Socially Awkward Math

A mathematician's random walk theorem explains the stark mathematical difference between drunk people and drunk birds

- JACK MURTAGH

MORE THAN 100 YEARS AGO Hungarian-born mathematician George Pólya found himself trapped in a loop of social awkwardness. A professor at the Swiss Federal Institute of Technology Zurich, he enjoyed solitary strolls through the woods outside the city. During one of these rambles, he walked by one of his students and the student’s fiancée. Then, sometime later, still roaming aimlessly, he bumped into the couple again. And then later he did so yet again.

Writing about the experience in an essay published in a 1970 book, Pólya recounted, “I don’t remember how many times [this happened], but certainly much too often and I felt embarrassed: It looked as if I was snooping around which was, I assure you, not the case.”

Desperate to clear his name as a lurker, Pólya did what any good mathematician would do: he generalized the problem. Are two wanderers mathematically destined to cross paths? His original formulation simplified the picture by considering only a single walker on an infinite grid. Every second, the walker chooses a compass direction at random, independent of previous steps. Pólya’s mathematical aim was to determine the probability that the walker would eventually return to their starting point. This answer turns out to be equivalent to the probability that two walkers who start at the same location will ever meet again. He found that if a walker roams forever, they will return to their starting place.

The answer not only absolved him but also revealed a fundamental divide in how the laws of chance interact with physical space. Pólya’s calculations showed that on a two-dimensional surface (such as a forest floor), a random walker is destined to return to their starting point—but in a three-dimensional space, that person is more likely to never return to the starting point. The discovery crops up across chemistry and biology, even explaining how certain molecules efficiently find the appropriate receptor on cell surfaces.