Bootstrap percolation in random geometric graphs

Following Bradonji\'c 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 $\theta a \log n$ infected neighbours. Here $p, \theta \in [0,1]$ are taken to be fixed constants. We show that if $\theta < (1+p)/2$, 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 $\theta > (1+p)/2$, even if one adversarially infects every vertex inside a ball of radius $O(\sqrt{\log n} )$, 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 $(a, p, \theta)$ 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.