We investigate generalizations of the classical percolation critical probabilities pc, pT and the critical probability pc defined by Duminil-Copin and Tassion  to bounded degree unimodular random graphs. We further examine Schramm’s conjecture in the case of unimodular random graphs: does pc(Gn) converge to pc(G) if Gn→ G in the local weak sense? Among our results are the following: • pc=pcholds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and pT < pc; i.e., the classical sharpness of phase transition does not hold. • We give conditions which imply lim pc(Gn) = pc(limGn). • There are sequences of unimodular graphs such that Gn→ G but pc(G) > lim pc(Gn) or pc(G) < lim pc(Gn) < 1. As a corollary to our positive results, we show that for any transitive graph with subexponential volume growth there is a sequence Tnof large girth bi-Lipschitz invariant subgraphs such that pc(Tn) → 1. It remains open whether this holds whenever the transitive graph has cost 1.
Bibliographical noteFunding Information:
Our work was partially supported by the ERC Consolidator Grant 648017 (DB), the Hungarian National Research, Development and Innovation Office, NKFIH grant K109684 (GP and ÁT), an MTA Rényi Institute “Lendület” Research Group (GP), and an EU Marie Curie Fellowship (ÁT).
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- Critical probability
- Local weak convergence
- Unimodular random rooted graphs