Critical phenomena in Percolation occurs at the critical value of the occupation probability corresponding to the Percolation phase transition, which separates the percolating and the non-percolating phases.

Critical phenomena

Percolation models at the critical point show several interesting critical phenomena:

• Symmetries
  • Scaling invariance.
  • Conformal invariance. This has lead to modelling them (as in many other examples of Critical phenomena) in the continuum limit, as a Conformal field theory.
• Fractal structure of the critical percolation clusters. Scaling invariance leads to the self-similarity characteristic of Fractals, and indeed the clusters have fractal geometry. Even for $p\neq p_c$, the clusters are fractal at length scales $l\ll\xi$, the correlation length, and non-fractal (Euclidean) at larger lengthscales. An argument using the scaling hypothesis, and the number of nodes (mass) of clusters, shows that the fractal dimension of the clusters is also universal, as it is related to other universal critical exponents (see pages 13-14 in Saberi's review). There are other fractals dimension that one can define, like the one for the minimum length path between points, or those for perimeter, backbone, dangling ends, and red sites (or bonds).

Scaling hypotheses

There are a number of scaling hypotheses for several quantities for percolation near criticality (see Renormalization group for origin of scaling hypotheses).

Upper critical dimension

$d_c$. It is believed that when $d \geq d_c$ , the percolation process behaves roughly in the same manner as percolation on an infinite regular tree and their critical exponents take on the corresponding values given by mean-field theory

Real-space renormalization group

Renormalization Group Theory - Percolation. In particular, see here.

A real-space renormalization group for site and bond percolation

See also here.

Scaling theory of percolation clusters