A one-dimensional chain is a tree in which every site has two neighbors. The Bethe lattice, or Cayley tree, generalizes it: every site has z neighbors and there are no loops, so any two sites are joined by exactly one path. That makes percolation exactly solvable. The threshold pc = 1/(z − 1) depends on z, but the critical exponents do not, and they are the mean-field values seen on random networks and on lattices above six dimensions.
Based on lecture notes by Sang Hoon Lee, following K. Christensen and N. R. Moloney, Complexity and Criticality (2005), Section 1.3. References
Sites of the center’s cluster in each generation ℓDots: this lattice. Dashed: the expected number N(ℓ) = z(z − 1)ℓ−1pℓ when the center is occupied. At pc it stays at z/(z − 1) in every generation.
Each site is occupied with probability p. Every site keeps its own random number, so raising p only adds sites.
A finite Cayley tree has a large share of its sites on the boundary, but in the infinite lattice every site is equivalent and the center is nothing special.
A cluster extends indefinitely only if a walk along it, never retracing its steps, can always continue. Each site reached offers z − 1 new branches, each occupied with probability p, so percolation needs p(z − 1) ≥ 1:
For z = 2 this gives pc = 1, the chain. The value depends on z, so pc is not universal. To find the probability P∞(p) that a site belongs to the infinite cluster, let Q∞ be the probability that a given branch does not connect to it. Either the branch’s first site is empty, or it is occupied and none of its z − 1 sub-branches connect:
For z = 3 the nontrivial root is Q∞ = (1 − p)/p above pc = 1/2. In general P∞ picks up linearly, P∞ ≈ [2z/(z − 2)](p − pc), so β = 1 for every z. The dots come from growing the center’s cluster generation by generation on an infinite lattice; a cluster still growing after thousands of generations, or past 200 000 sites in one generation, counts as infinite.
Let B be the contribution of one branch to the size of the center’s cluster. If the branch’s first site is empty it adds nothing; if occupied, it adds itself and B from each of its z − 1 sub-branches. So B = p[1 + (z − 1)B], and with the center itself,
χ diverges as (pc − p)−1, so γ = 1 whatever z is. For z = 2 it reduces to (1 + p)/(1 − p), the chain result. Above pc the dots show the mean size of the center’s cluster when it is finite.
For z > 2 clusters of the same size s come in many shapes, but on a tree they all have the same number of perimeter sites, t = 2 + s(z − 2). So n(s, p) = g(s, t)(1 − p)tps with one degeneracy factor, which Fisher and Essam counted: g = z[(z − 1)s]! / (s! [(z − 2)s + 2]!). Near pc it factorizes as
At pc, n(s, pc) ∝ s−τ. Requiring Σ s n(s, pc) to stay finite while Σ s² n(s, pc) diverges gives 2 < τ ≤ 3, and matching χ ∝ sξ3−τ with χ ∝ (pc − p)−1 and sξ ∝ (p − pc)−2 gives τ = 5/2.
Distances on the Bethe lattice are chemical distances: the number of steps ℓ along the unique path, like the hopping distance in network science. The probability that a site ℓ generations away is occupied and in the same cluster as an occupied site is g(ℓ) = pℓ. With z(z − 1)ℓ−1 sites in generation ℓ, the expected number of them in the cluster is
Summing over all sites recovers the average cluster size, 1 + Σℓ N(ℓ) = χ(p): the sum rule Σj g(i, j) = χ(p), a special case of the fluctuation–dissipation theorem.
The threshold and the amplitudes change with z, but every exponent is the same: the Bethe lattice is the mean-field limit of percolation. The same values describe Erdős–Rényi random networks and lattices above the upper critical dimension d = 6.
Key reference. K. Christensen and N. R. Moloney, Complexity and Criticality, Imperial College Press, London (2005). Section 1.3 covers percolation on the Bethe lattice, and Exercise 1.7 the order parameter for general z.