correspondence, enabling Bethe-ansatz based analysis of the critical free energy and distinguishing transition types by the scaling of certain crossing events. A major achievement is showing rotational invariance of the critical phase and universality on isoradial lattices, achieved via star-triangle transformations and track-exchanges that couple models across geometries, together with sophisticated control of mesoscopic cluster dynamics. The results illuminate how critical FK-percolation may exhibit conformal universality (via loop ensembles and CLE) and provide rigorous foundations for the scaling theory and universality classes of two-dimensional statistical mechanics models.

Abstract

The aim of these notes is to give a quick introduction to FK-percolation, focusing on certain recent results about the phase transition of the two dimensional model, namely its continuity or discontinuity depending on the cluster weight $q$, and the asymptotic rotational invariance of the critical phase (when the phase transition is continuous). As such, the main focus is on FK-percolation on $\mathbb Z^2$ with $q \geq 1$, but we do mention some important results valid for general dimension. To favour quick access to recent results, the style is minimal, with certain proofs omitted or left as exercises.

Figures (30)

• Figure 1: A piece of a square lattice (solid vertices and edges) and its dual (hollow vertices, dashed edges). The primal open edges of $\omega$ are red, those of $\omega^*$ are blue. Note that the cluster of the central primal vertex is finite and surrounded by a blue circuit.
• Figure 2: Left: An illustration of the argument in the proof of Lemma \ref{['lem:phiS_exp']}: once the cluster ${\sf C}$ of $0$ in $S$ has been explored, for $0$ to be connected to distance $n$, one of the red vertices needs to be connected to distance $n-{\rm diam}(S)$in the complement of $\sf C$. Right: The white region containing $0$ is the set of edges $S$. Here $0$ is connected to four vertices on $\partial S$. When proving Lemma \ref{['lem:phiS_supercrit']}, $S$ denotes the connected component of $0$ in the complement of the cluster of $\partial \Lambda_n$.
• Figure 3: Left: an $(n + 1) \times n$ rectangle is either crossed horizontally in the primal model or vertically in the dual. At the self-dual point, these two events have equal probability $1/2$. Right: By symmetry, the probability to connect the lower half of the left side ${\sf BottomLeft}$ to the right side of the square is at least $1/4$. The same holds for connections between the ${\sf TopRight}$ and the left side. Combining these two crossings with a vertical one ensures the existence of a connection between ${\sf BottomLeft}$ and ${\sf TopRight}$.
• Figure 4: In each square ${\sf BottomLeft}$ is connected to ${\sf TopRight}$. The left and right red paths are the topmost and bottommost, respectively, paths realising these connections in a configuration $\omega$. The grey paths are their reflections $\sigma$ with respect to the medial line; by construction, the red and grey paths intersect. When orienting the left red path from left to right, call $b$ its last intersection point with the left grey path, and $a$ its endpoint on the medial line. Let $d = \sigma(b)$ and $c$ be the endpoint of the lower red path on the medial line. The blue $\sigma$-symmetric domain $\mathcal{Q}$ is bounded by the arcs $(ab)$ and $(cd)$ of the red paths, and their grey images through $\sigma$. When sampling an independent configuration $\tilde{\omega}$ in $\mathcal{Q}$, the probability that $(ab)$ and $(cd)$ are connected is $1/2$. The explored regions (hashed) may intersect $\mathcal{Q}$, however the boundary of any such intersection is an open path. If one completes the explored configuration $\omega$ by pasting $\tilde{\omega}$ in the unexplored parts of $\mathcal{Q}$, then whenever $\mathcal{Q}$ is crossed in $\tilde{\omega}$, it is also crossed in $\omega$, leading ultimately to a left-right crossing of the $2n\times n$ rectangle.
• Figure 5: The red path is the leftmost top-bottom crossing of the rectangle. A point on it connected by a dual path to the right side of the rectangle is a pivotal for the top-bottom crossing event. When one such point exists, one may use concentric dyadic annuli to create a logarithmic number of pivotals.

Theorems & Definitions (118)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• proof : Proof of Theorem \ref{['thm:pc_nontrivial_perco']}
• Proposition 1.4
• proof
• Proposition 1.6
• proof
• Theorem 1.7
• Proposition 1.8: Russo's formula