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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.

The honeycomb lattice is the periodic degree-three planar graph consisting of the vertices and edges of a regular hexagonal tiling. It has two vertex classes; every edge joins opposite classes. Its planar dual graph is the triangular lattice.

Group the martini lattice into independent three-terminal cells. A cell has terminals $A,B,C$, an inner triangle with vertices $a,b,c$, and the three spokes $Aa,Bb,Cc$. All six bonds are independently open with parameter $p$. The cells arise from the vertices in the decorated class of the honeycomb lattice; they have disjoint bond sets and meet only at terminals. The terminal positions form a triangular lattice, with cells occupying one orientation of its triangular faces.

A cell induces one of five connection partitions of its terminals. Let $P_3(p)$ be the probability that all three terminals are connected, and $P_0(p)$ the probability that no pair is connected. Each of the three remaining pair partitions has equal probability by symmetry. The three-terminal cell partition duality gives a particularly useful exact criterion. Draw dual terminals on the three intervening boundary arcs. The dual cell has a triangle of these three terminals and a central vertex joined to all three: its outer bonds cross the primal spokes, and its inner spokes cross the primal triangle bonds. A primal connection separates the corresponding dual boundary arcs, and conversely a dual connection separates primal terminals. Consequently duality swaps the all-connected and all-separate partitions, and permutes the three pair partitions. For example, if $A,B$ connect and $C$ is isolated, the dual terminal between $A,B$ is isolated while the other two dual terminals connect. It follows that $P_3=P_0$ makes the entire symmetric cell partition law invariant under this duality.

The triangular arrangement of cells is itself self-dual: the dual terminals lie in the complementary triangular gaps, and reflection and translation identify their cell arrangement with the original one. The induced partitions of distinct cells are independent in both models. At a parameter satisfying $P_3=P_0$, the primal model at $p$ and the complementary dual bond percolation model at $1-p$ therefore have the same coarse connection law. Replacing a cell by its terminal partition preserves infinite connectivity, since the cells have uniformly bounded size. This gives a rigorous link between the finite calculation and the infinite model, rather than assuming that a symmetric-looking cell is critical.

For all three terminals to connect, all three spokes must be open and the inner triangle must be connected. The latter event requires two or three of its edges. ThereforeTo compute $P_0$, condition on the number of open spokes. Zero or one open spoke cannot connect a pair of terminals. If exactly two spokes are open, the two attached inner vertices must not connect: their direct triangle bond is closed, and the two-edge route through the third inner vertex is not fully open. These independent requirements have probability $(1-p)(1-p^2)$. If all three spokes are open, all three triangle bonds must be closed. HenceSubtracting and factoring givesThe second factor equals $1+p^2(p-1)(p-2)$ and is at least $1$ on $[0,1]$. Thus the unique equality parameter is $p_0=1/\sqrt{2}$.

Finally part (i) proves $c+c^{\ast }=1$. If $p_0>c$, the primal model percolates while $1-p_0<c^{\ast }$ makes the dual model subcritical, contradicting their identical coarse connection laws. If $p_0<c$, the same contradiction arises with the roles reversed. Therefore the critical probability isThe local partition transformation is the mechanism underlying this exact value; see Ziff's original cell–dual-cell paper.

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.