Locality of the critical point for percolation

I will sketch why the critical point for percolation on an infinite transitive graph G only depends on the geometry of G on small scales (except in the degenerate case when G is one-dimensional). This is based on joint work with Hutchcroft and was predicted by Schramm around 2008. Our techniques are inspired by a lot of previous progress on the problem, most significantly by recent work of Contreras, Martineau, and Tassion, who handled the case of graphs with (uniform) polynomial growth in 2022. Our proof uses random walks and the geometry of nilpotent groups in the service of percolation arguments.