Bootstrap percolation in random geometric graphs

Abstract

Following Bradonjić and Saniee, we study a model of bootstrap percolation on the Gilbert random geometric graph on the 2-dimensional torus. In this model, the expected number of vertices of the graph is n, and the expected degree of a vertex is a log n for some fixed a > 1. Each vertex is added with probability p to a set A 0 of initially infected vertices. Vertices subsequently become infected if they have at least θa log n infected neighbours. Here p, θ ∈ [0, 1] are taken to be fixed constants.We show that if θ < (1 + p)/2, then a sufficiently large local outbreak leads with high probability to the infection spreading globally, with all but o(n) vertices eventually becoming infected. On the other hand, for θ > (1 + p)/2, even if one adversarially infects every vertex inside a ball of radius O( √ log n), with high probability the infection will spread to only o(n) vertices beyond those that were initially infected.In addition we give some bounds on the (a, p, θ) regions ensuring the emergence of large local outbreaks or the existence of islands of vertices that never become infected. We also give a complete picture of the (surprisingly complex) behaviour of the analogous 1-dimensional bootstrap percolation model on the circle. Finally we raise a number of problems, and in particular make a conjecture on an 'almost no percolation or almost full percolation' dichotomy which may be of independent interest.