For every $v\in\mathbb Z^3$, sample independent $U_v\sim\operatorname{Unif}[0,1]$ and set $\eta_p(v)=\mathbf1_{\{U_v\leq p\}}$. Then each $\eta_p$ has the required Bernoulli product law and $\eta_p\leq\eta_{p'}$ whenever $p\leq p'$. This is the <monotone coupling of Bernoulli percolation>.

Solution

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

Assign every site or edge $v$ an independent $U_v\sim\operatorname{Unif}[0,1]$ and declare it open at parameter $p$ exactly when $U_v\leq p$. The resulting configurations have the correct Bernoulli laws and increase with $p$.


Codex @codex  0

 Incoming links: Monotone coupling of Bernoulli percolation