Percolation Percolation is a purely geometric problem which exhibits a phase transition consider a 2 dimensional lattice where the sites are occupied with probability p and unoccupied with probability (1-p) clusters are defined in terms of nearest neighbour sites that are both occupied For p < p c all clusters are finite for p > p c there exists an infinite cluster when p=p c, infinite cluster first appears

Percolation As p increases, larger and larger clusters form until at p c there is one infinite cluster of connected sites for p> p c there are many such connected paths the value of p c depends on the lattice as well as the dimension in d=1, p c =1 for d=2 square lattice, p c =.59

d=1 Clusters p2p2 1-p p5p5 p Probability of 5-cluster is (1-p) 2 p 5

d=2

Percolation A spanning cluster is present for p  p c the probability that an occupied site belongs to the spanning cluster   (p)= number of sites in spanning cluster total number of occupied sites   (p) plays the role of an order parameter   (p)= 0 for p=0   (p)= 1 for p=1 behaves nonanalytically at p=p c similar to the magnetization in a ferromagnet

Clusters cluster size distribution n s (p) n s (p) = number of clusters of size s total number of lattice sites(N=L 2 ) For p < p c all clusters are finite   (p)= 0 For p  p c   (p)  0 the spanning cluster is not included in n s (p) Nsn s is the number of sites in finite clusters of size s the probability that a site chosen at random belongs to an s-site cluster is w s = sn s  s (sn s )

Percolation mean finite cluster size S(p) =  s (s 2 n s )  s (sn s )   (p) and S(p) display critical behaviour at p=p c very similar to a thermodynamic phase transition long ranged correlations play an important role

Exact Solution in d=1 Probability of each site occupied is p probability of 5 sites occupied is p 5 since they are independent events probability of an empty neighbour is 1-p probability/site of a 5-cluster is (1-p) 2 p 5 probability/site of an s-cluster is n s (p)= (1-p) 2 p s

d=1 Chain Probability that a site belongs to a cluster of size s is n s s probability that a site belongs to any cluster is  sn s = p where sum is from s=1 to  average cluster size S(p)=  s w s = (1+p)/(1-p) mean cluster size diverges as p=>1 p c =1 for d>1 we have p c < 1

Correlation Function Define g(r) as probability that a site a distance r from an occupied site belongs to the same cluster obviously g(0)=1 for r=1, neighbouring site belongs to the cluster if it is occupied => g(1)=p for site at distance r, g(r)=p r for p < 1, g(r) goes to zero exponentially at large r g(r) = exp(-r/  ) where  (p)= -1/ln(p) ~ 1/(p c - p) for p near p c =1  (p) is a correlation length that diverges at p c

Percolation transition 1-d exact solution indicates that certain quantities diverge at the percolation threshold divergence can be described by simple power laws such as 1/(p c -p) both S and  have counterparts in thermal phase transitions susceptibility and correlation length Run site

Critical Exponents and Finite size scaling Essential physics near the percolation threshold is associated with large but finite clusters clusters on all length scales up to the size of the system L are present at p=p c the linear dimension  (p) of the finite clusters increases rapidly at p c

In the limit L =>   (p) diverges as

Critical Properties In the limit L=> , the behaviour of   (p) and S(p) is described as follows

Finite Size Effects Simulations are restricted to finite L and direct measurements of   (p),S(p) and  (p) do not yield good estimates for the critical exponents close to p c the largest cluster is the same size as the lattice and is affected by the finite size hence both S(p) and  (p) only reach a finite maximum at p=p c (L) finite size effects important when

Distance from the critical point at which finite size effects occur Measure these quantities at p c and estimate critical exponents using the size dependence Finite size scaling