None of the assumptions of Theorem3.1.1 can be removed. This is obvious for the hypothesis that pc(G) < 1. As for freeness, take G to be Z2 with two extra pendant edges attached to each vertex. The groupZ/2Zacts onGby swapping the two pendant edges at each vertex. Since Z2 is amenable, one has pu(G) = pc(G) – see [37] – and pu(H) =pc(H), so that freeness is also necessary in Theorem3.1.2.
For the hypothesis thatGis quasi-transitive, letGbe defined by taking two disjoint copies ofZ2 and putting an additional edge between the two copies of the origin. The groupG:=Z/2Zacts by swapping copies. As for quasi-transitivity ofH, take Gto be the square lattice Z2, and G to be Z/2Z acting via the reflection (x, y) 7→(x,1−y).
See [74] for the classical fact that pc(N×Z) = pc(Z2).
Still, we do not know what happens if freeness is relaxed to the absence of trivial G-orbit.
Question 3.6.1. Let G be a group acting on a graph G by graph automorphisms.
Assume that both G and H := G/G are quasi-transitive, and that for every vertex x∈V(G) there existsg ∈G such thatgx6=x. Is it necessarily the case thatpc(G)<1 implies pc(G)< pc(H)? If we assume further thatG is finite, is it necessarily the case that pu(G)<1 implies pu(G)< pu(H)?
Remark. An interesting particular case (which we also do not know how to solve) is when G is normal in a quasi-transitive subgroup of Aut(G). In that setting, H is automatically quasi-transitive, and the map π always has uniformly non-trivial fibres.
As for Theorem3.2.1and Corollary3.2.2, notice that the assumption that fibres are uniformly non-trivial cannot be simply replaced by non-triviality of the fibres (namely
∀u∈V(H), |π−1({u})| 6= 1), even ifπ is taken to be aclassical covering map. Indeed, take H to be a connected graph with bounded degree and pc < 1, and pick a non-cutting6 edge e in H. To define G, start with two copies of H, and denote by {x, y}
and {x0, y0} the two copies of e. Then replace these two edges by {x0, y} and {x, y0}, thus yielding a connected graph. Take π to be the natural projection from G toH.
We do not know how to answer the following question, which investigates a gener-alisation of Theorem3.2.1/Corollary 3.2.2.
Question 3.6.2. Let G and H be graphs of bounded degree. Assume that there is a weak covering map π:V(G)→V(H) with uniformly non-trivial fibres.
Is it necessarily the case that pc(G) < 1 implies pc(G) < pc(H)? If we assume further that π has bounded fibres and that both G and H are quasi-transitive, is it necessarily the case that pu(G)<1 implies pu(G)< pu(H)?
Notice that one cannot remove the assumption of boundedness of the fibres (or finiteness ofG, in the case of Theorem3.1.2) from Theorems3.2.3 and 3.2.4or Corol-lary 3.2.5. Indeed, without this assumption, it is even possible to have the strict inequality in the reverse direction. The following example shows that this is easy to obtain if one further relaxes the assumption that pu(G) < 1: take G to be the 2d-regular tree and H to be the d-dimensional hypercubic lattice, for some d ≥ 2, then we have pu(H) = pc(H) < 1 = pu(G) = 1. Notice that H indeed is a quotient of G:
as H is 2d-regular, one can realise G as the set of finite non-bactracking paths of H launched at 0, and mapping such a path to its final position yields a strong covering map from G to H. If one does not want to relax the assumption that pu(G) <1, one can take d to be large enough, Gd to be the product of the 2d-regular tree and the bi-infinite line Z, and Hd to be the (d+ 1)-dimensional hypercubic lattice. Indeed, pu(Gd)∼ √1
d but pu(Hd) = pc(Hd)∼ 2d1, see respectively [73] and [92].
Finally, it is natural to look for a milder condition than the finiteness of G in Theorem 3.1.2. A natural condition to consider is that of amenability. Recall that a group G is said to be amenable if there is a sequence (Fn)n of non-empty finite subsets of Gsuch that for every g ∈G, one has |Fn4gFn|=o(|Fn|).
Question 3.6.3. Does Theorem 3.1.2 still hold if G is only assumed to be amenable instead of finite?
6. i.e., an edgeesuch thatH \ {e} remains connected.
Chapter 4
Existence of phase transition
In this chapter, we prove that Bernoulli percolation on bounded degree graphs with isoperimetric dimension d > 4 undergoes a non-trivial phase transition (in the sense thatpc<1). As a corollary, we obtain that the critical point of Bernoulli percolation on infinite quasi-transitive graphs (in particular, Cayley graphs) with super-linear growth is strictly smaller than 1, thus answering a conjecture of Benjamini and Schramm.
The proof relies on a comparison through interpolation between Gaussian free field level-sets and Bernoulli percolation, as explained in Chapter 2.
This chapter is based on the article entitled “Existence of phase transition for per-colation using the Gaussian Free Field” (Duke Mathematical Journal) which is a joint work with Hugo Duminil-Copin, Subhajit Goswami, Aran Raoufi and Ariel Yadin.
4.1 Introduction
Motivation. Whether a model undergoes a non-trivial phase transition or not is one of the most fundamental questions in statistical physics. In [127], Peierls introduced a combinatorial technique, known asPeierls argument, to prove that the critical tem-perature of the Ising model is non-zero on Zd for d≥2, thus opening a new realm of questions concerning this important model of ferromagnetism. This argument found many applications to other models, including Potts models and models of random graphs such as Bernoulli percolation or the random-cluster model.
Peierls argument has two drawbacks. First, it often does not apply to continu-ous spin models, for instance the spin O(n) models. In this case, the technique may sometimes be replaced by two other techniques: Reflection Positivity and the Renor-malization Group. More precisely, Fr¨ohlich, Simon and Spencer proved that the spin O(n) model undergoes a non-trivial order/disorder phase transition onZd with d≥3 [66] using Reflection Positivity, and Balaban and coauthors (see [14] and references therein) proved delicate properties of the large β regime using the Renormalization Group. Let us mention that in the special case of the XY model (i.e., when n = 2), there are special proofs relying on the Coulomb gas [65, 87].
Another problem with Peierls argument is that it requires a precise understanding of so-called cut sets, i.e., sets of edges which disconnect certain sets of vertices from infinity. On planar graphs, this boils down to the understanding of circuits in the dual graph. On non-planar graphs, the question is a much more complex combinatorial problem and it is not completely understood in general.
61
In this chapter, we wish to present a new technique which we believe can be useful to prove the existence of a phase transition for various models. The object of interest of this chapter will be Bernoulli percolation.
Main results. Consider a connected graph G= (V, E) with vertex-set V and edge-set E. An edge with endpoints x and y will be denoted by xy. For every x ∈ V, let d(x) be the number ofysuch thatxy∈E. We will assume that the graph hasbounded degree, that is sup{d(x) :x∈V}<+∞.
Bernoulli percolationis a model of a random subgraph ofGwith vertex-setV. The subgraph is encoded by a function ω fromE into {0,1}. We use the notation ωxy to denote the value of ω at the edge xy and think of edges xy with ωxy = 1 as being the edges of the subgraph. These edges are called open, while those with ωxy = 0 are called closed. We are interested in the connectivity properties of the graph ω. We use the notation S ←→ T (resp. S ←→ ∞) to denote the event that there is a nearest neighbor open path in ω connecting a vertex in S to a vertex in T (resp. the event that S intersects an infinite connected component of ω). Also, let S ←→6 T (resp.
S ←→ ∞) denote the complement of the event6 S ←→T (resp. S ←→ ∞).
The Bernoulli bond percolation of parameter p = (pxy : xy ∈ E) ∈ [0,1]E is the probability measure on configurations ω for which the variables ωxy form a family of independent Bernoulli random variables of respective parameterspxy. We denote such a measure by Pp and, when pxy =p for every xy∈E, we simply writePp. The main question of interest is whether the critical parameter
pc(G) := inf
p∈[0,1] :Pp[x←→ ∞]>0
(here x ∈ V is a vertex chosen arbitrarily) is strictly smaller than 1 or not. Let us mention that proving pc(G) > 0 is a much simpler task: a simple comparison with branching process actually implies pc(G)≥ D1, whereD is the maximum degree ofG.
In order to state the main result, we need another notion. Given a graphG,simple random walk is the discrete-time Markov chain (Xn)n≥0 on V moving, at each time step, from its position to one of its neighbors inGchosen uniformly at random. Define the heat kernel and the Green function by the following formula: for every x, y ∈ V and n≥0,
pn(x, y) := P[Xn=y|X0 =x] and G(x, y) := 1 d(y)
∞
X
n=0
pn(x, y).
The main object of this chapter is the following result.
Theorem 4.1.1. Consider a graph G with bounded degree. Assume that there exist real numbers d >4 and c >0 such that
pn(x, x)≤ c
nd/2 ∀x∈V , ∀n≥1. (Hd)
Then, there exists p <1 such that for every finite set S ⊂V,
Pp[S←→ ∞]6 ≤exp[−12cap(S)], (4.1.1) where cap(S) :=P
x∈Sd(x)P[Xk ∈/ S,∀k ≥ 1|X0 =x] is the capacity of S. In partic-ular, since cap(S) > 0 iff G is transient (which is implied by (Hd), d > 2), one has pc(G)<1.
Let us mention a few applications of the theorem. We say that a graph Gsatisfies an isoperimetric inequality of dimension d if there exists some constant c > 0 such that
|∂K| ≥c|K|d−1d for all finite K ⊂V, (Id) where ∂K := {{x, y} ∈ E : x ∈ K, y /∈ K} is the edge boundary of K. The isoperimetric dimensionofG, which we denote by Dim(G), is defined as the supremum over alldsuch that (Id) holds. In their celebrated paper [27], Benjamini and Schramm asked whether Dim(G) > 1 necessarily implies pc(G) < 1. We give the following partial answer to this question:
Theorem 4.1.2. If a bounded-degree graph G satisfies Dim(G)>4, then pc(G)<1.
This theorem follows directly from Theorem 4.1.1 and a result of Varopoulos [167]
(see also [117] or [106, Corollary 6.32] for a proof relying on evolving sets) stating that (Id) implies (Hd).
An important application of Theorem 4.1.2 is the following result answering the first two conjectures of Benjamini and Schramm from [27]. The graph G is called quasi-transitiveif the action of the automorphism group Aut(G) onV has only finitely many orbits. Typical examples of quasi-transitive graphs to keep in mind are Cayley graphs of finitely generated groups. Let Br(x) be the ball of radius r centered at x with respect to the graph distance. We say that a graphG has super-linear growth if lim sup1r|Br(x)|= +∞.
Theorem 4.1.3. Let G be a quasi-transitive graph with super-linear growth, then pc(G)<1.
The previous result can be deduced from Theorem 4.1.2 as follows:
— If lim inf r1d|Br(x)|<+∞for somed >0, then a result of Trofimov [166] (see also [168, Theorem 5.11]) implies that the graph is quasi-isometric to a Cayley graph with polynomial growth. This fact together with super-linear growth classically implies thatpc(G)<1 (see the next section for details).
— If lim inf r1d|Br(x)|= +∞for everyd >0, then in particular lim infr1d|Br(x)|>0 for some d >4. Therefore [105, Lemma 7.2] (see also the proof of Corollary 7.3 of the same paper or [46] for the special case of Cayley graphs) implies that the graph satisfies (Id), which in turn gives thatpc(G)<1 by Theorem 4.1.2.
All the results in this chapter can be extended to site percolation, finite dependent percolation and random-cluster models via classical comparisons, see respectively [76], [101] and [71]. The coupling between random-cluster models and the Ising/Potts model implies that the results translate into results on the latter as well.
Existing results. Our results should be put in context with the previous partial results regarding the Benjamini-Schramm questions.
As already mentioned, Peierls proved [127] that the Ising phase transition is non-trivial for Z2 through bounding the number of cut sets of specific size disconnecting a vertex from infinity. Peierls’ proof also applies to Bernoulli percolation and can be easily extended to all Zd (d ≥ 2) using monotonicity arguments. See also Lebowitz and Mazel [99] and Balister and Bollob´as [15] for estimates on the number of cut sets
ofZd. The cut set method was extended to Cayley graphs of finitely presented groups by Babson and Benjamini [13] (see also the work of Timar [165]). As of today, a technique bounding the number of cut sets of a certain cardinality has not been found for general graphs.
Say that a graphGhaspolynomial growthif lim supr1d|Br(x)|<+∞for somed >0.
As a consequence of Gromov’s celebrated theorem [77], every infinite finitely generated group of polynomial growth, which is not virtuallyZ (in the group theoretical sense), contains a subgroup isomorphic to Z2 [106, Proposition 7.18]. Hence, there exists a Cayley graph of such groups that has a subgraph isomorphic toZ2. Since the property that pc(G) < 1 is stable under quasi-isometries [106, Theorem 7.15], all the Cayley graphs of such groups have non-trivial phase transitions. This method was also used by Muchnik and Pak in [118] to prove pc(G)<1 for Grigorchuk groups which are a class of groups with intermediate (i.e., faster than polynomial but slower than exponential) growth.
Lyons has proved [104] that every Cayley graph of exponential growth (i.e., sat-isfying that lim inf 1rlog|Br(x)| > 0) has a non-trivial phase transition. As noted in [106, Page 264], the fact that pc(G)< 1 for quasi-transitive graphs G with exponen-tial growth can also be easily obtained from the finiteness of the susceptibility for subcritical percolation – see [6, 115, 61].
In [26] Benjamini, Pemantle and Peres proved another criterion for pc(G) < 1:
the existence of an EIT measure for the graph. A measure on self-avoiding paths starting from a fixed vertex is an EIT measure if the number of intersections of two independent paths sampled according to the measure has an exponential tail. In [134], by constructing an EIT measure, it is proved that if G is a Cayley graph of a virtually indicable group which is not virtually Z, thenpc(G)<1. Virtually indicable groups not only contain groups of polynomial growth, they also include some groups of intermediate growth. It is worth mentioning that the EIT method proves that for p sufficiently close to 1, there exists a transient infinite connected component almost surely. Unfortunately, it is usually difficult to construct EIT measures on general graphs.
The question of pc(G) < 1 has also been approached by analyzing isoperimetric inequalities. Benjamini and Schramm proved in [27] that ifGsatisfies the isoperimetric inequality of “dimension∞”, i.e., if Gis non-amenable, then pc(G)<1. It was proved in [24] that every unimodular transitive non-amenable graph Ghas a thresholdα <1 such that any (not necessarily i.i.d.) automorphism invariant percolation measure on G with density higher than α has an infinite connected component with positive probability. Kozma proved in [95] that planar graphs of polynomial growth with no vertex accumulation points and isoperimetric dimension greater than 1 have non-trivial phase transition.
In [163], Teixeira showed that pc(G) < 1 for graphs G with polynomial growth and isoperimetric dimension greater than 1 for a stronger version of the isoperimet-ric inequality, called local isoperimetric inequality. Teixeira’s proof relies on a clever renormalization argument using in a crucial way the (essential) uniqueness of large connected components in a box. It is important to note that this property is not satis-fied under the sole assumption that Dim(G)>1, as exemplified by the graph made of two copies ofZdconnected to each other by a single edge between two of their vertices.
Also, in contrast to Teixeira’s proof, our methoddoes not rely on uniqueness: it works
for graphs on which there may be any number of infinite connected components. The method of [163] was further exploited in [38] to prove, without invoking Gromov’s the-orem, that pc(G)< 1 for quasi-transitive graphs G with super-linear but polynomial growth.
Let us conclude by pointing out that for a givenexplicit graphG, it is often not hard to find some particular structure inside it that allows one to verify that pc(G) < 1.
For instance, all known examples of Cayley graphs can be proved to have a phase transition without using the previous result. Nevertheless, groups not in the above known categories are discovered from time to time, see e.g. [121], and without our result they need a case by case analysis. This should be compared to many group-theoretical properties of Cayley graphs that can often be proved (or disproved) for every explicit groups, yet are tremendously difficult to verify for the whole family of graphs under study.
Discussion on the strategy of proof. The proof of Theorem 4.1.1relies on a new connection between the Gaussian Free Field (GFF) and Bernoulli percolation. The connection goes through the observation that conditionally on the absolute value of the GFF at every point, the distribution of the signs of the GFF is the one of an Ising model with certain coupling constants. Once the connection between the GFF and the Ising is made, we use the Edwards-Sokal coupling to relate the Ising model to Bernoulli percolation. As a result, it is possible to express the expectations of particu-lar observables of the GFF in terms of the probabilities of connections for a (annealed) percolation model on a random environment (i.e., random edge-parameters). Since the expectation of these observables can be explicitly computed in terms of the Green function and are therefore easy to study, one may bound from below the averaged probability of connections in this percolation model. One can also derive such results by exploiting connections from [102] and [154] between GFF on the metric graph ˜G and random interlacements, see remarks in the end of Section 4.2 for details. Let us mention that in recent years, GFF on the metric graph ˜G has efficiently explained a number of connections between GFF and percolation-type quantities, and that the previous statement is yet another illustration of the usefulness of this object.
The second step in the proof of Theorem 4.1.1 consists in integrating out the randomness of the environment in order to compare the probabilities of connection in the previous model to those in a Bernoulli percolation with fixed edge-parameter.
Since the environment is expressed in terms of the GFF, we will proceed step by step using a multi-scale decomposition of the GFF. More precisely, we will writeϕ=P
nϕn where the ϕn are independent Gaussian fields with finite-range correlations. We will then remove the ϕn one by one, while increasing an “independent” edge-parameter q in order to guarantee that the probabilities of connection keep increasing. At the end of the process, the randomness due to theϕn(and therefore to ϕ) will have completely disappeared, and we will be facing a percolation model with constant edge-density.
It is interesting to notice that we will not prove, in our second step, that a per-colation with some constant edge-parameter p < 1 stochastically dominates the one on the random environment, because this would be simply false. Rather, we will only compare the probabilities of connections, which is a weaker statement.
We believe that the present argument consisting in integrating out the long-range modes of the GFF will have further implications in the study of strongly correlated
percolation models. For instance, sharpness of the phase transition of the GFF super-level set percolation is obtained in the [56] and presented in Chapter 5; the key step is a comparison between percolation probabilities for level-sets of the GFF and that of a truncated (finite-range) version of it, which is obtained by implementing a strategy similar to the one of this chapter.
Open problems. The present results raise a number of interesting problems. The first natural one is to try to relax the requirements on the heat kernel decay. More precisely, we will see that in the first step of the proof (Proposition4.2.1) we only need the graph to be transient (which is true as soon as d > 2), so that we can consider the GFF in infinite volume. The assumption that d > 4 is used in the second step of the proof (Proposition 4.3.2) for (we believe) purely technical reasons. This naturally raises the following problem.
Problem 4.1.4. Prove Theorem 4.1.1 under the assumption that d >2.
The main step in which we need d > 4 is in the rewiring estimate of Step 3 in the proof of Lemma 4.3.6, where a competition takes place between the exponential rewiring cost and a super-exponential gain coming from the assumption d > 4. Re-placing the exponential cost by a polynomial one in the rewiring estimate would enable one to prove the result for d > 2. We believe that this problem is tractable in the case of quasi-transitive graphs. We chose not to present the proof since the result was already obtained by different means, but we wish to highlight the fact that the barrier d= 4 is not related to intersection properties of simple random-walks.
The proof uses the bounded degree assumption in one place only (in the last step of the proof of Lemma4.3.6 again). It is therefore natural to ask the following problem.
The proof uses the bounded degree assumption in one place only (in the last step of the proof of Lemma4.3.6 again). It is therefore natural to ask the following problem.