Some properties of geodesics in first-passage percolation

par Antonin Jacquet (Télécom Paris)

mardi 10 mars 2026, 09:50 → 10:50 Europe/Paris

Amphi Schwartz

In this talk, we study the classical first-passage percolation model on Z^d. In this model, we consider a family of nonnegative, independent and identically distributed random variables indexed by the set of edges of the graph Z^d, called passage times. The time of any finite path is defined as the sum of the passage times of each of the edges it uses. The geodesics are then the minimal-time paths. We consider a local property of the passage times, called a pattern, and we study the number of translates of this pattern used by a geodesic. The main result presented in this talk ensures, under reasonable assumptions, that apart from an event of exponentially small probability, this number is linear in the distance between the endpoints of the geodesic. The objective is to present the first-passage percolation model, to introduce the notion of patterns, to give some ideas of the proof of the main result and to illustrate how the patterns can be used to obtain certain results in the model.