Bootstrap percolation in random k-uniform hypergraphs

We investigate bootstrap percolation with infection threshold r>1 on the binomial k-uniform random hypergraph Hk(n,p) in the regime n−1≪nk−2p≪n−1/r, when the initial set of infected vertices is chosen uniformly at random from all sets of given size. We establish a threshold such that if there are less vertices in the initial set of infected vertices, then whp only a few additional vertices become infected, while if the initial set of infected vertices exceeds the threshold then whp almost every vertex becomes infected. In addition, for k=2, we show that the probability of failure decreases exponentially.