What is . . .Percolation?

Harry Kesten

Description

WHAT IS...?Percolation?Harry Kesten2Percolation is a simple probabilistic model which v∈Z as the collection of points connected to vexhibits a phase transition (as we explain below). by an open path. The clusters C(v) are the maxi-2The simplest version takes place on Z , which we mal connected components of the collection of2view as a graph with edges between neighboring open edges of Z , and θ(p) is the probability that2vertices. All edges of Z are, independently of eachC(0) is infinite. If pp, then θ(p)> 0 and there is acists an open path, i.e., a path all of whose edges are strictly positive probability that C(0) is infinite.open, from the origin to the exterior of the square An application of Kolmogorov’s zero-one law shows2S := [−n,n] ?” This question was raised by Broad-n that there is then with probability 1 some infinitebent in 1954 at a symposium on Monte Carlo meth- cluster. In fact, it turns out that there is a uniqueods. It was then taken up by Broadbent and Ham- infinite cluster. Thus, the global behavior of the sys-mersley, who regarded percolation as a model for tem is quite different for p

p.c ca random medium. They interpreted the edges of ...

Sujets

• Théorie des graphes
• Percolation

Informations

• Publié par: mtoledan
• Publié le: 24 août 2011
• Nombre de lectures: 58
• Langue: English

Extrait

572

? W H A T

I S. . .

Harry Kesten is emeritus professor of mathematics at Cor- nell University. His e-mail address iskesten@math. cornell.edu.

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site percolation, in contrast to the version we con- sidered so far, and which is calledbond percolation. Initially research concentrated on finding the pre- cise value of p c for various graphs. This has not been very successful; one knows p c only for a few planar lattices (e.g., p c = 1/2 for bond percolation 2 on Z and for site percolation on the triangular lat- tice). The value of p c depends strongly on geo- metric properties of G . Attention has therefore shifted to questions about the distribution of the number of vertices in C(0) and geometric proper- ties of the open clusters when p is close to p c . It is believed that a number of these properties are universal, that is, they depend only on the dimen- sion of G , and not on details of its structure. In particular, one wants to study the behavior of various functions as p approaches p c , or as some other parameter tends to infinity, while p is kept at p c . It is believed that many functions obey so-calledpower laws. For instance, it is believed that the expected number of vertices in C(0) , denoted −γ by χ(p) , behaves like (p c −p) as p↑p c , in the sense that −logχ(p)/log(p c −p)→γ for a suit- able constant γ . Similarly one believes that θ(p) be- β haves like (p−p c ) for some β as p↓p c , or that the probability that there is an open path from 0 −1/ρ to the exterior of S n for p=p c behaves like n for some ρ . Even though such power laws have been proven only for site percolation on the triangular lattice or on high-dimensional lattices, it is be- lieved that the exponents β, γ, ρ , etc. (usually called critical exponents), exist, and in accordance with the universality hypothesis mentioned above depend only on the dimension of G . For instance, bond and 2 site percolation on Z or on the triangular lattice should all have the same exponents. Physicists in- vented the renormalization group to explain and/or prove such power laws and universality, but this has not been made mathematically rigorous for per- colation. d Z for large d behaves in many respects like a regular tree, and for percolation on a regular tree one can easily prove power laws and compute the relevant critical exponents. For bond percolation d on Z with d≥19 Hara and Slade succeeded in proving power laws and in showing that the expo- nents agree with those for a regular tree. They have even shown that their theory applies down to d d >6 when one adds edges to Z between any two sites within distance L 0 of each other for some L 0 =L 0 (d) . Due to this theory we have a reasonable under- standing of high-dimensional percolation. In the last few years Lawler, Schramm, Smirnov, and Werner have proven power laws for site percolation on the triangular lattice and confirmed most of the val- ues for the critical exponents conjectured by physi- cists. Their proof rests on Schramm’s invention of the Stochastic Loewner Evolutions or Schramm

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Loewner Evolutions (SLE) and
(p) . (1,1) on Smirnov’s beautiful proof of the existence and confor- mal invariance properties of certain crossing probabilities. Roughly speaking this says the following: Let D be a “nice” 2 domain in R and let A and B be two arcs in the boundary. p p O c For λ >0 , let P λ (D, A, B) be the probability for p=p c that there exists an open path of Figure 1. Graph of θ . Many aspects site percolation on the trian- of this graph are still conjectural. gular lattice in λD from λA to λB . In fact it is neater to take P λ (D, A, B) as the probability at nection in D from A to B on ( angular lattice. Conformal in Q(D, A, B) :=li  m λ→∞ P λ (D, A, B) Q(D, A, B) = QΦ(D),Φ(A),Φ(B) mal map Φ from D onto Φ(D) . gredients are characterizatio Werner of some SLE process on of properties of its evolution boundary. Conformal invarian conjectured by physicists, and formula for Q(D, A, B) . Smirno orous proof of Cardy’s formula the triangular lattice. Further w Newman (2005)) also has led to limit (in a suitable sense) as λ tern of the random configurati criticality, i.e., for p=p c . Since t processes have led to exciting n ory in their own right, for insta for the intersection probabilitie ian motions (see Lawler (2005)) So far conformal invariance achieved only for site percolat lar lattice. It is perhaps the prin of the subject to prove conforma colation on other two-dimens other related major problem is l a w sa n du n i v e r s a l i t yf o r d -dimensional lattices with 2 unsolved problem of fifteen whether there is an infinite op d cal percolation on Z , d≥3 . I thank Geoffrey Grimmett suggestions.

Further Reading [1] F EDERICO C AMIA and C HARLES M. ing limit of two-dimensional arXiv:math.PR/0504036. [2] G EOFFREY G RIMMETT ,Percolation, se 1999. [3] G REGORY F. L AWLER ,Conformally I the Plane, Amer. Math. Soc., 20

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