Consider a large, finite graph. In bond percolation, each edge is independently set to “open” with probability p. In many cases, when we increase the parameter p across a narrow critical window, the subgraph of open edges undergoes a phase transition. With high probability, below the window, there are no giant components, whereas above the window, there is at least one giant component. We prove that for transitive graphs above the window, there is exactly one giant component, with high probability. This was conjectured to hold by Benjamini, but was only known for large tori and expanders, using methods specific to those cases.
The work that I will describe is joint with Tom Hutchcroft.