When physicists study a “random walk,” they’re investigating the path followed when you’re allowed to take each step in a randomly determined direction. It’s a useful approach to studying random motion, from molecules moving in a gas to stock-market fluctuations, and it’s something that SFI Professor Sidney Redner has been studying, from many angles, for decades. “I love random walks,” he says.
In a confined space, a random walker sometimes runs into obstacles and is forced to change course. But what might happen if the random walker could push confining obstacles out of the way? In a recent paper published in Physical Review Letters (PRL), Redner and collaborators described a model of just such a system and showed how those movable obstacles can significantly reshape the journey of a random walk. They found that in one dimension, a random walker that pushes aside obstacles forms a very slowly growing cavity. In two dimensions, the random walker can get trapped by the obstacles when their density is sufficiently high, but otherwise can move almost freely when the obstacle density is low.
Researchers have long been interested in modified random walks. For example, what happens when a random walker encounters fixed obstacles, like an ant in a labyrinth? The new work in PRL has its roots in Sokoban, a video game created in the 1980s in which a player navigates a labyrinth and tries to push single blocks into new locations. Two of Redner’s co-authors, Ofek Lauber Bonomo and Shlomi Reuveni, both from Tel Aviv University, had previously developed a theory of Sokoban random walks where “you can push one obstacle out of the way to try and escape, but you can't push more than one obstacle,” Redner says.
The question at the heart of the new work emerged when they described their work to Redner. The group arrived at a question that extend those results further: What happens if the walker can move multiple blocks at once?
This new work provides a theoretical framework for pushy random walks, and it has potential for real-world implications. In the field of glassy dynamics, researchers study systems in which moving bodies — molecules, particles, cells, microbes — propel themselves through a strongly confining disordered space. How the dynamics unfolds remains an area of rich debate even after decades of study, says Redner. But one thing that is clear is that sometimes those moving bodies have to move multiple objects out of the way to get by. As they do so, they can change the medium in non-trivial ways.
“As far as I know, nobody has really thought about all the implications of the medium being deformed as an active particle is pushing,” he says.
The new work serves as more of a jumping-off point for new investigations of pushy random walk systems, says Redner. “We think there are all kinds of interesting generalizations and extensions to his model.”
Read the paper "Pushy Random Walk: A Minimal Model for Transport in Deformable Media" in Physical Review Letters (July 13, 2026). DOI: 10.1103/7hjs-rx8d