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 < pc all clusters are finite

• for p > pc there exists an infinite cluster

• when p=pc, infinite cluster first appears

1/cp p L

Distance from the critical point at whichfinite size effects occur

/

/

( ) ( )

( )

c c

c c

p p p p L

S p p p p L

Measure these quantities at pc and estimate critical exponentsusing the size dependence

Finite size scaling

• At pc there are clusters of all sizes present

• the geometry of the infinite spanning cluster is self-similar under magnification

• the cluster is very tenuous and stringy

• fractals can be used to describe such objects

• the fractal dimension D can be used as a measure of the structure

Spanning cluster

Tenuous and not compact

Fractal Dimension

• Recall for a uniform circle or sphere

• M(R) ~ RD

• if D=d (Euclidean dimension) then the object is compact

• density (R)=M(R)/Rd ~ RD-d

• if D=d then the density is uniform

• if D < d then the density of the object is scale dependent and decreases at large length scales

Fractal Objects

• The percolation cluster is an example of a random or statistical fractal because the relation M(R) ~ RD must be averaged over many different origins in a given cluster and over many clusters

Regular Fractals

• Consider a line segment and divide it in half

• the scaling factor is b=2

• the number of units is now N=2

• N=b

Regular Fractals

• Again b=2 but now the number of new units is N=4

• N=b2

• in general dimension we have N=bd

• hence d = ln(N)/ln(b)

Regular Fractals

• Each line segment is divided into 3 parts and replaced by 4 parts

• hence N=4 and b=3

• D = ln(N)/ln(b) = ln(4)/ln(3) = 1.26

• the curve is self-similar (looks the same under magnification) and the dimension is non-integral

Koch CurveKoch

Spanning Cluster

• The spanning cluster at pc has a fractal structure

• we can measure the number of sites M within a distance r of the centre of mass of the cluster

• plot ln(M) versus ln(r) to determine D

Algorithm for generating the spanning cluster

• Occupy a single site at the center of the lattice

• the 4 nearest neighbours are perimeter sites

• for each perimeter site generate a random number r on the unit interval

• if r p the site is occupied and added to the cluster

• otherwise the site is not occupied and not tested again

• for each occupied site determine if there are new perimeter sites to be tested

• continue until there are no untested perimeter sites

Percolationclusters

Fractal Dimension• The fractal dimension of percolation clusters

satisfies a scaling law

• D=d -/• for two dimensions, =5/36 and =4/3

• D=91/48 ~ 1.896

• the mass of the spanning cluster is the probability of a site belonging to it multiplied by the total number of sites Ld

( ) ( ) dM L L L

Fractal Dimension1/

cp p L /

/

( ) ( )

( )

c c

c c

p p p p L

S p p p p L

/( ) d DM L L L L

Renormalization Group

• The critical exponents characterizing the geometrical phase transition can be obtained by simulating percolation clusters near pc and examining the various properties on different length scales L

• another method of examining the system on different length scales is the renormalization group

Scaling

• Consider a photograph of a percolation configuration generated at p=p0 <pc

• now consider viewing it from further distances

2'p p

For p < 1 , p’ => 0 at large length scales

For p=1 , p’ =1 at all length scales

pc = 1

1-d chain

RG

Percolation and RandomWalks

• diffusion and transport in disordered media• a walker can only move ‘up’,’down’,’left’ or ‘right’ if the

neighbouring site is an occupied site• how does <r2> depend on the number of steps N ?

• For p < pc all walkers are confined to finite clusters

• for p > pc most walkers have <r2> N

• “normal” diffusion

• at p=pc we have anomalous diffusion

Anomalous Diffusion• when p=pc, only a few walkers are not localized

• let a walker start on a cluster and walk

• the mean square displacement <R2>~ Na

• for p < pc all clusters are finite and thus a=0

• for p>pc we have normal diffusion and a=1

• at pc we have anomalous diffusion a=2/3

Disordered Magnets