Electronic Journal of Probability

On percolation critical probabilities and unimodular random graphs

Dorottya Beringer, Gábor Pete, and Ádám Timár

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 106, 26 pp.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 60B99: None of the above, but in this section 05C80: Random graphs [See also 60B20]

percolation critical probability local weak convergence unimodular random rooted graphs

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Beringer, Dorottya; Pete, Gábor; Timár, Ádám. On percolation critical probabilities and unimodular random graphs. Electron. J. Probab. 22 (2017), paper no. 106, 26 pp. doi:10.1214/17-EJP124. https://projecteuclid.org/euclid.ejp/1514430042

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