Locality of percolation critical probabilities: uniformly nonamenable case

Let {Gn}∞n=1 be a sequence of transitive infinite connected graphs with sup n≥1 pc(Gn) < 1, where each pc(Gn) is bond percolation critical probability on Gn. Schramm (2008) conjectured that if Gn converges locally to a transitive infinite connected graph G, then pc(Gn) → pc(G) as n→∞. We prove the conjecture when G satisfies two rough uniformities, and {Gn}∞n=1 is uniformly nonamenable.