The Burton-Keane theorem establishes the uniqueness of the infinite cluster in certain percolation models, and is presented in page 16-17 in this set of lecture notes by Hugo Duminil-Copin.
The first part of the proof shows that a finite number of infinite clusters a.s. implies only one infinite cluster. It uses the fact that we can choose a box $\Lambda_n$ with $n$ large enough such that $\phi[\mathcal{F}] \geq \frac{1}{2}\phi [\mathcal{E_{<\infty}}] > 0$, where $\mathcal{E_{<\infty}}$ is the event that there is a finite number of infinite clusters, and $\mathcal{F}$ is the event that all these clusters intersect the box*.
My questions are:
- Is the number $1/2$ just an arbitrary constant, or is there a particular reason this number is used?
- Why is it true that we can choose a box large enough for $\phi[\mathcal{F}] \geq \frac{1}{2}\phi [\mathcal{E_{<\infty}}]$ a priori? Does it have to do with the translation-invariance of the measure and positive association? What is the minimum condition that needs to hold for the underlying graph / measure if we only care about this inequality (e.g. would this also be true for random-cluster models with $q<1$).