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

**Johan Jonasson**

*(Chalmers University of Technology)*

#### Abstract

Two different problems are studied:

- 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 $P_c = \sup\{p_c(G): G \text{ transitive }, p_c(G)<1\}$. Is $P_c<1$?
- Let $G$ be transitive with $p_c(G)<1$, 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 $q<1$ such that $p > q$ 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 $P_d = \sup\{p_d(G): G \text{ transitive }, p_c(G) < 1\}$. Is the stronger statement $P_d < 1$ true? On the other hand: Is it always true that $p_d(G) > p_c (G)$?

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Pages: 1-13

Publication Date: June 15, 2001

DOI: 10.1214/EJP.v6-88

#### References

- I. Benjamini, R. Lyons, Y. Peres and
O. Schramm, Group-invariant percolation
on graphs,
*Geom. Funct. Analysis***9**(1999), 29-66. MR 99m:60149. - I. Benjamini and O. Schramm, Percolation
beyound
**Z**^d, many questions and a few answers,*Electr. Comm. Probab.***1**(1996), 71-82. MR 97j:60179. - J. van den Berg,
A uniqueness condition for Gibbs measures,
with application to the 2-dimensional Ising antiferromagnet,
*Commun. Math. Phys.*,**152**, 61-66. MR 94c:82040. - J. van den Berg and C. Maes,
Disagreement percolation in the
study of Markov fields,
*Ann. Probab.***22**(1994), 749-763. MR 95h:60154. - R. Grigorchuk and P. de la Harpe, Limit behaviour of exponential growth rates for finitely generated groups, Preprint 1999, link.
- O. Häggström, A note on disagreement percolation,
*Random Struct. Alg.***18**(2001), 267-278. MR 1824276. - O. Häggström, J. Jonasson and R. Lyons, Explicit isoperimetric constants and phase transitions in the random-cluster model, Preprint 2000, link.
- O. Häggström, Y. Peres and
J. STEIF, Dynamical percolation,
*Ann. Inst. H. Poincare, Probab. Statist.***33**(1997), 497-528. MR 98m:60153. - J. Jonasson, The random cluster model on a general graph and
a phase transition characterization of nonamenability,
*Stoch. Proc. Appl.***79**(1999), 335-354. MR 99k:60249. - H. Kesten, ``Percolation Theory for Mathematicians, '' Birkhäuser, Boston, 1982. MR 84i:60145.
- J. C. Wierman, Bond percolation on honeycomb and triangular
lattices,
*Adv. Appl. Probab.***13**(1981), 298-313. MR 82k:60216.

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