From stability to chaos in last‐passage percolation

Abstract

AbstractWe study the transition from stability to chaos in a dynamic last passage percolation model on with random weights at the vertices. Given an initial weight configuration at time 0, we perturb the model over time in such a way that the weight configuration at time is obtained by resampling each weight independently with probability . On the cube , we study geodesics, that is, weight‐maximizing up‐right paths from to , and their passage time . Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at . Indeed, as grows large, for small values of , the passage times at time 0 and time are highly correlated, while for large values of , the geodesics become almost disjoint.