Percolation

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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 lt pc all clusters are finite
• for p gt pc there exists an infinite cluster
• when ppc, infinite cluster first appears

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Percolation

• As p increases, larger and larger clusters form
  until at pc there is one infinite cluster of
  connected sites
• for pgt pc there are many such connected paths
• the value of pc depends on the lattice as well as
  the dimension
• in d1, pc 1
• for d2 square lattice, pc.59

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d1 Clusters
1-p
p2
1-p
p5
p
Probability of 5-cluster is (1-p)2p5
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d2
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Percolation

• A spanning cluster is present for p?pc
• 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 p0
  ??(p) 1 for p1
• behaves nonanalytically at ppc
• similar to the magnetization in a ferromagnet

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Clusters

• cluster size distribution ns(p)
• ns(p) number of clusters of size s
  total number of lattice sites(NL2)
• For p lt pc all clusters are finite
  ??(p) 0
• For p ? pc ??(p)? 0
• the spanning cluster is not included in ns(p)
• Nsns 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
• ws sns
  ?s(sns)

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Percolation

• mean finite cluster size S(p) ?s(s2ns)
  ?s(sns)
• ??(p) and S(p) display critical behaviour at
  ppc
• very similar to a thermodynamic phase transition
• long ranged correlations play an important role

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Exact Solution in d1

• Probability of each site occupied is p
• probability of 5 sites occupied is p5 since they
  are independent events
• probability of an empty neighbour is 1-p
• probability/site of a 5-cluster is (1-p)2p5
• probability/site of an s-cluster is ns(p)
  (1-p)2ps

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d1 Chain

• Probability that a site belongs to a cluster of
  size s is nss
• probability that a site belongs to any cluster is
  ?sns p where sum is from s1 to ?
• average cluster size S(p) ?s ws
  (1p)/(1-p)
• mean cluster size diverges as pgt1
• pc 1
• for dgt1 we have pc lt 1

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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 r1, neighbouring site belongs to the cluster
  if it is occupied gt g(1)p
• for site at distance r, g(r)pr
• for p lt 1, g(r) goes to zero exponentially at
  large r
• g(r) exp(-r/?)
• where ?(p) -1/ln(p) 1/(pc - p) for p near
  pc 1
• ?(p) is a correlation length that diverges at pc

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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/(pc-p)
• both S and ? have counterparts in thermal phase
  transitions
• susceptibility and correlation length

Run site
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Critical Exponents andFinite 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 ppc
• the linear dimension ?(p) of the finite clusters
  increases rapidly at pc

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In the limit L gt ? ?(p) diverges as
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Critical Properties

• In the limit Lgt ?, the behaviour of ??(p) and
  S(p) is described as follows

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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 pc 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 ppc(L)
• finite size effects important when

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