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Harmonic balls in Liouville quantum gravity

This paper establishes the existence and uniqueness of harmonic balls on Liouville quantum gravity surfaces via the Hele-Shaw flow obstacle problem, characterizes their geometric properties as distinct from Lipschitz and metric balls with Jordan curve boundaries, and conjectures their role as the scaling limit of internal diffusion limited aggregation on random planar maps.

Original authors: Ahmed Bou-Rabee, Ewain Gwynne

Published 2026-10-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ahmed Bou-Rabee, Ewain Gwynne

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 world where the very fabric of space is not smooth and uniform, but instead is a wild, crumpled landscape that changes its density from point to point. In this realm, the rules of geometry we learn in school no longer apply. This is the setting of Liouville quantum gravity, a theoretical framework used by physicists and mathematicians to describe the random, fractal nature of space-time at the smallest scales. It is a concept that emerged from the study of string theory, suggesting that the universe might look like a jagged, infinitely detailed surface when viewed through a powerful enough lens. On such a surface, the idea of a "ball" becomes complicated. In our everyday world, a ball is a perfect circle or sphere, defined by a fixed distance from a center. But on a crumpled, uneven surface, the shape of a ball depends on how you measure distance and how much "stuff" or mass is packed into different regions.

For decades, researchers have tried to understand how random growth processes behave on these strange surfaces. One such process is internal diffusion-limited aggregation, a model where particles wander randomly and stick together to form a growing cluster. On a flat, ordinary plane, these clusters eventually smooth out into perfect circles. However, on the crumpled surfaces of Liouville quantum gravity, the outcome was a mystery. The central question was: what shape do these clusters actually take when they grow on this wild, random terrain? Do they become the natural "metric balls" of the surface, or do they form something entirely new?

In a new study, mathematicians Ahmed Bou-Rabee and Ewain Gwynne have answered this question by constructing and analyzing a specific type of shape called a "harmonic ball." They proved that these shapes exist and are unique, meaning that for any given amount of mass, there is exactly one such shape centered at a specific point. However, the most surprising discovery is what these shapes are not. The researchers demonstrated that these harmonic balls are not the smooth, geometric objects one might expect. They are neither the standard metric balls defined by the distance on the surface, nor are they simple, smooth shapes with clean edges. Instead, they are highly irregular domains with boundaries that are jagged and complex, yet still possess a specific, hidden order.

The team arrived at this conclusion by treating the growth of these clusters as a fluid flow problem. They imagined injecting a fluid into the surface at a constant rate and watching it spread out. The shape the fluid takes is determined by an optimization problem: it spreads to fill the area where the "work" required to move the fluid is minimized. By solving this problem mathematically, they showed that the resulting shape satisfies a fundamental rule known as the mean-value property. This rule states that the value of a harmonic function at the center of the shape is exactly the average of its values over the entire shape. This property is what defines a harmonic ball.

The researchers found that these shapes are surprisingly rough. While they are connected and do not have holes, their boundaries are not smooth curves. In fact, the boundaries are so irregular that they fail to meet the standard criteria for "Lipschitz" domains, which are shapes that can be locally described by a graph that does not change too abruptly. This means that if you were to zoom in on the edge of one of these harmonic balls, you would not see a smooth line; you would see a fractal-like structure that is infinitely detailed and rough. Furthermore, the study proved that these shapes are distinct from the metric balls of the surface. A metric ball is defined by the shortest path distance from a center point, but the harmonic balls grow in a way that is dictated by the random distribution of mass on the surface, leading to a different, more complex geometry.

To understand the boundaries of these shapes, the authors showed that the edges of the separate pieces of the surrounding empty space are simple loops, known as Jordan curves. This means that while the boundary is rough and fractal, it does not cross itself or form complicated knots; it is a single, continuous line that separates the inside from the outside. This finding is crucial because it provides a precise geometric description of these random shapes, confirming that they are well-behaved in a topological sense even though they are rough in a geometric sense.

The significance of this work extends beyond pure mathematics. The researchers conjecture that these harmonic balls are the ultimate limit of the internal diffusion-limited aggregation process on random planar maps, which are discrete models of random surfaces used in computer science and physics. In a companion paper, they have already proven this connection for a specific type of map, but the general case remains an open challenge. By establishing the existence and properties of these harmonic balls, Bou-Rabee and Gwynne have provided the necessary tools to understand how random growth processes behave in the quantum realm. They have shown that nature, even in its most chaotic and fractal forms, follows strict rules that can be uncovered through careful mathematical analysis, revealing a hidden order within the apparent randomness of the quantum universe.

Technical Summary: Harmonic Balls in Liouville Quantum Gravity

Problem Statement
The paper addresses the construction and characterization of "harmonic balls" on Liouville Quantum Gravity (LQG) surfaces. A harmonic ball for a measure $\mu$ centered at $z$ is an open set $\Lambda(z)$ containing $z$ that satisfies the mean-value property for harmonic functions:
$f(z) = \frac{1}{\mu(\Lambda(z))} \int_{\Lambda(z)} f(x) \mu(dx)$
for all functions $f$ harmonic in a neighborhood of $\overline{\Lambda(z)}$ . The authors focus on the case where $\mu$ is the $\gamma$ -Liouville measure $\mu_h$ associated with a Gaussian Free Field (GFF) $h$ on $\mathbb{C}$ , where $\gamma \in (0, 2)$ .

The primary motivation is the conjecture that LQG harmonic balls represent the scaling limit of Internal Diffusion Limited Aggregation (IDLA) on random planar maps. While IDLA on the standard lattice $\mathbb{Z}^2$ converges to Euclidean balls, and on certain graphs to metric balls, the irregular nature of the LQG measure suggests the limit shape may be distinct. The authors aim to rigorously establish the existence and uniqueness of these objects and determine their geometric properties, noting that classical theories of harmonic balls (e.g., [Sak85, Fri88]) fail in the LQG setting because $\mu_h$ is singular with respect to Lebesgue measure and lacks the necessary regularity (boundedness from above and below).

Methodology
The authors construct harmonic balls via an obstacle problem formulation of the Hele-Shaw flow, avoiding the approximation or regularization procedures used in constructing the LQG metric or Liouville Brownian motion.

Obstacle Problem Construction: For a fixed time $t > 0$ and a domain (initially the unit ball $B_1$ ), they define a cluster $\Lambda_t$ as the support of the solution to a variational inequality involving the Green's function $G_{B_1}$ and the measure $\mu_h$ . Specifically, $\Lambda_t$ is the set where the "odometer" function $v_t$ (the least supersolution to $\Delta w \leq \mu_h - t\delta_0$ with $w \geq 0$ ) is positive.
Harnack-Type Estimate: The core technical innovation is a novel Harnack-type estimate (Proposition 6.1). This estimate asserts that if an annulus $A$ has a small intersection with the cluster relative to its total mass, the cluster cannot penetrate deeply into the inner part of the annulus. This is proven by combining potential theoretic techniques (harmonic comparison, maximum principle) with specific properties of the LQG measure, particularly the prevalence of "very good" annuli where the measure behaves regularly enough to control Brownian motion exit probabilities.
Regularity and Uniqueness: Using the Harnack estimate, the authors prove that the clusters grow continuously in time, their boundaries have zero $\mu_h$ -mass, and they satisfy the mean-value property for a specific class of functions (including those generated by signed Radon measures). Uniqueness is established by showing that any harmonic ball must coincide with the constructed cluster up to a set of zero mass, and the requirement that the domain equals the interior of its closure ensures exact uniqueness.
Geometric Analysis: The authors analyze the boundaries of the constructed sets to determine if they are Lipschitz domains or LQG metric balls. They utilize the Harnack estimate to show that the boundaries cannot satisfy the cone condition (ruling out Lipschitz regularity) and possess geometric features (approximate pinch points) that distinguish them from LQG metric balls.

Key Contributions and Results

Existence and Uniqueness (Theorem 1.1): On an event of probability one, for every $z \in \mathbb{C}$ , there exists a unique family of harmonic balls $\{\Lambda_t(z)\}_{t>0}$ satisfying the mean-value property. These domains are bounded, connected, increase continuously in $t$ , and satisfy $\mu_h(\Lambda_t(z)) = t$ .
Novelty of Shape (Theorem 1.2): For Lebesgue-almost every $t$ $t$ , the harmonic ball $\Lambda_t(0)$ $Λ_{t} (0)$ is neither a Lipschitz domain nor an LQG metric ball.
- Not Lipschitz: The boundaries fail the cone condition almost everywhere. This is proven by showing that the oscillation of the Liouville measure across sectors of annuli prevents the existence of interior or exterior cones at typical boundary points.
- Not Metric Balls: LQG metric balls can have "approximate pinch points" (regions of small Euclidean diameter but large LQG mass) that disconnect large masses from the origin. The authors show that LQG harmonic balls cannot possess such features due to the Harnack-type constraint, which forces the mass to be distributed in a way that precludes these pinch points.
Boundary Topology: The boundaries of the complementary connected components of the harmonic balls are Jordan curves (Proposition 10.6). This is derived from the Harnack estimate, which limits the number of times a cluster can cross an annulus, thereby preventing the boundary from having cut points or being non-locally connected.
Locality: The constructed harmonic balls are local sets for the underlying GFF (Proposition 1.3), meaning the event that a ball is contained in a domain $U$ is measurable with respect to the field restricted to $U$ .

Significance and Claims
The paper establishes the first rigorous construction of harmonic balls on LQG surfaces, filling a gap in the analysis of partial differential equations on fractal-like random geometries. The authors explicitly state that their construction does not rely on regularization, distinguishing it from other LQG constructions.

The primary significance claimed is the identification of a new geometric object intrinsic to LQG that serves as a candidate scaling limit for IDLA on random planar maps. The authors note that while they prove this scaling limit connection for "mated-CRT maps" in a companion paper [BRG22], the construction of the harmonic balls themselves is independent of that specific map model.

The paper modestly frames the novelty of these balls as a departure from standard Euclidean or metric ball geometries. It highlights that the irregularity of the LQG measure induces a specific type of roughness in the harmonic balls (non-Lipschitz) that is distinct from the roughness of LQG metric balls. The authors identify several open problems, including computing the Hausdorff dimension of the boundaries and extending the theory of "Hele-Shaw geodesics" to this setting, acknowledging that the lack of smooth boundaries in LQG harmonic balls presents a challenge for such extensions.

In summary, the paper provides a rigorous existence and uniqueness theory for harmonic balls on LQG, characterizes their geometric irregularity, and posits them as the natural continuum limit for aggregation processes on random planar maps, distinct from both Euclidean balls and LQG metric balls.

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