= Monotone coupling of Bernoulli percolation
Assign each <edge> $e$ an <independent> <uniform distribution> label $U_e$ on $[0,1]$. At parameter $p$, declare $e$ open exactly when $U_e\leq p$. Each open/closed decision then has the required <Bernoulli distribution>, and different <edge> decisions are <independent>. The same construction applies to <site percolation> with labels on <vertices>.
Keep those labels fixed when increasing $p$. For $p\leq q$, every $p$-open <edge> is also $q$-open, so every $p$-open <graph path> survives and each <percolation cluster> can only gain <vertices>. This <coupling of probability distributions> compares complete configurations on one <probability space>, rather than comparing two unrelated pictures. Any <increasing event>, such as an <open path> between specified <vertices>, consequently has a nondecreasing <probability>.
\Image[/percolation-monotone-grid.png]
{title=One set of random edge labels gives nested root clusters as the opening probability increases}
The three panels use one seeded collection of labels on a finite $17\times17$ <square lattice> box. Colored <edges> and dots show the central root's <percolation cluster>; gray <edges> are open but disconnected from it. A <percolation cluster> can grow by many <vertices> when one newly opened <edge> joins it to another large <percolation cluster>. The panels illustrate a nested construction, not an estimate of $\theta(p)$ or a proof of the <Harris-Kesten theorem>.
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