Electronic Journal of Probability

On Disagreement Percolation and Maximality of the Critical Value for iid Percolation

Johan Jonasson

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Two different problems are studied:

  • 1. For an infinite locally finite connected graph $G$, let $p_c(G)$ be the critical value for the existence of an infinite cluster in iid bond percolation on $G$ and let $P_c = \sup\{p_c(G): G \text{ transitive }, p_c(G) \lt 1\}$. Is $P_c \lt 1$?
  • 2. Let $G$ be transitive with $p_c(G) \lt 1$, take $p \in [0,1]$ and let $X$ and $Y$ be iid bond percolations on $G$ with retention parameters $(1+p)/2$ and $(1-p)/2$ respectively. Is there a $q \lt 1$ such that $p \gt q$ implies that for any monotone coupling $(X',Y')$ of $X$ and $Y$ the edges for which $X'$ and $Y'$ disagree form infinite connected component(s) with positive probability? Let $p_d(G)$ be the infimum of such $q$'s (including $q=1$) and let $P_d = \sup\{p_d(G): G \text{ transitive }, p_c(G) \lt 1\}$. Is the stronger statement $P_d \lt 1$ true? On the other hand: Is it always true that $p_d(G) \gt p_c (G)$?

It is shown that if one restricts attention to biregular planar graphs then these two problems can be treated in a similar way and all the above questions are positively answered. We also give examples to show that if one drops the assumption of transitivity, then the answer to the above two questions is no. Furthermore it is shown that for any bounded-degree bipartite graph $G$ with $p_c(G) \lt 1$ one has $p_c(G) \lt p_d(G)$. Problem (2) arises naturally from [6] where an example is given of a coupling of the distinct plus- and minus measures for the Ising model on a quasi-transitive graph at super-critical inverse temperature. We give an example of such a coupling on the $r$-regular tree, ${\bf T}_r$, for $r \gt 1$.

Article information

Electron. J. Probab., Volume 6 (2001), paper no. 15, 13 pp.

Accepted: 15 June 2001
First available in Project Euclid: 19 April 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82B26: Phase transitions (general) 82B43: Percolation [See also 60K35]

coupling Ising model random-cluster model transitive graph planar graph

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Jonasson, Johan. On Disagreement Percolation and Maximality of the Critical Value for iid Percolation. Electron. J. Probab. 6 (2001), paper no. 15, 13 pp. doi:10.1214/EJP.v6-88. https://projecteuclid.org/euclid.ejp/1461097645

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