In 2008, the mathematician Oded Schramm died in a hiking accident in the Cascade mountains some 50 miles east of Seattle. Though he was just 46 years old, he had constructed entirely new areas of mathematics.
“He was a fantastic mathematician,” said Itai Benjamini, a mathematician at the Weizmann Institute of Science and Schramm’s friend and collaborator. “Extremely creative, extremely elegant, extremely original.”
The questions he asked are still pushing the frontiers of probability theory and statistical physics. Many of these questions concern mathematical structures that have a phase transition — a sudden macroscopic change, like ice melting to water. Just as different materials have different melting points, phase transitions of mathematical structures also vary.
Schramm conjectured that the phase transition in a process called percolation can be estimated by using only a close-up view of the system — called the local perspective — for many important mathematical structures. Zooming all the way out and looking at the whole thing won’t significantly change the calculation. In the past 15 years, mathematicians have chipped away at small pieces of the conjecture, but until now, they have not been able to completely resolve it.
In a preprint posted in October, Tom Hutchcroft of the California Institute of Technology and his doctoral student Philip Easo proved Schramm’s locality conjecture. Their proof relies on major ideas from across probability theory and other areas of math, which they combined in a clever way.
“It’s a remarkable paper. It’s an accumulation of long work,” Benjamini said.
The word “percolation” originally referred to the movement of fluid through a porous medium, such as water flowing through coffee grounds or oil seeping through cracks in a rock.
In 1957, the mathematicians Simon Ralph Broadbent and John Michael Hammersley developed a mathematical model of this physical process. In the decades since, this model has become an object of study in its own right. Mathematicians study percolation because it strikes an important balance: The setup is simple, but it exhibits complex and puzzling features.
“It’s sort of a canonical model for mathematicians,” Hutchcroft said. “You can think of things visually. That makes it really nice to work with.”
Percolation starts with a graph, which is a collection of vertices (points) that can be connected by edges (lines). One of the simplest examples is a square grid, with vertices lined up to form the corners of squares and edges connecting some of them.
When inserting the edges, you can use a weighted coin, changing the odds that an edge connects two points. Imagine that the weight of the coin is controlled by a dial. Initially, the coin will always land on “no edge,” and the graph will consist entirely of disconnected vertices. As you turn the dial, the coin becomes more likely to land on “insert,” and more edges appear in the graph.
In physical percolation, the edges might represent cracks in a rock. In this case, you might look for connected clusters, which indicate regions of rock that oil can freely flow through.
Mathematicians are interested in how infinite clusters form within infinite graphs, such as a square grid extending in all directions. In this setting, they observe something surprising: a phase transition.
As you turn the dial, slowly changing the weight of the coin, the probability of finding an infinite cluster does not increase gradually. Instead, there is a specific point on the dial, known as the percolation threshold, where an infinite cluster appears. The percolation threshold depends on the underlying graph. For the square grid, it is the point where the coin is equally weighted. Below this point, there is a 0% chance of finding an infinite cluster, and above it, there is a 100% chance. It’s generally unknown what happens when the dial is exactly at the threshold. But when it’s even an infinitesimal amount past the threshold, an infinite cluster suddenly appears, just as water suddenly becomes steam at 100 degrees Celsius.
Look Local, See Global
In 1990, the mathematicians Geoffrey Grimmett and John Marstrand wondered if it was possible to calculate a percolation threshold by only examining relatively small parts of a graph. They studied percolation on slabs, which are square grids stacked on top of each other in layers. The number of layers is finite, but if you were to look at only part of the slab, narrowing your perspective, you would just assume it’s a three-dimensional grid — everything looks the same.
Each slab has a percolation threshold, which changes depending on the number of layers in the slab. Grimmett and Marstrand proved that as the number of layers grows, the percolation threshold edges toward the threshold for the infinite three-dimensional grid. They looked from a narrow perspective — a slice of slabs — and approximated the threshold for the entire graph. “This result is really important for the field,” said Barbara Dembin of the Swiss Federal Institute of Technology Zurich (ETH Zurich).
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.