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The conformally invariant metric on CLE $_4$ I: subsequential limits of the non-simple CLE graph metric

This paper establishes the existence of a non-trivial, conformally invariant, and local metric on the CLE $_4$ loop ensemble by proving that it arises as a subsequential limit of renormalized graph metrics on CLE $_\kappa$ as $\kappa \downarrow 4$ , thereby characterizing the metric ball growth via the uniform exploration of Werner and Wu.

Original authors: Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, Yi Tian

Published 2026-09-04

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CC BY 4.0

Original authors: Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, Yi Tian

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

In the vast landscape of mathematics, there is a branch dedicated to understanding how shapes and spaces behave when stretched, twisted, or squashed without tearing. This field, known as conformal geometry, treats the plane like a sheet of rubber that can be deformed in infinitely many ways, provided that tiny angles between intersecting lines remain unchanged. Within this flexible world, mathematicians study random patterns that emerge at the very edge of order and chaos. One such pattern is a collection of loops that fill a space, weaving around each other without ever crossing. These loops are not static drawings; they are dynamic, random structures that appear in models of physical systems at critical points, such as when a magnet loses its magnetism or when a fluid transitions from liquid to gas. For decades, researchers have been able to describe these loops and measure distances between them in many scenarios, but a specific, critical case remained elusive: the moment when the loops are just simple enough to avoid touching each other, yet complex enough to form a dense, intricate web.

For a long time, a fundamental question hung over this critical case. When the loops are simple and do not intersect, the usual way of measuring distance—by hopping from one loop to a neighboring one—breaks down because there are no neighbors to hop to. The loops are like islands in a sea, separated by water, with no bridges connecting them. Yet, the underlying mathematical structure suggests that a meaningful distance should still exist, one that respects the unique symmetries of the system. If one were to imagine the loops as cities and the space between them as a landscape, the challenge was to find a way to measure the travel time between cities even when no roads directly connected them, relying only on the invisible geometry of the terrain itself. This missing measurement was crucial for understanding how random surfaces and maps behave at the smallest scales, potentially linking abstract geometry to the physical world of materials and quantum physics.

In a new series of papers, a team of mathematicians has finally constructed this elusive measurement for the first time. They focused on a specific type of random loop pattern where the loops are simple and non-intersecting, a scenario that had previously resisted a clear definition of distance. The researchers did not simply invent a new formula; instead, they watched how the distance behaved as the loops were gradually changed from a state where they frequently crashed into one another to the state where they just barely avoided each other. By starting with a version of the loops that were allowed to intersect and then slowly turning a mathematical "dial" to make them less likely to touch, they observed how the network of connections evolved. As the loops became more sparse and the direct connections vanished, the researchers found that if they adjusted the scale of their measurement correctly, a stable, consistent distance emerged from the chaos.

The team proved that this new distance is not arbitrary; it possesses a set of special properties that make it the natural way to measure space in this context. Most importantly, the distance behaves consistently no matter how the entire pattern is stretched or rotated, a quality known as conformal invariance. They demonstrated that the growth of a "ball" of space expanding from the boundary of the domain follows a precise, predictable pattern, much like a wave spreading out from a shore. This wave discovers the loops in a specific, random order that had been hypothesized years ago but never rigorously proven to be the result of a true distance metric. The researchers showed that the time it takes for this wave to reach a particular loop is exactly the distance from the boundary to that loop.

To achieve this, the authors had to overcome a significant hurdle: proving that the limit they were observing was not just a fleeting artifact of their method, but a genuine, robust mathematical object. They showed that as the loops became simpler, the number of steps required to travel from the edge of the domain to the center grew infinitely large, requiring a careful rescaling of the measurement to keep it finite. Through a series of rigorous arguments, they established that this rescaled distance converges to a single, well-defined metric. This metric is "local," meaning that the distance between two loops depends only on the loops and the space immediately surrounding them, not on the entire global structure. They also proved that the metric is "geodesic," implying that there is a shortest path between any two points, even if that path is not a straight line in the traditional sense.

The significance of this work extends beyond the abstract beauty of the proof. The researchers anticipate that this new metric will serve as a key to understanding the scaling limits of random planar maps, which are mathematical models of random surfaces used in physics. Specifically, they expect this metric to describe the geometry of a specific type of random map known as a 3/2-stable map. These maps are thought to represent the structure of certain physical systems at critical points, and having a precise way to measure distance on them allows physicists and mathematicians to make concrete predictions about their behavior. The work also connects to the study of random planar maps with large faces, suggesting a deep duality between different types of random surfaces.

While the paper establishes the existence of this metric and proves its fundamental properties, the authors note that the full uniqueness of the limit—confirming that there is only one such metric and not a family of them—will be the subject of their subsequent work. For now, they have successfully navigated the transition from a world of intersecting loops to a world of non-intersecting ones, revealing a hidden geometry that was previously invisible. They have shown that even when the loops do not touch, the space they inhabit is not empty; it is filled with a rich, conformally invariant structure that dictates how distances are measured. This discovery provides a solid foundation for future explorations into the geometry of random surfaces, offering a new lens through which to view the intricate dance of randomness and order in the mathematical universe.

Title: The Conformally Invariant Metric on CLE4 I: Subsequential Limits of the Non-Simple CLE Graph Metric

Authors: Emmanuel Kammerer, Konstantinos Kavvadias, Jason Miller, and Yi Tian

1. Problem Statement and Motivation

The paper addresses the construction of a canonical, conformally invariant metric on the Conformal Loop Ensemble (CLE) with parameter $\kappa = 4$ . CLEs are random collections of non-crossing loops in simply connected domains, serving as the natural loop variants of Schramm–Loewner Evolution (SLE).

For $\kappa \in (4, 8)$ , CLE loops are non-simple; they intersect each other and the domain boundary. This intersection structure allows for a natural definition of a graph metric on the loops, where vertices are loops and edges exist between intersecting loops. However, for the critical case $\kappa = 4$ , the loops are simple and do not intersect each other or the boundary. Consequently, the standard graph metric is undefined (infinite) for distinct loops.

The central problem is to determine whether a meaningful, conformally invariant metric exists on CLE $_4$ that arises as the scaling limit of the graph metrics on CLE $_\kappa$ as $\kappa \downarrow 4$ . Previous work by Werner and Wu defined a "uniform exploration" process that discovers CLE $_4$ loops in a specific order, conjecturing that this exploration corresponds to the growth of metric balls from the boundary. This paper aims to rigorously construct such a metric and verify that it satisfies specific axioms, including locality and geodesic properties.

2. Methodology

The authors employ a subsequential limit approach. They consider a sequence $\kappa_n \downarrow 4$ and analyze the rescaled graph distance $a_{\kappa_n}^{-1} D_{\kappa_n}$ on the CLE $_{\kappa_n}$ loops, where $a_\kappa$ is a normalization factor diverging to infinity as $\kappa \downarrow 4$ (specifically, the reciprocal of the probability that a loop surrounding the origin touches the boundary).

The methodology proceeds in several stages:

Tightness and Subsequential Limits: The authors first establish the tightness of the rescaled distances and the associated metric balls (sets of loops within a certain distance) in the product topology. This ensures the existence of subsequential limits.
Construction via Boundary Identification: To simplify the initial construction, they first define a metric $D_\partial$ on the set of loops augmented with the boundary $\partial D$ treated as a single point. They prove that the scaling limit of the graph distance to the boundary corresponds to the uniform exploration of Werner and Wu. This relies on the convergence of branching SLE $_\kappa(\kappa-6)$ processes to the uniform exploration as $\kappa \downarrow 4$ .
Extension to the Full Metric: Using the convergence of $D_\partial$ and inversion invariance properties of CLE, they construct the metric $D$ on the loops themselves (where the boundary is not a single point). This involves showing that the metric balls from any loop can be described via the uniform exploration in the complement of that loop.
Verification of Axioms: The core technical challenge is proving that the subsequential limit satisfies a specific set of axioms defining a "weak geodesic CLE $_4$ $_{4}$ metric coupling." These axioms include:
- Weak Geodesic Property: Metric balls are connected, càdlàg (right-continuous with left limits) with respect to the radius, and satisfy a specific additivity property for distances.
- Locality: The metric restricted to a subdomain depends only on the loops within that domain and the boundary conditions imposed by the exterior.
- Conformal Invariance: The law of the metric is preserved under conformal automorphisms.
- Uniform Exploration: The distance from the boundary to a loop matches the discovery time in the uniform exploration.

A significant portion of the technical work involves proving that "large" CLE $_\kappa$ loops are "fat" (i.e., they surround a Euclidean ball of a certain radius) uniformly as $\kappa \downarrow 4$ . This prevents the existence of macroscopic but "skinny" loops that could disrupt the convergence of the metric balls in the Hausdorff topology. This is achieved by coupling CLE $_\kappa$ with CLE $_{16/\kappa}$ (simple loops) and utilizing Brownian loop soup constructions.

3. Key Contributions and Results

Main Theorem (Theorem 1.3):
For any sequence $\kappa_n \downarrow 4$ , there exists a subsequence and a weak geodesic CLE $_4$ metric coupling $D$ such that the rescaled graph distances $a_{\kappa_n}^{-1} D_{\kappa_n}$ converge in law to $D$ in the product topology on the set of rational points.

Specific Results:

Existence of the Limit: The paper proves that the subsequential limits of the renormalized graph metrics exist and define a non-trivial metric on the CLE $_4$ loops.
Characterization of Metric Balls: The metric balls growing from the boundary are shown to coincide exactly with the regions explored by the uniform exploration of Werner and Wu.
Verification of Axioms: The constructed metric satisfies:
- Conformal Invariance: The metric is invariant under conformal maps.
- Locality: The metric behaves consistently when restricted to subdomains.
- Geodesic Structure: The metric balls are connected, and the distance satisfies a weak geodesic property (the distance between two loops is the sum of the distance to an intermediate ball and the distance from that ball).
Fatness of Loops: Proposition 5.2 establishes that for any $\epsilon > 0$ , there exists $\delta > 0$ such that as $\kappa \downarrow 4$ , all CLE $_\kappa$ loops with diameter at least $\epsilon$ surround a Euclidean ball of radius $\delta$ with high probability. This is a crucial technical lemma ensuring the stability of the metric convergence.

4. Significance and Scope

The paper claims to provide the first rigorous construction of a conformally invariant metric on CLE $_4$ as a scaling limit of non-simple CLE graph metrics. This approach differs from previous unpublished work by Sheffield, Watson, and Wu, which used a distinct, indirect construction.

Significance:

Resolution of a Conjecture: It confirms the conjecture by Werner and Wu that the uniform exploration corresponds to the growth of metric balls for a conformally invariant metric on CLE $_4$ .
Random Planar Maps: The authors anticipate that this metric space describes the scaling limit of $3/2$ -stable random planar maps (maps with vertex weights of order $k^{-2}$ ). While these maps do not admit a Gromov-Hausdorff scaling limit, the distances to the root in these maps are expected to converge to the distances in the constructed CLE $_4$ metric.
Duality: The metric space is viewed as the "dual" of the $\alpha$ -stable carpet (specifically the $3/2$ -stable carpet) arising in the scaling limits of random maps with large faces.

Modesty and Future Work:
The authors explicitly state that this is the first paper in a series of three.

Subsequential vs. True Limit: This paper only proves the existence of subsequential limits. The uniqueness of the limit (and thus the existence of the limit along the full sequence $\kappa \downarrow 4$ ) is established in the subsequent papers [KKMT26a, KKMT26b].
Applications: The paper does not claim to have solved the scaling limit of random planar maps but rather provides the necessary metric structure that is expected to describe it. The connection to $3/2$ -stable maps is presented as an expectation based on the properties of the constructed metric.

In summary, the paper successfully constructs a candidate metric for CLE $_4$ via a scaling limit, proves its existence as a subsequential limit, and verifies that it possesses the fundamental geometric and probabilistic properties required of a conformally invariant metric, laying the groundwork for the uniqueness proofs in the sequel.

1. Problem Statement and Motivation

2. Methodology

3. Key Contributions and Results

4. Significance and Scope

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