Karen Rosemarie Johannson

(Mathematics, University of Memphis)

Bootstrap Percolation

Date Wednesday, April 25, 2012

Memphis, Tenn., USA

In the study of usual percolation, one looks for infinite components in a random subset of an infinite lattice. Bootstrap percolation, on the other hand, is a deterministic process on a graph G with a parameter r which changes the state of vertices from ‘uninfected’ to ‘infected’ by the follow- ing rule: infected vertices remain infected forever and an uninfected vertex becomes infected if it has at least r infected neighbours. A set of initially in- fected vertices is said to ‘percolate’ if all vertices eventually become infected. For a finite square grid in two dimensions, vertices are 1 × 1 squares, called sites, two sites are adjacent if they share an edge and two infected neighbours cause an uninfected site to become infected. The initially infected sites are chosen at random, independently with a fixed probability, and one asks when is percolation either likely or unlikely? Aizenman and Lebowitz first determined the critical probability for percolation up to a constant factor and Holroyd gave a sharp threshold for the critical probability. In this talk, I will discuss some of these results and give a sketch of new work on a modification of the bootstrap process in which infected sites may ‘recover’ from their infection.