Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 204 / 1 / a / Solution

In bond percolation on an undirected graph $G=(V,E)$, a configuration is $\omega \in \{0,1{\}}^E$. The edge $e$ is open when ${\omega }_e=1$ and closed when ${\omega }_e=0$. Under ${\mathbb{P}}_p$, the coordinates are independent and identically distributed random variables with the Bernoulli distribution of parameter $p$. For a countable edge set, use the product sigma-algebra and the product measureThe open edges form a random subgraph with the original graph vertices. Its connected components of a graph are the percolation clusters. Write $x\leftrightarrow y$ when an open graph path joins $x$ to $y$; a zero-length graph path is allowed, so $x\leftrightarrow x$ always holds. The randomness is in the independently open bonds; all vertices remain present.