Internal DLA on mated-CRT maps

In the study of how matter organizes itself, scientists often look to the boundary between order and chaos. One powerful way to understand this is by watching how particles spread out over time, a process that happens everywhere from salt dissolving in water to bacteria colonizing a petri dish. In mathematics, this spreading is modeled by a process called internal diffusion limited aggregation, or IDLA. Imagine a single starting point where particles are released one by one. Each particle wanders randomly until it finds an empty spot, where it stops and becomes part of a growing cluster. Over time, this cluster expands, and mathematicians have long wanted to know what shape it takes when it becomes very large. On a flat, regular grid like graph paper, the answer is well known: the cluster eventually looks like a perfect circle. But the universe is rarely so simple. In the world of quantum physics, space itself can be warped and irregular, described by a theory called Liouville quantum gravity. This theory suggests that at a fundamental level, space is a rough, fractal surface rather than a smooth plane. The question that has remained unanswered is: if you let particles spread out on this rough, quantum surface, what shape will they form?

A team of researchers has now answered this question by studying the behavior of these particles on a specific type of random network that mimics the geometry of a quantum surface. They discovered that the growing cluster does not form a circle, nor does it take the shape of a standard ball defined by distance. Instead, it forms a unique, irregular shape known as a harmonic ball. This shape is defined not by how far points are from the center, but by how they balance the average value of certain mathematical functions across the surface. The researchers proved that as the particles spread, the cluster they build converges to this specific harmonic shape with high certainty. This finding is significant because it reveals that the rules governing growth on a quantum surface are fundamentally different from those on flat ground, creating a new kind of geometric object that exists only in this rough, probabilistic environment.

To reach this conclusion, the scientists worked with a model called the mated-CRT map. This is a complex, random network of points and connections that serves as a digital approximation of a quantum surface. It is constructed using two related random paths that wiggle across a plane, creating a structure that looks locally like a triangle mesh but behaves globally like the warped space of quantum gravity. The researchers simulated the IDLA process on this network, sending out thousands of random walkers from a central point. Each walker moved along the connections of the network until it hit an unoccupied spot, where it stopped and became part of the cluster. Because the network is random and the paths are unpredictable, the resulting cluster is jagged and uneven, with no smooth edges.

The challenge was to determine what this jagged cluster looks like when viewed from a distance. To do this, the researchers compared the IDLA cluster to a theoretical shape called a harmonic ball. This shape was constructed in a separate, related study using a different method involving fluid dynamics and optimization. The researchers proved that as the number of particles in their simulation increased, the IDLA cluster on the mated-CRT map became indistinguishable from this harmonic ball. In other words, if you were to zoom out and look at the cluster, its outer boundary would match the boundary of the harmonic ball so closely that the difference between them would vanish. This result holds true for any parameter that defines the roughness of the quantum surface, suggesting a universal rule for how matter grows in this specific type of space.

The proof relied on a clever connection between the random IDLA process and a deterministic model called the divisible sandpile. In the sandpile model, instead of individual particles, a continuous amount of mass is placed at the center and redistributed according to strict rules until everything is stable. The researchers showed that the final shape of this stable sandpile is the same as the IDLA cluster. By analyzing the sandpile, they could use powerful mathematical tools to predict the shape of the random cluster. They demonstrated that the sandpile's mass distribution converges to the same harmonic ball, and because the two models are linked, the random cluster must do the same. This approach allowed them to bypass the difficulty of tracking every single random walker and instead focus on the overall flow of mass.

One of the most striking aspects of this discovery is what it rules out. In many other settings, such as on a flat grid or in a regular lattice, the growing cluster eventually becomes a perfect metric ball, meaning every point on the edge is the same distance from the center. The researchers explicitly showed that this is not the case here. The harmonic ball they found is neither a smooth, round shape nor a standard metric ball. It is a domain with a rough, fractal boundary that reflects the underlying irregularity of the quantum surface. This confirms that the geometry of the space fundamentally alters the way growth occurs, preventing the formation of simple, symmetric shapes.

The researchers also addressed how quickly this convergence happens. While they proved that the shapes eventually match, they noted that the rate at which they approach each other is likely different from what is seen on flat surfaces. On a flat grid, the fluctuations around the final shape are known to be very small, growing only logarithmically with the size of the cluster. However, on the mated-CRT map, the cell sizes vary wildly in a multi-fractal pattern, which suggests that the fluctuations might be larger and more complex. The paper establishes the final shape with certainty but leaves the precise rate of convergence as an open question for future study.

This work bridges the gap between abstract probability theory and the geometry of quantum gravity. By showing that a random growth process on a discrete approximation of a quantum surface converges to a specific continuum shape, the researchers have provided a concrete example of how quantum geometry influences physical processes. The harmonic ball they identified is a new mathematical object that exists at the intersection of random walks, fluid dynamics, and quantum geometry. It stands as a testament to the idea that even in a universe that is fundamentally rough and random, there are still precise, predictable patterns waiting to be discovered. The findings suggest that if we could observe matter spreading on a true quantum surface, it would not form a circle, but would instead settle into this unique, harmonious, and irregular shape.