Percolation theory

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In statistical physics and mathematics, percolation theory describes the behavior of

a network when nodes or links are added. This is a geometric type of phase
transition, since at a critical fraction of addition the network of small, disconnected
clusters merge into significantly larger connected, so-called spanning clusters. The
applications of percolation theory to materials science and in many other disciplines
are discussed here and in the articles network theory and percolation.

Contents

 1Introduction
 2History
 3Computation of the critical parameter
 4Universality
 5Phases
o 5.1Subcritical and supercritical
o 5.2Criticality
 6Different models
 7Applications
o 7.1In biology, biochemistry, and physical virology
o 7.2In ecology
 8See also
 9References
o 9.1Further reading
 10External links

Introduction[edit]

A three-dimensional site percolation graph

Bond percolation in a square lattice from p=0.3 to p=0.52

A representative question (and the source of the name) is as follows. Assume that

some liquid is poured on top of some porous material. Will the liquid be able to make
its way from hole to hole and reach the bottom? This physical question
is modelled mathematically as a three-dimensional network of n × n × n vertices,
usually called "sites", in which the edge or "bonds" between each two neighbors may
be open (allowing the liquid through) with probability p, or closed with probability 1
– p, and they are assumed to be independent. Therefore, for a given p, what is the
probability that an open path (meaning a path, each of whose links is an "open"
bond) exists from the top to the bottom? The behavior for large n is of primary
interest. This problem, called now bond percolation, was introduced in the
mathematics literature by Broadbent & Hammersley (1957),[1] and has been studied
intensively by mathematicians and physicists since then.
In a slightly different mathematical model for obtaining a random graph, a site is
"occupied" with probability p or "empty" (in which case its edges are removed) with
probability 1 – p; the corresponding problem is called site percolation. The question
is the same: for a given p, what is the probability that a path exists between top and
bottom? Similarly, one can ask, given a connected graph at what fraction 1 – p of
failures the graph will become disconnected (no large component).

A 3D tube network percolation determination

The same questions can be asked for any lattice dimension. As is quite typical, it is
actually easier to examine infinite networks than just large ones. In this case the
corresponding question is: does an infinite open cluster exist? That is, is there a path
of connected points of infinite length "through" the network? By Kolmogorov's zero–
one law, for any given p, the probability that an infinite cluster exists is either zero or
one. Since this probability is an increasing function of p (proof
via coupling argument), there must be a critical p (denoted by pc) below which the
probability is always 0 and above which the probability is always 1. In practice, this
criticality is very easy to observe. Even for n as small as 100, the probability of an
open path from the top to the bottom increases sharply from very close to zero to
very close to one in a short span of values of p.

Detail of a bond percolation on the square lattice in two dimensions with percolation probability p = 0.51

History[edit]
The Flory–Stockmayer theory was the first theory investigating percolation
processes.[2]
The history of the percolation model as we know it has its root in the coal industry.
Since the industrial revolution, the economical importance of this source of energy
fostered many scientific studies to understand its composition and optimize its use.
During the 30' and 40', the qualitative analysis by organic chemistry left more and
more room to more quantitative studies. [3]
In this context, the British Coal Utilisation Research Association (BCURA) was
created in 1938. It is a research association funded by the coal mines owners. In
1942, Rosalind Franklin, who then recently graduated in chemistry from the
university of Cambridge, joined the BCURA. She started research on the density and
porosity of coal. During the Second World War, coal was an important strategic
resource. It was used as a source of energy, but also was the main constituent of
gas masks.
Coal is a porous medium. To measure its 'real' density, one was to sink it in a liquid
or a gas whose molecule are small enough to fill its microscopic pores. While trying
to measure the density of coal using several gases (helium, methanol, hexane,
benzene) and as she found different values depending on the used gas, Rosalind
Franklin showed that the pores of coal are made of microstructures of various
lengths that act as a microscopic sieve to discriminate the gases. She also
discovered that the size of these structures depends on the temperature of
carbonation during the coal production. With these research, she obtained a PhD
degree and she left the BCURA in 1946. [4]
In the mid fifties, Simon Broadbent worked in the BCURA as a statistician. Among
other interests, he studied the use of coal in gas masks. One question is to
understand how a fluid can diffuse in the coal pores, modeled as a random maze of
open or closed tunnels. In 1954, during a symposium on Monte Carlo methods, he
asks questions to John Hammersley on the use of numerical methods to analyze this
model. [5]
Broadbent and Hammersley introduced in their article of 1957 a mathematical model
to model this phenomenon, that is percolation.

Computation of the critical parameter[edit]

For most infinite lattice graphs, pc cannot be calculated exactly, though in some
cases pc there is an exact value. For example:

 for the square lattice ℤ2 in two dimensions, pc = 1/2  for

bond percolation, a fact which was an open question for
more than 20 years and was finally resolved by Harry
Kesten in the early 1980s,[6] see Kesten (1982). For site
percolation, the value of pc is not known from analytic
derivation but only via simulations of large lattices. [7]  
 A limit case for lattices in high dimensions is given by
the Bethe lattice, whose threshold is at pc = 1/z − 1  for
a coordination number z. In other words: for the
regular tree of degree ,  is equal to .

 For a random tree-like network without degree-degree

correlation, it can be shown that such network can have
a giant component, and the percolation
threshold (transmission probability) is given by ,
where  is the generating function corresponding to
the excess degree distribution. So, for random Erdős–
Rényi networks of average degree , pc = 1/⟨k⟩ .[8][9][10]
 In networks with low clustering, , the critical point gets
scaled by  such that:
[11]

This indicates that for a given degree distribution, the clustering leads to a larger
percolation threshold, mainly because for a fixed number of links, the clustering
structure reinforces the core of the network with the price of diluting the global
connections. For networks with high clustering, strong clustering could induce the
core–periphery structure, in which the core and periphery might percolate at different
critical points, and the above approximate treatment is not applicable. [12]

Universality[edit]
The universality principle states that the numerical value of pc is determined by the
local structure of the graph, whereas the behavior near the critical threshold, pc, is
characterized by universal critical exponents. For example the distribution of the size
of clusters at criticality decays as a power law with the same exponent for all 2d
lattices. This universality means that for a given dimension, the various critical
exponents, the fractal dimension of the clusters at pc is independent of the lattice
type and percolation type (e.g., bond or site). However, recently percolation has
been performed on a weighted planar stochastic lattice (WPSL) and found that
although the dimension of the WPSL coincides with the dimension of the space
where it is embedded, its universality class is different from that of all the known
planar lattices.[13][14]

Phases[edit]
Subcritical and supercritical[edit]
The main fact in the subcritical phase is "exponential decay". That is, when p < pc,
the probability that a specific point (for example, the origin) is contained in an open
cluster (meaning a maximal connected set of "open" edges of the graph) of
size r decays to zero exponentially in r. This was proved for percolation in three and
more dimensions by Menshikov (1986) and independently by Aizenman & Barsky
(1987). In two dimensions, it formed part of Kesten's proof that pc = 1/2 .[15]
The dual graph of the square lattice ℤ2 is also the square lattice. It follows that, in two
dimensions, the supercritical phase is dual to a subcritical percolation process. This
provides essentially full information about the supercritical model with d = 2. The
main result for the supercritical phase in three and more dimensions is that, for
sufficiently large N, there is[clarification needed] an infinite open cluster in the two-dimensional
slab ℤ2 × [0, N]d − 2. This was proved by Grimmett & Marstrand (1990).[16]
In two dimensions with p < 1/2 , there is with probability one a unique infinite closed
cluster (a closed cluster is a maximal connected set of "closed" edges of the graph).
Thus the subcritical phase may be described as finite open islands in an infinite
closed ocean. When p > 1/2  just the opposite occurs, with finite closed islands in an
infinite open ocean. The picture is more complicated when d ≥ 3 since pc < 1/2 , and
there is coexistence of infinite open and closed clusters for p between pc and 1 − pc.
Criticality[edit]

Zoom in a critical percolation cluster (Click to animate)

Percolation has a singularity at the critical point p = pc and many properties behave
as of a power-law with , near . Scaling theory predicts the existence of critical
exponents, depending on the number d of dimensions, that determine the class of
the singularity. When d = 2 these predictions are backed up by arguments
from conformal field theory and Schramm–Loewner evolution, and include predicted
numerical values for the exponents. Most of these predictions are conjectural except
when the number d of dimensions satisfies either d = 2 or d ≥ 6. They include:

 There are no infinite clusters (open or closed)

 The probability that there is an open path from some
fixed point (say the origin) to a distance
of r decreases polynomially, i.e. is on the order of rα for
some α
o α does not depend on the particular lattice chosen,
or on other local parameters. It depends only on the
dimension d (this is an instance of
the universality principle).
o αd decreases from d = 2 until d = 6 and then stays
fixed.
o α2 = −5/48
o α6 = −1.
 The shape of a large cluster in two dimensions
is conformally invariant.
See Grimmett (1999).[17] In 11 or more dimensions, these facts are largely proved
using a technique known as the lace expansion. It is believed that a version of the
lace expansion should be valid for 7 or more dimensions, perhaps with implications
also for the threshold case of 6 dimensions. The connection of percolation to the lace
expansion is found in Hara & Slade (1990).[18]
In two dimensions, the first fact ("no percolation in the critical phase") is proved for
many lattices, using duality. Substantial progress has been made on two-
dimensional percolation through the conjecture of Oded Schramm that the scaling
limit of a large cluster may be described in terms of a Schramm–Loewner evolution.
This conjecture was proved by Smirnov (2001)[19] in the special case of site
percolation on the triangular lattice.

Different models[edit]
 Directed percolation that models the effect
of gravitational forces acting on the liquid was also
introduced in Broadbent & Hammersley (1957),[1] and
has connections with the contact process.
 The first model studied was Bernoulli percolation. In this
model all bonds are independent. This model is called
bond percolation by physicists.
 A generalization was next introduced as the Fortuin–
Kasteleyn random cluster model, which has many
connections with the Ising model and other Potts
models.
  Bernoulli (bond) percolation on complete graphs is an
example of a random graph. The critical probability
is p = 1/N , where N is the number of vertices (sites) of
the graph.
 Bootstrap percolation removes active cells from clusters
when they have too few active neighbors, and looks at
the connectivity of the remaining cells.[20]
 First passage percolation.
 Invasion percolation.

Applications[edit]
In biology, biochemistry, and physical virology[edit]
Percolation theory has been used to successfully predict the fragmentation of
biological virus shells (capsids),[21][22] with the fragmentation threshold of Hepatitis
B virus capsid predicted and detected experimentally.[23] When a critical number of
subunits has been randomly removed from the nanoscopic shell, it fragments and
this fragmentation may be detected using Charge Detection Mass Spectroscopy
(CDMS) among other single-particle techniques. This is a molecular analog to the
common board game Jenga, and has relevance to the broader study of virus
disassembly. Interestingly, more stable viral particles (tilings with greater
fragmentation thresholds) are found in greater abundance in nature. [21]
In ecology[edit]
Percolation theory has been applied to studies of how environment fragmentation
impacts animal habitats[24] and models of how the plague bacterium Yersinia
pestis spreads.[25]

See also[edit]
 Continuum percolation theory
 Critical exponent – Parameter describing physics near
critical points
 Directed percolation – Physical models of filtering under
forces such as gravity
 Erdős–Rényi model – Two closely related models for
generating random graphs
 Fractal – Infinitely detailed mathematical structure
 Giant component – Large connected component of a
random graph
 Graph theory – Area of discrete mathematics
 Interdependent networks – Subfield of network science
 Invasion percolation
 Kahn–Kalai conjecture – Mathematical proposition
 Network theory – Study of graphs as a representation of
relations between discrete objects
 Network science – Academic field
  Percolation threshold – Threshold of percolation theory
models
 Percolation critical exponents – Mathematical parameter
in percolation theory
 Scale-free network – Network whose degree distribution
follows a power law
 Shortest path problem – Computational problem of
graph theory

References[edit]
1. ^ Jump up to:a b Broadbent, Simon;  Hammersley, John (1957).
"Percolation processes I. Crystals and mazes". Mathematical
Proceedings of the Cambridge Philosophical Society.  53  (3): 629–
641.  Bibcode:1957PCPS...53..629B. doi:10.1017/S03050041000
32680. ISSN 0305-0041.
2. ^ Sahini, M.; Sahimi, M. (2003-07-13).  Applications Of Percolation
Theory. CRC Press.  ISBN  978-0-203-22153-2.
3. ^ van Krevelen, Dirk W (1982). "Development of coal research—a
review". Fuel.  61  (9): 786–790.
4. ^ The rosalind franklin papers - the holes in coal: Research at
BCURA and in Paris, 1942-
1951. https://profiles.nlm.nih.gov/spotlight/kr/feature/coal.
Accessed: 2022-01-17.
5. ^ Hammersley, JM; Welsh, DJA (1980). "Percolation theory and
its ramifications".  Contemporary Physics. 21 (6): 593–605.
6. ^ Bollobás, Béla; Riordan, Oliver (2006). "Sharp thresholds and
percolation in the plane". Random Structures and
Algorithms.  29  (4): 524–
548.  arXiv:math/0412510. doi:10.1002/rsa.20134. ISSN 1042-
9832.  S2CID 7342807.
7. ^ MEJ Newman; RM Ziff (2000). "Efficient Monte Carlo algorithm
and high-precision results for percolation". Physical Review
Letters.  85  (19): 4104–
4107.  arXiv:cond-mat/0005264. Bibcode:2000PhRvL..85.4104N. 
doi:10.1103/physrevlett.85.4104.  PMID  11056635. S2CID  74766
5.
8. ^ Erdős, P. & Rényi, A. (1959). "On random graphs I.". Publ.
Math. (6): 290–297.
9. ^ Erdős, P. & Rényi, A. (1960). "The evolution of random
graphs". Publ. Math. Inst. Hung. Acad. Sci. (5): 17–61.
10. ^ Bolloba's, B. (1985). "Random Graphs".  Academic.
11. ^ Berchenko, Yakir; Artzy-Randrup, Yael; Teicher, Mina; Stone,
Lewi (2009-03-30). "Emergence and Size of the Giant Component
in Clustered Random Graphs with a Given Degree
Distribution". Physical Review Letters.  102  (13):
138701. doi:10.1103/PhysRevLett.102.138701.  ISSN  0031-9007.
12. ^ Li, Ming; Liu, Run-Ran; Lü, Linyuan; Hu, Mao-Bin; Xu, Shuqi;
Zhang, Yi-Cheng (2021-04-25). "Percolation on complex
networks: Theory and application". Physics Reports. Percolation
on complex networks: Theory and application.  907: 1–
68.  doi:10.1016/j.physrep.2020.12.003. ISSN 0370-1573.
13. ^ Hassan, M. K.; Rahman, M. M. (2015). "Percolation on a
multifractal scale-free planar stochastic lattice and its universality
class". Phys. Rev. E. 92 (4):
040101. arXiv:1504.06389. Bibcode:2015PhRvE..92d0101H. doi:
10.1103/PhysRevE.92.040101. PMID 26565145.  S2CID 1191122
86.
14. ^ Hassan, M. K.; Rahman, M. M. (2016). "Universality class of site
and bond percolation on multi-multifractal scale-free planar
stochastic lattice".  Phys. Rev. E.  94  (4):
042109. arXiv:1604.08699. Bibcode:2016PhRvE..94d2109H. doi:
10.1103/PhysRevE.94.042109. PMID 27841467.  S2CID 2259302
8.
15. ^ Kesten, Harry  (1982). Percolation Theory for Mathematicians.
Birkhauser. doi:10.1007/978-1-4899-2730-9. ISBN 978-0-8176-
3107-9.
16. ^ Grimmett, Geoffrey; Marstrand, John (1990). "The Supercritical
Phase of Percolation is Well Behaved". Proceedings of the Royal
Society A: Mathematical, Physical and Engineering
Sciences.  430  (1879): 439–
457.  Bibcode:1990RSPSA.430..439G. doi:10.1098/rspa.1990.010
0. ISSN 1364-5021.  S2CID 122534964.
17. ^ Grimmett, Geoffrey  (1999). Percolation. Grundlehren der
mathematischen Wissenschaften. Vol. 321. Berlin:
Springer. doi:10.1007/978-3-662-03981-6. ISBN 978-3-642-
08442-3.  ISSN  0072-7830.
18. ^ Hara, Takashi; Slade, Gordon (1990). "Mean-field critical
behaviour for percolation in high dimensions".  Communications in
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391.  Bibcode:1990CMaPh.128..333H.  doi:10.1007/BF02108785. 
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19. ^ Smirnov, Stanislav (2001). "Critical percolation in the plane:
conformal invariance, Cardy's formula, scaling limits".  Comptes
Rendus de l'Académie des Sciences. I. 333 (3): 239–
244.  arXiv:0909.4499.  Bibcode:2001CRASM.333..239S.  CiteSeer
X 10.1.1.246.2739. doi:10.1016/S0764-4442(01)01991-7.  ISSN  0
764-4442.
20. ^ Adler, Joan (1991), "Bootstrap percolation", Physica A:
Statistical Mechanics and Its Applications, 171 (3): 453–
470,  Bibcode:1991PhyA..171..453A, doi:10.1016/0378-
4371(91)90295-n.
21. ^ Jump up to:a b Brunk, Nicholas E.; Twarock, Reidun
(2021).  "Percolation Theory Reveals Biophysical Properties of
Virus-like Particles". ACS Nano. American Chemical Society
(ACS).  15  (8): 12988–
12995. doi:10.1021/acsnano.1c01882.  ISSN  1936-0851. PMC  83
97427. PMID 34296852.
22. ^ Brunk, N. E.; Lee, L. S.; Glazier, J. A.; Butske, W.; Zlotnick, A.
(2018).  "Molecular Jenga: the percolation phase transition
(collapse) in virus capsids".  Physical Biology. 15 (5):
056005. Bibcode:2018PhBio..15e6005B. doi:10.1088/1478-
3975/aac194.  PMC 6004236. PMID 29714713.
23. ^ Lee, L. S.; Brunk, N.; Haywood, D. G.; Keifer, D.; Pierson, E.;
Kondylis, P.; Zlotnick, A. (2017).  "A molecular breadboard:
Removal and replacement of subunits in a hepatitis B virus
capsid".  Protein Science. 26 (11): 2170–
2180.  doi:10.1002/pro.3265. PMC  5654856.  PMID  28795465.
24. ^ Boswell, G. P.; Britton, N. F.; Franks, N. R. (1998-10-
22).  "Habitat fragmentation, percolation theory and the
conservation of a keystone species".  Proceedings of the Royal
Society of London B: Biological Sciences.  265  (1409): 1921–
1925.  doi:10.1098/rspb.1998.0521. ISSN 0962-8452.  PMC 1689
475.
25. ^ Davis, S.; Trapman, P.; Leirs, H.; Begon, M.; Heesterbeek, J. a.
P. (2008-07-31). "The abundance threshold for plague as a critical
percolation phenomenon".  Nature.  454  (7204): 634–
637.  Bibcode:2008Natur.454..634D. doi:10.1038/nature07053. hd
l:1874/29683.  ISSN  1476-4687. PMID 18668107.  S2CID 442520
3.
  Aizenman, Michael; Barsky, David (1987), "Sharpness
of the phase transition in percolation
models", Communications in Mathematical
Physics, 108 (3): 489–
526, Bibcode:1987CMaPh.108..489A, doi:10.1007/BF0
1212322, S2CID 35592821
 Menshikov, Mikhail (1986), "Coincidence of critical
points in percolation problems", Soviet Mathematics -
Doklady, 33: 856–859
Further reading[edit]
 Austin, David (July 2008). "Percolation: Slipping through the Cracks".
American Mathematical Society.
 Bollobás, Béla; Riordan, Oliver (2006). Percolation. Cambridge
University Press. ISBN 978-0521872324.
 Kesten, Harry (May 2006). "What Is ... Percolation?"  (PDF).  Notices of
the American Mathematical Society. 53 (5): 572–573. ISSN 1088-
9477.

External links[edit]
 PercoVIS: a Mac OS X program to visualize percolation
on networks in real time
 Interactive Percolation
 Nanohub online course on Percolation Theory
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ck–Karasinski

ck–Scholes

an–Karolyi–Longstaff–Sanders (CKLS)

en

nstant elasticity of variance (CEV)

x–Ingersoll–Ross (CIR)

man–Kohlhagen

ath–Jarrow–Morton (HJM)

ston

–Lee

l–White

BOR market

ndleman–Bartter

BR volatility

šíček

kie

hlmann

mér–Lundberg

k process

rre–Anderson

id

neralized queueing network

G/1

M/1

M/c

dlàg paths

ntinuous

ntinuous paths

odic
changeable

ler-continuous

uss–Markov

rkov

xing

cewise deterministic

dictable

gressively measurable

f-similar

tionary

me-reversible

ntral limit theorem

nsker's theorem

ob's martingale convergence theorems

odic theorem

her–Tippett–Gnedenko theorem

ge deviation principle

w of large numbers (weak/strong)

w of the iterated logarithm

ximal ergodic theorem

ov's theorem

o–one laws (Blumenthal, Borel–Cantelli, Engelbert–Schmidt, Hewitt–Savage, Kolmogorov, Lévy)

kholder–Davis–Gundy

ob's martingale

ob's upcrossing

nita–Watanabe

rcinkiewicz–Zygmund

meron–Martin formula

nvergence of random variables

éans-Dade exponential

ob decomposition theorem

ob–Meyer decomposition theorem

ob's optional stopping theorem

nkin's formula

nman–Kac formula

ration

sanov theorem
nitesimal generator

integral

s lemma

hunen–Loève theorem

mogorov continuity theorem

mogorov extension theorem

vy–Prokhorov metric

lliavin calculus

rtingale representation theorem

ional stopping theorem

khorov's theorem

adratic variation

lection principle

orokhod integral

orokhod's representation theorem

orokhod space

ll envelope

chastic differential equation 

naka

pping time

atonovich integral

form integrability

ual hypotheses

ener space 
ssical

stract

uarial mathematics

ntrol theory

onometrics

odic theory

reme value theory (EVT)

ge deviations theory

thematical finance

thematical statistics

bability theory

eueing theory

newal theory
 n theory

nal processing

tistics

chastic analysis

me series analysis

chine learning

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