Extremal and probabilistic bootstrap percolation

Abstract

In this dissertation we consider several extremal and probabilistic problems in bootstrap percolation on various families of graphs, including grids, hypercubes and trees. Bootstrap percolation is one of the simplest cellular automata. The most widely studied model is the so-called r-neighbour bootstrap percolation, in which we consider the spread of infection on a graph G according to the following deterministic rule: infected vertices of G remain infected forever and in successive rounds healthy vertices with at least r already infected neighbours become infected. Percolation is said to occur if eventually every vertex is infected. In Chapter 1 we consider a particular extremal problem in 2-neighbour bootstrap percolation on the n × n square grid. We show that the maximum time an infection process started from an initially infected set of size n can take to infect the entire vertex set is equal to the integer nearest to (5n − 2n)/8. In Chapter 2 we relax the condition on the size of the initially infected sets and show that the maximum time for sets of arbitrary size is 13n/18 +O(n). In Chapter 3 we consider a similar problem, namely the maximum percolation time for 2-neighbour bootstrap percolation on the hypercube. We give an exact answer to this question showing that this time is bn/3c. In Chapter 4 we consider the following probabilistic problem in bootstrap percolation: let T be an infinite tree with branching number br(T ) = b. Initially, infect every vertex of T independently with probability p > 0. Given r, define the critical probability, pc(T, r), to be the value of p at which percolation becomes likely to occur. Answering a problem posed by Balogh, Peres and Pete, we show that if b ≥ r then the value of b itself does not yield any non-trivial lower bound on pc(T, r). In other words, for any e > 0 there exists a tree T with branching number br(T ) = b and critical probability pc(T, r) 0 such that if T is a Galton–Watson tree with branching number br(T ) = b ≥ r then pc(T, r) > cr b e− b r−1 . We also show that this bound is sharp up to a factor of O(b) by describing an explicit family of Galton–Watson trees with critical probability bounded from above by Cre b r−1 for some constant Cr > 0.