Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 204 / 2 / a / Solution

In site percolation on the three-dimensional cubic lattice, each vertex $v\in {\mathbb{Z}}^3$ is independently open with probability $p$ and closed with probability $1-p$. Thus ${\mathbb{P}}_p$ is the Bernoulli product measure on $\{0,1{\}}^{{\mathbb{Z}}^3}$, and open paths are nearest-neighbour paths all of whose vertices are open.