Bootstrap percolation is a process in which an initially infected set of vertices in a graph spreads the infection to uninfected vertices according to some rule. In r-neighbour bootstrap percolation, a vertex becomes infected if it has at least r infected neighbours. A central question in the study of bootstrap percolation concerns the likely behaviour of a randomly selected set of initially infected vertices. This version of the problem has been studied extensively in a variety of lattice-like graphs. In particular, Balogh, Bollobás, and Morris identified a sharp threshold function for 2-neighbour bootstrap percolation in the hypercube. A related bootstrap percolation variant, Froböse percolation, has also been studied on lattice-like graphs, particularly the grid. In this talk, I will examine the Froböse percolation process on the hypercube and present a sharp threshold function for Froböse percolation in the hypercube.
The talk is based on joint work with Fabian Burghart and Mihyun Kang.