Caltech
The Caltech team that solved the “random walk” problem
A team of researchers combined algebra and geometry to solve the ancient “random walk” problem, which has influenced areas as diverse as biology and video games. The team identified a fundamental geometric concept that unites certain random walks and distinguishes others.
In 2019, mathematicians at the California Institute of Technology solved an ancient problem related to a mathematical process called “random walk“, a model that describes a trajectory composed of a succession of steps random and independent, starting from a starting point.
The team, which also had the collaboration of a researcher from Ben-Gurion University, in Israel, came up with the solution at great speedin a night of inspiration, tells .
The solution was presented in a published at the time in the magazine Annals of Mathematics.
The main author of the article, Omer Tammuzwhich is dedicated to the study of economics and mathematics, explains that the team resorted to probability theory and used the ergodic theory as link — a progressive, interdisciplinary approach that Abel Prize-winning mathematicians (the “Nobel of Mathematics”) helped pave the way.
In one published at the time by Caltech, Tamuz explained that he had shared a possible discovery with his students and that, the next day, he learned that they they had gone ahead and solved the problem.
“I remember talking to the students about a conclusion we had reached about this problem, and the next morning I learned that they had stayed up late and managed to solve it“, stated Tammuz.
The discovery, and the problem it helped the team unravel, is again based on ability to identify a link between ways of thinking apparently disparate.
“Random walk” is a common expression to designate a method of construction of a route based on decisions made randomly in each fork.
Five eight-step random walks from a central point
Anyone who has ever played an algorithmically generated video game, such as Minecraft, Diablo or No Man’s Sky, has come across a random walk, in the form of a dungeon or from a generated terrain this way by programming.
The principles and ideas of random walk theory are applied across numerous disciplines.
Os biologists resort to random walks to model behavior and animal movements. You physical use them to describe and model the particle behavior. And learning how to implement versions of this process has been an ongoing project for computer scientists.
Some random walks seem to behave depending on the routes already taken, which is called “path dependency“.Others seem ignore their “past” and end up converging with other routes that have different histories.
It is precisely this difference that Tamuz and his colleagues Joshua Frisch, Yair Hartman and Pooya Vahidi Ferdowsi dug in and found a solution. In the words of Tamuz, in the statement:
“Let us imagine two societies: one of them makes a technological advance while the other suffers a natural catastrophe. These differences will persist forever, or will eventually disappear and we will fall into oblivion that one day there was an advantage?”, says Tamuz in the statement.
“In random walks, it has long been known that there are groups that keep these memories, while In other groups, memories are eraseds. But it was not at all clear which groups had this property and which did not—i.e. what makes a group have memory? That’s what we discovered”, details the mathematician.
In other words: why some random walks regress to the mean and others not?
When programming using random walk ideas, it is possible to simply set a limitation or parameter to ensure that the paths converge or not. But in pure theory, the problem is much more difficult to explain.
The team’s secret consisted of combine ideas from algebra with ideas from geometry in describing random walks, and in exploring this connection to investigate the phenomenon.
Based on this idea, researchers concluded that random walks that satisfy a certain criterion based on vector geometry are those that converge with all the remaining. “In the end, we were very pleased to have solved a long-standing open problem in mathematics,” concludes Ferdowsi.
Simple, isn’t it?
