Existence and Region of Critical Probabilities in Bootstrap Percolation on Inhomogeneous Periodic Trees

Abstract

Bootstrap percolation is a growth model inspired by cellular automata. At the initial time $t=0$, the bootstrap percolation process starts from an initial random configuration of active vertices on a given graph, and proceeds deterministically so that a node becomes active at time $t=1,2,\dots$ if sufficiently many of its neighbors are already active at the previous time $t-1$. In the most basic model, all vertices have the same initial probability of being active in the initial configuration. One of the main questions is to determine the percolation threshold (if it exists) with the property that all nodes in the given graph become active asymptotically almost surely (a.a.s.) for the initial probability above this threshold, while this is not the case below the threshold. In this work, we study a scenario where the nodes do not all receive the same probabilities, but to keep the problem tractable, we impose conditions on the shape of the graph and the initial probabilities. Specifically, we consider infinite periodic trees, in which the degrees and initial probabilities of nodes on a path from the root node are periodic, with a given periodicity. Instead of the simple percolation threshold, we now obtain an entire region of possible probabilities for which all nodes in the tree become a.a.s. active. We show: (i) that the unit cube, as the support of the initial probabilities, can be partitioned into two regions, denoted by $W_0$ and $\overline{W}_0$, such that the tree becomes (does not become) a.a.s. fully active for any initial probability vector that belongs to $\overline{W}_0$ (resp. $W_0$); (ii) for every node in the tree, we provide the probability that the node becomes eventually active, for any initial probability vector that belongs to $W_0$; (iii) further, we specify the boundary of $W_0$ and show how it can be numerically computed.

S. Wagner—Supported by the National Research Foundation of South Africa under grant number 96236.