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

**2.1.2 Percolation beyond Z d**

Until the mid 90’s, the study of percolation was mostly focused on the simplest
possible geometry: the Euclidean hypercubic latticeZ^{d}. It was in 1996 that Benjamini
and Schramm [27] started a systematic study of Bernoulli percolation on general graphs
in their seminal paper entitled “Percolation beyond Z^{d}, many questions and a few
answers”. As its title says, the paper provides a few results, but most importantly,
many questions and conjectures that have inspired a substantial part of the research
in percolation theory since then. The main goal in this line of research is to relate the
behavior of percolation on graphs to their geometric and algebraic properties.

The questions and conjectures from [27] have a special emphasis on Cayley graphs
and, more generally, almost transitive graphs. Given an infinite group Γ and a finite
generating set S, the associated Cayley graph G = G(Γ, S) = (V, E) is constructed
by setting V = Γ and {g, h} ∈ E if and only if g^{−1}h or its inverse h^{−1}g belongs to
S. A graph G is called transitive if its group of automorphisms has a single orbit,
i.e., for any two vertices u,v in G, there is an automorphism of G mapping u ontov.

In particular, Cayley graphs are necessarily transitive. Similarly, a graph G is called almost transitive (or quasi-transitive) if its group of automorphisms has finitely many orbits.

Unlike in the Euclidean case, answering question Q1 is quite challenging in this
general context. As before, it is easy to prove thatp_{c}(G)>0 for every almost transitive
graphG(actually, for any graph with bounded degree). As forp_{c}(G)<1, one could try
to use Peierls’ argument mentioned above. However, this argument relies on bounding
the number of cut-sets of given size, which is a tractable problem for graphs with
simple geometry likeZ^{d}, but turns out to be very hard for the more general context of
almost transitive (or Cayley) graphs. Also, recall that it is possible to havep_{c}(G) = 1,
as exemplified by G=Z.

As an answer to this fundamental question, Benjamini and Schramm propose a precise characterization of almost transitive and Cayley graphs having a non-trivial phase transition.

Conjecture 2.1.6([27], Conjecture 1). A Cayley graphG=G(Γ, S)satisfiesp_{c}(G)<

1 if and only if Γ is not a finite extension of Z.

Conjecture 2.1.7([27], Conjecture 2). An almost transitive graphGsatisfiesp_{c}(G)<

1 if and only if G has ball volume growth faster than linear.

Many partial results towards these conjectures have been established in the liter-ature, see Chapter 4 for an account on that. Let us just mention here that, due to a sufficiently good understanding of their geometry, transitive graphs with either poly-nomial or exponential ball volume growth were already known to satisfy pc <1. The geometry of (almost) transitive graphs with intermediate growth on the other hand is poorly understood. Conjecture 2.1.6has also been solved for finitely presented groups [13].

Beyond the framework of almost transitive graphs, Benjamini and Schramm also
ask whether isoperimetric inequalities are sufficient to guarantee p_{c} < 1. One says
that a graph G(not necessarily almost transitive) satisfies an isoperimetric inequality
of dimension d≥1 if there exists c=c(G, d)>0 such that

|∂S| ≥c|S|^{d−1}^{d} , for all finite S ⊂V. (2.1.7)
The isoperimetric dimension ofG, denoted by Dim(G), is defined as the supremum of
d such that (2.1.7) is satisfied.

Question 2.1.8 ([27], Question 2). Does Dim(G)>1 imply p_{c}(G)<1?

In the same paper, Benjamini and Schramm proved thatp_{c}(G)<1 for every graph
Gsatisfying an isoperimetric inequality of “dimension ∞”, i.e., if Gis non-amenable.

Teixeira proved in [163] thatp_{c}(G)<1 wheneverGhas polynomial growth and satisfy
a certainlocal isoperimetric inequality of dimensiond >1. However, besides requiring
the graph to have polynomial growth, the local notion of isoperimetry considered in
[163] is much stronger than the one defined above. Except for these results, very little
was proved regarding Question 2.1.8.

In Chapter 4, we provide the following partial answer to Question 2.1.8.

Theorem 2.1.9 ([55]). If G has bounded degree and Dim(G)>4, then pc(G)<1.

As mentioned above, all the graphs for which Conjectures 2.1.6and 2.1.7 were not known, had super-polynomial growth. It is known that any such graph satisfies an isoperimetric inequality of dimension d for every d ≥ 1, and we directly deduce the following.

Corollary 2.1.10 ([55]). Conjectures 2.1.6 and 2.1.7 hold.

The first result from [27] concerns a monotonicity property for p_{c} under covering
maps, which is related to question Q2. Given a graph G and a group Γ of
automor-phisms of G, one can consider the quotient graph, denoted by G/Γ, with vertex set
{Γv : v ∈ V(G)} and an edge between Γu and Γv whenever uv ∈ E(G). Given two
graphs G and H, one says that G covers H if there exists a group of automorphisms
Γ acting freely on V(G) such that G/Γ is isomorphic to H. In this case, the
canon-ical projection map π : G → H is called a covering map. By a simple exploration
argument, one can prove (see Theorem 1 from [27]) that for any G and Γ one has
p_{c}(G)≤p_{c}(G/Γ). Benjamini and Schramm then asked the following question.

Question 2.1.11([27], Question 1). When does the strict inequality p_{c}(G)< p_{c}(G/Γ)
hold? For instance, if both G and H are connected almost transitive graphs, G covers
but is not isomorphic to H and pc(H)<1, does it imply pc(G)< pc(H)?

In Chapter 3, we aim at answering the first part of Question 2.1.11 and confirm, in particular, the second part of it.

Theorem 2.1.12 ([110]). Let G and H be connected almost transitive graphs. If G
covers but is not isomorphic to H and p_{c}(H)<1, then p_{c}(G)< p_{c}(H).

In fact, we provide a more general result, which guarantees the strict inequality
p_{c}(G) < p_{c}(G/Γ) under very mild assumptions on Γ. We also provide examples for
which p_{c}(G) = p_{c}(G/Γ), thus showing that our assumptions are essentially sharp. In
addition, we investigate an analogous question for the uniqueness critical parameter
p_{u} defined in (2.1.9) below, see Chapter 3 for details.

It turns out that the proofs of subcritical sharpness from [6,115] are also valid for
general almost transitive graphs, which has again many consequences regarding the
subcritical phase p < p_{c} and can be seen as an answer to question Q3.

Theorem 2.1.13 ([6, 115]). For every almost transitive graphG and p < p_{c}(G) there
exist c >0 such that for every N ≥1

P_{p}[x←→∂B_{N}(x)]≤e^{−cN}. (2.1.8)
Answering question Q4 – i.e., understanding the supercritical phase – for general
almost transitive graphs is a harder task. Again, one would like to understand both
the infinite and finite clusters. The argument of Burton and Keane mentioned above
still works to prove uniqueness of infinite clusters whenever the graph is amenable.

One calls a graph G amenable if one can find a sequence (Fn)n of finite subsets of V
such that ^{|∂F}_{|F}^{n}^{|}

n| →0. In other words, a graph is amenable when it does not satisfy an
isoperimetric inequality of “dimension∞” – recall (2.1.7). The infinite (k+ 1)-regular
tree Tk, k ≥ 2, is a non-amenable transitive graph and, in fact, one can easily prove
that in this case there exist infinitely many infinite clusters at any p∈(pc,1). As for
the product graphTk×Z(which is also non-amenable), one can show the existence of
another critical pointp_{u} ∈(p_{c},1) such that there exists infinitely many infinite cluster
at anyp∈(pc, pu), while forp∈(pu,1] there is a unique infinite cluster – see [73]. For
a general almost transitive graphsG, one can define

p_{u}(G) := inf{p∈[0,1] : P_{p}[there is a unique infinite cluster]>0}. (2.1.9)
Remark 2.1.14. It turns out that, for every almost transitive graph G, the number
of infinite clusters is almost surely ∞ for all p ∈ (p_{c}, p_{u}) and 1 for all p ∈ (p_{u},1].

This is not obvious since, unlike the existence of an infinite cluster, uniqueness is not
an increasing event (i.e., uniqueness for ω does not necessarily implies uniqueness for
every ω^{0} ≥ω). However, one can prove that uniqueness at p implies uniqueness at p^{0}
for every p^{0} ≥p – see [78, 143].

As mentioned above, p_{c}(G) = p_{u}(G) for every amenable graph G, while p_{c}(Tk)<

pu(T^{k}) = 1 andpc(T^{k}×Z)< pu(T^{k}×Z)<1. The following conjecture arises naturally.

Conjecture 2.1.15 ([27], Conjecture 6). If G is a non-amenable almost transitive
graph, then p_{c}(G)< p_{u}(G).

Conjecture 2.1.15 remains open, but Pak and Smirnova-Nagnibeda [126] showed
that p_{c} < p_{u} for the Cayley graph of non-amenable groups provided the set of
genera-tors is properly chosen.

Similarly to the example of the regular tree Tk mentioned above, one can easily
prove thatp_{u}(G) = 1 for every almost transitive graph with infinitely many ends – see
[84] for a definition. One can then ask the following question, which remains widely
open.

Question 2.1.16 ([27], Question 3). Give general conditions that guarantee p_{u} < 1.

For example, is p_{u} <1 for any transitive graph with one end?

Still concerning question Q4, we now turn to the study of finite clusters in the
supercritical phasep > p_{c} of general almost transitive graphs. Similarly to the case of
Z^{d} seen above, one expects the diameter of a finite cluster to have exponential tail for
any almost transitive graph.

Conjecture 2.1.17. For every almost transitive graph G and p > p_{c}(G), there exists
c >0 such that for every N ≥1,

P_{p}[x←→∂B_{N}(x), x←→ ∞]6 ≤e^{−cN}. (2.1.10)
As for the volume of a finite cluster, one expects the tail to behave according to
the isoperimetric profile of the graph. Given a connected graph G, its isoperimetric
profile is defined as

ψ(t) = ψ_{G}(t) := inf

|∂K|: K ⊂V, t ≤ |K|<∞ . (2.1.11)
Conjecture 2.1.18. For every almost transitive graph G and p > p_{c}(G), there exists
c >0 such that for every N ≥1,

Pp[N ≤ |Cx|<∞]≤e^{−cψ(cN)}. (2.1.12)
As we already mentioned, Conjectures 2.1.17and 2.1.18 are proved for the
hyper-cubic lattice Z^{d} as a consequence of [75]. More recently, Hermon and Hutchcroft [83]

proved both conjectures for the case of non-amenable graphs. Except for these results, Conjectures 2.1.17 and 2.1.18 remain widely open.

A related and famous conjecture is the so-called “locality conjecture”, which is
due to Schramm and was first stated in [25]. It is inspired by the intuition that one
can always witness the existence of an infinite cluster by only observing a (sufficiently
large) finite ball. The aforementioned result of Grimmett and Marstrand [75] can be
seen as an evidence of this intuition for the case of Z^{d}. One says that a sequence of
transitive graphs Gn locally converges to a transitive graphGif for every m≥1 there
exists n_{0} = n_{0}(m) sufficiently large such that the ball of radius m in G_{n} and G are
isomorphic for every n≥n_{0}.

Conjecture 2.1.19. Let (G_{n})_{n} and G be transitive graphs such that G_{n} converges to
G locally. If sup_{n}pc(Gn)<1, then pc(Gn)→pc(G).

Some progress towards this conjecture was made by Martineau and Tassion [111]

for Cayley graphs of Abelian groups, and by Hutchcroft [86] for transitive graphs of exponential growth.

Our understanding of the critical and near-critical regimes of percolation on almost transitive graphs is rather limited. Similarly to the Euclidean case discussed in the previous subsection, these regimes are expected to be extremely interesting, but very few conjectures are explicitly stated in the literature. In [27], the only conjecture concerning this is a generalization of Conjecture 2.1.5 above.

Conjecture 2.1.20 ([27], Conjecture 4). For almost transitive graph G with p_{c}(G)<

1, one has θ(p_{c}) = 0.

Let us mention that this conjecture has been confirmed for non-amenable Cayley graphs by Benjamini, Lyons, Peres and Schramm [23], and recently proved for al-most transitive graphs of exponential growth by Hutchcroft [85]. Continuity of phase transition is also much more tractable in the planar case due to duality and the Russo-Seymour-Welsh theory – see [82, 139, 145].