The original model: independent percolation on Z d

In this section we will discuss the most classical and well studied setting in per-colation theory: Bernoulli (or independent) perper-colation on the hypercubic lattice Z^d. This is the simplest example of percolation model from both geometric (Euclidean) and probabilistic (independent) aspects. As mentioned above, it was introduced by Broadbent and Hammersley in 1957 [36] and is defined by simply taking G=Z^d and P_p to be product measure with marginals Ber(p). For simplicity, let us consider only bond percolation here. We refer the interested reader to the textbook [74] for more about this classical model.

It turns out that answering question Q1 is relatively simple in this case. First, an easy counting argument gives p_c(Z^d) >0 for every d ≥ 1. It is also easy to convince oneself that pc(Z) = 1. A combinatorial argument based on bounding the number cut-sets, which originally used by Peierls [127] in the study of the Ising model, can be used to prove that p_c(Z^d) < 1 for every d ≥ 2. In summary, the phase transition is non-trivial if and only if d >1.

Question Q2 is a bit more subtle, but for d = 2 one has a powerful extra tool, calledduality. For any bond configurationω on a planar graphG, one can associate a dual configuration ω^∗ on its dual graph G^∗ by the relation ω^∗(e^∗) := 1−ω(e), where the edgee^∗ is the dual ofe. This implies that “the complement” ofP_p is distributed as P1−p on G^∗. Since the dual of Z² is itself, one could naturally conjecture that p_c(Z²) is the solution of 1−p = p, i.e., pc(Z²) = 1/2. It was only in 1980 that Kesten [88]

came up with a proof of this conjecture. As for d ≥ 3, there is no reason to believe thatp_c(Z^d) could be explicitly computed. However, some estimates can be proved, for instancepc(Z^d)∼1/2d as d→ ∞ [92].

QuestionQ3 was only answered satisfactorily in the mid 80’s, when Aizenman and Barsky [6] and Menshikov [115] independently showed that when p < p_c the clusters are not only finite almost surely, but typically very small. More precisely, they proved

the following exponential bound for the tail of the diameter of a cluster.

Theorem 2.1.1 ([6, 115]). For every d ≥ 1 and p < p_c(Z^d) there exists c > 0 such that for every N ≥1,

P_p[0←→∂B_N]≤e^−cN. (2.1.2) One can further enhance this result to obtain the same bound for the volume of a cluster (see [74] for a proof). Given a vertex x, we denote byCx the cluster of x.

Theorem 2.1.2. For every d ≥ 1 and p < p_c(Z^d) there exists c > 0 such that for every N ≥1,

P_p[|C₀| ≥N]≤e^−cN. (2.1.3) These results are often referred to as subcritical sharpness, and have many con-sequences. It readily follows from Theorems 2.1.1 and 2.1.2 that the largest cluster inside B_N has size (either diameter or volume) of order logN with high probability.

Another consequence is that the expected cluster size p7→χ(p) :=E_p[|C₀|] is analytic on [0, p_c) – see [89].

Understanding the supercritical phase p > p_c – and thus answering questionQ4 – is more difficult. Obviously, in this case clusters can be of two types: infinite or finite.

It was proved by Aizenman, Kesten and Newman [10] that the infinite cluster is almost surely unique for alld ≥2. An alternative and beautiful proof was later obtained by Burton and Keane [37]. This result implies, for instance, that the percolation density function

θ(p) := P_p[0←→ ∞] (2.1.4)

is continuous on (p_c,1] (notice that θ(p) = 0 for all p < p_c, while θ(p) > 0 for all p > p_c). In the planar case, one can use duality to extract information about the supercritical phase (p > p_c = 1/2) out of the subcritical phase (p < p_c = 1/2). This readily implies that, for d = 2, the diameter of a finite cluster also has exponential tail. The key progress in understanding the supercritical phase in dimensions d ≥ 3 came with the work of Grimmett and Marstrand [75]. Their result states that for every d≥3 and p > p_c(Z^d), there exists M =M(d, p)≥ 0 sufficiently large such that there is an infinite cluster at pinside Z²×[−M, M]^d−2. It is then not very hard to deduce the following (see [74] for a proof).

Theorem 2.1.3. For every d ≥ 2 and p > p_c(Z^d), there exists c > 0 such that for every N ≥1,

P_p[0←→∂B_N, x ←→ ∞]6 ≤e^−cN. (2.1.5) As in the subcritical phase, one can also study the tail distribution for the volume of a finite cluster, but unlike the subcritical case, the decay is just stretched exponential (see [74] for a proof).

Theorem 2.1.4. For every d ≥ 2 and p > p_c(Z^d) there exist C, c > 0 such that for every N ≥1,

e^−CN

d−1

d ≤P_p[N ≤ |C₀|<∞]≤e^−cN

d−1

d . (2.1.6)

These results are often referred to assupercritical sharpness. It is again a straight-forward consequence of these theorems that, in the supercritical phasep > p_c(Z^d), the largest finite cluster inside BN will typically have diameter of order logN, but volume of order (logN)^d−1d . Using renormalization techniques, one can deduce more informa-tion about the supercritical phase by relying on the result of Grimmett and Marstrand.

For instance, one can prove that for all p > p_c, the chemical (i.e., intrinsic) distance in the (unique) infinite cluster C_∞^p is comparable to the Euclidean one [12] and that the simple random walk onC_∞^p satisfies a quenched invariance principle [146, 31, 112]

as well as quenched Gaussian bounds for its heat kernel [17]. One can also study large finite clusters, proving large deviation results and (rescaled) convergence to a deterministic shape, known as Wulff crystal [40]. Very recently, Georgakopoulos and Panagiotis [68] also relied on the result of Grimmett and Marstrand in order to prove that the percolation density θ is analytic on (pc,1] for all d≥2.

Questions Q5 and Q6 are substantially more delicate. The first natural question related toQ5 is whether there exists an infinite cluster atp_cor not. This leads to the following conjecture

Conjecture 2.1.5. For every d≥2, one has θ(p_c) = 0.

This is certainly one of the most famous conjectures in probability theory, and it is often referred to as “θ(pc) = 0 conjecture” or “continuity of phase transition”. Indeed, θ(p_c) = 0 is equivalent to continuity of the function θ since it is already known to be continuous on (p_c,1] (even analytic, as mentioned above), identically 0 on [0, p_c) and right continuous at pc – see [74].

For d = 2, it was proved by Harris [82] that θ(1/2) = 0, which combined with Kesten’s result [88] that p_c(Z²) = 1/2, settles the Conjecture 2.1.5 in this case. Hara and Slade [81] proved Conjecture 2.1.5 for d≥ 19 by using a technique known under the name of lace expansion. Their ideas were further exploited, and the same result is now known for all d≥11 – see [63]. However, their approach consists in proving that critical percolation exhibits a so-called mean-field behavior, which is only expected to hold for d ≥ 6. Therefore, a full solution to Conjecture 2.1.5 would necessarily require different techniques, with dimensions 3, 4 and 5 being the most interesting and challenging cases. Let us mention that, for everyd≥2, it was proved by Barsky, Grimmett and Newman [18] that θ(p_c) = 0 for the half-spaceN×Z^d−1, but enhancing this result to the full spaceZ^d seems to be very difficult.

Although there should be no infinite cluster atpc, the (finite) clusters are expected to have a substantially different behavior than in subcritical phase. Indeed, while con-nection probabilities decay exponentially fast for p < p_c (see Theorem 2.1.1 above), the same quantities are expected to exhibit polynomial decay atpc. The exponents in those algebraic decays are calledcritical exponents. Their values are of extreme impor-tance from the physical point of view. Critical percolation models (in fact, statistical physics models in general) are believed to have a non-trivial scaling limit related to Conformal Field Theories (CFTs) whenever its phase transition is continuous. The value of critical exponents are key traits of their “universality class”, and therefore directly related to their CFT scaling limits.

In the mathematical side, great progress have been made in the planar case. In his celebrated paper [147], Smirnov proved that critical Bernoulli site percolation on the triangular lattice T has a conformal invariant scaling limit. Having Smirnov’s

work as a starting point, many other fine properties of planar critical percolation were derived, including the calculation of its critical exponents [149]. An important object in the study of critical models on the plane is the so-called Schramm Lowner Evolution (SLE), introduced by Schramm in [144], which is a one-parameter family of conformally invariant random curves that arise as the scaling limit of interfaces in critical planar models. For arbitrary dimensions (in particular the “physical dimension” d = 3), the mathematical understanding of critical phase is rather limited, but recently some amazing progress has been made in the physics community with the development of a method known as Conformal Bootstrap [130].

It turns out that quantities in near-critical percolation (i.e., p approaching p_c) also exhibit polynomial decay, leading to the so-called near-critical exponents. These quantities are very much related to critical exponents, as demonstrated by Kesten [91].

We refer the interested reader to [124] for more information about the extremely rich world of (near-)critical phenomena in planar statistical physics.