Julien Brémont
Collège de France
The memory of geometry in random explorations : From self-interacting walks to universal flip statistics
Most complex systems, from migrating cells to time series, explore their phase space in ways that strongly depend on their past. Such non-Markovian explorations remain notoriously difficult to analyze, even at the level of single-time statistics: even basic observables such as propagators, first-passage statistics or splitting probabilities are often inaccessible.
In this seminar, I will first present a framework that overcomes these difficulties for a broad family of self-interacting random walks, yielding exact results for key observables including propagators, persistence exponents, splitting probabilities, and the size of the visited territory. Building on these results, I will then introduce flips, defined as reversals in the direction of exploration, and show that they obey strikingly universal statistics across non-Markovian processes. This universality is model-agnostic and is observed in real datasets of widely different origins.
This universality invites a shift in perspective: the natural clock for exploration is the territory size, not physical time. This paradigm allows one to characterize space exploration far beyond the traditional framework of single-time exploration properties of Markovian walks, and opens new directions for modeling complex memory-driven dynamics in physics, biology, and beyond.
