On percolation critical probabilities and unimodular random graphs

Dorottya Beringer; Gábor Pete; Ádám Timár

paper

On percolation critical probabilities and unimodular random graphs

arXiv:1609.07043

Abstract

We investigate generalisations of the classical percolation critical probabilities $p_{c}$ , $p_{T}$ and the critical probability $p_{c} ~$ 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. $p_{c} = p_{c} ~$ 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.

28 pages, 1 figure. The paper is reorganized, the theorem numbering is completely changed