Percolation probability ($\theta (p)$)

For independent bond percolation on an infinite graph with a specified root $o$, let $C_o$ be its percolation cluster. The percolation probability isIt concerns the root, rather than the existence of an infinite percolation cluster somewhere else. Even above the percolation critical probability of the square lattice, an isolated root has probability $(1-p{)}^4$, because its four incident edges can all be closed. Thus $\theta (p)\le 1-(1-p{)}^4<1$ when $p<1$. Positive percolation probability does not mean every vertex is connected to infinity.

On the square lattice, the events that $o$ is joined by an open path to the boundary of $[-n,n{]}^2$ decrease as $n$ increases. Their probabilities converge to $\theta (p)$ by continuity from above of a measure. This relates an infinite question to finite observations, without equating any one finite boundary connection with survival forever. The monotone coupling of Bernoulli percolation shows that $\theta$ is a monotone function of $p$.

If an infinite percolation cluster exists almost surely, the probability that a large square box intersects one tends to one. The four rotationally symmetric exterior-side arm events are increasing. By the Harris-FKG inequality, their common probability is at least $1-(1-P(D_N){)}^{1/4}$, where $D_N$ is their union. Thus a pair of specified opposite sides both has an infinite exterior arm with probability at least $1-2(1-P(D_N){)}^{1/4}$, tending to one. The rays start at a boundary graph vertex and immediately leave the graph vertex box.

Suppose a planar bond percolation model has critical one-arm probability at most $C{n}^{-\alpha }$ with $\alpha >0$, and a finite-radius connection event depends on $O(n^2)$ edges. The finite-box comparison for percolation parameters gives $\theta (p_c+\epsilon )\le C{n}^{-\alpha }+C^{\prime }\epsilon n^2$. Balancing at $n\asymp {\epsilon }^{-1/(\alpha +2)}$ gives $\theta (p_c+\epsilon )\le A{\epsilon }^{\alpha /(\alpha +2)}$. For the square lattice use $p_c=1/2$ and the polynomial critical one-arm upper bound.

For independent site percolation or bond percolation on a rooted locally finite graph, $\theta (p)$ is right continuous on $[0,1)$. Finite-radius connection events have polynomial probabilities in $p$ and decrease to infinite connection. Thus $\theta$ is their infimum by continuity from above of a measure. This infimum of continuous functions is upper semicontinuous; parameter monotonicity from the monotone coupling of Bernoulli percolation then gives right continuity.

Independent site percolation and bond percolation on the cubic lattice have continuous percolation probability on $(p_c,1]$. Fix $p>p_c$ and an intermediate $q\in (p_c,p)$. By uniqueness of the infinite percolation cluster, every origin in the infinite $p$-cluster has a finite $p$-open graph path to the infinite $q$-cluster. In the uniform-label monotone coupling of Bernoulli percolation, the finitely many labels on this graph path are strictly below fixed $p$ almost surely, so it persists at some parameter below $p$. The coupled onset parameter for origin percolation has no attained atom there, giving left continuity, including at $1$. Combine this with right continuity of percolation 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.

Independent bond percolation on the cubic lattice has exponential connection decay at every $p<p_c$ and positive percolation probability at every $p>p_c$. The finite-set criterion for percolation sharpness gives decay below its auxiliary threshold. The Margulis–Russo formula and exploration from a finite box boundary give $g_n^{\prime }(p)\ge (1-g_n(p)){\inf }_{S\ni 0}{\phi }_p(S)/(p(1-p))$, with the infimum over interior finite sets. Above the auxiliary threshold the infimum is at least one; integrating gives positive percolation probability and identifies the two thresholds. This theorem does not identify the value of $p_c$ for a particular lattice.

For independent bond percolation on a transitive locally finite lattice, let $c_n$ count rooted self-avoiding walks and let $\mu =\lim c_n^{1/n}$ be the connective constant. An infinite open percolation cluster supplies an open walk of every length. The union bound gives $\theta (p)\le c_n{p}^n$, which tends to zero when $p\mu <1$. Thus $p_c\ge 1/\mu$.

The percolation critical probability of independent bond percolation on the infinite square lattice is exactlyIn this model, $\theta (p)=0$ for $p\le 1/2$ 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: ^[ref]. These are claims about the infinite square lattice, rather than deductions from a simulated finite sample.

Uniform fixed-aspect rectangle crossing in bond percolation bounds at the self-dual parameter yield a positive lower bound for closed dual circuits in geometrically spaced disjoint annuli. The independent annular barriers for percolation imply that an origin-to-infinity open graph path has probability zero. Parameter monotonicity gives zero percolation probability for every $p\le 1/2$.

At parameter $1/2$, primal and dual bond percolation on the square lattice have the same probability distribution. Suppose an infinite percolation cluster exists. Large boxes meet it with probability tending to one. The square-root trick for positively associated events and quarter-turn symmetry imply that each side has an infinite exterior open arm with probability tending to one. The same holds for dual arms. A union bound then gives positive probability of primal top and bottom arms together with dual left and right arms.

Use a primal box $[-r,r{]}^2$ and a dual box with sides at $\pm (r+1/2)$. Open only primal edges whose endpoints both lie in the primal box, preserving all outward and exterior arm edges. The two primal arms are joined inside. Planarity prevents the two alternating dual arms from joining: a proposed dual connection together with the inner boundary arc separates one primal infinite arm from infinity. Thus the modified configuration has two infinite dual clusters. Its positive probability by finite modification of Bernoulli percolation contradicts uniqueness of the infinite percolation cluster. Consequently the percolation probability at $1/2$ is zero. This is often called Zhang's argument.

For independent bond percolation on a periodic plane lattice with three-fold rotation symmetry, the primal model at $p$ and the dual bond percolation model at $1-p$ cannot both percolate. To adapt Zhang's argument, divide a large boundary circle into six arcs, repeating the division in each $120$-degree sector. Vary the division point until the failure probabilities $a,b$ of the two alternating primal arc types are comparable within a factor $(1-p{)}^{-2}$. Moving past a single crossing edge changes a failure probability by at most a factor $(1-p{)}^{-1}$, by the FKG inequality. If the probability of no primal exterior arm is at most $\epsilon$, then $a^3{b}^3\le \epsilon$ and both failures are at most $(1-p{)}^{-1}{\epsilon }^{1/6}$. For the dual model, at least one arc type has failure at most ${\epsilon }^{1/6}$; its three rotations have the same bound. Four alternating primal/dual arms now occur with positive probability, contradicting the alternating exterior-arm separation lemma and uniqueness after a finite modification. This extends the four-fold argument to the martini lattice.

Four infinite exterior arms entering a square in alternating primal-open and dual-open order force the two primal arms to be disjoint outside the square. If they met, their finite connecting crosscut together with a boundary arc would enclose one dual arm, preventing its escape to infinity. Connecting the two primal arms through the square creates a proper doubly infinite open path, separating the two remaining dual infinite tails. finite-energy property of Bernoulli percolation can then turn this configuration into two infinite dual clusters, contradicting uniqueness. Care is needed to preserve the exterior dual tails even if the finite modification destroys their original boundary attachments.