We prove two results concerning percolation on general graphs.
- We establish the converse of the classical Peierls argument: if the critical parameter for (uniform) percolation satisfies p_c<1, then the number of minimal cutsets of size n separating a given vertex from infinity is bounded above exponentially in n. This resolves a conjecture of Babson and Benjamini from 1999.
- We prove that p_c<1 for every uniformly transient graph. This solves a problem raised by Duminil-Copin, Goswami, Raoufi, Severo and Yadin, and provides a new proof that p_c<1 for every transitive graph of superlinear growth.
