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?WHAT IS... Percolation? Harry Kesten Percolation is a simple probabilistic model which v ∈ Z2 as the collection of points connected to v exhibits a phase transition (as we explain below). by an open path. The clusters C(v) are the maxi- 2 The simplest version takes place on Z , which we mal connected components of the collection of view as a graph with edges between neighboring open edges of Z2, and θ(p) is the probability that Z2 vertices. All edges of are, independently of each C(0) is infinite. If p<pc, then θ(p)=0by defini- other, chosen to be open with probability p and tion, so that C(0) is finite with probability 1. It is closed with probability 1 − p. A basic question in not hard to see that in this case all open clusters this model is “What is the probability that there ex- are finite. If p>pc, then θ(p) > 0 and there is a ists 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 shows 2 Sn := [−n, n] ?” This question was raised by Broad- that there is then with probability 1 some infinite bent in 1954 at a symposium on Monte Carlo meth- cluster. In fact, it turns out that there is a unique ods. 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<pc and for p>pc. a random medium. They interpreted the edges of Such a sharp transition in global behavior of a sys- Z2 as channels through which fluid or gas could tem at some parameter value is called a phase tran- flow if the channel was wide enough (an open edge) sition or a critical phenomenon by statistical physi- and not if the channel was too narrow (a closed cists, and the parameter value at which the edge). It was assumed that the fluid would move transition takes place is called a critical value. wherever it could go, so that there is no random- There is an extensive physics literature on such phe- ness in the behavior of the fluid, but all random- nomena. Broadbent and Hammersley proved that ness in this model is associated with the medium. 2 0 <pc < 1 for percolation on Z , so that there is We shall use 0 to denote the origin. A limit as indeed a nontrivial phase transition. Much of the →∞ n of the question raised above is “What is the interest in percolation comes from the hope that probability that there exists an open path from 0 one will be better able to analyze the behavior of to infinity?” This probability is called the percola- various functions near the critical point for the tion probability and denoted by θ(p). Clearly simple model of percolation, with all its built-in in- θ(0) = 0 and θ(1) = 1, since there are no open edges dependence properties, than for other, more com- at all when p =0 and all edges are open when plicated models for disordered media. Indeed, per- p =1. It is also intuitively clear that the function colation is the simplest one in the family of the p → θ(p) is nondecreasing. Thus the graph of θ as so-called random cluster or Fortuin-Kasteleyn mod- a function of p should have the form indicated in els, which also includes the celebrated Ising model Figure 1, and one can define the critical probabil- for magnetism. The studies of percolation and ran- ity by p = sup{p : θ(p)=0}. c dom cluster models have influenced each other. Why is this model interesting? In order to answer Percolation can obviously be generalized to per- this we define the (open) cluster C(v) of the vertex colation on any graph G, even to (partially) directed Harry Kesten is emeritus professor of mathematics at Cor- graphs. One can also consider the model in which nell University. His e-mail address is kesten@math. the vertices are independently open or closed, but cornell.edu. all edges are assumed open. This version is called 572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 site percolation, in contrast to the version we con- Loewner Evolutions (SLE) and θ (p) . (1,1) sidered so far, and which is called bond percolation. on Smirnov’s beautiful proof Initially research concentrated on finding the pre- of the existence and confor- cise value of pc for various graphs. This has not mal invariance properties of been very successful; one knows pc only for a few certain crossing probabilities. planar lattices (e.g., pc =1/2 for bond percolation Roughly speaking this says the on Z2 and for site percolation on the triangular lat- following: Let D be a “nice” 2 tice). The value of pc depends strongly on geo- domain in R and let A and B G metric properties of . Attention has therefore be two arcs in the boundary. p p O c shifted to questions about the distribution of the For λ>0, let Pλ(D,A,B) be number of vertices in C(0) and geometric proper- the probability for p = pc that there exists an open path of ties of the open clusters when p is close to pc. It Figure 1. Graph of θ. Many aspects site percolation on the trian- is believed that a number of these properties are of this graph are still conjectural. universal, that is, they depend only on the dimen- gular lattice in λD from λA to sion of G, and not on details of its structure. λB. In fact it is neater to take In particular, one wants to study the behavior Pλ(D,A,B) as the probability at pc of an open con- D of various functions as p approaches pc, or as nection in from A to B on (1/λ) times the tri- some other parameter tends to infinity, while p is angular lattice. Conformal invariance says that D D kept at pc. It is believed that many functions obey Q( ,A,B) := limλ→∞ Pλ( ,A,B) exists and that so-called power laws. For instance, it is believed that Q(D,A,B)=Q Φ(D), Φ(A), Φ(B) for every confor- the expected number of vertices in C(0), denoted mal map Φ from D onto Φ(D). Further crucial in- −γ gredients are characterizations by Lawler and by χ(p), behaves like (pc − p) as p ↑ pc , in the Werner of some SLE process on a domain by means sense that − log χ(p)/ log(pc − p) → γ for a suit- able constant γ. Similarly one believes that θ(p) be- of properties of its evolution before it hits the β boundary. Conformal invariance had earlier been haves like (p − pc ) for some β as p ↓ pc , or that the probability that there is an open path from 0 conjectured by physicists, and Cardy had given a −1/ρ formula for Q(D,A,B). Smirnov’s work gives a rig- to the exterior of Sn for p = pc behaves like n for some ρ. Even though such power laws have been orous proof of Cardy’s formula for percolation on proven only for site percolation on the triangular the triangular lattice. Further work (see Camia and lattice or on high-dimensional lattices, it is be- Newman (2005)) also has led to a description of the →∞ lieved that the exponents β, γ, ρ, etc. (usually called limit (in a suitable sense) as λ of the full pat- critical exponents), exist, and in accordance with the tern of the random configuration of open paths at universality hypothesis mentioned above depend criticality, i.e., for p = pc. Since their discovery, SLE only on the dimension of G. For instance, bond and processes have led to exciting new probability the- site percolation on Z2 or on the triangular lattice ory in their own right, for instance, to power laws should all have the same exponents. Physicists in- for the intersection probabilities of several Brown- ian motions (see Lawler (2005)). vented the renormalization group to explain and/or So far conformal invariance results have been prove such power laws and universality, but this achieved only for site percolation on the triangu- has not been made mathematically rigorous for per- lar lattice. It is perhaps the principal open problem colation. of the subject to prove conformal invariance for per- Zd for large d behaves in many respects like a colation on other two-dimensional lattices. An- regular tree, and for percolation on a regular tree other related major problem is to establish power one can easily prove power laws and compute the laws and universality for percolation on relevant critical exponents. For bond percolation d-dimensional lattices with 2 ≤ d ≤ 6. Finally, an on Zd with d ≥ 19 Hara and Slade succeeded in unsolved problem of fifteen years’ standing is proving power laws and in showing that the expo- whether there is an infinite open cluster for criti- nents agree with those for a regular tree. They cal percolation on Zd,d ≥ 3. have even shown that their theory applies down to I thank Geoffrey Grimmett for several helpful d>6 when one adds edges to Zd between any two suggestions. sites within distance L0 of each other for some L0 = L0(d) .

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