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Directed distances in spanning-tree-decorated planar maps: exact exponent, scaling limit and universality

This paper establishes that directed distances in spanning-tree-decorated planar maps scale as $n^{1/3}$ and converge to a $3/2$ -stable Lévy process, thereby determining the scaling dimension for a hypothetical directed version of the $\sqrt{2}$ -Liouville quantum gravity metric and extending these universality results to related random planar map models.

Original authors: Jacopo Borga, Ewain Gwynne, Yuanzheng Wang

Published 2026-09-22

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CC BY 4.0

Original authors: Jacopo Borga, Ewain Gwynne, Yuanzheng Wang

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 vast, shifting landscape made entirely of triangles, squares, and other shapes, all glued together to form a single, continuous surface. In the world of mathematics, these are called planar maps, and they are more than just abstract drawings; they are a way to model the geometry of space itself. For decades, scientists have been fascinated by how these maps behave when they grow infinitely large, trying to understand if they settle into a predictable, smooth shape or if they remain chaotic and jagged. This question touches on a deep theory in physics known as quantum gravity, which attempts to describe how space and time might look at the tiniest possible scales. While the standard version of this theory has been studied extensively, a specific, more complex version involving "directed" paths—where you can only travel in certain directions, like a one-way street system—has remained a mystery. Understanding how far you can travel in such a constrained, directed world is crucial for testing the limits of our physical theories.

A team of researchers has now cracked this specific puzzle, focusing on a particular type of map decorated with a spanning tree. Think of a spanning tree as a network of roads that connects every single point on the map without any loops, ensuring you can get from anywhere to anywhere else, but never by circling back on yourself. The researchers took these maps and assigned a specific direction to every road based on a natural, winding path that explores the entire tree. This created a world where movement is strictly governed by the flow of this exploration, turning the map into a directed system. The central question they asked was simple yet profound: if you start at one point and try to reach another using only these allowed directions, how far away is it? Does the distance grow slowly, or does it explode as the map gets bigger?

The answer they found is surprisingly precise and elegant. By analyzing maps with a specific number of edges, the researchers discovered that the typical distance between two points grows at a rate proportional to the cube root of the total number of edges. In other words, if you double the size of the map, the distance you can travel doesn't double; it increases by a much smaller, specific factor. This result is significant because it provides the first exact solution for this kind of directed distance in this specific class of maps. The team didn't just guess this relationship; they proved it mathematically, showing that as the maps become enormous, the behavior of these distances converges to a known, stable pattern described by a specific type of random process. This convergence acts like a fingerprint, confirming that the underlying geometry of these directed maps is consistent and predictable.

To reach this conclusion, the researchers had to invent a new way of measuring distance to infinity within these maps. They constructed a mathematical tool, essentially a function that assigns a value to every point representing how far it is from a starting point when traveling toward the infinite edge of the map. They showed that this function behaves in a very specific way: its changes are independent of one another and follow a predictable statistical distribution. This allowed them to treat the complex, tangled geometry of the map as a simpler, one-dimensional journey that could be analyzed with powerful statistical tools. The proof relied on a clever connection between the map and a random walk, a mathematical model of a path taken by someone moving randomly step by step. By translating the geometry of the map into the language of this random walk, they could calculate the exact scaling of the distances.

The implications of this work extend far beyond the specific maps they studied. The researchers demonstrated that this same cube-root scaling rule applies to several other different models of random maps that belong to the same broad family of geometric structures. This suggests a deep universality: no matter how you construct the map, as long as it belongs to this specific family, the directed distances will always grow in the same way. This finding offers a new window into a hypothetical version of quantum gravity where space has a preferred direction. While the standard theory of quantum gravity describes a world where space is the same in all directions, this work explores what happens when that symmetry is broken. The results suggest that even in this directed, one-way universe, the geometry remains orderly and follows a strict mathematical law.

The study also clarifies the relationship between these discrete maps and the continuous, smooth surfaces they are believed to approximate. The researchers showed that the scaling dimension of these directed distances is exactly three, meaning that the "volume" of space in this directed world grows in a specific, predictable manner relative to the distance. This is a direct counterpart to the known dimensions of undirected space, but with a twist that reflects the constraints of the one-way travel. By proving that these directed distances are exactly solvable, the team has provided a rare, concrete example of how complex geometric systems can be understood without approximation. Their work does not just offer a number; it provides a complete framework for understanding how directionality shapes the very fabric of random geometric spaces, offering a new benchmark for theories that attempt to describe the universe at its most fundamental level.

Technical Summary: Directed Distances in Spanning-Tree-Decorated Planar Maps

Problem Statement
This paper investigates the scaling behavior of directed distances (lengths of shortest directed paths) in random planar maps decorated with a spanning tree (ST). While the scaling limits of undirected distances in random planar maps are well-understood to converge to Liouville Quantum Gravity (LQG) metrics, the behavior of directed distances remains less explored, particularly for models in the $\sqrt{2}$ -LQG universality class. The authors define a natural orientation on ST-decorated maps where edges are oriented to align with the contour exploration of the spanning tree. The primary objective is to determine the scaling exponent for these directed distances, construct the associated Busemann function for the infinite volume limit, and establish universality across different models within the $\sqrt{2}$ -LQG class.

Methodology
The core of the methodology relies on the Mullin bijection, which encodes spanning-tree-decorated planar maps as two-dimensional simple random walks (SRW) on $\mathbb{Z}^2$ with increments in $\{(1,0), (-1,0), (0,1), (0,-1)\}$ , conditioned to stay in the first quadrant.

Infinite Volume Analysis (UIST): The authors analyze the Uniform Infinite Spanning-Tree-decorated map (UIST) as the Benjamini-Schramm local limit of finite maps. They utilize the encoding by a bi-infinite SRW to define a "future" submap (FUIST) bounded by specific infinite branches.
Busemann Function Construction: A Busemann function $X: \mathbb{Z} \to \mathbb{Z}$ is constructed to measure directed distances to infinity along a natural interface (the left boundary of the FUIST). The existence and uniqueness of this function are proven by identifying "boundary cut vertices" (simultaneous running minima of the encoding walk) which force the coalescence of infinite directed geodesics.
Probabilistic Properties: The increments of the Busemann function are shown to be independent and stationary. The authors derive tail asymptotics for these increments, proving they follow a distribution with a heavy tail of order $x^{-3/2}$ and an exponential negative tail.
Scaling Limits: Using a heavy-tailed functional central limit theorem, the rescaled Busemann function is shown to converge to a $3/2$ -stable Lévy process.
Universality via Strong Coupling: To extend results to other models (uniform meandric systems and mated-CRT maps), the authors employ the KMT strong coupling theorem. They compare the directed distances in the Mullin maps to those in discrete mated maps and continuous mated-CRT maps, showing that the scaling exponents are preserved up to polylogarithmic factors.

Key Contributions and Results

Exact Solvability of Directed Exponents: Unlike undirected distances in general LQG classes where the Hausdorff dimension $d_\gamma$ $d_{γ}$ is unknown for $\gamma \neq \sqrt{8/3}$ $γ \neq = 8/3$ , this paper establishes an exact scaling exponent for directed distances in the $\sqrt{2}$ $2$ -LQG class.
- Theorem 1.6: The rescaled Busemann function converges in law to a $3/2$ -stable Lévy process.
- Theorem 1.8: The increments of the Busemann function have mean zero and tail decay $P[X \ge x] \sim c x^{-3/2}$ .
Scaling Dimension: For finite maps with $n$ $n$ edges, the typical directed distance scales as $n^{1/3}$ $n^{1/3}$ .
- Theorem 6.2 & 6.4: In uniform spanning-tree-decorated maps and directed Mullin maps with $n$ edges, the maximum directed distance is bounded above and below by $n^{1/3}$ (up to polylogarithmic factors). This identifies $1/3$ as the directed counterpart to the reciprocal LQG dimension $1/d_{\sqrt{2}}$ .
Universality: The $n^{1/3}$ $n^{1/3}$ scaling is shown to be universal for other models in the $\sqrt{2}$ $2$ -LQG class:
- Theorem 1.10: Uniform meandric systems of size $n$ exhibit directed distances of order $n^{1/3}$ .
- Theorem 1.12: Induced $\sqrt{2}$ -mated-CRT maps of size $n$ exhibit directed distances of order $n^{1/3}$ .
Conditional Independence: A crucial technical contribution is the proof of a conditional independence property for directed Boltzmann Mullin maps (Lemma 2.19). This states that the left and right geodesic maps cut by a leftmost directed geodesic are conditionally independent given the geodesic's length. This property is essential for proving the independence of the Busemann function increments.

Significance and Claims
The paper claims to provide the first exact solvability results for directed distances on random planar maps that do not belong to the $\sqrt{8/3}$ -LQG universality class. The results suggest that the scaling dimension for a hypothetical "directed $\sqrt{2}$ -LQG metric" is 3 (since distance scales as area $^{1/3}$ ).

The authors emphasize that their proof strategy, inspired by previous work on bipolar-oriented triangulations, requires significant new ideas due to the structural differences in spanning-tree-decorated maps. Specifically, the orientation induced by a single Hamiltonian path (the contour exploration) allows for the application of strong coupling arguments that fail for other orientations (like bipolar orientations) where the orientation is not induced by a single path.

The paper posits that these results support a conjecture regarding the minimal Hausdorff dimension of directed paths in the continuum $\sqrt{2}$ -LQG metric, suggesting it is $d_{\sqrt{2}}/3$ . However, the authors note that the KPZ formula is not directly applicable here because the minimizing path depends on the underlying random field.

Limitations and Scope
The authors explicitly state that the universality results for other discrete models in the $\sqrt{2}$ -LQG class with local constraints (e.g., fixed vertex degrees) are expected to hold but are not proven in this work for brevity. Furthermore, the paper focuses on the $\sqrt{2}$ case; establishing similar universality for $\gamma \neq \sqrt{2}$ is noted as significantly harder. The results for finite maps include polylogarithmic error factors ( $(\log n)^k$ ), which the authors attribute to technical issues in the strong coupling comparison, though they expect the true scaling to be exactly $n^{1/3}$ .

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