Open Access
2017 On percolation critical probabilities and unimodular random graphs
Dorottya Beringer, Gábor Pete, Ádám Timár
Electron. J. Probab. 22: 1-26 (2017). DOI: 10.1214/17-EJP124


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 ${p_T}<{p_c}$; i.e., the classical sharpness of phase transition does not hold.

  • We give conditions which imply $\lim{p_c} (G_n)= {p_c}(\lim G_n)$.

  • 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 $\mathcal{T} _n$ 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.


Download Citation

Dorottya Beringer. Gábor Pete. Ádám Timár. "On percolation critical probabilities and unimodular random graphs." Electron. J. Probab. 22 1 - 26, 2017.


Received: 26 September 2016; Accepted: 6 November 2017; Published: 2017
First available in Project Euclid: 28 December 2017

zbMATH: 06827083
MathSciNet: MR3742403
Digital Object Identifier: 10.1214/17-EJP124

Primary: 05C80 , 60B99 , 60K35 , 82B43

Keywords: critical probability , Local weak convergence , percolation , unimodular random rooted graphs

Vol.22 • 2017
Back to Top