We 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 t is obtained by resampling each weight independently with probability t. On the cube [0, ^n]d, we study geodesics, that is, weight-maximizing up-right paths from (0,0,⋯,0) to (n,n,⋯,n), and their passage time T. Under mild conditions on the weight distribution, we prove a phase transition between stability and chaos at t≍ Var(T). Indeed, as n grows large, for small values of t, the passage times at time 0 and time t are highly correlated, while for large values of t, the geodesics become almost disjoint.

We consider random-access networks where nodes represent servers with a queue and can be either active or inactive. A node deactivates at unit rate, while it activates at a rate that depends on its queue length, provided none of its neighbors is active. We consider arbitrary bipartite graphs in the limit as the initial queue lengths become large and identify the transition time between the two states where one half of the network is active and the other half is inactive. The transition path is decomposed into a succession of transitions on complete bipartite subgraphs. We formulate a randomized greedy algorithm that takes the graph as input and gives as output the set of transition paths the network is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rates, we identify three regimes of behavior.

We study competing first passage percolation on graphs generated by the configuration model with infinite-mean degrees. Initially, two uniformly chosen vertices are infected with a type 1 and type 2 infection, respectively, and the infection then spreads via nearest neighbors in the graph. The time it takes for the type 1 (resp. 2) infection to traverse an edge e is given by a random variable X₁(e) (resp. X₂(e)) and, if the vertex at the other end of the edge is still uninfected, it then becomes type 1 (resp. 2) infected and immune to the other type. Assuming that the degrees follow a power-law distribution with exponent τ ∈ (1, 2), we show that with high probability as the number of vertices tends to infinity, one of the infection types occupies all vertices except for the starting point of the other type. Moreover, both infections have a positive probability of winning regardless of the passage-time distribution. The result is also shown to hold for the erased configuration model, where self-loops are erased and multiple edges are merged, and when the degrees are conditioned to be smaller than n^α for some α > 0.