1. Introduction to Percolation
산돌광수체
Introduction to Percolation
: basic concept and something else
Seung-Woo Son
Complex System and Statistical Physics Lab.

2. Index Basic Concept of the Percolation Lattice and Lattice animals
Bethe Lattice ( Cayley Tree )
Percolation Threshold
Cluster Numbers & Exponents
Small Cell Renormalization
Continuum Percolation

3. What is Percolation? -_-; ? Giant cluster Square lattice
percolation n.     1 여과; 삼출, 삼투 2 퍼컬레이션 ((퍼컬레이터로 커피 끓이기))
사전적 의미(?)
-_-; ?
Cluster
Giant cluster
Square
lattice
The number and properties
of clusters ?
통계물리학
Percolation
- First discussed by Hammersley in 1957

4. Other fun example
Let's consider a 2D network as shown in left figure. The communication network, represented by a very large square-lattice network of interconnections, is attacked by a crazed saboteur who, armed with wire cutters, proceeds to cut the connecting links at random.
Q. What fraction of the links(or bonds) must be cut in order to electrically isolate the two boundary bars?
A. 50%

5. Threshold concentration
P = 0.6
P = 0.5
Threshold concentration ( ) = ( 2D square site )

6. Examples of percolation in real world
Water molecule in a coffee percolator
Oil in a porous rock & ground water
Forest fires
Gelation of boiled egg & hardening of cement
Insulator - conductor transition

7. Forest fires L  L lattice Tree Burned tree Burning tree Empty hole
A green tree is ignited and becomes red if
it neighbors another red tree which at that
time is still burning.
Thus a just-ignited tree ignites its right and
bottom neighbor within same sweep through
the lattice, its top and left neighbor tree at
the next sweep.
Average termination time for forest fires, as simulated on a square lattice.
The center curve corresponds to the simplest case. p =

8. Oil fields and Fractals
Percolation can be used as an idealized simple model for the
distribution of oil or gas inside porous rocks in oil reservoirs.
The average concentration of oil concentration of oil in the rock
is represented by the occupation probability p. ( porosity )
bad investment !
광수생각
p < pc
It will most probably hit a small cluster.
They must take out rock samples from the well !!
5~10 cm diameter long rock logs sample  extrapolate to the reservoir scale.
M(L) - how many points within this frame belong to the same cluster  L2
Average density of points P = M(L)/L2 is independent of L.
But near pc …
M(L)  L1.9  fractal dimension D = 1.9 is not equal to Euclidean dimension 2.
So…
Average density decays as L For 100km size, (106)-0.1 ~ 0.25
Remaining 75% can’t directly extract.

9. Bond percolation & site percolation
Site percolation is dealt more frequently, even though bond percolation historically came first.
Site-bond percolation(?)

10. Lattice & dimension
Square lattice, triangular lattice, honeycomb lattice – 2D
Simple cubic, body-centered cubic, face-centered cubic, diamond lattice -3D
Hypercubic lattice – higher than 3

11. Percolation thresholds
In finite systems as simulated on a computer one does not have in general a sharply defined threshold; any effective threshold values obtained numerically or experimentally need to be extrapolated carefully to infinite system size.
Thermodynamic limit - physicist
Mathematically exact ? ^^;

12. Exact solution It’s very simple example.
1D case
the number of s-clusters per lattice site (normalized cluster number)
: the probability that an arbitrary site is part of an s-cluster
for one dimension. It’s trivial.
average cluster size
correlation function (pair connectivity)
- the probability that a site a distance r apart from an occupied site belongs to the
same cluster. ex)
correlation(connectivity) length
The correlation length is proportional to a typical cluster diameter.
Unfortunately the higher dimension,
the more complicated.

13. Animals in d Dimensions
Animals in d Dimensions
= fixed polyomino 1 2 3 4 5
2-dimension square lattice animals
monomino
domino
It is nice exercise to find
all 63 configurations for s=5.
^^;
triomino
tetromino
pentomino
For s=4, 19 possible configurations
exponentially increase !

14. Perimeter It is difficult to sum over all possible perimeter t.
Perimeter – the number of empty neighbors of a cluster. ( t ) c.f. cluster surface
- the number of lattice animals (cluster configurations) with size s and
perimeter t
It is difficult to sum over all possible perimeter t.
Perimeter polynomial
There seems to be no exact solution for general t and s available at present.
Asymptotic result…
The perimeter t, averaged over all animals with a given size s, seems to be
proportional to s for s  .
It is appropriate to classify different animals of the same large size s by the
ratio a = t/s . If a is smaller than (1-pc)/pc , then gst varies as

15. Bethe lattice ( Cayley tree )
z = 3
Bethe lattice ( Cayley tree )
A tree in which each non-leaf graph vertex has a constant number of branches n is called an n-Cayley tree. 2-Cayley trees are path graphs. The unique n-Cayley tree on nodes is the star graph.
star graph
Path graph

16. Exact percolation threshold Pc
Bethe lattice ( with z branch ) =
1 D chain = 1
square bond percolation = 1/2
triangular site percolation = 1/2
triangular bond percolation =
honeycomb bond percolation =
honeycomb site percolation  1/2
For square site percolation and 3D percolation, no plausible guess for exact result.
Next will be more serious calculation… -_-;
It will border you and me.

17. Power law behavior near Pc
briefly~!
: density of clusters of size s
number of clusters of size s per lattice site
For
1st moment of cluster size
: probability that any given site belongs to the infinite cluster

18. Power law behavior near Pc
briefly~!!
: average cluster size S (Percolation susceptibility)
2nd moment of cluster size
Percolation specific heat
zeroth moment of cluster size
Consider the Gibbs free energy as the singular part of the zeroth moment of
cluster size distribution.
Percolation correlation length

19. Scaling relation Near p = pc
Exact results on a Bethe Lattice ( Cayley tree )
These are the results in the limit of d   !!

20. Exponents
Universality !!

21. Small cell renormalization
Rescale bb cell into 11 cell
Spanning probability :
bb cell
11 cell b
Recursion
relation b
Fixed point :
Correlation length
bb cell :
11 cell :
1D case…
fixed point

22. Small cell renormalization
33 triangular lattice
Recursion relation
Fixed point
22 square lattice bond percolation (?)

23. Continuum percolation
Fully penetrable sphere model
Swiss cheese model
Inverse Swiss cheese model
Equi-sized particles of diameter σ are
distributed randomly in a system of
side L σ.
Penetrable concentric shell model
Particles of diameter σ contain
impenrable core of diameter λσ
Randomly bonded percolation
bonding probability
Adhesive sphere model

24. Universality Class All exponents of the continuum
percolation models with short-
range interactions were found to
be the same as for the lattice
percolation.
For overlapping disks :
For interacting particles :

25. Summary Basic Concept of the Percolation Lattice and Lattice animals
Bethe Lattice ( Cayley Tree )
Percolation Threshold
Cluster Numbers & Exponents
Small Cell Renormalization
Continuum Percolation
Dynamics

26. Reference
Dietrich Stauffer and Amnon Aharony, Introduction to Percolation Theory 2nd (1994)
Hoshen-Kopelman algorithm
J. Hoshen and R. Kopelman, PRB 14, 3438 (1976)
Review of the renomalization
M. E. Fisher, Rev. Mod. Phys. 46, 597 (1974)
S. K. Ma, Rev. Mod. Phys. 45, 589 (1973)
M. E. Fisher, Lecture notes in Physics (1983)
Renormalization for percolation
P. J. Reynolds, Ph. D. Thesis (MIT)
P. J. Reynolds, H. E. Stanley, and W. Klein, Phys. Rev. B 21, 1223 (1980)
For continuum percolation models
D. Y. Kim et al. PRB 35, 3661 (1987)
I. Balberg, PRB 37, 2361 (1988)
Lee and Torquato, PRA 41, 5338 (1990)

28. Finite size scaling

29. Dynamics ?