# Mathematics Colloquia and Seminars

Bootstrap percolation is a discrete growth process on a graph, in which vertices become occupied as soon as they have at least $\theta$ occupied neighbors. The initial set of occupied sites is given by a product measure with density $p$, and the graph is said to be spanned if the initial set of occupied sites leads to all vertices becoming occupied. Given a sequence of graphs of increasing size, we are first interested in the asymptotic location of the critical probability, which is the smallest value of $p$ for which spanning becomes likely. Second, we are interested in the nature of the phase transition. On many finite lattices, the critical probability is known, and the phase transition sharp. I will present a class of graphs, given by the Cartesian products of finite lattices and complete graphs, on which bootstrap percolation may exhibit sharp, gradual or hybrid phase transitions. Based on joint works with Janko Gravner, Chris Hoffman and James Pfeiffer.