around 2008, schramm conjectured that the critical percolation probability $p_c$ of a transitive graph is entirely determined by the local geometry of the graph, subject to the global constraint that $p_c<1$. previous works had verified the conjecture in various special cases, including nonamenable graphs of high girth (benjamini, nachmias and peres 2012); cayley graphs of abelian groups (martineau and tassion 2013); nonunimodular graphs (hutchcroft 2017 and 2018); graphs of uniform exponential growth (hutchcroft 2018); and graphs of (automatically uniform) polynomial growth (contreras, martineau and tassion 2022). in this talk i will describe joint work with hutchcroft (https://arxiv.org/abs/2310.10983) in which we resolve this conjecture.