On percolation critical probabilities and unimodular random graphs

We investigate generalisations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p_c}$ defined by Duminil-Copin and Tassion (2015) 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 $G_n\to G$ in the local weak sense? Among our results are the following: 1. $p_c=\tilde{p_c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and $p_T < p_c$; i.e., the classical sharpness of phase transition does not hold. 2. We give conditions which imply $\lim p_c(G_n) = p_c(\lim G_n)$. 3. There are sequences of unimodular graphs such that $G_n\to G$ 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 sub-exponential volume growth there is a sequence $T_n$ of large girth bi-Lipschitz invariant subgraphs such that $p_c(T_n)\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.