Title: Sharp and gradual phase transitions in bootstrap percolation
Speaker: David Sivakoff (OSU)
Abstract: 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.