Random trees and branching processes

In problems involving graphs and their properties, it is often helpful to first consider graphs without cycles, i.e., trees.

Percolation on regular branching trees

DEFINITION 10.1 (k-regular trees and branching trees) A k-regular tree (or Bethe lattice) is a tree where all vertices have degree k > 0. A (k − 1)-branching tree is obtained from a k-regular tree by declaring one of the vertices the root vertex and removing one of its edges.

Necessarily, k-regular trees contain an infinite number of vertices (and thus no leaf nodes). Two-regular trees, for example, are isomorphic to the graph with vertex set V = ℤ with edges between vertices i, j if i − j = 1. The root vertex in a (k − 1)-branching tree has degree k − 1. Also, if we delete an arbitrary edge in a k-regular tree, it splits into two (k − 1)-branching trees. For the purposes of branching and percolation, branching trees are the more natural objects to study than regular trees. We will denote the k-branching tree by k. The 3-regular tree and 2-branching tree are shown in Fig. 10.1.

Let us define a percolation model on a k-branching tree Tk = (V, E) by keeping each edge of the tree with probability p > 0 (we delete it otherwise), independently of all other edges. What is left is a random, possibly infinite, subgraph of k. Natural questions include the following.