Bootstrap Percolation on Periodic Trees

Abstract

We study bootstrap percolation with the threshold parameter θ ≥ 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 a given node, called root, all nodes at the same graph distance from the root have the same degree. We show the existence of the critical threshold pf(θ) ∈ (0, 1) such that with high probability, (i) if p > pf(θ) then the periodic tree becomes fully active, while (ii) if p < pf(θ) then a periodic tree does not become fully active. We also derive a system of recurrence equations for the critical threshold pf(θ) and compute these numerically for a collection of periodic trees and various values of θ, thus extending previous results for regular (homogeneous) trees.