[Math.PR] 11 Jul 2005 Percolation Theory

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[0 = then , : stesto l pnvrie hc are which vertices open all of set the is = = C , v P ]on 1] { p ( and 0 p p ya pnpath: open an by v u oe if -open { c ) { ↔ } ( 0 p G A = oe etcsi non-decreasing is vertices -open ∞} ↔ ) θ ↔ G { = ( θ u Let : p (1) v B snndcesn swell. as non-decreasing is ) sup ∈ } sdfie as defined is : U en htteeeit an exists there that means The . V = = G ( { { v oee,st perco- site However, . 1. : p P U ) P p u : ≤ p ( p p {| v A θ pncutrC cluster open hs transition phase ↔ c o each for ) < C ( p ∈ , osm etxof vertex some to u p tews tis it otherwise , ( v v ) 0 [0 p ↔ } ∈ ) = c | . , = V ] uhthat such 1], and θ 0 v G } } ∞} ilstand will sanon- a is . p sequiv- is u , einde- be ∈ p ] De- 1]. p . and -open ( [0 > v , of ) , i.e. p 1]. v c . , θ(p) 1.1 The sub-critical regime 1 When p < pc, all open clusters are finite almost surely. One of the greatest challenges in percolation theory has been to prove that χ(p) := Ep{|C(v)|} is finite if p < pc (Ep stands for the ex- pectation with respect to Pp). For that one can p define another critical probability as the threshold 0 value for the finiteness of the expected cluster size pc 1 of a fixed vertex:

Figure 1: The behavior of θ(p) around the critical pT(G):= sup{p : χ(p) < ∞}. point (for bond-percolation) It was an important step in the development of the theory to show that pT(G) = pc(G). The fun- damental estimate in the subcritical regime, which By definition, when p < pc the open cluster of the is a much stronger statement than pT(G) = pc(G), origin is Pp-a.s. finite, hence all the clusters are also is the following: finite. On the other hand, for p > pc there is a strictly positive Pp-probability that the cluster of Theorem 1 (Aizenman and Barsky, Menshikov) the origin is infinite. Thus, from Kolmogorov’s Assume that G is periodic. Then for p < pc there exist zero-one law it follows that constants 0 < C1, C2 < ∞, such that

P {|C v | ≥ n} ≤ C e−C2n. Pp{|C(v)| = ∞ for some v ∈ V} = 1 for p > pc. p ( ) 1 The last statement can be sharpened to a “local Therefore, if the intervals [0, pc)and(pc, 1] are both limit theorem” with the help of a subadditivity ar- non-empty, there is a phase transition at pc. gument : For each p < pc there exists a constant Using a so-called Peierls argument it is easy to 0 < C3(p) < ∞, such that see that pc(G) > 0 for any graph G of bounded de- gree. On the other hand, Hammersley proved that 1 = = d lim − log Pp{|C(v)| n} C3(p). pc(Z ) < 1 for bond percolation as soon as d ≥ 2, n→∞ n and a similar argument works for site percolation and various periodic graphs as well. But for some 1.2 The super-critical regime graphs G it is not so easy to show that pc(G) < 1. One says that the system is in the subcritical (resp. Once an infinite open cluster exists, it is natural to ask how it looks like, and how many infinite supercritical) phase if p < pc (resp. p > pc). open clusters exist. It was shown by Newman and It was one of the most remarkable moments in Schulman that for periodic graphs, for each p, ex- the history of percolation when Kesten proved [10] actly one of the following three situations prevails: that the critical parameter for bond-percolation on If N ∈ Z+ ∪ {∞} is the number of infinite open Z2 is equal to 1/2. Nevertheless the exact value of clusters, then P (N = 0) = 1, or P (N = 1) = 1, or p (G) is known only for a handful of graphs, all of p p c P (N = ∞) = 1. them periodic and two-dimensional — see below. p Aizenman, Kesten and Newman showed that the third case is impossible on Zd. By now several proofs exist, perhaps the most elegant proof of that d 1 Percolation in Z is due to Burton and Keane, who prove that indeed there cannot be infinitely many infinite open clus- The graph on which most of the theory was origi- ters on any amenable graph. However, there are nally built is the cubic lattice Zd, and it was not be- some graphs, such as regular trees, on which coex- fore the late 20th century that percolation was se- istence of several infinite clusters is possible. riously considered on other kinds of graphs (such The geometry of the infinite open cluster can be as e.g. Cayley graphs), on which specific phenom- explored in some depth by studying the behavior ena can appear, such as the coexistence of multi- of a random walk on it. When d = 2, the random ple infinite clusters for some values of the param- walk is recurrent, and when d ≥ 3 is a.s. transient. eter p. In all this section, the underlying graph is In all dimensions d ≥ 2 the walk behaves diffu- thus assumed to be Zd for d ≥ 2, although most sively, and the Central Limit Theorem and the In- of the results still hold in the case of a periodic d- variance principle were established in both the an- dimensional lattice. nealed and quenched cases.

2 Wulff droplets and simultaneously showed that the slab percolation and half-space percolation thresholds coin- In the supercritical regime, aside from the infinite cide. This was complemented by Grimmett and open cluster, the configuration contains finite clus- Marstrand showing that ters of arbitrary large sizes. These large finite open clusters can be thought of as droplets swimming d pc(slab) = pc(Z ). in the areas surrounded by an infinite open cluster. The presence at a particular location of a large finite cluster is an event of low probability, namely, 1.3.2 Critical exponents Zd on , d ≥ 2, for p > pc, there exist positive con- In the sub-critical regime, exponential decay of the stants 0 < C4(p), C5(p) < ∞, such that correlation indicates that there is a finite correlation length ξ(p) associated to the system, and defined 1 C (p) ≤ − log Pp{|C(v)| = n} ≤ C (p) (up to constants) by the relation 4 n(d−1)/d 5 nϕ(x) for all large n. This estimate is based on the fact Pp(0 ↔ nx) ≈ exp − that the occurrence of a large finite cluster is due to ξ(p) a surface effect. The typical structure of the large where ϕ is bounded on the unit sphere (this is finite cluster is described by the following theo- known as Ornstein-Zernike decay). The phase tran- rem: sition can then also be defined in terms of the di- vergence of the correlation length, leading again Theorem 2 Let d ≥ 2, and p > pc. There exists a bounded, closed, convex subset W of Rd containing to the same value for pc; the behavior at or near the origin, called the normalized Wulff crystal of the the critical point then has no finite characteristic Bernoulli percolation model, such that, under the condi- length, and gives rise to scaling exponents (conjec- d turally in most cases). tional probability Pp{· | n ≤ |C(0)| < ∞}, the random measure The most usual critical exponents are defined 1 as follows, if θ(p) is the percolation probability, C ∑ δ d x/n the cluster of the origin, and ξ(p) the correlation n x∈C(0) length: (where δx denotes a Dirac mass at x) converges weakly 3 in probability towards the random measure θ(p)½ (x − ∂ W E [|C|−1] ≈ |p − p |−1−α M)dx (where M is the rescaled center of mass of the ∂ p3 p c cluster C(0)). The deviation probabilities behave as θ(p) ≈ (p − p )β exp{−cnd−1} (i.e. they exhibit large deviations of sur- c + f −γ face order). χ (p):= Ep[|C|½|C|<∞] ≈ |p − pc| P [|C| = n] ≈ n−1−1/δ This result was proved in dimension 2 by pc 2−d−η Alexander, Chayes and Chayes [1], and in dimen- Ppc [x ∈ C] ≈ |x| ν sions 3 and more by Cerf [5]. ξ(p) ≈ |p − pc| = −1−1/ρ Ppc [diam(C) n] ≈ n 1.3 Percolation near the critical point k+1 Ep[|C| ½|C|<∞] −∆ k ≈ |p − pc| 1.3.1 Percolation in slabs Ep[|C| ½|C|<∞] The main macroscopic observable in percolation is These exponents are all expected to be univer- θ(p), which is positive above p , 0 below p , and c c sal, i.e. to depend only on the dimension of the continuous on [0, 1] \ {p }. Continuity at p is an c c lattice, although this is not well understood at the open question in the general case; it is known to mathematical level; the following scaling relations hold in two dimensions (cf. below) and in high between the exponents are believed to hold: enough dimension (at the moment d ≥ 19 though the value of the critical dimension is believed to be 2 − α = γ + 2β = β(δ + 1), ∆ = δβ, γ = ν(2 − η). 6) using lace expansion methods. The conjecture = that θ(pc) 0 for 3 ≤ d ≤ 18 remains one of the In addition, in dimensions up to dc = 6, two addi- major open problems. tional hyperscaling relations involving d are strongly Efforts to prove that led to some interesting and conjectured to hold: important results. Barsky, Grimmett and New- man solved the question in the half-space case, dρ = δ + 1, dν = 2 − α,

3 while above dc the exponents are believed to take of pc in some cases, and for the derivation of a pri- their mean-ﬁeld value, i.e. the ones they have for ori bounds for the probability of crossing events percolation on a regular tree: at or near the critical point, leading to the fact that θ(pc) = 0. On another front, the scaling limit of crit- α = −1, β = 1, γ = 1, δ = 2, ical site-percolation on the two-dimensional triangular lattice can be described in terms of SLE pro- 1 1 η = 0, ν = , ρ = , ∆ = 2. cesses. 2 2

Not much is known rigorously on critical exponents in the general case. Hara and Slade ([9]) proved that mean ﬁeld behavior does hap- pen above dimension 19, and the proof can likely be extended to treat the case d ≥ 7. In the two- dimensional case on the other hand, Kesten ([13]) showed that, assuming that the exponents δ and ρ exist, then so do β, γ, η and ν, and they satisfy the scaling and hyperscaling relations where they appear.

1.3.3 The incipient infinite cluster When studying long-range properties of a critical model, it is useful to have an object which is infinite at criticality, and such is not the case for percolation clusters. There are two ways to condition the cluster of the origin to be infinite when p = pc: The first one is to condition it to have diameter at least n (which happens with positive probability) and take a limit in distribution as n goes to infinity; the second one is to consider the model for parameter p > pc, condition the cluster of 0 to be Figure 2: Two large critical percolation clusters in infinite (which happens with positive probability) a box of the square lattice (first: bond-percolation, and take a limit in distribution as p goes to pc. The second: site-percolation) limit is the same in both cases, it is known as the incipient infinite cluster. As in the super-critical regime, the structure of 2.1 Duality, exact computations and the cluster can be investigated by studying the be- RSW theory havior of a random walk on it, as was suggested by de Gennes; Kesten proved that in two dimensions, Given a planar lattice L, define two associated the random walk on the incipient infinite cluster graphs as follows. The dual lattice L ′ has one ver- is sub-diffusive, i.e. the mean square displacement tex for each face of the original lattice, and an edge after n steps behaves as n1−ε for some ε > 0. between two vertices if and only if the correspond- The construction of the incipient infinite clus- ing faces of L share an edge. The star graph L∗ is ter was done by Kesten in two dimensions [12], obtained by adding to L an edge between any two and a similar construction was performed recently vertices belonging to the same face (L∗ is not pla- in high dimension by Van der Hofstad and Jarai nar in general; (L, L∗) is commonly known as a ([20]). matching pair). Then, a result of Kesten is that, under suitable technical conditions, pbond L + pbond L ′ = psite L + psite L∗ = . 2 Percolation in two dimensions c ( ) c ( ) c ( ) c ( ) 1 Two cases are of particular importance: The lat- As is the case for several other models of statistical tice Z2 is isomorphic to its dual; the triangular lat- physics, percolation exhibits many specific prop- tice T is its own star graph. It follows that erties when considered on a two-dimensional lat- 1 pbond(Z2) = psite(T ) = . tice: Duality arguments allow for the computation c c 2

4 The only other critical parameters that are known 2. The limit is conformally invariant, in the follow- bond Φ Ω exactly are pc (T ) = 2 sin(π/18) (and hence also ing sense: If is a conformal map from to bond ′ bond Ω′ = Φ Ω ′ pc for T , i.e. the hexagonal lattice), and pc some other domain ( ), and maps a to a , ′ ′ ′ Ω for the bow-tie lattice which is a root of the equa- b to b , ctoc and d to d , then f0(ab, cd; ) = 5 3 2 ′ ′ ′ ′ Ω′ tion p − 6p + 6p + p − 1 = 0. The value of the f0(a b , c d ; ); critical parameter for site-percolation on Z2 might on the other hand never be known, it is even pos- 3. In the particular case when Ω is an equilateral tri- sible that it is “just a number” without any other angle of side length 1 and vertices a, b and c, and signification. if d is on (ca) at distance x ∈ (0, 1) from c, then f (ab, cd; Ω) = x. Still using duality, one can prove that the prob- 0 ability, for bond-percolation on the square lattice Point 3. in particular is essential since it allows to with parameter p = 1/2, that there is a connected compute the limiting crossing probabilities in any component crossing an (n + 1) × n rectangle in the conformal rectangle. In the original work of Cardy, longer direction is exactly equal to 1/2. This and he made his prediction in the case of a rectangle, clever arguments involving the symmetry of the for which the limit involves hypergeometric func- lattice lead to the following result, proved inde- tions; the remark that the equilateral triangle gives pendently by Russo and by Seymour and Welsh rise to nicer formulae is originally due to Carleson. and known as the RSW theorem: To precisely state the convergence of percola- Theorem 3 (Russo [15]; Seymour-Welsh [17]) For tion to its scaling limit, define the random curve every a, b > 0 there exist η > 0 and n0 > 0 such that known as the percolation exploration path (see fig. 3) for every n > n0, the probability that there is a cluster as follows: In the upper half-plane, consider a site- crossing an ⌊na⌋ × ⌊nb⌋ rectangle in the first direction percolation model on a portion of the triangular is greater than η. lattice and impose the boundary conditions that on the negative real half-line all the sites are open, The most direct consequence of this estimate is while on the other half-line the sites are closed. that the probability that there is a cluster going The exploration curve is then the common bound- around an annulus of a given modulus is bounded ary of the open cluster spanning from the nega- below independently of the size of the annulus; tive half-line, and the closed cluster spanning from in particular, almost surely there is some annulus the positive half-line; it is an infinite, self-avoiding around 0 in which this happens, and that is what random curve in the upper half-plane. allows to prove that θ(pc) = 0 for bond-percolation on Z2.

2.2 The scaling limit RSW-type estimates give positive evidence that a scaling limit of the model should exist; it is indeed essentially sufficient to show convergence of the crossing probabilities to a non-trivial limit as n goes to infinity. The limit, which should depend only on the ration a/b, was predicted by Cardy using conformal fields theory methods. A most celebrated result of Smirnov is the proof of Cardy’s formula in the case of site-percolation on the trian- Figure 3: A percolation exploration path gular lattice T :

Theorem 4 (Smirnov [18]) Let Ω be a simply con- As the mesh of the lattice goes to 0, the ex- nected domain of the plane with four points a, b, c, d (in ploration curve then converges in distribution to that order) marked on its boundary. For every δ > 0, the trace of an SLE process, as introduced by = consider a critical site-percolation model on the inter- Schramm, with parameter κ 6 — see ﬁg. 4. The limiting curve is not simple anymore (which cor- section of Ω with δT and let fδ(ab, cd; Ω) be the probability that it contains a cluster connecting the arcs ab responds to the existence of pivotal sites on large and cd. Then: critical percolation clusters), and it has Hausdorff dimension 7/4. For more details on SLE processes, Ω Ω 1. fδ(ab, cd; ) has a limit f0(ab, cd; ) as δ → 0; see e.g. the related entry in the present volume.

5 3.1 Percolation on non-amenable graphs The first modification of the model one can think of is to modify the underlying graph and move away from the cubic lattice; phase transition still occurs, and the main difference is the possibility for infinitely many infinite clusters to coexist. On a regular tree, such is the case whenever p ∈ (pc, 1), the first non-trivial example was produced by Grimmett and Newman as the product of Z by a tree: There, for some values of p the infinite cluster is unique, while for others there is coexistence of infinitely many of them. The corresponding definition, due to Benjamini and Schramm, is then the following: If N is as above the number of infinite open clusters, = = = Figure 4: An SLE process with parameter κ = 6 pu : inf p : Pp(N 1) 1 ≥ pc. (infinite time, with the driving process stopped at The main question is then to characterize graphs time 1) on which 0 < pc < pu < 1. A wide class of interesting graphs is that of Cay- As an application of this convergence result, one ley graphs of infinite, finitely generated groups. can prove that the critical exponents described in There, by a simultaneous result by Häggström and the previous section do exist (still in the case of the Peres and by Schonmann, for every p ∈ (pc, pu) triangular lattice), and compute their exact values, there are Pp-a.s. infinitely many infinite cluster, except for α, which is still listed here for complete- while for every p ∈ (pu, 1] there is only one — note ness: that this does not follow from the definition since new infinite components could appear when p is 2 5 43 91 α = − , β = , γ = , δ = , increased. It is conjectured that pc < pu for any 3 36 18 5 Cayley graph of a non-amenable group (and more generally for any quasi-transitive graph with pos- 5 4 48 91 η = , ν = , ρ = , ∆ = . itive Cheeger constant), and a result by Pak and 24 3 5 36 Smirnova is that every infinite, finitely generated, These exponents are expected to be universal, in the non-amenable group has a Cayley graph on which sense that they should be the same for percolation pc > pu; this is then expected not to depend on the on any two-dimensional lattice; but at the time of choice of generators. In the general case, it was this writing this phenomenon is far from being un- recently proved by Gaboriau that if the graph G derstood on a mathematical level. is unimodular, transitive, locally finite and sup- The rigorous derivation of the critical exponents ports non-constant harmonic Dirichlet functions for percolation is due to Smirnov and Werner [19]; (i.e. harmonic functions whose gradient is in ℓ2), the dimension of the limiting curve was obtained then indeed pc(G) < pu(G). by Beffara [2]. For reference and further reading on the topic, the reader is advised to refer to the review paper by Benjamini and Schramm [3], the lecture notes of 3 Other lattices and percolative Peres[14], and the more recent article of Gaboriau systems [6].

Some modiﬁcations or generalizations of standard 3.2 Gradient percolation Bernoulli percolation on Zd exhibit an interesting behavior and as such provide some insight into the Another possible modiﬁcation of the original original process as well; there are too many mathe- model is to allow the parameter p to depend on matical objects which can be argued to be percola- the location; the porous medium may for instance tive in some sense to give a full account of all of have been created by some kind of erosion, so that them, so the following list is somewhat arbitrary there will be more open edges on one side of a and by no means complete. given domain than on the other. If p still varies

6 smoothly, then one expects some regions to look the passage time T(π) of π as subcritical and others to look supercritical, with interesting behavior in the vicinity of the critical T(π):= ∑ T(e). e∈π level set {p = pc}. This particular model was introduced by Sapoval et al. [16] under the name of The ﬁrst passage time a(x, y) between vertices x and gradient percolation; see ﬁg. 5. y is given by

a(x, y) = inf{T(π): π a path from x to y};

and we can deﬁne

W(t):= {x ∈ Zd : a(0, x) ≤ t},

the set of vertices reached by the liquid by time t. It turns out that W(t) grows approximately linearly as time passes, and that there exists a non-random limit set B such that either B is compact and 1 (1 − ε)B ⊆ W(t) ⊆ (1 + ε)B, eventually a.s. t e for all ε > 0, or B = Rd, and 1 {x ∈ Rd : |x| ≤ K} ⊆ W(t), eventually a.s. t e for all K > 0. Here W(t) = {z + [−1/2, 1/2]d : z ∈ W(t)}. Figure 5: Gradient percolation in a square. In black e is the cluster spanning from the bottom side of the Studies of first passage percolation brought square. many fascinating discoveries, including King- man’s celebrated sub-additive ergodic theorem. In recent years interest has been focused on study of The control of the model away from the critical fluctuations of the set W(t) for large t. In spite of zone is essentially the same as for usual Bernoulli huge effort and some partial results achieved, it percolation, the main question being how to esti- e still remains a major task to establish rigorously mate the width of the phase transition. The main conjectures predicted by Kardar-Parisi-Zhang the- idea is then the same as in scaling theory: If the ory about shape fluctuations in first passage per- distance between a point v and the critical level colation. set is less than the correlation length for parameter p(v), then v is in the phase transition domain. This of course makes sense only asymptotically, say in 3.4 Contact processes a large n × n square with p(x, y) = 1 − y/n as is the Introduced by Harris and conceived with biolog- case in the figure: The transition then is expected ical interpretation, the contact process on Zd is a to have width of order na for some exponent a > 0. continuous-time process taking values in the space of subsets of Zd. It is informally described as follows: Particles are distributed in Zd in such a way 3.3 First-passage percolation that each site is either empty or occupied by one particle. The evolution is Markovian: Each parti- First passage percolation (also known as Eden or cle disappears after an exponential time of param- Richardson model) was introduced by Hammer- eter 1, independently from the others; at any time, sley and Welsh in 1965 [8] as a time-dependent each particle has a possibility to create a new parti- model for the passage of fluid through a porous cle at any of its empty neighboring sites, and does medium. To define the model, with each edge e ∈ so with rate λ > 0, independently of everything E(Zd) is associated a random variable T(e), which else. can be interpreted as being the time required for The question is then whether, starting from a fi- fluid to flow along e. T(e) are assumed to be in- nite population, the process will die out in finite dependent non-negative random variables having time or whether it will survive forever with posi- common distribution F. For any path π we define tive probability. The outcome will depend on the

7 value of λ, and there is a critical value λc, such that more speciﬁc topics are given at the end of each for λ ≤ λc process dies out, while for λ > λc indeed section. there is survival, and in this case the shape of the population obey a shape theorem similar to that of ﬁrst-passage percolation. See also

The analogy with percolation is strong, the cor- Introductory article: Statistical mechanics; 2D responding percolative picture being the follow- Ising model; Wulff droplets; Stochastic Loewner Zd+1 ing: In + , each edge is open with probability evolutions. p ∈ (0, 1), and the question is whether there exists an infinite oriented path π (i.e. a path along which the sum of the coordinates is increasing), com- References posed of open edges. Once again, there is a critical parameter customarily denoted by ~pc, at which no [1] K. ALEXANDER, J. T. CHAYES, AND such path exists (compare this to the open ques- L. CHAYES, The Wulff construction and tion of the continuity of the function θ at pc in di- asymptotics of the finite cluster distribution for mensions 3 ≤ d ≤ 18). This variation of percolation two-dimensional Bernoulli percolation, Comm. lies in a different universality class than the usual Math. Phys., 131 (1990), pp. 1–50. Bernoulli model. [2] V. BEFFARA, Hausdorff dimensions for SLE6, Ann. Probab., 32 (2004), pp. 2606–2629. 3.5 Invasion percolation [3] I. BENJAMINI AND O.SCHRAMM, Percolation Let X(e): e ∈ E be independent random variables beyond Zd, many questions and a few answers, indexed by the edge set E of Zd, d ≥ 2, each hav- Electron. Comm. Probab., 1 (1996), pp. no. 8, ing uniform distribution in [0, 1]. One constructs 71–82 (electronic). a sequence C = {Ci, i ≥ 1} of random connected subgraphs of the lattice in the following iterative [4] S. R. BROADBENT AND J. M. HAMMERSLEY, way: The graph C0 contains only the origin. Hav- Percolation processes, I and II, Proc. Cambridge ing defined Ci, one obtains Ci+1 by adding to Ci an Philos. Soc., 53 (1957), pp. 629–645. edge ei+1 (with its outer lying end-vertex), chosen [5] R. CERF, Large deviations for three dimensional from the outer edge boundary of C so as to min- i supercritical percolation, vol. 267 of Astérisque, imize X(e + ). Still very little is known about the i 1 SMF, 2000. behavior of this process. An interesting observation, relating θ(pc) of [6] D. GABORIAU, Invariant percolation and har- usual percolation with the invasion dynamics, monic Dirichlet functions, Geometric And comes from C.M. Newman: Functional Analysis, (2005). To appear.

θ(pc) = 0 ⇔ P{x ∈ C}→ 0 as |x|→∞. [7] G. GRIMMETT, Percolation, vol. 321 of Grundlehren der Mathematischen Wis- senschaften, Springer-Verlag, Berlin, sec- Further reading ond ed., 1999.

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Keywords

Percolation, random medium, porous medium, random graph, phase transition, critical exponent, shape theorem, correlation length, scaling, scaling relation, scaling limit, exploration path, Cayley graph, contact process.