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

Abstract

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 $P_c = \sup\{p_c(G): G \text{ transitive }, p_c(G)<1\}$. Is $P_c<1$?
2. 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 $p_d(G) > p_c (G)$?

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 $p_c(G) < 1$ one has $p_c(G) < p_d(G)$. 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 $r > 1$.