**1.2 Les m´ ecanismes d’interpolation**

**2.1.1 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 Z^{2} is itself, one could naturally conjecture that p_{c}(Z^{2})
is the solution of 1−p = p, i.e., pc(Z^{2}) = 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}[|C_{0}| ≥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}[|C_{0}|] 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 Z^{2}×[−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 ≤ |C_{0}|<∞]≤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−1}^{d} . 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_{c}or 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}(Z^{2}) = 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.