 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 $U_e\le p$. 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 $p\le q$, 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.

The three panels use one seeded collection of labels on a finite $17\times 17$ 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.