= Percolation critical probability
{title2=$p_c$}
= Critical probability
{synonym}
For the rooted infinite model defining <percolation probability>, the percolation critical probability is
$$
p_c=\inf\{p:\theta(p)>0\}.
$$
Below $p_c$, 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.
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