From stability to chaos in last-passage percolation

We study the transition from stability to chaos in a dynamic last passage percolation model on (Formula presented.) 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 (Formula presented.) is obtained by resampling each weight independently with probability (Formula presented.). On the cube (Formula presented.), we study geodesics, that is, weight-maximizing up-right paths from (Formula presented.) to (Formula presented.), and their passage time (Formula presented.). Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at (Formula presented.). Indeed, as (Formula presented.) grows large, for small values of (Formula presented.), the passage times at time 0 and time (Formula presented.) are highly correlated, while for large values of (Formula presented.), the geodesics become almost disjoint.