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The conformally invariant metric on CLE $_4$ II: existence of geodesics

This paper establishes the existence of geodesics within the conformally invariant metric uniquely determined by the CLE $_4$ ensemble, proving that these geodesics are supported on the loops themselves and do not intersect the domain boundary, while also providing sharp quantitative estimates for the metric's geometry.

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, invisible landscape of probability, mathematicians often study how random shapes fill a space. Imagine a collection of loops, like rubber bands, floating inside a circle. In some versions of this random arrangement, the loops are simple and separate, never touching each other or the edge of the circle. In others, they are tangled, touching themselves and their neighbors. There is a specific, critical point where the behavior of these loops changes from one state to the other. At this precise threshold, the loops are simple and disjoint, yet they are packed so tightly that they form a complex, fractal-like structure. This arrangement, known as a conformal loop ensemble, is a fundamental object in the study of random geometry, appearing as the limit of many physical systems at the edge of chaos, such as magnets cooling down or fluids flowing through porous rock.

For years, researchers have known how to measure the distance between these loops if they were allowed to touch. But when the loops are strictly separated, as they are at this critical point, the usual way of measuring distance breaks down. You cannot simply walk from one loop to another because there is no path connecting them directly; you must jump across the empty space between them. The question that has puzzled mathematicians is whether a meaningful, consistent way to measure distance exists in this empty space, and if so, whether the shortest path between two loops—a geodesic—actually exists and what it looks like. This is not just a theoretical curiosity; understanding these paths helps define the very geometry of the random world these loops inhabit.

In a new paper, a team of researchers has finally proven that such a shortest path not only exists but is also remarkably well-behaved. They show that for any two loops in this specific random arrangement, there is a continuous curve that connects them with the minimum possible distance. This curve is not a straight line cutting through the void, nor is it a jagged, random walk. Instead, the path is supported almost entirely by the loops themselves. The researchers demonstrated that the geodesic travels along the boundaries of the loops, hopping from one to the next, and only deviates from the loops for a set of points so small that it has no measurable size. Furthermore, the path never touches the outer boundary of the domain, staying strictly within the interior of the random structure.

The team achieved this by constructing a detailed map of the distances between loops. They began by establishing that the distance across a rectangular region of this random space behaves in a predictable way, with the likelihood of a very long distance dropping off rapidly. They then used this knowledge to analyze what happens when two "metric balls"—regions of space explored outward from a starting loop—grow toward each other. They proved that the probability of these two growing regions remaining separate until they hit a tiny, specific point is extremely low, dropping off so fast that it effectively forces the paths to merge. This merging behavior is the key to proving that a continuous, unbroken path exists between any two loops.

By combining these estimates with a multi-scale construction, the researchers built a sequence of intermediate loops that act as stepping stones between the start and end points. They showed that as they refined the scale of these stepping stones, the sequence converged to a single, continuous curve. This curve satisfies the definition of a geodesic: its length equals the distance between the two loops. The work also confirms that the distance along this path is continuous, meaning there are no sudden jumps or gaps in the measurement as one travels along the curve.

This result is a crucial step in a larger project to fully define the geometry of this random loop system. The existence of these geodesics allows mathematicians to treat the collection of loops as a proper geometric space with a well-defined metric. While the first paper in their series established the existence of the metric itself, this second paper proves that the geometry is rich enough to support shortest paths. A forthcoming third paper will use these findings to prove that the metric is unique and determined entirely by the arrangement of the loops themselves. The discovery confirms that even in a world of random, disjoint loops, there is an underlying order that allows for the definition of a shortest path, revealing a hidden structure within the chaos.

Technical Summary: The Conformally Invariant Metric on CLE4 II: Existence of Geodesics

Problem Statement
This paper addresses the construction and structural analysis of a conformally invariant metric on the set of loops of the Conformal Loop Ensemble (CLE) with parameter $\kappa = 4$ . While CLE $_4$ consists of simple, disjoint loops that do not touch each other or the domain boundary, defining a metric on these loops is non-trivial because the loops themselves are disconnected. In the first paper of this series [KKMT26a], the authors constructed such a metric as a subsequential scaling limit of renormalized graph metrics on CLE $_\kappa$ as $\kappa \downarrow 4$ . However, the existence of geodesics (shortest paths) between loops in this limiting metric was not guaranteed by the construction, as the topology of the limit differs from the graph topology where geodesics naturally exist for $\kappa > 4$ .

The central problem of this work is to prove that for any "weak geodesic CLE $_4$ metric coupling" (a metric satisfying specific axioms of locality, conformal invariance, and uniform exploration), geodesics between any two loops actually exist. Furthermore, the paper aims to characterize the geometric properties of these geodesics, specifically whether they are supported on the loops of the ensemble and whether they intersect the domain boundary.

Methodology
The authors employ a multi-scale probabilistic analysis combining several distinct perspectives on CLE $_4$ and Schramm–Loewner Evolution (SLE):

Metric Geometry and Uniform Exploration: The metric is analyzed via the "uniform exploration" process introduced by Werner and Wu [WW13b], where metric balls grow from the domain boundary. The paper utilizes the locality property (Axiom II), which states that conditionally on the metric balls outside a subdomain, the metric inside is independent and distributed as the metric of a CLE $_4$ in that subdomain.
Quantitative Estimates:
- Rectangle Distances: The authors establish sharp bounds on the distance across conformal rectangles. They prove that the distance between the top and bottom of an $r \times 1$ rectangle is stochastically dominated by a geometric random variable scaled by $1/r$ (Proposition 3.2), providing exponential tail bounds.
- Non-Intersection of Metric Balls: A critical component is Proposition 4.10, which estimates the probability that two metric balls grown from distinct loops remain disjoint until they hit a small Euclidean ball of radius $\epsilon$ . The authors show this probability is bounded by $\epsilon^{2+o(1)}$ . This is achieved by translating the event into a requirement for four macroscopic arms in a bichordal SLE $_4$ configuration, utilizing four-arm exponent estimates by Zhan [Zha20, Zha21].
- Vertical vs. Horizontal Distances: Lemma 5.1 establishes that with positive probability, the horizontal distance across a rectangle can be strictly smaller than the vertical distance, a property crucial for constructing paths that "cut corners."
Multi-Scale Construction of Geodesics: To prove existence, the authors construct a sequence of finite families of loops that serve as "waypoints" between two target loops. Using the non-intersection estimates and the locality property, they iteratively refine these families at finer scales. They show that these sequences converge almost surely to a continuous path whose length equals the metric distance, thereby establishing the existence of a geodesic.
Coupling with Gaussian Free Field (GFF): The analysis relies heavily on the coupling between CLE $_4$ and the GFF via level lines (Section 2.7), allowing the use of tools from SLE theory, such as the interaction rules of level lines and the properties of local sets.

Key Contributions and Results

Existence of Geodesics (Theorem 1.4): The primary result is that every weak geodesic CLE $_4$ metric coupling is a "geodesic CLE $_4$ metric coupling." Specifically, for any two loops $\mathcal{L}_1, \mathcal{L}_2$ in a CLE $_4$ , there exists an admissible path $P$ connecting them such that the path's length equals the metric distance $D(\mathcal{L}_1, \mathcal{L}_2)$ .
Support and Dimension of Geodesics (Theorem 1.5): The paper proves three key structural properties of these geodesics:
1. No Distance Gaps: The set of distances from the starting loop to loops intersected by the geodesic forms a dense subset of the interval $[0, D(\mathcal{L}_1, \mathcal{L}_2)]$ .
2. Support on Loops: The geodesic is supported on the loops of the CLE $_4$ . The set of points on the geodesic that do not lie on any loop has Hausdorff dimension zero.
3. Boundary Avoidance: Almost surely, no geodesic connecting two loops touches the domain boundary $\partial U$ .
Internal Metric Equivalence (Corollary 6.6): The authors show that the internal metric restricted to a subdomain (defined by paths staying within the subdomain) coincides almost surely with the metric induced by the CLE $_4$ restricted to that subdomain. This confirms the consistency of the metric under the locality axiom.
Sharp Estimates: The paper provides new quantitative estimates, including exponential tail bounds for rectangle distances and multi-scale four-arm non-intersection bounds for metric balls, which are of independent interest for the geometry of CLE $_4$ .

Significance and Claims
The authors position this work as the second in a three-part series. The significance of this paper lies in establishing the existence of geodesics, a property that is not automatic in the subsequential limit construction used in the first paper [KKMT26a].

The paper explicitly states that these results are foundational for the third paper in the series [KKMT26b], where the uniqueness of the CLE $_4$ metric will be proven. The existence and structural properties of geodesics (specifically their support on loops and boundary avoidance) are described as "crucial tools" for establishing that the metric is uniquely determined by the CLE $_4$ configuration.

The authors note that while a CLE $_4$ metric was previously constructed by Sheffield, Watson, and Wu in unpublished work via a different approach, that construction did not investigate the existence of geodesics. This paper fills that gap, showing that geodesics exist and possess specific geometric regularity, regardless of the specific approximation scheme used, provided the metric satisfies the natural axioms of locality, conformal invariance, and uniform exploration.

The paper does not claim to solve the uniqueness problem itself, nor does it propose new experimental applications. Its contribution is strictly theoretical, providing the necessary geometric framework to complete the characterization of the CLE $_4$ metric in the subsequent work.

Technical Summary: The Conformally Invariant Metric on CLE4 II: Existence of Geodesics

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