Fixing Ideas: Percolation on a Tree and Branching Random Walk

Abstract

These lecture notes focus on percolation in high dimensions, where space is so vast that faraway pieces of percolation clusters are close to being independent. Our main purpose of is to make this heuristic precise. One reflection of this heuristic is that critical percolation clusters in high dimensions have relatively few cycles. On a tree, there are no cycles, making different clusters precisely independent. The above heuristic thus suggests that percolation on the high-dimensional hypercubic lattice is close to percolation on a tree. In this chapter, we explicitly calculate various critical exponents on a tree. We discuss branching random walk, which we consider to be the proper mean-field model for percolation in high dimensions, as contrary to the tree, it does contain geometry.