A Better Understanding of the Dance
A Colby professor and his student have proved something new about random walk theory
Associate Professor of Mathematics Evan Randles has been spending a lot of time lately pondering a century-old mathematical probability problem in random walk theory—a way of looking at the world that has applications in such diverse fields as biology, chemistry, economics, physics, and even card shuffling.
All that thinking has led to a major mathematical breakthrough for Randles and Yutong “Tony” Yan ’25, a physics and mathematics double major who is collaborating with Randles. It’s the subject of a paper they’re readying for publication, “The Predictable Dance of Random Walk,” that proves something new in the theory.
When Randles realized they had discovered a theorem—then proved it—it was an “aha” moment for the professor and the student.
“It’s the greatest feeling in the world when you really, truly learn something. When you truly know something,” Randles said. “And when a mathematician is able to discover and then prove a theorem, there’s nothing better.”
A primer on random walk theory
The Random Walk Theory was first posited by George Pòlya, a Hungarian-American mathematician who was teaching in Switzerland at ETH Zurich in 1919 when something happened that piqued his curiosity. Pòlya liked to walk in a small forest by the campus, and one day he was walking there when he bumped into one of his graduate students, out strolling with his fiancée. They said hello to each other, separated, then ran into each other again, and again.
It’s impossible to know exactly what Pòlya thought about this repeated chance meeting, but Randles imagines he felt embarrassed—as if his student and fiancée might have thought he was intentionally following them around the forest. Still, in true mathematician fashion, Pòlya turned the experience into a rigorous problem: if two groups, or two objects, are moving at random, what is the probability that they run into each other again, and again, and again?
To illustrate, Randles sketched a simple picture of a long straight line. Imagine you are standing on this line—perhaps a city street—and with every step, you flip a coin to determine whether you move forward or backward.
“One basic question you can ask is what is the probability that the walker comes back to the starting point again and again and again, or in mathematics, we would say ‘ad infinitum,’ or infinitely often?” Randles asked.
This question can be just as easily posed for random walks on two-dimensional city grids, or grids in three dimensions where, say, birds fly randomly. In all of these settings, you can ask what is the probability that the random walker returns to where they started infinitely often.
It turns out that this question, in two dimensions, is equivalent to Pòlya’s conundrum of repeatedly running into the couple. Pòlya proved that, for random walks along a line, like a city street, or in a plane, like a city grid, the probability that a random walker returns to where they started, infinitely often, is 100 percent. Birds that fly randomly, however, will always fly away from where they started, with 100-percent probability of doing so.
Randles, who works primarily in probability and mathematical analysis, said that Pòlya created a mathematical model to untangle the process. “This is really the realm of studying an applied math problem,” the professor said. “He figured out a formula to understand this.”
Learning how to dance
Most mathematicians who study random walks make two standard assumptions about them. The first is that the walk is “aperiodic,” or not stuck in a cycle, and the second asks that the walk is “irreducible,” meaning that a random walker is able to reach any position over time. In other words, if a person flips a coin at every step to determine their direction, over time, they might go just about anywhere, and once they get there, there is a probability that they’ll stay.
There is a large-scale pattern that shows that the probability of a random walker staying closer to where they started is higher than it is for them to be far away. The precise pattern demonstrates the nature of diffusion, Randles said, adding that if the probability of these outcomes is illustrated by a graph, it will take the shape of a bell curve.
“It turns out that when you assume irreducibility and aperiodicity, the analysis becomes more tractable,” he said. “The theoretical difficulties in some sense clear up and one only sees the diffusion, or the bell curve, emerge.”
But making those assumptions also means that mathematicians disregard something cool about random walk theory—that, despite the randomness, there can be interesting patterns found in the steps taken. For example, imagine that a random walker is stepping between even- and odd-numbered bricks in a sidewalk, and every time they flip the coin, they move two steps rather than one. If they start on an even-numbered brick, they will never land on an odd-numbered one.
“If you’re watching hundreds of random walkers doing it at the same time, there’s something that looks like a dance,” Randles said.
“Understanding the diffusion, which is the bell curve, is a really well-known and well-studied problem. Everybody has known about this forever,” Randles said. “I wanted to focus not on the diffusion, but instead on the dance. … I wanted to understand the nature of the dance.”
He and Yan have spent months developing a theory that simultaneously describes the dance and the diffusion and does away with the usual assumptions.
“And we’ve done it,” Randles exulted.
A new discovery
The paper they are readying for publication, “The Predictable Dance of Random Walk,” shows that over time, the position of the random walk is described as a product of two things, one that describes the diffusion and the other that describes the dance. It also shows that the dance can be described by a function that tells the walker where they can be and when, which Randles believes to be a new discovery.
It felt wonderful when they realized they had cracked the code.
“In some ways, there are two reasons why I’m a mathematician,” the professor said. “One of them is because I get so much pleasure out of helping others learn something and have an aha moment. And the other is for me to have my own aha moment.”
It was an “absolute pleasure” to work on the project with Yan, who came to the professor at the beginning of last summer and asked to be a research assistant. Randles, who always has a few problems he’s thinking about, proposed this one and gave Yan some relevant background.
“He hit the ground running,” the professor said. “At Colby, we have such amazing students. And Tony really is a fantastic student. He’s a great budding mathematician. … Tony worked very, very hard at the beginning of the summer and was able to prove some really nice results.”
For Yan, who is planning to study mathematics in graduate school, the project was extremely rewarding—although certainly not without its challenges. He bumped into unforeseen obstacles as he reviewed existing literature and worked to prove the theorem. When he read other sources, he found that mathematicians seemed to have inconsistent ways of looking at things.
“A lot of effort had to be put into just making sense of all these difficult nuances,” Yan said.
That could be frustrating. But Randles helped him persevere.
“He is very nice to work with. I think the major thing is that he is very motivated all the time,” Yan said of his professor. “Even when I felt frustrated, he was able to push me and help me to get through a lot of these difficulties. And that certainly made me grow a lot in both my mathematical abilities and also just resilience.”
As Randles pondered their results, he determined that the duo had a general theory in the works.
“I was able to basically scaffold an argument together. I gave Tony a number of results and said, ‘I think this is true, and you should try to see if it is true, and prove it.’ And we built this argument together,” he said. “It’s just little building blocks, and it’s done through deductive reasoning, and we just prove the hell out of everything. We came up with a very clear and coherent argument that we get to put together into this great thing.”
