Bootstrap Percolation on Periodic Trees

We study bootstrap percolation with the threshold parameter theta geq 2 and the initial probability p on infinite periodic trees that are defined as follows. Each node of a tree has degree selected from a finite predefined set of non-negative integers and starting from any node, all nodes at the same graph distance from it have the same degree. We show the existence of the critical threshold p_f(theta) in (0,1) such that with high probability, (i) if p > p_f(theta) then the periodic tree becomes fully active, while (ii) if p