Past exam of the mathematics course of the University of Cambridge / 2016 / iii / Paper 204 / 3 / c / Solution

Use the probability space $([0,1{]}^{{\mathbb{Z}}^d},\mathcal{F},\mathbb{P})$ with the product measure of the uniform distribution on $[0,1]$. Its coordinate random variables $U_v$ are independent and identically distributed random variables. Define simultaneously for every parameterFor $r\le s$, the indicator function inequality $X_r(v)\le X_s(v)$ holds pointwise at every graph vertex. For fixed $p$, each coordinate has the Bernoulli distribution with parameter $p$, and the coordinates are independent because they are functions of distinct independent $U_v$. Thus every marginal configuration is exactly the required site percolation model. This is a continuum-indexed family of processes, all on one probability space, and is the standard monotone coupling of Bernoulli percolation.