Abstract
We investigate generalizations of the classical percolation critical probabilities $p_c$, $p_T$ and the critical probability $\tilde{p} _c$ 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 $G_n\to G$ in the local weak sense? Among our results are the following: ${p_c}={\tilde{p} _c}$ holds for bounded degree unimodular graphs. However, there are unimodular graphs with sub-exponential volume growth and ${p_T} \lim{p_c} (G_n)$ 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 $\mathcal{T} _n$ of large girth bi-Lipschitz invariant subgraphs such that ${p_c}(\mathcal{T} _n)\to 1$. It remains open whether this holds whenever the transitive graph has cost 1.
Highlights
1.1 Motivation and resultsThere are several definitions of the critical probability for percolation on the lattices Zd, which have turned out to be equivalent on Zd, and in the more general context of arbitrary transitive graphs [28, 1, 16, 4, 11, 12]
One of our goals is to investigate the relationship between these different definitions when the graph G is an ergodic unimodular random graph [9, 2], which is the natural extension of transitivity to the disordered setting
We examine the generalisations of pc = sup{p : Pp(there is an infinite cluster) = 0}, pT = sup {p : Ep(|Co|) < ∞} and pc, defined by Duminil-Copin and Tassion [11]
Summary
There are several definitions of the critical probability for percolation on the lattices Zd, which have turned out to be equivalent on Zd, and in the more general context of arbitrary transitive graphs [28, 1, 16, 4, 11, 12]. We discuss this family of graphs in more detail in Example 3.2 This example motivates our investigations on the locality of the critical probability in the class of unimodular random graphs and it shows that it is natural to restrict one’s attention to bounded degree graphs. Let Gn be uniformly bounded degree unimodular random graphs converging to G in the local weak sense, in a uniformly sparse way: there is a positive integer k such that for each n there is a coupling νn of μG and μGn such that G ⊆ Gn and there is a sequence of positive integers rn → ∞ that satisfies |(E(Gn) \ E(G)) ∩ BGn(o, rn)| ≤ k νn-almost surely. If G is a bounded degree unimodular random tree with uniformly subexponential volume growth, all five critical percolation densities equal 1, and G is uniformly good. A key conclusion of our work seems to lie in the counterexamples: there appears to be no perfect definition of a “critical density” that would make locality a robust phenomenon, true for a large class of unimodular random graphs and possibly more accessible for a proof in the transitive case
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