🌀 nonlinear sciences

Rotational invariance in critical planar lattice models

This paper proves that the large-scale properties of the critical random-cluster model on the square lattice with cluster-weight $1 \le q \le 4$ , including Bernoulli percolation and critical Potts models, exhibit rotational invariance.

Original authors: Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Mendes Oulamara

Published 2026-07-01

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Hugo Duminil-Copin, Karol Kajetan Kozlowski, Dmitry Krachun, Ioan Manolescu, Mendes Oulamara

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, infinite checkerboard made of tiny tiles. On this board, we play a game of chance where we randomly decide whether to keep the lines between the tiles open or close them. This creates a patchwork of connected islands (clusters) and empty spaces. This is the Random-Cluster Model, a mathematical way to describe how things like electricity, magnetism, or water might flow through a material.

For decades, physicists and mathematicians have suspected that if you zoom out far enough—looking at the board from a helicopter rather than a microscope—the specific shape of the grid (square, hexagonal, or distorted) shouldn't matter. They believed that at a "critical" tipping point, the patterns formed by these islands would look the same no matter how you rotated your view. They thought the system possessed rotational invariance: if you spun the board, the statistical laws governing the islands wouldn't change.

However, proving this was incredibly hard. Most previous proofs relied on the board having very specific, perfect symmetries (like a perfect square grid). This paper by Hugo Duminil-Copin and his team breaks new ground by proving this "rotational invariance" holds true even for a wide family of these models, specifically when the "cluster weight" (a parameter that controls how likely islands are to merge) is between 1 and 4.

Here is a breakdown of their journey and findings using everyday analogies:

1. The Problem: The "Distorted" Map

Imagine you have a map of a city. Usually, city blocks are perfect squares. But in this paper, the authors look at a city where the blocks are stretched into long rectangles or tilted at weird angles.

The Old Way: Previous proofs could only handle the "perfect square" city. They used special mathematical tools (like "discrete holomorphic observables") that only worked on perfect grids.
The New Approach: The authors realized they didn't need the grid to be perfect. They used a trick called the Star-Triangle transformation.
- Analogy: Imagine you have a traffic intersection with three roads meeting at a point (a star). You can mathematically swap this for a triangle of roads connecting the same three points without changing the overall traffic flow between the outer cities.
- The authors showed that you can take a distorted grid and, by repeatedly swapping these "stars" and "triangles," slowly morph it into a different distorted grid. Crucially, this swapping process preserves the connections between the islands.

2. The Journey: The "Shape-Shifting" Walk

To prove that the distorted grid behaves like the square grid, the authors created a "time-lapse" movie of the grid transforming.

The Process: They started with a grid tilted at one angle and slowly, step-by-step, swapped tracks to tilt it toward a different angle.
The Drift: As they did this, they noticed the "islands" (clusters) didn't just stay put; they slowly drifted or stretched.
- Analogy: Imagine a rubber band with a knot in it. If you stretch the rubber band in one direction, the knot moves. The authors calculated exactly how much the knot moves on average.
The Surprise: They proved that for these specific models, the "drift" (the average movement of the islands) is actually zero. The islands might wiggle, but they don't systematically stretch or shrink in a way that changes the overall shape of the universe.

3. The Main Result: The "Magic Mirror"

The paper's headline result is Theorem 1.2.

What it says: If you take the critical random-cluster model on a square lattice and rotate it by any angle (say, 30 degrees), the statistical properties of the rotated system are indistinguishable from the original system when you look at them from far away.
The Metaphor: Imagine looking at a cloud formation. If you walk around the cloud, the shape of the cloud looks different from your new angle. But if the cloud is "rotationally invariant," it means that the laws governing how the cloud forms are the same regardless of your angle. The authors proved that for these specific lattice models, the "cloud" of connections looks statistically identical whether you view it straight on or from a tilted angle.

4. Why This Matters (According to the Paper)

The authors state that this result is a "pillar" for proving something even bigger: that these models converge to a Gaussian Free Field (a mathematical object describing a random, wavy surface, like a vibrating drumhead or a rough sea).

The Connection: To prove the model becomes this smooth, wavy surface, you first need to prove it doesn't care about the direction you look (rotational invariance). This paper provides that missing piece of the puzzle.
Specific Models: This result confirms that the Potts model (a model used to describe magnets with multiple states) with 2, 3, or 4 colors behaves rotationally invariant at large scales. It also covers Bernoulli percolation (simple random connectivity) as a special case.

5. What They Did Not Do

It is important to stick to what the paper claims:

They did not prove that the system is "conformally invariant" (invariant under any squashing or stretching, not just rotation). They only proved rotation and scale invariance.
They did not apply this to real-world medical treatments or engineering designs. The paper is purely mathematical, focusing on the behavior of abstract lattice models.
They did not solve the problem for all possible models, only for the specific range of parameters where the "cluster weight" is between 1 and 4.

Summary

In simple terms, this paper proves that for a large class of random grid models, the "big picture" looks the same no matter how you rotate your head. The authors achieved this by showing that you can mathematically morph one distorted grid into another without breaking the connections, and that during this morphing, the patterns don't get stretched or skewed in a biased way. This confirms a long-held belief in physics that at the critical point, nature's patterns are perfectly symmetrical, regardless of the underlying grid's shape.

Technical Summary: Rotational Invariance in Critical Planar Lattice Models

Problem Statement
The paper addresses a fundamental question in the statistical mechanics of two-dimensional lattice models: the rotational invariance of their large-scale properties at criticality. While the Renormalization Group (RG) formalism predicts that critical fixed points should exhibit translation, rotation, and scale invariance, and Conformal Field Theory (CFT) suggests full conformal invariance, rigorous mathematical proofs have historically been limited to specific models (e.g., site percolation on the triangular lattice, the Ising model, dimers, and uniform spanning trees). These proofs typically rely on discrete holomorphic observables, a strategy that is highly dependent on specific discrete symmetries and difficult to generalize.

The authors focus on the random-cluster model (Fortuin-Kasteleyn percolation) on the square lattice with cluster-weight $q \in [1, 4]$ . This range includes Bernoulli percolation ( $q=1$ ) and the critical Potts models for $q=2, 3, 4$ . The central problem is to prove that the scaling limit of these models is rotationally invariant, thereby establishing a crucial step toward proving full conformal invariance without relying on model-specific holomorphic observables.

Methodology
The authors employ a strategy inspired by field theory, utilizing the star-triangle transformation (or Yang-Baxter equation) as the primary tool. Unlike previous approaches that computed specific eigenvalues or partition functions using the Bethe ansatz, this work relies on the structural properties of the star-triangle transformation in the setting of isoradial graphs.

The core methodology involves:

Isoradial Embeddings: The authors consider the random-cluster model on a family of distorted square lattices, denoted $L(\alpha)$ , which are isoradial rectangular lattices. These lattices are constructed such that the inhomogeneity of edge parameters (horizontal vs. vertical) is compensated by the geometry of the embedding.
Track-Exchange Transformations: They define a sequence of transformations that gradually deform a lattice $L(\beta)$ into $L(\alpha)$ by exchanging "tracks" (sequences of faces in the diamond graph) using the star-triangle transformation. This process preserves connectivity between vertices but alters the geometry of clusters.
Coupling and Resampling: To compare the measures on $L(\beta)$ and $L(\alpha)$ , the authors construct a coupling of configurations $(\omega_t)$ . A key technical innovation is the resampling of extrema. To ensure that the local environment around the extrema of large clusters behaves independently of the global configuration (mimicking the Incipient Infinite Cluster or IIC), they resample the configuration in small boxes surrounding the cluster extrema at each step of the transformation.
Drift Analysis: The evolution of the extremal coordinates (top, bottom, left, right) of mesoscopic clusters is analyzed. The authors show that these coordinates evolve according to a linear transformation $M_{\beta, \alpha}$ determined by the "drifts" (expected increments) of the extrema.
Symmetry Arguments: To prove that the linear transformation is the identity (implying universality), the authors utilize a symmetry argument. They show that the critical measure on the standard square lattice $L(\pi/2)$ is asymptotically invariant under a group of transformations generated by reflections. By analyzing the eigenvalues of these transformations, they deduce that the drifts must vanish, forcing $M_{\beta, \alpha}$ to be the identity map.

Key Contributions and Results

Theorem 1.2 (Rotational Invariance): The authors prove that the critical random-cluster model on the square lattice with $q \in [1, 4]$ is asymptotically rotationally invariant. Specifically, the distance between the measure on the lattice $\delta \mathbb{Z}^2$ and the measure on the rotated lattice $e^{i\alpha}\delta \mathbb{Z}^2$ vanishes as $\delta \to 0$ in both the Camia-Newman ( $d_{CN}$ ) and Schramm-Smirnov ( $d_{SS}$ ) topologies.
Theorem 1.7 (Universality on Isoradial Rectangular Graphs): They establish that the random-cluster model on any isoradial rectangular lattice $L(\alpha)$ (with $q \in [1, 4]$ ) is asymptotically equivalent to the model on the standard square lattice $L(\pi/2)$ . This confirms that the isoradial embedding is the "correct" geometric compensation for the model's inhomogeneity to ensure a conformally invariant scaling limit.
Theorem 1.9 (Universality up to Linear Transformation): A weaker intermediate result showing that the model on $L(\beta)$ and $L(\alpha)$ are related by an invertible linear map $M_{\beta, \alpha}$ . The proof of the main theorems relies on showing that for the critical case, this map is the identity.
Corollaries for Potts Models: Using the Edwards-Sokal coupling, the authors deduce that the spin-spin correlations of the critical $q$ -state Potts models ( $q \in \{2, 3, 4\}$ ) are rotationally invariant at large scales.
Corollaries for Crossing Probabilities: The results imply that crossing probabilities for quads in simply connected domains are invariant under rotation in the scaling limit.

Significance and Claims
The paper claims that its results are instrumental in proving the convergence of the full-plane six-vertex model to the Gaussian Free Field (GFF) in the range corresponding to the critical random-cluster model ( $1 \le q \le 4$ ), a result detailed in a separate paper [22].

The authors emphasize that their strategy is general and applies to a broad class of integrable planar systems satisfying the Yang-Baxter equation, potentially extending to models like the Ashkin-Teller model and certain loop models. By avoiding the need for discrete holomorphic observables and instead relying on the star-triangle transformation and mixing properties (RSW), the paper provides a more robust pathway to proving conformal invariance for a wider class of models. The work bridges the gap between the heuristic predictions of physics (RG and CFT) and rigorous mathematical proof for a significant family of lattice models.

Modesty and Limitations
The authors note that the quantitative version of their results (specifically the rate of convergence in the rotational invariance) is not provided, as a quantitative proof would require a substantially more technical argument. They also restrict their domain considerations to $C^1$ -smooth boundaries to focus on the core mechanism, noting that the result likely extends to Jordan domains. Furthermore, while they prove rotational invariance for spin-spin correlations, they explicitly state that deducing the rotation invariance of energy correlations requires additional work and is not covered in this paper.