Percolation on finite transitive graphs

Tom Hutchcroft and I have been working to develop a general theory of percolation on arbitrary finite transitive graphs. This extends from percolation on local approximations to infinite graphs, such as a sequence of tori, to percolation on the complete graphs - the Erdős-Rényi model. I will summarise our progress on the basic questions: When is there a phase transition for the emergence of a giant cluster? When is the giant cluster unique? How does this relate to percolation on infinite graphs? I will then sketch our proof that for finite transitive graphs with uniformly bounded vertex degrees, the supercritical giant cluster is unique, verifying a conjecture of Benjamini from 2001.