Supercritical percolation on finite transitive graphs

In Bernoulli bond percolation, each edge of some graph are chosen to be either deleted or retained independently at random with retention probability p. For many large finite graphs, there is a phase transition such that if p is sufficiently large then there exists a giant cluster whose volume is proportional to that of the graph with high probability. We prove that in this phase the giant cluster must be unique with high probability: this was previously known only for tori and expander graphs via methods specific to those cases. The work that I will describe is joint with Philip Easo.

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