# Critical Phenomena and Percolation Theory: I

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1 Critical Phenomena and Percolation Theory: I Kim Christensen Complexity & Networks Group Imperial College London Joint CRM-Imperial College School and Workshop Complex Systems Barcelona 8-13 April 2013

2 Outline Critical Phenomena & Percolation Theory 1 Critical Phenomena & Percolation Theory 2 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size 3

3 Critical Phenomena & Percolation Theory Aim: Study connections between macroscopic quantities and the underlying microscopic world in a model displaying a phase transition. Objective: Gain qualitative and quantitative understanding of critical phenomena and associated concepts such as scale-free behaviour, scaling theory, and universality.

4 L L Each site in a (regular) lattice is occupied randomly and independently with occupation probability p, 0 p 1. A cluster is a group of nearest-neighbour occupied sites. The size s of a cluster is the number of sites in the cluster. The critical occupation probability p c is the occupation probability p at which an infinite cluster appears for the first time in an infinite lattice L =.

5 Percolation deals with the number of the clusters formed properties of the clusters formed when occupying randomly and independently each site in a lattice with probability p.

6 Quantities of interest Onset of percolation critical occupation probability, p c. Probability that a site belongs to the infinite cluster, P (p). Geometry of the infinite cluster at p = p c and p > p c. Excluding the infinite cluster: Average cluster size, χ(p). Typical size of the largest cluster, s ξ (p). Typical radius (linear size) of the largest cluster, ξ(p).

7 For fixed lattice size L, there is only one parameter, the occupation probability p, 0 p 1. p = 0: Empty lattice. No clusters. 0 < p < 1: Percolation is a random{ process. No. of different realisations 2 L L 10 6,773 for L=150; ,370 for L=600. p = 1: Fully occupied lattice. One cluster of size s = L 2. A percolating cluster is one that spans the lattice from left to right, top to bottom, or both.

8 p = 0.10 p = 0.55 p = 0.58 p = p c p = 0.65 p = 0.90

9 Typically expects no percolating cluster for small p. Typically expects a percolating cluster for large p. Consider these two probabilities at occupation probability p in a lattice of size L: Π (p; L) = prob. that percolating cluster exists. P (p; L) = prob. that a site belongs to a percolating cluster = fraction of volume covered by percolating cluster

10 Prob. that percolating cluster exists at occupation probability p Π (p, L = ) p

11 Prob. that percolating cluster exists at occupation probability p Π (p, L = ) p

12 Prob. that percolating cluster exists at occupation probability p Π (p, L = ) p

13 1 0.8 Π (p, L = ) No percolating cluster Sub-critical p<p c Phase transition Critical p=p c Percolating cluster Super-critical p>p c p Critical point

14 Π (p; L = ) = { 0 for p < p c 1 for p > p c The critical occupation probability p c is the occupation probability above which a percolating (infinite) cluster appears for the first time in an infinite lattice. Onset of percolation is a geometrical phase transition: When increasing p from 0 towards 1, there is a phase transition at p = p c from a lattice with no percolating infinite cluster for p < p c to a lattice with a percolating infinite cluster for p > p c. For two-dimensional square lattice p c = For two-dimensional triangular lattice p c = 1/2. For three-dimensional simple cubic lattice p c =

15 Prob. that site belongs to percolating cluster at occ. prob. p. 1 P (p; L= ) picks up abruptly for p>p c P 0.8 (p; L= ) = 0 for p p c P (p, L = ) No percolating cluster Percolating cluster p

16 P (p; L = ) = = { 0 for p p c nonzero for p > p c { 0 for p p c A (p p c ) β for p p c +. The critical exponent β characterises the abrupt pick-up of the order parameter P (p) for p p + c.

17 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) 0 p 0 p c 1

18 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) 0 p 0 p c 1

19 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) 0 p 0 p c 1

20 Excluding the percolating (infinite) cluster, what is the typical size of the largest cluster s ξ (p)? s ξ (p) s ξ (p) p p c 1/σ for p p c Critical exponent σ = 36 in d = σ is independent of lattice details Example of universality 0 p 0 p c 1

21 Excluding the percolating (infinite) cluster, what is the typical radius (linear size) of the largest cluster ξ(p)? ξ(p) ξ(p) p p c ν for p p c Critical exponent ν = 4 in d =2 3 ν is independent of lattice details Example of universality 0 p c 1 p

22 Excluding the percolating (infinite) cluster, what is the average cluster size to which an occupied site belongs, χ(p)? χ(p) χ(p) p p c γ for p p c Critical exponent γ = 43 in d = γ is independent of lattice details Example of universality 0 p 0 p c 1

23 l 10 8 l M (pc; l) l Mass of the percolating cluster at p = p c increases with window size l: M (p c ; l) l D for l 1. Critical exponent D is the fractal dimension of cluster. D = 91 in d = D is independent of lattice details. Example of universality.

24 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size Can be solved analytically. Many of the characteristic features encountered are present for percolation in d > 1. L What is the critical occupation probability p c?

25 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size Prob. that site belongs to percolating infinite cluster P (p) For d = 1, p c = p

26 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size Calculate the cluster size frequency N(s, p; L) probabilistically. Empty site s consecutive sites occupied s Empty site Cluster number density = number of s-clusters per lattice site: n(s, p) = lim L N(s,p:L) L =(prob. empty site) (prob. s occupied sites) (prob. empty site) = (1 p)p s (1 p) = (1 p) 2 p s

27 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size 10 0 n(s, p) p = 0.4 p = p = 0.99 p = p = s

28 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size n(s, p) = (1 p) 2 p s with the characteristic cluster size s ξ (p)= 1 ln p = = (1 p) 2 exp (ln p s ) = (1 p) 2 exp (s ln p) = (1 p) 2 exp ( s/s ξ ), 1 ln (1 [1 p]) (1 p) 1 = (p c p) 1 for p p c. For d = 1, the critical exponent σ = 1.

29 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size sξ(p) p

30 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size sξ(p) (1 p)

31 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size The probability that a site belongs to an s-cluster: sn(s, p). Given an occupied site - how large, on average, is its cluster? s=1 χ(p) = s2 n(s, p) s=1 sn(s, p) = 1 + p (see page 11 in notes) 1 p 2(1 p) 1 for p 1

32 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size χ(p) p

33 Onset of percolation: Critical occupation probability p c Cluster number density & characteristic cluster size Average cluster size χ(p) (1 p)

34 General pattern for the exact solution to d = 1 percolation: Characteristic cluster size s ξ (p) = 1 ln p (p c p) 1 for p p c, σ = 1. Average cluster size χ(p) = 1+p 1 p 2 (p c p) 1 for p p c, γ = 1. Asymptotically close to p c, the divergence is characterized by a power-law in (p c p), the distance away from the critical point. Special for d = 1 is that the phase-transition can only be approached from below, p p c.

35 When increasing occupation probability p from 0 towards 1, there is a phase transition at p = p c from a lattice with no percolating infinite cluster for p < p c to a lattice with a percolating infinite cluster for p > p c. Sub-critical behaviour for p < p c where ξ <. Critical behaviour for p = p c where ξ =. Super-critical behaviour for p > p c where ξ <. Order parameter picks up abruptly at p = p c : P (p) (p p c ) β for p p + c. Quantities of interest diverges at p = p c : Characteristic cluster size: s ξ (p) p p c 1/σ for p p c. Average cluster size: χ(p) p p c γ for p p c. Typical radius of largest cluster: ξ(p) p p c ν for p p c.

36 Thank you for listening! For a comprehensive introduction to percolation, please see K. Christensen and N.R. Moloney, Complexity and Criticality, Imperial College Press (2005), Chapter 1. Access to animations, please visit

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