Abstract: Consider a large, finite, connected, transitive graph. In bond percolation, each edge is independently set to “open” with probability 
. As we increase the parameter 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.
In the first half of the talk, I will prove that 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. In the second half of the talk, I will prove that the volume of this unique giant component is concentrated. This proof also yields new results for bond percolation on infinite, transitive graphs.
The work that I will describe is joint with Tom Hutchcroft.