Imagine that a grid of hexagons, honeycomb-like, stretches before you. Some hexagons are empty; others are filled by a 6-foot tall column of solid concrete. The result is a maze of sorts. For over half a century, mathematicians have posed questions about such randomly generated mazes. How big is the largest web of cleared paths? What are the chances that there is a path from one edge to the center of the grid and back out again? How do those chances change as the grid swells in size, adding more and more hexagons to its edges?

These questions are easy to answer if there is either a lot of empty space or a lot of concrete. Say every hexagon is assigned its state at random, independent of all the other hexagons, with a probability that is constant across the entire grid. There could be, say, a 1% chance that each hexagon is empty. Concrete crowds the grid, leaving only small pockets of air in between, making the chance of finding a path to the edge effectively zero. On the other hand, if there is a 99% chance that each hexagon is empty, there is just a thin sprinkling of concrete walls, punctuating swaths of open space — not much of a maze. Finding a path from the center to the edge in this case is a near-certainty.

For large grids, there is a remarkably sudden change when the probability hits 1/2. Just as ice melts into liquid water at exactly zero degrees Celsius, the character of the maze changes drastically at this transition point, called the critical probability. Below the critical probability, most of the grid will lie underneath concrete, while empty paths invariably come to dead ends. Above the critical probability, massive tracts are left empty, and it’s the concrete walls that are sure to peter out. If you stop exactly at the critical probability, concrete and emptiness will balance one another, with neither able to dominate the maze.

“At the critical point, what emerges is a higher degree of symmetry,” said Michael Aizenman, a mathematical physicist at Princeton University. “That opens the door to a huge body of mathematics.” It also has practical applications to everything from the design of gas masks to analyses of how infectious diseases spread or how oil seeps through rocks.

In a paper posted last fall, four researchers have finally calculated the chance of finding a path for mazes at the critical probability of 1/2.

An Arms Race

As a doctoral student in France in the mid-2000s, Pierre Nolin studied the critical probability scenario in great detail. The random maze, he thinks, is “a really beautiful model, maybe one of the simplest models you can invent.” Near the end of his doctoral studies, which he finished in 2008, Nolin became captivated by a particularly challenging question about how a hexagonal grid at the critical probability behaves. Say you build a grid around a central point, so that it approximates a circle, and you randomly build your maze from there. Nolin wanted to explore the chance that you’ll be able to find an open path that reaches from the edge to the center and back out, without retracing itself. Mathematicians calls this a monochromatic two-armed path, because both the inward and outward “arms” are on open paths. (Sometimes such grids are equivalently thought of as made of two different colors, say light blue and dark blue, rather than of open and closed cells.) If you increase the size of the maze, the length of the needed path will grow as well, and the chance of finding such a path will get smaller and smaller. But how quickly do the odds diminish, as the maze grows arbitrarily large?

Simpler related questions were answered decades ago. Calculations from 1979 by Marcel den Nijs estimated the chance that you can find one path, or arm, from the edge to the center. (Contrast this with Nolin’s requirement that there be one arm in and a separate one out.) Den Nijs’ work predicted that the chance of finding one arm in a hexagonal grid is proportional to $latex 1/n^{5/48}$, where n is the number of tiles from the center to the edge, or the radius of the grid. In 2002, Gregory Lawler, Oded Schramm and Wendelin Werner finally proved that the one-arm prediction was correct. To succinctly quantify the diminishing probability as the size of the grid grows, researchers use the exponent from the denominator, 5/48, which is known as the one-arm exponent.

Nolin wanted to compute the more elusive monochromatic two-arm exponent. Numerical simulations in 1999 showed that it was very close to 0.3568, but mathematicians failed to pin down its exact value.

It was much easier to compute what’s known as the polychromatic two-arm exponent, which characterizes the chance that, starting in the center, you can find not only an “open” path to the perimeter, but also a separate “closed” path. (Think of the closed path as one that traverses the tops of the concrete walls of the maze.) In 2001, Stanislav Smirnov and Werner proved that this exponent was 1/4. (Because 1/4 is substantially larger than 5/48, $latex 1/n^{1/4}$ shrinks more quickly than $latex 1/n^{5/48}$ as n grows. The chance, then, of a polychromatic two-arm structure is a lot lower than the chance of one arm, as one might expect.)

That computation had leaned heavily on knowledge about the shape of clusters in the graph. Imagine that a maze at the critical probability is extremely large — made up of millions and millions of hexagons. Now find a cluster of empty hexagons and trace the edge of the cluster with a thick black Sharpie. This probably won’t result in a simple, round blob. From miles in the air, you’d see a wriggling curve that constantly doubles back, often seeming as if it’s about to cross itself but never quite committing.

This is a type of curve called an SLE curve, introduced by Schramm in a 2000 paper that redefined the field. A mathematician studying the chances of finding one open path and one closed path knows that those paths must sit inside larger clusters of open and closed sites, which eventually meet along an SLE curve. The mathematical properties of SLE curves then translate to invaluable information about paths within the maze. But if mathematicians are searching for multiple paths of the same type, SLE curves lose much of their effectiveness.

By 2007, Nolin and his collaborator Vincent Beffara had created numerical simulations showing that the monochromatic two-arm exponent was about 0.35. This was suspiciously close to 17/48 — the sum of the one-arm exponent, 5/48, and the polychromatic two-arm exponent, 1/4 (or 12/48). “17/48 is really striking,” Nolin said. He began to suspect that 17/48 was the true answer — meaning there was a simple link between the different kinds of exponents. You could just add them together. “We said, OK, it’s too good to be false; it has to be true.”