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For critical site percolation on the triangular lattice, scaling-limit crossing probabilities transform naturally under conformal maps between planar domains. This rigorously established property supports descriptions of interfaces by SLE and calculations of planar percolation critical exponents. Transferring it to a different lattice or to bond percolation requires a separate universality argument.

For a random field $(Y_x)$ on a countable locally finite graph, declare $x$ open when $Y_x\ge a$. Increasing $a$ can only remove open vertices, so the event that an infinite percolation cluster exists is decreasing in $a$. One convention for the critical threshold for level-set percolation isThe value may be infinite, and the definition does not decide what happens at $a=a_c$. On a general graph, a specified-root threshold can differ from a global existence threshold, so the convention matters.

For independent and identically distributed random variables $Y_x$ with uniform distribution on $[0,1]$, the open probability is $1-a$ for $0\le a\le 1$. The resulting model is site percolation, and reversing the parameter gives $a_c=1-p_c$, where $p_c$ is its percolation critical probability. This example separates the direction of the level threshold from the usual increasing opening-density parameter.

In bond percolation on a directed graph, independently open each directed edge with probability $p$. The out-cluster of $x$ consists of vertices reachable by open directed paths. In site percolation, independently open vertices and require every vertex of the directed path, including $x$, to be open. The same definitions allow a multigraph. The orientation matters: an infinite underlying connected graph need not contain an infinite out-cluster from a specified root.

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.

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.

Color the vertices of a periodic graph independently red or blue with equal probabilities, and declare a vertex of its line graph open exactly when the corresponding edge has equal-colored endpoints. The resulting site percolation has marginal $1/2$ and is a finite-range dependent random field: sets at line-graph distance greater than one use disjoint color inputs. Red and blue open clusters cannot join. If the original graph has an infinite red component and an infinite blue component, its line graph consequently has at least two infinite percolation clusters. One explicit amenable graph with that property is obtained by replacing each square-lattice vertex by a complete graph on ten vertices, joining neighboring blocks completely. Blocks containing both colors have independent probability $511/512$; a Peierls argument proves that they percolate, and their infinite cluster carries both a red and a blue component. Thus finite-range dependence, even with all site marginals nondegenerate, does not substitute for finite energy in percolation uniqueness.

Assign each edge $e$ an independent uniform distribution label $U_e$ on $[0,1]$. At parameter $p$, declare $e$ open exactly when $U_e\le p$. Each open/closed decision then has the required Bernoulli distribution, and different edge decisions are independent. The same construction applies to site percolation with labels on vertices.

Keep those labels fixed when increasing $p$. For $p\le q$, every $p$-open edge is also $q$-open, so every $p$-open graph path survives and each percolation cluster can only gain vertices. This coupling of probability distributions compares complete configurations on one probability space, rather than comparing two unrelated pictures. Any increasing event, such as an open path between specified vertices, consequently has a nondecreasing probability.

The three panels use one seeded collection of labels on a finite $17\times 17$ square lattice box. Colored edges and dots show the central root's percolation cluster; gray edges are open but disconnected from it. A percolation cluster can grow by many vertices when one newly opened edge joins it to another large percolation cluster. The panels illustrate a nested construction, not an estimate of $\theta (p)$ or a proof of the Harris-Kesten theorem.

Take a directed graph with vertices $v_0,v_1,\dots$ and $n+1$ parallel bonds from $v_n$ to $v_{n+1}$. This is a locally finite graph, although the incoming degrees are unbounded. For bond percolation, ${\theta }_b(p)={\prod }_{n\ge 0}(1-(1-p{)}^{n+1})>0$ whenever $p>0$: the failure probabilities are summable, so the infinite product is positive. For site percolation, survival requires every vertex of the ray to be open, which has probability zero for $p<1$. Thus the survival and susceptibility thresholds for directed percolation satisfy $p_H^b=0$ and $p_H^s=1$.

In bond percolation on ${\mathbb{Z}}^d$, edges are independently open with probability $p$ and closed otherwise. An open percolation cluster is a connected component using only open edges. In site percolation it is the graph vertices that are independently declared open. The percolation probability $\theta (p)=P_p(0\leftrightarrow \infty )$ measures the chance that the origin lies in an infinite percolation cluster, and the percolation critical probability is $p_c=\inf \{p:\theta (p)>0\}$. Monotone coupling of Bernoulli percolation by independent uniform distribution makes $\theta$ nondecreasing.

The sharpness of the percolation transition says that below $p_c$ connection probabilities on ${\mathbb{Z}}^d$ decay exponentially with distance, while above $p_c$ the percolation probability is positive. There is almost surely a unique infinite percolation cluster in the supercritical regime. On the square lattice, the Harris-Kesten theorem gives bond threshold $1/2$, and there is no infinite percolation cluster at percolation critical probability. The emergence of connections on arbitrarily large scales is the geometric meaning of the phase transition.

Several observables describe this singular behavior. The percolation susceptibility is $\chi (p)=E_p|C(0)|$ for $p<p_c$. A directional correlation length can be defined through exponential two-point decay, $\xi (p{)}^{-1}=-{\lim }_n{n}^{-1}\log P_p(0\leftrightarrow n{e}_1)$. Percolation critical exponents describe leading powers, conveniently defined logarithmically rather than requiring exact asymptotic amplitudes:The first limit of a sequence approaches from above, the other two from below. At percolation critical probability, percolation two-point connection probability is expected to have leading power $|x{|}^{-(d-2+\eta )}$, and the number per site of percolation clusters of size $s$ has leading power $s^{-\tau }$. This cluster-count probability distribution differs from the size-biased probability distribution seen by choosing a site. Scaling relations for critical exponents such as $\gamma =(2-\eta )\nu$ and $2\beta +\gamma =d\nu$ connect the exponents in their applicable scaling regime; hyperscaling relation is not valid above the upper critical dimension of percolation in ordinary mean-field regimes.

The percolation universality hypothesis predicts that dimension and large-scale symmetry, rather than microscopic graph details, determine these exponents and scaling functions. Thus square-lattice bond and triangular-lattice site models are expected to share the planar graph universality class despite different microscopic definitions. Universality does not mean identical critical probabilities, amplitudes or finite-lattice distributions. Exact values and existence of exponents require model-specific theorems, not merely this prediction.

Two dimensions have an additional structure: conformal maps preserve angles, and critical crossing probabilities are expected to be unchanged under conformal transport of the domain and its marked boundary arcs. A crossing probability then depends only on the conformal shape of that marked quadrilateral. The Cardy boundary crossing formula gives this dependence in terms of its boundary cross ratio. For critical triangular-lattice site percolation, conformal invariance and the resulting exponents are established; examples are $\beta =5/36$, $\gamma =43/18$, $\nu =4/3$, with the rigorous statements in leading-power form. These values are summarized in Smirnov and Werner's original paper. Their universality extension to arbitrary planar lattices should not be presented as the same theorem.

An exploration path separating the two colors, with opposite boundary colors on two arcs, has a domain Markov property of a percolation exploration: conditional on the explored portion, the unexplored colors retain their independent laws with the newly exposed boundary colors. Conformal invariance, this domain Markov property and suitable regularity lead to Schramm–Loewner evolution. In the upper half-plane its chordal equation iswhere $B_t$ is standard Brownian motion and time is normalized by half-plane capacity. Critical percolation corresponds to $\kappa =6$. Thus the complicated discrete interface becomes a conformally natural random curve driven by one real Brownian motion. Its crossing and arm probabilities supply critical exponents and, through scaling relations for critical exponents, near-critical singularities. This makes conformal methods predictive tools for the transition, rather than only descriptions of an interface's appearance.

Each marginal is a Bessel process of dimension $\delta =1+2a\in (1,2)$, but finite lifetime can also be proved directly. Apply the Itô formula before $X$ reaches zero:With clock $u(t)={\int }_0^t{X}_s^{-2}ds$, the Itô integral is a Brownian motion $W_u$. The inverse clock satisfiesThe clock runs through all $u\ge 0$ up to the maximal lifetime: a finite terminal clock would leave $X$ at a finite positive limit, allowing continuation, or, at an infinite physical lifetime, would make its clock grow at a positive rate, a contradiction. There is no finite-time explosion to infinity; for example $d{X}_t^2=2X_{t}d{B}_t+(1+2a)dt$ and localization bounds the probability of reaching arbitrarily large levels on a fixed time interval. Since $W_u/u\to 0$ almost surely and $1-2a>0$, the last exponential integral has finite limit, while $X_{t(u)}\to 0$. ThereforeThe same argument applies to $Y$, whose noise $-B$ is also Brownian motion, so its lifetime is finite too. No independence between the two lifetimes is being claimed.

Set $S=X+Y$ and $U=Y/S$ before $T=\zeta \wedge \tau$. Brownian motion terms cancel in the sum:Thus simultaneous extinction is impossible. At the finite first lifetime, precisely one coordinate is zero and the other remains positive. Because $S$ has finite variation, differentiating the ratio givesAfter clock change $v={\int }_0^t{S}_s^{-2}ds$, its generator isA scale function of a one-dimensional diffusion $H$ solves $\mathcal{L}H=0$, soBoth endpoint singularities are integrable since $a<1/2$. Normalize $H(0)=0,H(1)=1$ to obtainThe Itô formula makes $H(U_{t\wedge T})$ a bounded martingale, first by localization away from the endpoints and then by bounded convergence theorem. Its terminal value is one on $\{\zeta <\tau \}$, when $U_T=1$, and zero on the opposite event, when $U_T=0$. optional stopping theorem therefore proves the oppositely driven Bessel exit probability

Here $\mathrm{B}$ is the beta function and $\Gamma$ is the Gamma function.

The integral printed in the question is incorrect for general $a$. Its first exponent is $2-4a$, whereas the two coupled equations give $2a$. At $a=1/4$, which lies in the allowed interval, its integral from zero diverges for every positive upper limit; no finite normalizing constant gives a probability. Moreover exchanging $X$ and $Y$ while changing $B$ to $-B$ shows that at $x=y$ the probability must be $1/2$. The corrected symmetric integral has this value for every $a\in (0,1/2)$, whereas the printed asymmetric normalized integral generally does not. For $a>1/4$, even when the printed integral converges, its derivative density $H_{\mathrm{print}}^{\prime }(u)\propto u^{-(2-4a)}(1-u{)}^{-2a}$ satisfiesso it is harmonic for the required exit problem only at $a=1/3$. The two exponents agree exactly at that value.

To relate that value to percolation, write the two SLE boundary gaps as $g_t(x)-{\xi }_t$ and ${\xi }_t-g_t(-y)$. In Brownian-time units $s=\kappa t$, they have the displayed coupled equations with $a=2/\kappa$, after reversing the Brownian motion sign. Hence $a=1/3$ corresponds to $\kappa =6$. The event is that the positive marked point $x$ is swallowed before the negative marked point $-y$.

For a quadrilateral with boundary points in the order $(-y,0,x,\infty )$, its conformal cross-ratio coordinate is $u=y/(x+y)$. In the critical percolation exploration, wire the arcs $(-y,0)$ and $(x,\infty )$ blue and the other two arcs yellow. Exploration from zero, stopped when one of the two marked sides is swallowed, detects a blue crossing between those two blue arcs exactly in the event just computed. The continuum exploration has the SLE6 law; its Locality property of SLE lets one stop the ordinary chordal exploration before the distant boundary colors affect it. Thus the crossing probability isthe Cardy boundary crossing formula. This is the rigorous scaling-limit formula for critical site percolation on the triangular lattice; applying it to other critical planar models requires the corresponding universality assumption. At this percolation value of $a$, the printed integral is already the correct one.

No. The parallel-bond ray with different site and bond thresholds supplies a locally finite graph counterexample. Take vertices $v_0,v_1,\dots$ and $n+1$ parallel directed edges from $v_n$ to $v_{n+1}$. Its underlying multigraph is connected and every individual degree is finite, but there is no uniform incoming-degree bound.

For bond percolation rooted at $v_0$, survival means that every successive group of parallel bonds contains at least one open bond. Independence givesFor $p>0$, the sum of the failure probabilities is ${\sum }_{n\ge 0}(1-p{)}^{n+1}=(1-p)/p<\infty$. Eventually these terms are at most $1/2$, and $\log (1-t)\ge -2t$ for $0\le t\le 1/2$. The tail of the infinite product is therefore bounded below by a positive exponential; its finitely many initial factors are positive as well. Thus ${\theta }_b(p)>0$ for every $p>0$.

For site percolation, however, every vertex of the unique underlying ray must be open. For $p<1$, the probability that its first $N$ vertices are all open is $p^N\to 0$. At $p=1$ the ray is open. Consequently

Use the incoming-edge coupling of site and bond percolation. Start with independent random variables indicating open bonds, each with probability $q$. For a vertex $v$ of incoming degree $d_v$, let $J_v$ indicate that at least one incoming bond is open. ThenThe indicators $J_v$ are independent: the sets of bonds entering different vertices are disjoint. Using an additional independent coin at each vertex, turn some zeros into ones so that the resulting indicators $S_v$ are independent with common probability $r$. Explicitly, when $J_v=0$, set $S_v=1$ with conditional probability $(r-\mathbb{P}(J_v=1))/(1-\mathbb{P}(J_v=1))$.

Every vertex other than $x$ in an open bond percolation out-cluster has an open incoming bond, and is consequently an open site. On $\{S_x=1\}$, the whole bond out-cluster is therefore contained in the site percolation out-cluster. The event of infinite bond reachability does not depend on bonds entering $x$: delete cycles from any finite open directed path starting at $x$. It is thus independent of $S_x$, including its additional coin. HenceFor every $q>p_H^b$ below $1$, monotonicity and the definition of the survival threshold give ${\theta }_b(q)>0$. Letting $q\downarrow p_H^b$ yields the quantitative boundIn particular, if $p_H^b<1$, the right-hand side is strictly less than $1$. If $p_H^b=1$, no separation from $1$ is implied or generally possible.

Let $C_x^b$ and $C_x^s$ be the out-clusters in directed percolation, with the site percolation cluster empty when $x$ is closed. Write ${\theta }_b(p)={\mathbb{P}}_p(|C_x^b|=\infty )$ and ${\chi }_b(p)={\mathbb{E}}_p|C_x^b|$, and similarly for site percolation. The survival and susceptibility thresholds for directed percolation are the infima of the parameters at which these quantities become positive and infinite, respectively.

We construct a coupling of probability distributions that compares the whole clusters. First condition the site percolation root to be open. Explore its out-cluster, using a fixed order of the finitely many outgoing bonds at each vertex. When a previously untested vertex $v$ is first encountered along a bond $u\to v$ from an already reached vertex, use a fresh Bernoulli random variable of parameter $p$ both for the state of $v$ and for the state of this particular bond. If $v$ is open, add it to the exploration; if it is closed, mark it as tested and never add it. Assign fresh independent random variables of the same parameter to every other bond, including bonds leading to vertices already tested. The choice of the next unassigned bond depends only on previously revealed information. Its assigned coin is therefore independent of that information, so the resulting bond configuration has exactly the product bond percolation law. Parallel bonds cause no difficulty; loops may be discarded.

Every reached site has an open parent bond leading from a previously reached site. Consequently the conditional site percolation cluster is contained in the bond percolation cluster. The connected graph is countable because it is a locally finite graph; taking increasing finite stages of the exploration proves the same containment for infinite clusters. Removing the conditioning on the root givesThus positive site percolation survival implies positive bond percolation survival at the same parameter, and infinite percolation susceptibility for site percolation implies infinite percolation susceptibility for bond percolation. Taking the infima in the definitions proves both comparisons:

We use Zhang's argument, obtaining the critical nonpercolation conclusion from uniqueness rather than assuming it as part of the Harris-Kesten theorem. For independent bond percolation on ${\mathbb{Z}}^2$, apply part (ii) to independent site percolation on its line graph; open sites correspond to open bonds. The line graph is amenable and of finite type. Therefore an infinite bond cluster, if it exists, is almost surely unique. The same holds for the dual bond percolation model.

Assume for contradiction that $\theta (1/2)>0$. The planar dual graph is another square lattice, and its open bonds correspond to closed primal bonds, also with parameter $1/2$. Both models then have an infinite percolation cluster almost surely, by part (i). As a disc $D_R$ expands, the probability that it meets an infinite cluster tends to one, for each model. Choose $R$ avoiding all primal and dual vertices and all intersections of either graph with the four endpoints of the following arcs. Divide the boundary circle into four equal arcs $A_1,A_2,A_3,A_4$ in cyclic order. Let $L_i$ be the event of an infinite primal open ray whose last exit from $D_R$ is in $A_i$; define $L_i^{\ast }$ similarly for an infinite dual open ray. After that exit the entire ray stays outside the disc. The existence of a ray follows by finite branching in an infinite locally finite graph.

Each $L_i$ is increasing in the primal bond states, and their union has probability tending to one. Four-fold rotation symmetry makes their probabilities equal. The FKG inequality for their decreasing complements gives the square-root trick for positively associated eventsThe left side tends to zero, so $\mathbb{P}(L_i)\to 1$. Apply the same argument to the dual states to get $\mathbb{P}(L_i^{\ast })\to 1$. A union bound, requiring no independence between the models, now shows that for sufficiently large $R$ the alternating eventhas positive probability.

The alternating exterior-arm separation lemma says that the two primal rays on this event cannot be joined by a primal open path lying outside $D_R$. Indeed, such a joining path contains a simple crosscut of the exterior between the two primal exit points. Join that crosscut to a chord inside the disc. The Jordan curve theorem puts one of the alternating dual exit points on the bounded side. Its dual ray, which is restricted to the exterior of the disc and runs to infinity, would have to cross the open primal crosscut. A dual open bond cannot cross a primal open bond. This is impossible.

Let $F$ be the finite set of primal bonds whose drawn edges meet $D_R$, and let $E$ be the projection of the alternating-arm event onto the states outside $F$. Because $F$ is finite, this projection is a finite union of measurable events; $\mathbb{P}(E)>0$. Every configuration in $E$ admits an assignment on $F$ witnessing the original four arms. Now close all bonds of $F$. The infinite tails of the two primal rays beyond their first exterior vertices survive; any path joining the surviving tails would now avoid the disc. Appending their original exterior segments back to their exit points would give the forbidden exterior connection. Thus the modified configuration contains at least two infinite percolation clusters. By independence, or the finite-energy property of Bernoulli percolation, $\mathbb{P}(E\cap \{F\text{ all closed}\})=2^{-|F|}\mathbb{P}(E)>0$. This contradicts the uniqueness obtained from part (ii). Therefore

We first specify the standard hypotheses of percolation uniqueness on an amenable quasi-transitive graph. A locally finite graph has finite degree at every vertex; a quasi-transitive graph, or graph of finite type, has finitely many vertex orbits under the acting group $\Gamma$ of graph automorphisms. Here the site percolation law is invariant under $\Gamma$ and independent, with one parameter $p_j$ for each orbit. Initially suppose $0<p_j<1$. This gives the finite-energy property of Bernoulli percolation: conditional on all states outside any finite set, every assignment inside that set has positive probability. An amenable graph has finite sets $W_n$ with $|{\partial }^{+}{W}_n|/|W_n|\to 0$, where ${\partial }^{+}{W}_n$ denotes the external vertex boundary. Finite type bounds the degrees uniformly, so internal and external boundary conventions are equivalent here.

Let $N$ be the number of infinite percolation clusters, taking values in $\{0,1,2,\dots ,\infty \}$. Each event $\{N=j\}$ is invariant. Part (i), together with countability of these possibilities, shows that $N$ is an almost-sure constant.

Suppose first that this constant is a finite integer $j\ge 2$. Some finite connected set $F$ meets all $j$ infinite percolation clusters with positive probability, since increasing finite sets exhaust the graph. Make every vertex of $F$ open. This joins all the infinite clusters. Opening a finite set cannot produce a new infinite cluster from finite clusters: only finitely many clusters meet its finite boundary, and their finite union is finite. By the finite-energy property the modified event has positive probability, contradicting the almost-sure value $j$. Thus a finite $N$ must be $0$ or $1$.

Suppose instead that $N=\infty$ almost surely. Some finite connected set $F$ meets three distinct infinite percolation clusters with positive probability. Open all of $F$. Its vertices belong to one open cluster, and deleting $F$ from that cluster leaves at least three infinite components. This is an encounter box in percolation. We use boxes rather than assume that the local geometry permits a single trifurcation vertex in percolation.

Here is the boundary-counting contradiction in detail. Translates of $F$ have the same positive encounter probability. Finite type and bounded degree allow a deterministic packing of mutually disjoint such translates whose centers have positive lower density in the interiors of the Følner sets $W_n$: every vertex is within a fixed distance of one chosen orbit, and a bounded-radius greedy packing discards only a bounded number of possible centers per chosen center. Boundary layers of any fixed thickness have size $o(|W_n|)$. The expected number of encounter boxes wholly inside $W_n$ is consequently at least $c|W_n|$ for a fixed $c>0$ and all large $n$.

On the other hand, contract the open encounter boxes in each cluster to vertices and retain the part inside $W_n$. Each contracted encounter box separates at least three branches that run to infinity, and each branch must leave $W_n$. Choose a minimal forest connecting all these boundary exits. Every encounter junction has degree at least three in that forest: its three infinite branches remain in different components after deleting the junction, so none can be omitted. The leaves are boundary exits. For each nontrivial tree, the degree-sum identity gives at most the number of leaves minus two vertices of degree at least three. Thus the number of encounter boxes is at most a constant times $|{\partial }^{+}{W}_n|$, where the constant accounts for the bounded boundary degrees. Taking expected values contradicts amenability. This excludes $N=\infty$ and provesThere is a genuine endpoint qualification if different orbit parameters are allowed to be $0$ or $1$. Subdivide each horizontal edge of the square lattice once. Open every original vertex deterministically and close every new midpoint deterministically. This is an invariant independent law on an amenable graph that is also a quasi-transitive graph, but its infinite open clusters are the distinct vertical columns. Thus the finite-energy property must be included when interpreting a general vector $\mathbf{p}$. The homogeneous endpoint laws themselves cause no problem: they give zero or one infinite cluster.

The conclusion is false for $k$-independent laws, even with all site marginals equal to $1/2$. Here is a fully specified monochromatic line-graph percolation counterexample. Form a graph $B$ by replacing each vertex of ${\mathbb{Z}}^2$ by a complete graph on ten vertices and putting every possible edge between the blocks of two neighboring lattice vertices. Every vertex of $B$ has degree $9+4\cdot 10=49$. The graph is connected, periodic and amenable: a square of blocks has volume of order $n^2$ and boundary of order $n$. Assign independent fair red/blue colors to the vertices of $B$.

Call a block good when it contains both colors. Good-block indicators are independent, withThey percolate. To verify this without presupposing a site threshold, put $q=1-g=1/512$. A finite nearest-neighbor good-block cluster around a good origin is surrounded by a circuit of bad blocks in the matching square graph, where horizontal, vertical and diagonal neighbors are allowed. A surrounding circuit of length $\ell$ has an anchor within $[-\ell ,\ell {]}^2$. Overcounting its choices by $9{\ell }^2{8}^{\ell }$ gives the Peierls argumentThe sum bound follows directly from ${\sum }_{\ell \ge 1}{\ell }^2{r}^{\ell }=r(1+r)/(1-r{)}^3$, with $r=1/64$. An infinite good-block cluster therefore exists with positive probability, and by part (i) it exists almost surely. Inside each of its blocks the red vertices are connected to one another, as are the blue vertices; between adjacent good blocks the complete bipartite joining edges connect each color. Hence $B$ has both an infinite red component and an infinite blue component almost surely.

Now use site percolation on $\Gamma =L(B)$, the line graph of $B$: its vertices are the edges $\{u,v\}$ of $B$, two adjacent when they share an endpoint. Declare such a site open exactly when $u,v$ have the same color. Each site has open probability $1/2$. Sets of sites at graph distance greater than one involve disjoint sets of underlying color variables, so this is a translation-invariant $1$-independent law. The line graph is again connected, locally finite, amenable and of finite type. The infinite red and blue components of $B$ yield infinite open clusters in $L(B)$. An open path in $L(B)$ cannot change color at a shared endpoint, so those two clusters are distinct. Thus $N\ge 2$ almost surely. Local color constraints destroy the finite-energy property, precisely the ingredient missing from the uniqueness proof.

Independence or another ergodicity hypothesis is necessary. Read literally for an arbitrary translation-invariant site percolation measure, the assertion is false: the mixture assigning probability $1/2$ to the all-open configuration and $1/2$ to the all-closed configuration is translation-invariant, but the invariant event of an infinite percolation cluster has probability $1/2$. We prove the zero–one law for the invariant independent law used in part (ii); the proof also works for the stated finite-range independent alternative.

Let $\Gamma$ be the group of translations, or the specified group of graph automorphisms, and let $A$ be invariant under it, up to null sets. An event determined by the states in a finite vertex set is a cylinder set. For every $\epsilon >0$, approximation in the product sigma-algebra provides a cylinder set $B$, supported on a finite set $F$, with $\mathbb{P}(A△B)<\epsilon$. Choose $g\in \Gamma$ so that $F$ and $gF$ are disjoint. Then $B$ and $gB$ are independent events, have equal probabilities, and $\mathbb{P}(A△gB)<\epsilon$. HenceandLetting $\epsilon \downarrow 0$ proves the zero–one law $\boxed{\mathbb{P}(A)\in \{0,1\}}$. This is translation ergodicity of Bernoulli percolation.

For a quasi-transitive graph, the required displacement exists: put $F$ in a ball about a vertex $v$; its orbit is infinite, so an automorphism can move $v$, and therefore that whole ball, beyond any prescribed finite neighborhood. A $k$-independent site percolation law means that the sigma-algebras generated by vertex sets at graph distance greater than $k$ are independent. Move $F$ far enough that $\mathrm{dist}(F,gF)>k$ and the same proof applies. Thus finite-range independence does give the zero–one law, although it will not give uniqueness.

Independent bond percolation turns a deterministic graph into a random network: each edge is open with probability $p$, independently, and a percolation cluster is a connected component of its open subgraph. Site percolation instead randomizes vertices. On the cubic lattice the percolation probability $\theta (p)=P_p(0\leftrightarrow \infty )$ is the order parameter and $p_c=\inf \{p:\theta (p)>0\}$ is the critical probability. A monotone coupling, using independent uniform edge labels and opening labels below $p$, makes the growth of connectivity with $p$ transparent.

For $d\ge 2$, $0<p_c<1$: self-avoiding walk counting gives $p_c\ge 1/\mu (d)$, and the planar bound proved above gives an upper bound below one. Below $p_c$, clusters are finite and connection probabilities decay exponentially, a percolation sharpness theorem rather than a consequence of the threshold definition alone. Above $p_c$ there is almost surely a unique infinite percolation cluster. At criticality, the presence of an infinite percolation cluster must be addressed for the particular model. In two dimensions there is none. The transition replaces a finite characteristic length by scale-free geometry, followed by a macroscopic connected component.

Percolation critical exponents quantify different aspects of this change. Use power-law notation for leading exponents; it may suppress amplitudes and logarithmic corrections. From the subcritical side, define the percolation susceptibility $\chi (p)={\mathbb{E}}_p|C(0)|$ and an exponential correlation length $\xi (p)=1/\lambda (p)$, using a fixed norm for the one-arm rate. Above criticality the unrestricted expectation is infinite, so a finite-cluster percolation susceptibility must exclude the infinite percolation cluster. The principal observables are:

The last distribution is not the distribution seen from a uniformly chosen site: the latter is weighted by cluster size. In the scaling description this accounts for $1/\delta =\tau -2$. Under the usual below-upper-critical-dimension scaling hypotheses, further scaling relations for critical exponents includeThey organize the exponents into a small number of independent quantities; they are not automatic identities following only from their definitions. A percolation cluster fractal dimension $D_f$ describes the mass of a large critical cluster of radius $r$ as roughly $r^{D_f}$, with the scaling relation for critical exponents $D_f=d-\beta /\nu$ in this regime.

The percolation universality hypothesis is that ordinary short-range independent models in the same spatial dimension share these exponents and their continuum behavior despite differing microscopic lattices or using bonds rather than sites. The threshold itself is not universal. Long-range connections, correlations or changes of geometry can change the universality class, so the hypothesis does not include every random graph called percolation.

Two dimensions have a particularly strong geometric structure. Closed planar dual graph circuits obstruct primal open paths. The Harris-Kesten theorem fixes the square lattice bond percolation threshold at $p_c=1/2$, using planar duality and crossing estimates. Self-duality alone is not a proof; uniform rectangle-crossing control is a crucial extra ingredient. The Russo-Seymour-Welsh theorem keeps critical crossing probabilities of rectangles of fixed aspect ratio bounded away from zero and one at all scales. Closed dual circuits on disjoint annular scales then rule out an infinite critical cluster and reveal why arbitrarily large finite structures persist.

Critical site percolation on the triangular lattice has rigorously established conformal invariance of planar percolation for crossing limits and an SLE description of interfaces. These results are specific to this model. The exact planar valuesare rigorously established there; its one-arm probability has exponent $5/48$. In the corresponding scaling description these values give $\eta =5/24$, $\delta =91/5$, $\tau =187/91$ and $D_f=91/48$. The percolation universality hypothesis predicts the same planar exponents for square lattice bond percolation, but the triangular-lattice theorem by itself does not prove that transfer. This distinction separates exact model-specific mathematics from a broader physical prediction.

Dimension changes the importance of loops and correlations between growing branches. The mean-field percolation exponents come from approximating critical clusters by branching processes. Their values are $\beta =1$, $\gamma =1$, $\delta =2$, $\nu =1/2$, $\eta =0$ and $\tau =5/2$. The predicted upper critical dimension of percolation is six. One way to see its role is the triangle diagram appearing in the percolation triangle condition: if the long-distance Fourier connectivity behaves like $|k{|}^{-2}$, its infrared contribution is proportional to ${\int }_0^{\epsilon }{r}^{d-7}dr$. It is finite above six and logarithmically divergent at six. This motivates mean-field behavior above six and logarithmic corrections at six, without turning that heuristic into a proof for every lattice.

Lace expansion makes the mean-field picture rigorous in suitable high-dimensional regimes. For nearest-neighbor percolation, sufficiently high dimensions are covered; sufficiently spread-out models have mean-field results for every $d>6$. Between two and six the exponents depend nontrivially on dimension, and exact three-dimensional values are not supplied by the planar theory. Hyperscaling relation illustrates the change: mean-field values give $2\beta +\gamma =3$, while $d\nu =d/2$, so the naive equality fails above six. In $d=1$, $p_c=1$ and no infinite percolation cluster exists for $p<1$; this is a degenerate endpoint case rather than the ordinary interior transition.

The cubic lattice has graph vertices ${\mathbb{Z}}^d$ and an edge between $x,y$ when ${\sum }_i|x_i-y_i|=1$. In independent site percolation, every graph vertex is open with probability $p$ and closed with probability $1-p$, independently. The open subgraph contains only open graph vertices and the edges between them. Let $C_p(0)$ be the open connected component of a graph containing $0$, with $C_p(0)=\varnothing$ when $0$ is closed. The percolation probability isSince the cubic lattice is a locally finite graph, the König infinity lemma identifies an infinite open connected component of a graph with the existence of an infinite open graph ray from the origin. In particular, the origin itself must be open. The percolation critical probability is $p_c=\inf \{p:\theta (p)>0\}$.

Use the probability space $([0,1{]}^{{\mathbb{Z}}^d},\mathcal{F},\mathbb{P})$ with the product measure of the uniform distribution on $[0,1]$. Its coordinate random variables $U_v$ are independent and identically distributed random variables. Define simultaneously for every parameterFor $r\le s$, the indicator function inequality $X_r(v)\le X_s(v)$ holds pointwise at every graph vertex. For fixed $p$, each coordinate has the Bernoulli distribution with parameter $p$, and the coordinates are independent because they are functions of distinct independent $U_v$. Thus every marginal configuration is exactly the required site percolation model. This is a continuum-indexed family of processes, all on one probability space, and is the standard monotone coupling of Bernoulli percolation.

Use the following general result, including its site version: independent site percolation on ${\mathbb{Z}}^d$ has almost surely at most one infinite percolation cluster at each fixed parameter; for every fixed $r>p_c$ it has exactly one almost surely. This is the uniqueness of the infinite percolation cluster, also called the Burton-Keane theorem. For $d\ge 2$, the additional general fact $p_c<1$ ensures a choice of a parameter strictly between $p_c$ and $1$. In dimension one, $p_c=1$ and the asserted interval is empty.

Fix $p\in (p_c,1]$ and choose $q\in (p_c,p)$. Let ${\mathcal{C}}_q$ be the unique infinite open connected component of a graph at $q$. Under the monotone coupling of Bernoulli percolation, it is contained in the unique infinite open connected component of a graph at $p$. On $I_p$, the origin and ${\mathcal{C}}_q$ therefore lie in that same connected component of a graph, and a finite $p$-open graph path connects $0$ to some graph vertex of ${\mathcal{C}}_q$.

For this fixed deterministic $p$, the countable set of uniforms satisfies $U_v\ne p$ at every graph vertex almost surely. Thus all uniforms on that finite graph path are strictly less than $p$, including its endpoints. If $t$ is their maximum, choose $r$ with $\max (q,t)<r<p$. The graph path is then $r$-open, and its endpoint remains connected to infinity through ${\mathcal{C}}_q$. Consequently $I_r$ occurs and $M<p$. We have provedPart (d) gives left continuity at this $p$; part (b) supplies right continuity when $p<1$. The same finite-path proof gives left continuity at $p=1$, since all uniforms are strictly less than $1$ almost surely. Thus $\theta$ is continuous on $(p_c,1]$, without asserting continuity at $p_c$. This is supercritical continuity of percolation probability.