Bond percolation

In independent bond percolation with parameter $p$, every edge of a graph is open with probability $p$ and closed with probability $1-p$. The choices are independent random variables with a Bernoulli distribution. All vertices remain present; only the usable connections change. The percolation clusters are the connected components of a graph formed by the open edges, including isolated vertices.

A specified simple graph path with $m$ edges is an open path with probability $p^m$, by independence. Connecting its endpoints is a different event: other graph paths may also connect them. The four-edge example in percolation theory makes that distinction explicit. Interpreting open edges as usable channels makes the model useful for connectivity questions; it does not by itself assign a travel time, flow rate, or capacity to a channel.

For the induced square lattice subgraph in $[0,m]\times [0,n]$, $h_p(m,n)$ is the probability of an open graph path from its left boundary to its right boundary under independent bond percolation of density $p$. Width and height count lattice spacings. Dual bond percolation exchanges failure with a top-to-bottom dual crossing of dimensions $(m-1)\times (n+1)$, with the usual exterior boundary vertices.

Take the increasing event that a finite square torus graph contains some long horizontal or vertical rectangle crossing. Translations and a quarter turn make this a transitive increasing event on its bonds. A positive critical crossing bound and the Friedgut-Kalai sharp threshold theorem make its supercritical probability tend to one. A bounded family of shorter, thicker rectangles intercepts every possible long crossing. The square-root trick for positively associated events transfers the high probability to each fixed intercepting rectangle. Using every possible translate would lose the useful bounded exponent.

A path in the underlying graph all of whose edges are open. Connectivity of open paths determines the percolation clusters. A ray in a graph with all edges open is an infinite ray in a graph.

If independent edges of a locally finite graph receive fair orientations, exploring outward from a root tests each fresh reached-to-unreached boundary edge with success probability $1/2$. Deferred decisions give exactly the exploration law of the root cluster in bond percolation of density $1/2$. Thus the two reachable graph vertex sets have the same distribution. An infinite out-cluster contains an infinite simple directed ray by the König infinity lemma. Geometrically biased orientations do not generally give the same uniform edge-success law.

Here $\tau (x,y)=P_{p_c}(x\leftrightarrow y)$ is critical two-point connectivity. The condition controls the intersection effects that spoil a branching approximation. It is a sufficient criterion for important mean-field percolation exponents, including the order-parameter, percolation susceptibility and cluster-tail exponents. Establishing detailed spatial asymptotics is a further task, commonly addressed by lace expansion.

The two-point percolation two-point connection probability ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ is the probability that two fixed graph vertices belong to the same percolation cluster. On a translation-invariant lattice it depends on their displacement. Restricting the permitted connecting graph paths to a subgraph gives the corresponding confined percolation two-point connection probability. The Harris-FKG inequality implies ${\tau }_p(x,z)\ge {\tau }_p(x,y){\tau }_p(y,z)$.

For bond percolation on the square lattice, write $h_n={\mathbb{P}}_p(0\leftrightarrow (n,0))$ and $g_k={\mathbb{P}}_p(0\leftrightarrow \partial [-k,k{]}^2)$. A maximal-probability boundary graph vertex can be placed at $(k,j)$ by symmetry. Reflection about $x=k$ identifies its connection probability from $0$ with that from $(2k,0)$. The Harris-FKG inequality yields $(g_k/(8k){)}^2\le h_{2k}\le g_{2k}$ for $k\ge 1$.

For nearest-neighbour bond percolation with $0<p<1$ on the strip $T_k=\mathbb{Z}\times \{-k,\dots ,k\}$, put $q_k(n)={\mathbb{P}}_p((0,0)\leftrightarrow (n,0)\text{ in }{T}_k)$. The Harris-FKG inequality and translation invariance imply $q_k(n+m)\ge q_k(n)q_k(m)$. Thus $-\log q_k(n)$ is a subadditive sequence, and the Fekete lemma givesThe direct horizontal graph path gives $0\le f_k(p)\le -\log p$. Enlarging the strip increases every percolation two-point connection probability, so $f_k(p)$ is nonincreasing in $k$ and has a nonnegative limit of a sequence.

Let $q(n)$ be the unrestricted planar two-point percolation two-point connection probability. Every finite connecting graph path is contained in some strip, so $q_k(n)\uparrow q(n)$ for each fixed $n$. ThereforeThe equality uses commuting infima and monotonicity, not an unjustified interchange of two general limits. The right-hand side is the whole-plane connection decay rate by the Fekete lemma.

A vertical cut between consecutive columns of $T_k$ contains $2k+1$ edges. All are closed with probability $(1-p{)}^{2k+1}$. Cuts in different columns use disjoint edges, so their closed-cut events are independent. A connection from column zero to column $n$ must cross each of the intervening $n$ cuts. ConsequentlyThis finite-width obstruction persists even when the unrestricted planar percolation is supercritical.

The independent edge law of bond percolation on ${\mathbb{Z}}^d$ is ergodic under lattice translations: every translation-invariant event has probability zero or one.

In bond percolation, a percolation cluster is a connected component of a graph formed by the open edges, with every graph vertex retained. In site percolation, it is a connected component of a graph of the subgraph induced by the open graph vertices. Write $C(x)$ for the cluster containing $x$; for site percolation set $C(x)=\varnothing$ when $x$ is closed.

Suppose the one-arm probability on the cubic lattice satisfies ${\beta }_r=\exp (-\lambda r+o(r))$ with $\lambda >0$. Define $M_n$ as the largest maximum-norm radius of any cluster rooted in $[-n,n{]}^d$. Then $M_n/\log n\to d/\lambda$ convergence in probability. An upper bound is a union bound over $O(n^d)$ roots. For a lower bound, pack $\Theta ((n/\log n{)}^d)$ disjoint radius-$O(\log n)$ boxes and use the independent first-hit connection events inside them. Clusters are finite almost surely under the positive-rate hypothesis.

An infinite percolation cluster is an open connected component of a graph with infinitely many graph vertices. On a countable translation-invariant lattice, a zero root percolation probability implies that no infinite percolation cluster exists almost surely, by the countable union over all graph vertices. A positive root probability implies almost-sure existence by translation ergodicity of Bernoulli percolation; uniqueness is an additional theorem.

For independent site percolation or bond percolation on the cubic lattice ${\mathbb{Z}}^d$, there is almost surely at most one infinite percolation cluster at every fixed parameter. Above the percolation critical probability, there is exactly one almost surely. This theorem is called the Burton-Keane theorem. Existence uses translation ergodicity together with positive percolation probability; uniqueness is a separate conclusion. At $p=1$ uniqueness is deterministic.

For a connected infinite locally finite graph that is also an amenable graph and a quasi-transitive graph, an invariant independent site percolation law with every orbit parameter strictly between $0$ and $1$ has either no infinite percolation cluster or exactly one, almost surely. Invariance and independence give a zero–one law. The finite-energy property of Bernoulli percolation excludes a fixed finite number greater than one by opening a finite joining region. Infinitely many clusters would create an encounter box in percolation with positive probability; translated disjoint boxes then have positive density, contradicting the boundary count in a Følner sequence. The finite-energy hypothesis matters if some orbit parameters are $0$ or $1$.

Independent bond percolation on ${\mathbb{Z}}^2$ has at most one infinite percolation cluster at every parameter, including a possible critical parameter. The number of infinite percolation clusters is constant almost surely by translation ergodicity of Bernoulli percolation. A finite constant larger than one is impossible: a box meeting two clusters can be made entirely open, joining them and decreasing that number with positive probability by finite modification of Bernoulli percolation.

If infinitely many clusters existed, a box would meet three with positive probability. Preserve one infinite exterior arm from each and replace the finitely many interior and boundary edges by a three-armed tree joining them, closing the remaining edges. Its branch graph vertex becomes a trifurcation vertex in percolation. Translation invariance therefore gives a positive density $q$. But the trifurcation boundary-counting lemma gives $q|W|\le |{\partial }^{+}W|$ for every large box, contradicting vanishing boundary-to-volume ratio. This proves uniqueness without assuming absence of an infinite percolation cluster at criticality.

An encounter box is a finite connected set of open vertices whose removal from its infinite percolation cluster leaves at least three infinite connected components. Unlike a trifurcation vertex in percolation, it allows branching to occur across several vertices, which is useful for site percolation on graphs with local cycles. Contract disjoint encounter boxes to junctions. In a finite region, each junction must connect at least three different exits. Pruning to a forest gives at most as many junctions as boundary exits, by the identity ${\sum }_v(\mathrm{deg}(v)-2)=-2$ for a finite tree. Positive density of encounter boxes is therefore incompatible with an amenable graph.

A trifurcation graph vertex is a graph vertex of an infinite percolation cluster whose deletion separates that cluster into at least three infinite connected components of a graph. In a finite modification argument one may make it have exactly three open incident edges leading to three different infinite components. On a translation-invariant lattice, a positive probability of trifurcation at one graph vertex gives a positive expected density of trifurcations.

For a finite graph vertex set $W$ in a locally finite graph, let ${\partial }^{+}W$ be its exterior graph neighbours. The number of trifurcation vertices in percolation lying in $W$ is at most $|{\partial }^{+}W|$.

For each cluster meeting $W$, contract its connected components of a graph outside $W$ to terminal graph vertices, keeping only those adjacent to $W$. The resulting incidence graph is finite and connected. Each trifurcation graph vertex in $W$ separates at least three groups of terminals, because every infinite branch must leave the finite set $W$. Take a minimal subtree connecting all terminals. Each such trifurcation is unavoidable in that subtree and has degree of a vertex at least three. All leaves are terminals. The tree identity ${\sum }_v(\mathrm{deg}(v)-2)=-2$ bounds the number of these branching graph vertices by the number of leaves minus two, hence by the number of terminals. Different terminals, even across different clusters, can be assigned different exterior graph neighbours. Summing proves the bound. For square boxes in ${\mathbb{Z}}^2$, the boundary has order $n$ and the volume has order $n^2$.

Delete a finite set of degree-three trifurcation vertices in percolation in each infinite percolation cluster and contract the remaining components to nodes. The incidence graph is a tree: a cycle would bypass an original trifurcation vertex in percolation. All leaf components are infinite, otherwise a graph vertex would have a finite branch after deletion. The tree degree identity gives at least two more infinite leaves than degree-three graph vertices. Distinct infinite components escape a containing box through distinct outer boundary graph vertices. Hence the number of counted trifurcations is at most the outer boundary size.

The percolation susceptibility is the expected size of the open cluster containing a specified root:

For translation-invariant independent bond percolation with finite percolation susceptibility $\chi ={\sum }_y\mathbb{P}(0\leftrightarrow y)$, the BK inequality gives $\mathbb{E}|C_0{|}^k\le (2k-3)!!{\chi }^{2k-1}$ for $k\ge 1$, with $(-1)!!=1$. Prune a connecting open tree for the $k+1$ labelled vertices, suppress unmarked degree-two vertices, and split higher branching into binary branching with zero-length connections. There are $(2k-3)!!$ binary tree topologies and $2k-1$ connections. The connections have disjoint bond witnesses, so the BK inequality bounds their joint probability by a product of two-point connection probabilities. Summing successive leaves contributes $\chi$ for each connection. Since $(2k-3)!!\le 2^{k-1}(k-1)!$, this proves $\mathbb{E}{e}^{t|C_0|}\le 1-(2\chi {)}^{-1}\log (1-2t{\chi }^2)$ for $0<t<(2{\chi }^2{)}^{-1}$. Finite percolation susceptibility therefore gives an exponential cluster-volume tail.

The percolation susceptibility is $\chi (p)=1+{\sum }_{m\ge 1}{\sum }_{y\in \partial {\Lambda }_m}{\tau }_p(0,y)$. Finiteness gives a boundary sum $a<1$. The weighted BK boundary-splitting estimate then gives $g_k\le a^{\lfloor k/m\rfloor }$. The strict bound $g_1=1-(1-p{)}^{2d}<1$ absorbs the finitely many smaller radii into a positive exponential rate with unit prefactor. At $p=0$, all positive-radius connection probabilities are zero.

At critical planar percolation, the probability of an open crossing of a rectangle of any fixed aspect ratio stays bounded away from zero and one uniformly over its scale.

For independent bond percolation on the square lattice, a positive uniform lower bound for square crossings implies a positive uniform lower bound for rectangle crossings of every fixed aspect ratio $\rho >1$. The RSW reflection extension lemma first extends a square to aspect ratio $3/2$. Overlapping rectangles and transverse overlap-square crossings then glue using the Harris-FKG inequality. The required constant depends on $\delta$ and $\rho$, not on scale. At a self-dual parameter, dual crossings also give a uniform upper bound below one.

For independent bond percolation on the square lattice at $p=1/2$, square crossing probabilities bounded below by self-duality extend to a uniform positive bound for rectangles of aspect ratio $3/2$. Expose an extremal vertical square crossing, reflect its geometry, and attach it to an exterior side through an unexposed independent region. Reflection symmetry bounds the attachment probability below. Two attachments and a horizontal square crossing join by the Harris-FKG inequality. Overlapping rectangles and transverse overlap-square crossings then give all fixed aspect ratios. Reflecting a geometrical graph path does not require its reflected edges to be open.

In a rectangle $[0,m]\times [0,2n]$, $m\ge n$, let $X$ require an open vertical crossing of the corner square $[0,n{]}^2$ attached to the rectangle's right side. Explore the leftmost vertical crossing of the square without inspecting edges to its right. Reflect its geometry in the line $y=n$, obtaining a deterministic separator. Every horizontal rectangle crossing reaches this separator from the right; reflection divides these possible attachments into two equally probable choices, so the lower-half attachment has probability at least half the horizontal crossing probability. The required edges remain unexposed and independent of the explored crossing. Averaging proves the estimate. Reflected geometry need not be open.

The one-arm probability $\pi (r)$ is the probability that a specified vertex has an open path to distance $r$.

For independent bond percolation on the cubic lattice, define ${\tau }_p(x,y)={\mathbb{P}}_p(x\leftrightarrow y)$ and $g_n={\mathbb{P}}_p(0\leftrightarrow \partial {\Lambda }_n)$, with $g_0=1$. Split an open self-avoiding walk at its first radius-$m$ boundary vertex $y$. The remaining segment reaches distance at least $n-m$ from $y$, using disjoint edges. The BK inequality, translation invariance and the union bound yield the displayed estimate. Keeping the individual connection weights can be stronger than replacing their sum by $|\partial {\Lambda }_m|g_m$.

At $p=1/2$ for independent bond percolation on the square lattice, there exist $C<\infty$ and $\alpha >0$ with ${\mathbb{P}}_{1/2}(0\leftrightarrow \partial [-n,n{]}^2)\le C{n}^{-\alpha }$. The Russo-Seymour-Welsh theorem and the Harris-FKG inequality give a uniform positive probability of a closed dual graph cycle around each geometrically spaced square ring. The independent annular barriers for percolation give the polynomial bound and imply zero critical percolation probability. This does not determine the exact critical exponent.

Closed dual graph cycles in disjoint square rings depend on disjoint primal edge sets. In independent bond percolation their barrier events are independent. If each event has probability at least $\delta >0$, an open graph path from the origin across $k$ rings has probability at most $(1-\delta {)}^k$. A geometric sequence of radii turns this estimate into the polynomial critical one-arm upper bound.

For bond percolation on the square lattice, put $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$. Splitting an open self-avoiding walk at its first visit to the radius-$m$ boundary gives disjoint certificates for reaching a boundary graph vertex and then reaching radius $n$ about that graph vertex. The Van den Berg-Kesten inequality and the union bound give $g_{m+n}\le 8m{g}_m{g}_n$ for positive $m,n$. With $g_0=1$ and $|\partial {\Lambda }_0|=1$, the boundary-cardinality form holds for zero indices as well.

For bond percolation on the square lattice, the one-arm probability $g_n={\mathbb{P}}_p(0\leftrightarrow \partial [-n,n{]}^2)$ has a root limit of a sequence $\gamma (p)\in [p,1]$. The BK boundary-splitting estimate implies that $32n^2{g}_n$ is submultiplicative for positive integers. Applying the Fekete lemma to its logarithm proves existence when $p>0$; at $p=0$ the rate is zero. The same rate is the root limit of the percolation two-point connection probability along a coordinate axis, by the reflection lower bound for two-point percolation.

For independent bond percolation of density $0<p<1$ on the cubic lattice in dimension $d\ge 2$, let ${\beta }_n$ be the one-arm probability for reaching maximum-norm distance $n$. The weighted BK boundary-splitting estimate gives ${\beta }_{m+n}\le |\partial {\Lambda }_n|{\beta }_m{\beta }_n$. Since the logarithm of this boundary size is $o(n)$, the asymmetrically almost-subadditive sequence argument applies to $\log {\beta }_n$. The rate exists and lies in $[0,-\log p]$; positive rate gives exponential decay.

The Russo-Seymour-Welsh theorem and the Harris-FKG inequality imply $\pi (2r)\ge c\pi (r)$ for a scale-independent constant $c>0$.

RSW rectangle crossings and positive association join two separated one-arm events with probability bounded below uniformly over scale.

For increasing events $A,B$ in a product percolation measure, their disjoint occurrence satisfies $\mathbb{P}(A\square B)\le \mathbb{P}(A)\mathbb{P}(B)$.

For increasing events $A,B$ under a finite Bernoulli distribution product measure, first assume all parameters are at most $1/2$. Write $A_0\subseteq A_1$ and $B_0\subseteq B_1$ for last-coordinate sections. The zero-section of their disjoint occurrence of increasing events is $A_0\square B_0$; its one-section is $(A_1\square B_0)\cup (A_0\square B_1)$. The intersection in this union contains $A_0\square B_0$. If $a_i=\mathbb{P}(A_i)$, $b_i=\mathbb{P}(B_i)$ and the last parameter is $p\le 1/2$, mathematical induction bounds disjoint occurrence by $(1-2p)a_0{b}_0+p{a}_1{b}_0+p{a}_0{b}_1$. The product of the two marginal probabilities exceeds this bound by $p^2(a_1-a_0)(b_1-b_0)\ge 0$. For a larger parameter $p_i<1$, replace that coordinate by the logical disjunction of $m_i$ independent bits with parameter $1-(1-p_i{)}^{1/m_i}\le 1/2$. Disjoint original witnesses lift to disjoint bit witnesses. Apply the small-parameter inequality and then pass to degenerate parameters by continuity.

At density $1/2$, planar duality gives crossing probability $1/2$ for the $2n$-by-$(2n-1)$ lattice rectangle. A self-avoiding left-right crossing passes through one of its $2n$ middle-column vertices. At that vertex its two halves are disjoint edge witnesses for two copies of the radius-$n$ connection event. The BK inequality and the union bound give $1/2\le 2n{\mathbb{P}}_{1/2}(0\leftrightarrow \partial [-n,n{]}^2{)}^2$, proving the estimate.

A finite open coordinate set witnesses an increasing event when prescribing those coordinates open forces the event, regardless of the remaining configuration. The disjoint occurrence of increasing events $A\square B$ means that $A$ and $B$ have disjoint finite witnesses. For finite-coordinate events this is the usual disjoint-occurrence definition. The Van den Berg-Kesten inequality bounds its probability by $\mathbb{P}(A)\mathbb{P}(B)$ under an independent Bernoulli distribution product measure.