In Raiders of the Lost Ark, Indiana Jones must find a secret chamber that contains the legendary Ark of the Covenant. To identify its exact location, Indy must uncover a special map that’s only visible when the sun shines through a special crystal in a certain room at a certain time of day.
This idea — that essential information can be revealed when circumstances are just right — occurs in many myths. It also appears in mathematics, sometimes in unlikely settings. Now three mathematicians have proved that when a certain type of randomness is tuned perfectly, intricate geometric shapes emerge in plain sight — like a map revealed on an ordinary floor.
These shapes are checkerboard-like designs arranged at random across grids that are themselves constructed by a random process. You’d think this piling of randomness atop randomness would produce a mess. Yet as it turns out, in the same way that every snowflake is unique, but all snowflakes are snowflakes, the disorder converges to a universal form — provided the conditions are just right.
The Tipping Point
Everyone knows that mathematicians study shapes. Most of those shapes follow deterministic rules: If I give you instructions for constructing a sphere, you’ll construct the exact same sphere every time.
But mathematicians also study shapes constructed by random processes, such as the random walk — the path traced if, at every step, you move in a random direction. Besides the random walk, there are other kinds of random geometric objects, such as random two-dimensional surfaces (picture a landscape where the hills and valleys are pitched at random) and random maps (collections of random points connected by lines).
No two of these shapes are identical. Yet mathematicians have discovered that these random processes converge to certain canonical forms. All random walks, for example, converge to the form of Brownian motion if you let the random process run long enough. In recent years, mathematicians have identified the canonical forms of other random processes — and have won some of the highest honors in the field as a result.
The new proof is about understanding the deep properties of another random process.
This process begins with the construction of a random surface. First, glue together triangles along their edges. Next, fit them together in any fashion, as long as the final shape closes up like wrapping paper around a gift (so no holes or openings). If you start with a specific number of triangles, you’ll have lots of possibilities. A few of the “triangulations” will produce nearly smooth surfaces, like that of a ball. Most will be far wilder — extreme surfaces that resemble bulging mountain ranges.
“It won’t look like a regular sphere. It will have these big spikes,” said Olivier Bernardi, a mathematician at Brandeis University and co-author of the work along with Nicolas Curien of Paris-Sud University and Grégory Miermont of École Normale Supérieure in Lyon, France.
Miermont and another mathematician, Jean-François Le Gall, established many of the properties of these random triangulations in earlier work. This new proof goes further, by adding a second layer of randomness atop the random triangulation.
To add this additional randomness, mark every place where the corners of the triangles meet with a point — what mathematicians call a vertex. Color each of the vertices white or black at random. You could do this by flipping a coin, though it doesn’t have to be a fair coin — it could be weighted to produce more of one color than the other.
Once you’ve colored your vertices, you can ask various questions about the pattern you’ve created. One of the most basic is: Can you move far across the surface by only traveling on black vertices? This process, of moving along connected vertices of the same color, is known as a percolation. It provides a mathematical way to study the physical phenomenon of the same name, in which a fluid flows through a porous medium.
Percolation will be easy (or not) depending on the weight of the coin: If the coin is weighted heavily toward black vertices, percolation will be nearly guaranteed; if it favors white vertices, percolation will almost certainly be impossible.
Bernardi, Curien and Miermont study the case in between these extremes — the tipping point in the weight of the coin where percolation switches from almost impossible to nearly guaranteed. They refer to this point as the “critical threshold.” It’s an example of a phase transition, like the magic moment when hot water suddenly becomes steam.
“The critical threshold means if I move my parameter a little, the behavior of my system goes from something dramatic to something dramatic in the opposite direction,” Curien said.
Physicists are interested in phase transitions because many of the most important phenomena in nature occur on the cusp. Mathematicians are interested in phase transitions, too, because important mathematical features often emerge at precisely those points.
“We know right at 212 degrees, water is boiling, it’s making crazy patterns, there’s steam pouring out,” said Scott Sheffield, a mathematician at the Massachusetts Institute of Technology. “It’s this wild and crazy behavior that’s somehow very interesting. Right at this phase transition it feels like it’s calling out to be understood.”
In the new paper, the three mathematicians prove that similar wild behaviors occur right at the percolation phase transition. They show that at that critical threshold a unique geometric shape emerges — unique, but also universal.
Hidden Order in the Randomness
The first part of the paper determines how to weight a coin so that the coloring of the vertices sits on the threshold between percolation and no percolation. Confirming intuition, they prove that the critical value is a perfectly fair coin — one that gives a 50 percent chance for black and a 50 percent chance for white.
“That’s the first part of the paper. We prove that interesting stuff happens at one-half,” Bernardi said.
The second part examines that interesting stuff. When you color vertices white or black using a fair coin, you’ll end up with a good balance between clusters of black vertices and white ones. The clusters will grow around each other, like clumps of weeds vying for space in an overgrown garden, creating complicated geometric shapes that don’t appear when the vertices are predominantly one color.
“When you choose the critical parameter, you find you’ll have some big clusters,” Sheffield said. “But they’re not everything and they’re not tiny.”
Because the underlying surface is chosen at random, and the process of coloring the vertices is random, the largest cluster on one surface will always be different from the largest cluster on another. But the mathematicians prove that across all surfaces and all possible ways of coloring the vertices on those surfaces, the largest clusters have traits in common. The first thing they prove is an exact probability distribution for the sizes of the largest black clusters across all surfaces. They establish that there’s a certain intermediate size cluster that occurs most often, and that the frequency with which larger or smaller clusters appear decreases exponentially as you move away from that intermediate size.
They also prove that the largest clusters all scale to a single canonical form known as the stable map. The stable map is to these clusters as Brownian motion is to the random walk. This means that as you zoom out on individual clusters — so that each random step within a cluster becomes less prominent in the geometry of the overall shape — the clusters increasingly take on a common form. They’re like snowflakes: They appear unique up close, but they’re clearly all of a kind when you take a step back.
“They find that this stable map, this natural scaling limit, is what comes about when you take the limit of one percolation cluster,” Sheffield said.
The paper extends an understanding of random shapes and processes that has been building in mathematics in recent years. It also reveals that at the precise moment when a random system seems most chaotic, exquisite geometric order peers through.