Bond percolation

In bond percolation with parameter $p$, every edge of a graph is independently open with probability $p$ and closed with probability $1-p$.

The percolation susceptibility is the expected size of the open cluster containing a specified root:

At critical planar percolation, the probability of an open crossing of a rectangle of any fixed aspect ratio stays bounded away from zero and one uniformly over its scale.

The one-arm probability $\pi (r)$ is the probability that a specified vertex has an open path to distance $r$.

The Russo-Seymour-Welsh theorem and the Harris-FKG inequality imply $\pi (2r)\ge c\pi (r)$ for a scale-independent constant $c>0$.

RSW rectangle crossings and positive association join two separated one-arm events with probability bounded below uniformly over scale.

For increasing events $A,B$ in a product percolation measure, their disjoint occurrence satisfies $\mathbb{P}(A\square B)\le \mathbb{P}(A)\mathbb{P}(B)$.