 Percolation critical probability

For the rooted infinite model defining percolation probability, the percolation critical probability isBelow $p_c$, the root has probability zero of belonging to an infinite percolation cluster. Above $p_c$, that probability is positive, by the definition and the monotone coupling of Bernoulli percolation. The definition alone does not determine behavior at $p_c$, or the numerical value of $p_c$: these require results specific to the graph and the model.

A finite graph with $M$ edges has $2^M$ open/closed configurations. The probability of any connectivity event is a finite sum of terms $p^k(1-p{)}^{M-k}$, hence a polynomial in $p$. The square's $2p^2-p^4$ is an example. These finite probabilities vary smoothly, although an infinite-system limit can distinguish a whole interval with $\theta (p)=0$ from one with $\theta (p)>0$. A steep finite connection curve and a mathematically proved percolation critical probability are different kinds of information.