Percolation Universality Class
Overview
Percolation describes a geometric phase transition: sites (or bonds) on a lattice are occupied independently with probability $p$. At $p = p_c$ an infinite connected cluster first appears. Near $p_c$ the cluster statistics exhibit power-law behaviour governed by critical exponents that depend only on the spatial dimension $d$.
Unlike thermal phase transitions, percolation has no Hamiltonian and no temperature. The control parameter is the occupation probability $p$ and the "order parameter" is the probability $P_\infty$ that a randomly chosen site belongs to the infinite cluster.
Symmetry: Permutation symmetry of the $q \to 1$ limit of the $q$-state Potts model (Kasteleyn-Fortuin mapping).
$d = 2$ –- Exact Critical Exponents
In $d = 2$, the percolation exponents are known exactly via the Coulomb gas mapping and SLE (Schramm-Loewner evolution).
| Exponent | Value | Physical meaning | Reference |
|---|---|---|---|
| $\beta$ | $5/36$ | Infinite-cluster density: $P_\infty \sim (p - p_c)^\beta$ | Nienhuis (1982) J. Stat. Phys. 34, 731 |
| $\nu$ | $4/3$ | Correlation length: $\xi \sim \lvert p - p_c\rvert^{-\nu}$ | den Nijs (1979); Nienhuis (1982) |
| $\gamma$ | $43/18$ | Mean cluster size: $\langle s \rangle \sim \lvert p - p_c\rvert^{-\gamma}$ | Scaling relation $\gamma = \nu(2 - \eta)$ |
| $\eta$ | $5/24$ | Pair connectivity: $g(r) \sim r^{-(d-2+\eta)}$ at $p_c$ | Coulomb gas |
| $\alpha$ | $-2/3$ | Cluster-number density | $\alpha = 2 - d\nu = 2 - 8/3$ |
| $\delta$ | $91/5$ | $P_\infty \sim h^{1/\delta}$ at $p_c$ with ghost field | Scaling relation |
| $\sigma$ | $36/91$ | Cluster-size distribution: $n_s \sim s^{-\tau}$ with $\tau = 1 + 1/(\sigma\beta\delta)$ | — |
| $\tau$ | $187/91$ | Fisher exponent: $n_s \sim s^{-\tau} f(s/s_\xi)$ | — |
Scaling Relations
The exponents satisfy all standard scaling relations, verified algebraically in QAtlas (see calculation note for the general framework):
\[\alpha + 2\beta + \gamma = -\tfrac{2}{3} + \tfrac{5}{18} + \tfrac{43}{18} = 2 \quad\checkmark\]
\[\gamma = \nu(2 - \eta) = \tfrac{4}{3}\left(2 - \tfrac{5}{24}\right) = \tfrac{43}{18} \quad\checkmark\]
$d = 3$ –- Numerical
In $d = 3$, no exact solution is known. The best estimates come from large-scale Monte Carlo simulations.
| Exponent | Value | Reference |
|---|---|---|
| $\beta$ | $0.4181(8)$ | Wang, Zhou, et al. (2013) Phys. Rev. E 87, 052107 |
| $\nu$ | $0.8765(12)$ | " |
| $\gamma$ | $1.793(3)$ | " |
| $\eta$ | $-0.046(8)$ | " |
$d \geq 6$ –- Mean-Field
The upper critical dimension for percolation is $d_c = 6$. For $d \geq 6$ the exponents take mean-field values on a Bethe lattice:
\[\beta = 1, \quad \nu = 1/2, \quad \gamma = 1, \quad \eta = 0, \quad \alpha = -1\]
Note that $\alpha < 0$ (no specific-heat divergence) and $\beta = 1$ (linear onset of $P_\infty$) differ from the thermal mean-field values, reflecting the geometric nature of the transition.
QAtlas API
using QAtlas
# d = 2: exact (Rational{Int})
e = QAtlas.fetch(Universality(:Percolation), CriticalExponents(); d=2)
# (β = 5//36, ν = 4//3, γ = 43//18, η = 5//24, ...)
# d = 3: numerical (Float64 + _err)
e3 = QAtlas.fetch(Universality(:Percolation), CriticalExponents(); d=3)
# d ≥ 6: mean-field
e6 = QAtlas.fetch(Universality(:Percolation), CriticalExponents(); d=6)References
- B. Nienhuis, "Exact critical point and critical exponents of O(n) models in two dimensions", Phys. Rev. Lett. 49, 1062 (1982).
- M. P. M. den Nijs, "A relation between the temperature exponents of the eight-vertex and q-state Potts model", J. Phys. A 12, 1857 (1979).
- D. Stauffer, A. Aharony, Introduction to Percolation Theory (Taylor & Francis, 1994).
- G. Grimmett, Percolation (Springer, 1999).
Connections
- Potts $q \to 1$: The Kasteleyn-Fortuin mapping relates bond percolation to the $q \to 1$ limit of the Potts model.
- Scaling relations: verified in calc/ising-scaling-relations.md (same algebraic framework, different exponent values).