A Close-Up View Reveals the ‘Melting’ Point of an Infinite Graph

Shortly before his death, Schramm conjectured that Grimmett and Marstrand’s theorem could be generalized. He thought that the percolation threshold is determined entirely by the close-up, or “microscopic,” perspective for a large class of graphs known as transitive graphs.

In 2009, Benjamini, Asaf Nachmias and Yuval Peres proved Schramm’s locality conjecture, as it’s now known, for a specific type of transitive graph that resembles a tree. Schramm, however, had postulated that it would hold for all transitive graphs (with an exception for one-dimensional graphs).

In a transitive graph, all of the vertices look similar. A two-dimensional grid is one example. If you pick any two vertices, you can always find a symmetry that moves one vertex to the other.

This relationship holds for any transitive graph. Because of these symmetries, if you zoom in and look at any two equal-size patches of a transitive graph, they will look the same. For this reason, Schramm believed that the close-up perspective was sufficient to allow mathematicians to calculate the percolation threshold for all transitive graphs.

Transitive graphs can take many shapes and forms. They can be a simple grid, made up of squares, triangles, hexagons or some other shape. Or they can form a more complex object, like a “3-regular tree,” where one central point connects to three vertices, and each vertex then branches to create two new ones ad infinitum, the first few steps of which are seen here:

The variety of transitive graphs contributed to the difficulty of proving Schramm’s locality conjecture. In the 15 years between Schramm’s conjecture and Easo and Hutchcroft’s proof, various groups of mathematicians proved the conjecture for specific types of graphs, but their ideas never extended to the general case.

“The space of all possible geometries is just so vast, and there are always weird things lurking,” Hutchcroft said.

Widening the Lens

Easo and Hutchcroft weren’t initially looking for a solution to Schramm’s locality conjecture, which applies to infinite graphs. They were instead studying percolation on finite graphs. But they had an idea that suddenly shifted their attention to the conjecture.

“We came up with this new tool, and we thought, oh, this seems like the kind of thing that could be helpful to attack locality,” Easo said.

To prove the conjecture, they needed to show that the microscopic perspective gives an accurate snapshot of the percolation threshold. When you view just part of a graph and observe a big connected cluster, you might assume that the graph has an infinite cluster and is therefore above the percolation threshold. Easo and Hutchcroft set out to prove it.

They relied on a technique that can be thought of as “widening the lens.” Start at a single vertex. Then zoom out to view all vertices that are just one edge away on the original graph. On the square grid, you will now be able to see five total vertices. Widen the lens again to see all vertices within a distance of two edges, and then a distance of three edges, four edges, and so on.

Easo and Hutchcroft set the dial that determines how many links there are close to where they saw a large cluster. They then widened the lens, watching more and more edges gather in their large cluster. As they did so, they had to increase the probability that links would be present, which makes it easier to show that the graph has a large connected component. This is a delicate balancing act. They needed to widen the field of view quickly enough and add links slowly enough to reveal the full infinite graph without dramatically changing the position of the dial.

They were able to show that large clusters grow faster than smaller ones, so that, as Easo put it, “your cluster grows faster and faster as it gets bigger and bigger, just like when you’re rolling a snowball.”

For the square grid, the vertex count grows relatively slowly. It’s roughly the width of your lens squared. After 10 steps, you’ll find around 100 vertices. But a 3-regular tree grows exponentially faster — roughly 2 raised to the power of your lens width. After 10 steps, you’ll see approximately 1,024 vertices. The illustration below shows how the 3-regular tree is much bigger after only seven steps, even though the square grid has more vertices at first. In general, graphs can have different growth rates at different scales — they might start out fast, and then slow down.

Back in 2018, Hutchcroft used a similar idea to prove the locality conjecture for fast-growing graphs like the 3-regular tree. But it didn’t work for slow-growth graphs like the square grid, or for graphs that grow at intermediate speed, meeting neither the mathematical criteria for fast growth nor those for slow growth.

“This is where things get really frustrating for like three years,” Hutchcroft said.

Structure Versus Expansion

For graphs that mix growth rates at different scales, you have to use a variety of techniques.

One very helpful fact is that, as Easo explained, “if a graph looks slow-growth at some scale, then it gets stuck.” It will continue to grow slowly at larger scales. Because slow-growth graphs have additional structure determined by a branch of mathematics called group theory, it was also known that if you zoom out far enough, slow-growth graphs display geometry that is mathematically tame.

In 2021, Sébastien Martineau of Sorbonne University in Paris, working with Daniel Contreras and Vincent Tassion of ETH Zurich, was able to use this property to prove Schramm’s locality conjecture for graphs that eventually grow slowly.

At this point, the two groups of mathematicians had successfully tackled the conjecture from different directions: fast-growth and slow-growth. But this left sizeable gaps. For one, there is an intermediate-growth category that wasn’t covered by Easo and Hutchcroft’s technique or by Contreras, Martineau and Tassion’s proof. Another problem was that the arguments still didn’t apply to graphs with changing growth rates — only ones that stayed fast or stayed slow. For the Contreras, Martineau and Tassion argument to be applied to arbitrary graphs, it wasn’t enough that the geometry eventually looks tame when you zoom out, Easo explained: “We need it to look tame now, near the current scale.”

The Middle of Nowhere

Transitive graphs of intermediate growth are very mysterious. Mathematicians have never found an example of a transitive graph whose growth falls in this range. It’s possible that they don’t even exist. But mathematicians haven’t proved they don’t exist, so any complete proof of Schramm’s locality conjecture must address them. Adding to the challenge, Easo and Hutchcroft needed to address graphs which might only briefly have intermediate growth at a particular length scale, even if they grow faster or slower when you zoom in or out.

Easo and Hutchcroft spent much of the past year working to extend their results to apply to graphs that weren’t covered by any of the earlier methods.

First, they modified the 2018 technique that Hutchcroft had applied to fast-growing graphs to work on graphs that change growth levels at different scales. They then tackled the slow-growth case, in a 27-page paper they shared in August that expanded on the work on Contreras, Martineau, and Tassion. Finally, in their October preprint, they devised another argument using the theory of random walks — lines that wiggle randomly through space — to handle the intermediate-growth case. With the trichotomy complete, they had proved Schramm’s locality conjecture.

“We had to throw everything we knew at the problem,” Hutchcroft said.

The solution gives mathematicians a better insight into what happens above the percolation threshold, where the chance of an infinite cluster is 100%, and below it, where the chance is 0%. But mathematicians are still stumped by what happens exactly at the threshold for most graphs, including the three-dimensional grid. “That’s probably the most famous, most basic open question in percolation theory,” said Russell Lyons of Indiana University.

The two-dimensional grid is one of the few cases where mathematicians have proved what happens exactly at the threshold: infinite clusters don’t form. And after Grimmett and Marstrand proved a version of the locality conjecture for big slabs, Grimmett and collaborators showed that if you slice a 3D grid in half horizontally, creating a floor, and tune the dial exactly to the percolation threshold, no infinite clusters appear. Their result hints that the full three-dimensional grid, like its two-dimensional counterpart, might not have an infinite cluster at the percolation threshold.

In 1996, Benjamini and Schramm conjectured that the chance of finding an infinite cluster right at the threshold is zero for all transitive graphs — just as it is for the 2D grid or for the 3D grid sliced in half. Now that the locality conjecture has been settled, an understanding of what happens right at the point of transition might be just a little bit closer.

Correction: December 18, 2023
The number of nodes within n links of a starting node on a 3-regular graph grows as roughly 2n, not 3n as this article originally stated. The article has been corrected.

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