On percolation critical probabilities and unimodular random graphs

We investigate generalizations of the classical percolation critical probabilities p_c, pT and the critical probability pc defined by Duminil-Copin and Tassion [11] to bounded degree unimodular random graphs. We further examine Schramm’s conjecture in the case of unimodular random graphs: does p_c(G_n) converge to p_c(G) if G_n→ G in the local weak sense? Among our results are the following: • p_c=p_cholds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and pT < p_c; i.e., the classical sharpness of phase transition does not hold. • We give conditions which imply lim p_c(G_n) = p_c(limG_n). • There are sequences of unimodular graphs such that G_n→ G but p_c(G) > lim p_c(G_n) or p_c(G) < lim p_c(G_n) < 1. As a corollary to our positive results, we show that for any transitive graph with subexponential volume growth there is a sequence T_nof large girth bi-Lipschitz invariant subgraphs such that pc(T_n) → 1. It remains open whether this holds whenever the transitive graph has cost 1.