In Bernoulli bond percolation, one defines a random subgraph of a given connected graph G by deleting or retaining edges of G independently at random, each edge being retained with the same probability p. When G is infinite, a central and classical question is whether there exists a choice of p strictly less than 1 such that this random subgraph has infinite connected components. When G is finite, a natural analogue is to ask whether there exists a choice of p bounded away from 1 such that the random subgraph contains a connected component containing at least half (or 1% or 99%) of the vertices of G. Tom Hutchcroft and I recently showed that such a p exists provided G is not close to a cycle in some sense, confirming most cases of a conjecture of Benjamini (2001). I will give an overview of the proof, and go into some detail on certain aspects of it that have a distinctly additive combinatorial flavour.
This video is part of the Webinar in Additive Combinatorics series, and this is
their YouTube channel.
