# Math Graduate Student Seminar

*,*Department of Mathematics

*,*Caltech

*,*

In bond percolation, we build a random subgraph of a graph G by indepedently choosing to retain each edge with probability p. We call these retained edges "open" and the rest "closed". For many examples of G, 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, it contains no giant components, whereas above the window, it contains at least one giant component. Now take G to be a large, finite, connected, transitive graph with bounded vertex degrees. We prove that above the window, there is exactly one giant component with high probability. This was conjectured to hold by Benjamini in 2001 but was previously only known for large torii and expanders, using methods specific to those cases.

The work that I will describe is joint with Tom Hutchcroft.