= Harris-Kesten theorem
{c}
{title2=$p_c(\mathbb Z^2)=1/2$}
The <percolation critical probability> of independent <bond percolation> on the infinite <square lattice> is exactly
$$
\boxed{p_c(\mathbb Z^2)=\frac12.}
$$
In this model, $\theta(p)=0$ for $p\leq1/2$ and $\theta(p)>0$ for $p>1/2$. The particular <graph>, independent <edge> decisions, and distinction between <bond percolation> and <site percolation> matter: this value is not a universal threshold for arbitrary random networks. The equality of open and closed <edge> <probabilities> at $1/2$ suggests a relation to <dual bond percolation>, but that symmetry alone is not a proof of long-distance connectivity.
Harris established the lower bound, including absence of an <infinite percolation cluster> at $p=1/2$, in 1960. Kesten completed the equality in 1980. His original paper states the threshold result as Theorem 1 and the positive/zero <percolation probability> alternatives as Theorem 2: https://math.bme.hu/~balint/oktatas/perkolacio/percolation_papers/kesten_1.pdf[Harry Kesten, The critical probability of bond percolation on the square lattice equals 1/2]{ref}. These are claims about the infinite <square lattice>, rather than deductions from a simulated finite sample.
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