1. 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

2. 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

3. Finite size scaling At p c 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

4. Spanning cluster Tenuous and not compact

5. Fractal Dimension Recall for a uniform circle or sphere M(R) ~ R D if D=d (Euclidean dimension) then the object is compact density  (R)=M(R)/R d ~ R D-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

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

7. 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

8. Regular Fractals Again b=2 but now the number of new units is N=4 N=b 2 in general dimension we have N=b d hence d = ln(N)/ln(b)

9. 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 Curve Koch

10. Spanning Cluster The spanning cluster at p c 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

11. 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 Percolation clusters

12. 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 L d

13. Fractal Dimension

14. Renormalization Group The critical exponents characterizing the geometrical phase transition can be obtained by simulating percolation clusters near p c and examining the various properties on different length scales L another method of examining the system on different length scales is the renormalization group

15. Scaling Consider a photograph of a percolation configuration generated at p=p 0 <p c now consider viewing it from further distances

17. For p 0 at large length scales For p=1, p’ =1 at all length scales p c = 1 1-d chain

18. RG

19. Percolation and Random Walks 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 depend on the number of steps N ? For p < p c all walkers are confined to finite clusters for p > p c most walkers have  N “normal” diffusion at p=p c we have anomalous diffusion

20. Anomalous Diffusion when p=p c, only a few walkers are not localized let a walker start on a cluster and walk the mean square displacement ~ N a for p < p c all clusters are finite and thus a=0 for p>p c we have normal diffusion and a=1 at p c we have anomalous diffusion a=2/3

21. Disordered Magnets