Locality of percolation for abelian Cayley graphs
LLocality of percolation for abelian Cayley graphs
Sébastien
Martineau ∗ , Vincent Tassion ∗ November 11, 2018
Abstract
We prove that the value of the critical probability for percolation on anabelian Cayley graph is determined by its local structure. This is a partialpositive answer to a conjecture of Schramm: the function p c defined onthe set of Cayley graphs of abelian groups of rank at least 2 is continuousfor the Benjamini-Schramm topology. The proof involves group-theoretictools and a new block argument. In the paper [3], Benjamini and Schramm launched the study of percolation inthe general setting of transitive graphs. Among the numerous questions thathave been studied in this setting stands the question of locality: roughly, “doesthe value of the critical probability depend only on the local structure of theconsidered transitive graph ?” This question emerged in [2] and is formalizedin a conjecture attributed to Oded Schramm. In the same paper, the particularcase of (uniformly non-amenable) tree-like graphs is treated.In the present paper, we study the question of locality in the context ofabelian groups. • Instead of working in the geometric setting of transitive graphs, we employthe vocabulary of groups — or more precisely of marked groups , aspresented in section 2. This allows us to use additional tools of algebraicnature, such as quotient maps, that are crucial to our approach. Thesetools could be useful to tackle Schramm’s conjecture in a more generalframework than the one presented in this paper, e.g. Cayley graphs ofnilpotent groups. • We extend renormalization techniques developed in [10] by Grimmett andMarstrand for the study of percolation on Z d (equipped with its standardgraph structure). The Grimmett-Marstrand theorem answers positivelythe question of locality for the d -dimensional hypercubic lattice. Withlittle extra effort, one can give a positive answer to Schramm’s conjec-ture in the context of abelian groups, under a symmetry assumption. Ourmain achievement is to improve the understanding of supercritical bondpercolation on general abelian Cayley graphs: such graphs do not have ∗ Both autors have been supported by the ANR grant MAC2 (ANR-10-BLAN-0123). a r X i v : . [ m a t h . P R ] D ec nough symmetry for Grimmett and Marstrand’s arguments to apply di-rectly. The techniques we develop here may be used to extend other resultsof statistical mechanics from symmetric lattices to lattices which are notstable under any reflection. The following paragraph presents the vocabulary needed to state Schramm’sconjecture (for more details, see [2]).
Transitive graphs
We recall here some standard definitions from graph the-ory. A graph is said to be transitive if its automorphism group acts transitivelyon its vertices. Let G denote the space of (locally finite, non-empty, connected)transitive graphs considered up to isomorphism. By abuse of notation, we willidentify a graph with its isomorphism class. Take G ∈ G and o any vertexof G . Then consider the ball of radius k (for the graph distance) centeredat o , equipped with its graph structure and rooted at o . Up to isomorphism ofrooted graphs, it is independent of the choice of o , and we denote it by B G ( k ).If G , H ∈ G , we set the distance between them to be 2 − n , where n := max { k : B G ( k ) ’ B H ( k ) } ∈ N ∪ {∞} . This defines the
Benjamini-Schramm distance on the set G . It was intro-duced in [4] and [2]. Locality in percolation theory
We will use the standard definitions frompercolation theory and refer to [9] and [12] for background on the subject. Toany
G ∈ G corresponds a critical parameter p c ( G ) for i.i.d. bond percolation.One can see p c as a function from G to [0 , Question 1 (Locality of percolation) . Consider a sequence of transitive graphs ( G n ) that converges to a limit G .Does the convergence p c ( G n ) −−−−→ n →∞ p c ( G ) hold? With this formulation, the answer is negative. Indeed, for the usual graphstructures, the following convergences hold: • ( Z /n Z ) −−−−→ n →∞ Z , • Z /n Z × Z −−−−→ n →∞ Z .In both cases, the critical parameter is constant equal to 1 all along the se-quence and jumps to a non trivial value in the limit. The following conjecture,attributed to Schramm and formulated in [2], states that Question 1 shouldhave a positive answer whenever the previous obstruction is avoided. Conjecture 1.1 (Schramm) . Let G n −−−−→ n →∞ G denote a converging sequence oftransitive graphs. Assume that sup n p c ( G n ) < . Then p c ( G n ) −−−−→ n →∞ p c ( G ) .
2t is unknown whether sup n p c ( G n ) < c ( G n ) < n . In other words, we do not know if 1 is an isolated point in the set of criticalprobabilities of transitive graphs. Besides, no geometric characterization of theprobabilistic condition p c ( G ) < The following theorem, proved in [10], is an instance of locality result. It was animportant step in the comprehension of the supercritical phase of percolation.
Theorem 1.2 (Grimmett-Marstrand) . Let d ≥ . For the usual graph struc-tures, the following convergence holds: p c (cid:0) Z × {− n, . . . , n } d − (cid:1) −→ n →∞ p c (cid:0) Z d (cid:1) . Remark.
Grimmett and Marstrand’s proof covers more generally the case ofedge structures on Z d that are invariant under both translation and reflection.The graph Z × {− n, . . . , n } d − is not transitive, so the result does not fitexactly into the framework of the previous subsection. However, as remarkedin [2], one can easily deduce from it the following statement:p c Z × (cid:18) Z n Z (cid:19) d − ! −→ n →∞ p c (cid:0) Z d (cid:1) . (1)Actually, after having introduced the space of marked abelian groups, we willsee in section 2.3 that one can deduce from the Grimmett-Marstrand theorema statement that is much stronger than convergence (1). We will be able toprove that p c ( Z d ) = lim p c ( G n ) for any sequence of abelian Cayley graphs G n converging to Z d with respect to the Benjamini-Schramm distance. In this paper we prove the following theorem, which provides a positive answerto Question 1 in the particular case of Cayley graphs of abelian groups (seedefinitions in section 2).
Theorem 1.3.
Consider a sequence ( G n ) of Cayley graphs of abelian groupssatisfying p c ( G n ) < for all n . If the sequence converges to the Cayley Graph G of an abelian group, then p c ( G n ) −−−−→ n →∞ p c ( G ) . (2)We now give three examples of application of this theorem. Let d ≥
2, fix agenerating set S of Z d , and denote by G the associated Cayley graph of Z d . Example 1:
There exists a natural Cayley graph G n of Z × (cid:0) Z n Z (cid:1) d − that iscovered by G . For such sequence, the convergence (2) holds, and general-izes (1). 3 xample 2: Consider the generating set of Z d obtained by adding to S all the n · s , for s ∈ S . The corresponding Cayley graph H n converges to theCartesian product G × G , and we getp c ( H n ) −−−−→ n →∞ p c ( G × G ) . Example 3:
Consider a sequence of vectors x n ∈ Z d such that lim | x n | = ∞ ,and write G n the Cayley graph of Z d constructed from the generating set S ∪ { x n } . Then the following convergence holds:p c ( G n ) −−−−→ n →∞ p c ( G × Z ) . The content of Example 2 was obtained in [11] when G is the canonicalCayley graph of Z d , based on Grimmett-Marstrand theorem. In the statementabove, G can be any Cayley graph of Z d , and Grimmett-Marstrand theoremcannot be applied without additional symmetry assumption. In this paper, we work with abelian groups because their structure is very wellunderstood. An additional important feature is that the net formed by largeballs of an abelian Cayley graph has roughly the same geometric structure asthe initial graph. Since nilpotent groups also present these characteristics, thefollowing question appears as a natural step between Theorem 1.3 and Ques-tion 1.
Question 2.
Is it possible to extend Theorem 1.3 to nilpotent groups?
This question can also be asked for other models of statistical mechanicsthan Bernoulli percolation. In questions 3 and 4, we mention two other naturalcontexts where the locality question can be asked.Theorem 2.1 of [5] states that locality holds for the critical temperature ofthe Ising model for the hypercubic lattice. This suggests the following question.
Question 3.
Is it possible to prove Theorem 1.3 for the critical temperature ofthe Ising model instead of p c ? Define c n as the number of self-avoiding walks starting from a fixed root ofa transitive graph G . By sub-multiplicativity, the sequence c /nn converges to alimit called the connective constant of G . In this context, the following questionwas raised by I. Benjamini [1]: Question 4.
Does the connective constant depend continuously on the consid-ered infinite transitive graph?
Section 2 presents the material on marked abelian groups that will be neededto establish Theorem 1.3. In section 2.4, we explain the strategy of the proof,which splits into two main lemmas. Sections 3 and 4 are each devoted to theproof of one of these lemmas.We drive the attention of the interested reader to Lemma 3.6. Togetherwith the uniqueness of the infinite cluster, it allows to avoid the construction of“seeds” in Grimmett and Marstrand’s approach.4
Marked abelian groups and locality
In this section, we present the space of marked abelian groups and show howproblems of Benjamini-Schramm continuity for abelian Cayley graphs can be re-duced to continuity problems for marked abelian group. Then, we provide a firstexample illustrating the use of marked abelian groups in proofs of Benjamini-Schramm continuity. Finally, section 2.4 presents the proof of Theorem 2.3,which is the marked group version of our main theorem.General marked groups are introduced in [8]. Here, we only define markedgroups and Cayley graphs in the abelian setting, since we do not need a higherlevel of generality.
Let d denote a positive integer. A ( d -)marked abelian group is the data ofan abelian group together with a generating d -tuple ( s , . . . , s d ), up to isomor-phism . We write G d the set of the d -marked abelian groups. Elements of G d will be denoted by [ G ; s , . . . , s d ] or G • , depending on whether we want to insiston the generating system or not. Finally, we write G the set of all the markedabelian groups: it is the disjoint union of all the G d ’s. Quotient of a marked abelian group
Given a marked abelian group G • =[ G ; s , . . . , s d ] and a subgroup Λ of G , we define the quotient G • / Λ by G • / Λ = [ G/ Λ; s , . . . , s d ] , where ( s , . . . , s d ) is the image of ( s , . . . , s d ) by the canonical surjection from G onto G/ Λ. Quotients of marked abelian groups will be crucial to define andunderstand the topology of the set of marked abelian groups. In particular, forthe topology defined below, the quotients of a marked abelian group G • formsa neighbourhood of it. The topology
We first define the topology on G d . Let δ denote the canonicalgenerating system of Z d . To each subgroup Γ of Z d , we can associate an elementof G d via the mapping Γ [ Z d ; δ ] / Γ . (3)One can verify that the mapping defined by (3) realizes a bijection from theset of the subgroups of Z d onto G d . This way, G d can be seen as a subsetof { , } Z d . We consider on G d the topology induced by the product topologyon { , } Z d . This makes of G d a Hausdorff compact space. Finally, we equip G with the topology generated by the open subsets of the G d ’s. (In particular, G d is an open subset of G .)Let us illustrate the topology with three examples of converging sequences: • [ Z /n Z ; 1] converges to [ Z ; 1]. • [ Z ; 1 , n, . . . , n d ] converges to [ Z d ; δ ]. • [ Z ; 1 , n, n + 1] converges to [ Z ; δ , δ , δ + δ ]. ( G ; s , . . . , s d ) and ( G ; s , . . . , s d ) are isomorphic if there exists a group isomorphism from G to G mapping s i to s i for all i . ayley graphs Let G • = [ G ; s , . . . , s d ] be a marked abelian group. Its Cay-ley graph, denoted Cay ( G • ), is defined by taking G as vertex-set and declaring a and b to be neighbours if there exists i such that a = b ± s i . It is is uniquelydefined up to graph isomorphism. We write B G • ( k ) ⊂ G the ball of radius k in Cay ( G • ), centered at 0. Converging sequences of marked abelian groups
In the rest of the paper,we will use the topology of G through the following proposition, which gives ageometric flavour to the topology. In particular, it will allow to do the connectionwith the Benjamini-Schramm topology through corollary 2.2. Proposition 2.1.
Let ( G • n ) be a sequence of marked abelian groups that con-verges to some G • . Then, for any integer k , the following holds for n largeenough:1. G • n is of the form G • / Λ n , for some subgroup Λ n of G , and2. Λ n ∩ B G • ( k ) = { } .Proof. Let d be such that G • ∈ G d . For n large enough, we also have G • n ∈ G d .Let Γ (resp. Γ n ) denote the unique subgroup of Z d that corresponds to G • (resp. G • n ) via bijection (3). The group Γ is finitely generated: we consider F a finite generating subset of it. Taking n large enough, we can assume thatΓ n contains F , which implies that Γ is a subgroup Γ n . We have the followingsituation Z d ϕ −→ Z d / Γ ψ n −−→ Z d / Γ n . Identifying G with Z d / Γ and taking Λ n = ker ψ n = Γ n / Γ, we obtain the firstpoint of the proposition.By definition of the topology, taking n large enough ensures that Γ n ∩ B Z d ( k ) = Γ ∩ B Z d ( k ). We have B Z d / Γ ( k ) ∩ Λ n = ϕ ( B Z d ( k ) ∩ Γ n )= ϕ ( B Z d ( k ) ∩ Γ)= { } . This ends the proof of the second point.
Corollary 2.2.
The mapping
Cay from G to G that associates to a markedabelian group its Cayley graph is continuous. Via its Cayley graph, we can associate to each marked abelian group G • acritical parameter p c • ( G • ) := p c ( Cay ( G • )) for bond percolation. If G • is amarked abelian group, then p c • ( G • ) < G is at least 2.(We commit the abuse of language of calling rank of an abelian group the rankof its torsion-free part.) This motivates the following definition:˜ G = { G • ∈ G : rank ( G ) ≥ } . In the context of marked abelian groups, we will prove the following theorem:6 heorem 2.3.
Consider G • n −→ G • a converging sequence in ˜ G . Then, p c • ( G • n ) −−−−→ n →∞ p c • ( G • ) . Theorem 2.3 above states that p c • is continuous on ˜ G . It seems a prioriweaker than Theorem 1.3. Nevertheless, the following lemma allows us to deduceTheorem 1.3 from Theorem 2.3. Lemma 2.1.
Let G • be an element of ˜ G . Assume it is a continuity point ofthe restricted function p c • : ˜ G −→ (0 , . Then its associated Cayley graph
Cay ( G • ) is a continuity point of the restrictedfunction p c : Cay ( ˜ G ) −→ (0 , . Proof.
Assume, by contradiction, that there exists a sequence of marked abeliangroups G • n in ˜ G such that Cay ( G • n ) converges to some Cay ( G • ) and p c • ( G • n ) staysaway from p c • ( G • ). Define d to be the degree of Cay ( G • ). Considering n largeenough, we can assume that all the G • n ’s lie in the compact set S d ≤ d G d . Upto extraction, one can then assume that G • n converges to some marked abeliangroup G •∞ . This group must have rank at least 2. Since Cay is continuous,
Cay ( G • ) = Cay ( G •∞ ) and Theorem 2.3 is contradicted by the sequence ( G • n )that converges to G •∞ .We will also use the following theorem, which is a particular case of theorem3.1 in [3]. Theorem 2.4.
Let G • be a marked abelian group and Λ a subgroup of G . Then p c • ( G • / Λ) ≥ p c • ( G • ) . In this section, we will prove Proposition 2.5, which is a particular case ofTheorem 1.3. We deem interesting to provide a short independent proof of it.This proposition epitomizes the scope of Grimmett-Marstrand results in ourcontext. It also illustrates how marked groups can appear as useful tools todeal with locality questions. More precisely, Lemma 2.1 reduces some questionsof continuity in the Benjamini-Schramm space to equivalent questions in thespace of marked abelian groups, where the topology allows to employ methodsof algebraic nature.
Proposition 2.5.
Let ( G • n ) be a sequence in ˜ G . Assume that G • n −−−−→ n →∞ [ Z d ; δ ] ,where δ stands for the canonical generating system of Z d . Then p c • ( G • n ) −−−−→ n →∞ p c • ([ Z d ; δ ]) . Proof.
Since G d is open, we can assume that G • n belongs to it. It is thus aquotient of [ Z d ; δ ], and Theorem 2.4 giveslim inf p c • ( G • n ) ≥ p c • ([ Z d ; δ ]) .
7o establish the other semi-continuity, we will show that the Cayley graph of G • n eventually contains Z × { , . . . , K } as a subgraph (for K arbitrarily large),and conclude by applying Grimmett-Marstrand theorem.Let us denote Γ n the subgroup of Z d associated to G • n via bijection (3). Wecall coordinate plane a subgroup of Z d generated by two different elements ofthe canonical generating system of Z d . Lemma 2.2.
For any integer K , for n large enough, there exists a coordinateplane Π satisfying (Π + B Z d (0 , K + 1)) ∩ Γ n = { } . Proof of Lemma 2.2.
To establish Lemma 2.2, we proceed by contradiction. Up toextraction, we can assume that there exists some K such thatfor all Π, (Π + B Z d (0 , K + 1)) ∩ Γ n = { } . (4)We denote by v Π n a non-zero element of (Π + B Z d (0 , K + 1)) ∩ Γ n . Up to extraction,we can assume that, for all Π, the sequence v Π n / k v Π n k converges to some v Π . (Thevector space R d is endowed with an arbitrary norm k k .) Since Γ n converges pointwiseto { } , for any Π, the sequence k v Π n k tends to infinity. This entails, together withequation (4), that v Π is contained in the real plane spanned by Π. The incompletebasis theorem implies that the vector space spanned by the v Π ’s has dimension at least d −
1. By continuity of the minors, for n large enough, the vector space spanned byΓ n as dimension at least d −
1. This entails that, for n large enough, Γ n has rank atleast d −
1, which contradicts the hypothesis that Z d / Γ n has rank at least 2. For any K , provided that n is large enough, one can see Z ×{− K, . . . , K } d − as a subgraph of Cay ( G • n ). (Restrict the quotient map from Z d to G • n to theΠ + B Z d (0 , K ) given by Lemma 2.2 and notice that it becomes injective.) Itresults from this thatlim sup p c • ( G • n ) ≤ p c ( Z × {− K, . . . , K } d − ) . The right-hand side goes to p c • ([ Z d ; δ ]) as K goes to infinity, by Grimmett-Marstrand theorem. This establishes the second semi-continuity. Remark.
Proposition 2.5 states exactly what Grimmett-Marstrand theorem im-plies in our setting. Together with Lemma 2.1, it entails that the hypercubiclattice is a continuity point of p c on Cay ( ˜ G ). Without additional idea, onecould go a bit further: the proof of Grimmett and Marstrand adjusts directlyto the case of Cayley graphs of Z d that are stable under reflections relativeto coordinate hyperplanes. This statement also has a counterpart analog toProposition 2.5. Though, we are still far from Theorem 2.3, since Grimmett-Marstrand theorem relies heavily on the stability under reflection. In the restof the paper, we solve the locality problem for general abelian Cayley graphs.We do so directly in the marked abelian group setting, and do not use a “slabresult” analog to Grimmett-Marstrand theorem. The purpose of this section is to reduce the proof of Theorem 2.3 to the proofof two lemmas (Lemma 2.3 and Lemma 2.4). These are respectively establishedin sections 3 and 4. 8s in section 2.3, it is the upper semi-continuity of p c • that is hard to estab-lish: given G • and p > p c • ( G • ), we need to show that the parameter p remainssupercritical for any element of ˜ G that is close enough to G • . To do so, we willcharacterize supercriticality by using a finite-size criterion , that is a propertyof the type “ P p [ E N ] > − η ” for some event E N that depends only on the statesof the edges in the ball of radius N . The finite-size criterion we use is denotedby FC ( p, N, η ) and characterizes supercriticality through lemmas 2.3 and 2.4.Its definition involving heavy notation, we postpone it to section 3.4.First, we work with a fixed marked abelian group G • . Assuming that p > p c • ( G • ), we construct in its Cayley graph a box that exhibits nice connectionproperties with high probability. This is formalized by Lemma 2.3 below, whichwill be proved in section 3. Lemma 2.3.
Let G • ∈ ˜ G . Let p > p c • ( G • ) and η > . Then, there exists N such that G • satisfies the finite-size criterion FC ( p, N, η ) . Then, take H • = G • / Λ a marked abelian group that is close to G • . Since Cay ( G • ) and Cay ( H • ) have the same balls of large radius, the finite criterionis also satisfied by H • . This enables us to prove that there is also percolationin Cay ( H • ). As in Grimmett and Marstrand’s approach, we will not be ableto prove that percolation occurs in Cay ( H • ) for the same parameter p , but wewill have to slightly increase the parameter. Here comes a precise statement,established in section 4. Lemma 2.4.
Let G • ∈ ˜ G . Let p > p c • ( G • ) and δ > . Then there exists η > such that the following holds: if there exists N such that G • satisfies the finite-size criterion FC ( p, N, η ) , then p c ( H • ) < p + δ for any marked abelian group H • close enough to G • . Assuming these two lemmas, let us prove Theorem 2.3.
Proof of Theorem 2.3.
Let G • n −−−−→ n →∞ G • denote a converging sequence of ele-ments of ˜ G . Our goal is to establish that p c • ( G • n ) −−−−→ n →∞ p c • ( G • ).For n large enough, G • n is a quotient of G • . (See Proposition 2.1.) ByTheorem 2.4, for n large enough, p c • ( G • ) ≤ p c • ( G • n ). Hence, we only need toprove that lim sup p c • ( G • n ) ≤ p c • ( G • ) . Take p > p c and δ >
0. By Lemma 2.3, we can pick N such that FC ( p, N, η )is satisfied. Lemma 2.4 then guarantees that, for n large enough, p c • ( G • n ) ≤ p + δ ,which ends the proof. Through the entire section, we fix:- G • ∈ ˜ G a marked abelian group of rank greater than two,- p ∈ (p c • ( G • ) , η > G • under the form [ Z r × T ; S ], where T is a finite abelian group. Let G = ( V, E ) = ( Z r × T, E ) denote the Cayley graph associated to G • . Paths andpercolation will always be considered relative to this graph structure.9 .1 Setting and notation An element of Z r × T will be written x = ( x free , x tor ) . For the geometric reasonings, we will use linear algebra tools. (The vertex set— Z r × T — is roughly R r .) Endow R r with its canonical Euclidean structure.We denote by k k the associated norm and B ( v, R ) the closed ball of radius R centered at v ∈ R r . If the center is 0, this ball may be denoted by B ( R ). Set R S := max s ∈ S k s free k . In G , we define for k > B ( k ) := { x : k x free k ≤ kR S } = ( B ( kR S ) ∩ Z d ) × T. Up to section 3.4, we fix an orthornomal basis e = ( e , . . . , e d ) of R r . Define π e : R r −→ R P ri =1 x i e i ( x , x ) . We now define the function
Graph , which allows us to move between the con-tinuous space R and the discrete set V . It associates to each subset X of R the subset of V defined by Graph ( X ) := (cid:0)(cid:0) π − e ( X ) + B ( R S ) (cid:1) ∩ Z r (cid:1) × T. (5)In section 3.4, we will have to consider different bases. To insist on the depen-dence on e , we will write Graph e .If a and b belong to R , we will consider the segment [ a, b ] and the parallel-ogram [ a, b, − a, − b ] spanned by a and b in R , defined respectively by[ a, b ] = { λa + (1 − λ ) b ; 0 ≤ λ ≤ } and[ a, b, − a, − b ] = { λa + µb ; | λ | + | µ | ≤ } Write then L ( a, b ) := Graph ([ a, b ]) and R ( a, b ) := Graph ([3 a, b, − a, − b ]) thecorresponding subsets of V .The following lemma illustrates one important property of the function Graph connecting continuous and discrete.
Lemma 3.1.
Let X ⊂ R . Let γ be a finite path of length k in G . Assumethat γ ∈ Graph ( X ) and γ k Graph ( X ) . Then the support of γ intersects Graph ( ∂X ) .Proof. It suffices to show that if x and y are two neighbours in G such that x ∈ Graph ( X ) and y / ∈ Graph ( X ), then x belongs to Graph ( ∂X ). By definitionof Graph , we have x free ∈ π − ( X ) + B ( R S ), which can be restated as π ( B ( x free , R S )) ∩ X = ∅ . (6)By definition of R S , we have y free ∈ B ( x free , R S ) and our assumption on y impliesthat π ( y free ) / ∈ X , which gives π ( B ( x free , R S )) ∩ c X = ∅ . (7)10ince π ( B ( x free , R S )) is connected, (6) and (7) implies that π ( B ( x free , R S )) ∩ ∂X = ∅ which proves that x belongs to Graph ( ∂X ). We denote by P p the law of independent bond per-colation of parameter p ∈ [0 ,
1] on G . Connections
Let A , B and C denote three subsets of V . The event “thereexists an open path intersecting A and B that lies in C ” will be denoted by“ A C ←→ B ”. The event “restricting the configuration to C , there exists a uniquecomponent that intersects A and B ” will be written “ A ! C ! ←−→ B ”. The event“there exists an infinite open path that touches A and lies in C will be denotedby “ A C ←→ ∞ ”. If the superscript C is omitted, it means that C is taken to bethe whole vertex set.This paragraph contains the percolation results that will be needed to proveTheorem 2.3. The following lemma, sometimes called “square root trick”, is astraightforward consequence of Harris-FKG inequality. Lemma 3.2.
Let A and B be two increasing events. Assume that P p [ A ] ≥ P p [ B ] . Then, the following inequality holds: P p [ A ] ≥ − (1 − P p [ A ∪ B ]) / . The lemma above is often used when P p [ A ] = P p [ B ], in a context wherethe equality of the two probabilities is provided by symmetries of the underly-ing graph (see [9]). This slightly generalized version allows to link geometricproperties to probabilistic estimates whithout any symmetry assumption, asillustrated by the following lemma. Lemma 3.3.
Let a and b be two points in R . Let A ⊂ V be a subset of verticesof G . Assume that P p [ A ←→ L ( a, b )] > − ε for some ε > . (8) Then, there exists u ∈ [ a, b ] such that both P p [ A ←→ L ( a, u )] and P p [ A ←→ L ( u, b )] exceed − ε .Remark. The same statement holds when we restrict the open paths to lie in asubset C of V . Proof.
We can approximate the event estimated in inequality (8) and pick k large enough such that P p [ A ←→ L ( a, b ) ∩ B ( k )] > − ε . The set L ( a, b ) ∩ B ( k ) being finite, there are only finitely many different setsof the form L ( a, u ) ∩ B ( k ) for u ∈ [ a, b ]. We can thus construct u , u . . . , u n ∈ [ a, b ] such that u = a and u n = b , and for all 1 ≤ i < n ,11. [ a, u i ] is a strict subset of [ a, u i +1 ],2. L ( a, b ) ∩ B ( k ) is the union of L ( a, u i ) ∩ B ( k ) and L ( u i +1 , b ) ∩ B ( k ).Assume that for some i , the following inequality holds: P p [ A ←→ L ( a, u i ) ∩ B ( k )] ≥ P p [ A ←→ L ( u i +1 , b ) ∩ B ( k )] . (9)Lemma 3.2 then implies that P p [ A ←→ L ( a, u i ) ∩ B ( k )] > − ε. If inequality (9) never holds (resp. if it holds for all possible i ), then A isconnected to L ( { a } ) (resp. to L ( { b } )) with probability exceeding 1 − ε . In thesetwo cases, the conclusion of the lemma is trivially true. We can assume thatwe are in none these two situations, and define j ∈ { , . . . , n − } to be thesmallest possible i such that inequality (9) holds. We will show the conclusionof Lemma 3.3 holds for u = u j . We already have P p [ A ←→ L ( a, u j ) ∩ B ( k )] > − ε, and inequality (9) does not hold for i = j −
1. Once again, Lemma 3.2 impliesthat P p [ A ←→ L ( u j , b ) ∩ B ( k )] > − ε. Lemma 3.4.
Bernoulli percolation on G at a parameter p > p c ( G ) producesalmost surely a unique infinite component. Moreover, any fixed infinite subsetof V is intersected almost surely infinitely many times by the infinite component. The first part of the lemma is standard (see [6] or [9]). The second partstems from the 0-1 law of Kolmogorov.
In this section, we aim to prove that a set connected to infinity with highprobability also has “good” local connections. To formalize this, we need a fewadditionnal definitions. We say that ( a, b, u, v ) ∈ (cid:0) R (cid:1) is a good quadruple if 1. u = a + b ,2. v ∈ [ − a, b ] and3. [ a, b, − a, − b ] contains the planar ball of radius R S .Property 3 ensures that the parallelogramm [ a, b, − a, − b ] is not too degenerate.To each good quadruple ( a, b, u, v ), we associate the following four subsets ofthe graph G : Z ( a, b, u, v ) = { L ( a, u ) , L ( u, b ) , L ( b, v ) , L ( v, − a ) } . S ba − a − b uv Figure 1: A good quadruple
Lemma 3.5.
Let A be a finite subset of V containing and such that − A := {− x ; x ∈ A } = A. Let k ≥ be such that B := B ( k ) contains A . Assume the following relation tohold for some ε ∈ (0 , : P p [ A ←→ ∞ ] > − ε . Then there exists a good quadruple ( a, b, u, v ) such that for any Z ∈ Z ( a, b, u, v ) (i) B ∩ Z = ∅ ,(ii) P p (cid:20) A R ( a,b ) ←−−−→ Z (cid:21) > − ε .Proof. Let ( n, h, ‘ ) ∈ N × R × R + . Define a := ( n, h − ‘ ), b := ( n, h + ‘ ) and thethree following subsets of V illustrated on Figure 2: C ( n, h, ‘ ) := Graph ([ a, b, − a, − b ]) LR ( n, h, ‘ ) := Graph ([ a, b ] ∪ [ − a, − b ]) = L ( a, b ) ∪ L ( − a, − b ) UD ( n, h, ‘ ) := Graph ([ − a, b ] ∪ [ − b, a ]) = L ( − a, b ) ∪ L ( − b, a )Let us start by focusing on the geometric constraint (i), which we wish totranslate into analytic conditions on the triple ( n, h, ‘ ). We fix n B large enoughsuch that B ∩ Graph (cid:0) R \ ( − n B + 1 , n B − (cid:1) = ∅ . (10)This way, any set defined as the image by the function Graph of a planar setin the complement of ( − n B + 1 , n B − will not intersect B . In particular,defining for n > n B and h ∈ R ‘ B ( n, h ) = n B (cid:18) | h | n (cid:19) , n ‘LR ( n,h,‘ ) UD ( n,h,‘ ) C ( n,h,‘ ) Figure 2: Pictures of the planarsets defining C ( n, h, ‘ ), UD ( n, h, ‘ )and LR ( n, h, ‘ ) hnn B ‘ B ( n,h ) Figure 3: Definition of ‘ B ( n, h )the set UD ( n, h, ‘ ) does not intersect B whenever ‘ ≥ ‘ B −
1. (See Figure 3.)Suppose that A intersects the infinite cluster. By Lemma 3.4, V \ C ( n, h, ‘ )— which is infinite — intersects the infinite cluster almost surely. Thus thereexists an open path from A to V \ C ( n, h, ‘ ). By Lemma 3.1, A is connected to UD ( n, h, ‘ ) ∪ LR ( n, h, ‘ ) within C ( n, h, ‘ ), which gives the following inequality: P p (cid:20)(cid:18) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:19) ∪ (cid:18) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:19)(cid:21) > − ε . (11)The strategy of the proof is to work with some sets C ( n, h, ‘ ) that are bal-anced in the sense that P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) and P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) are close, and conclude with Lemma 3.2. We shall now prove two facts, which en-sure that the inequality between the two afore-mentioned probabilities reversesfor some ‘ between ‘ B ( n, h ) and infinity. Fact 1.
There exists n > n B such that, for all h ∈ R , when ‘ = ‘ B ( n, h ) P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) < P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) . Proof of fact 1.
For n > n B + R S , define the following sets, illustrated on Fig-ure 4: X = Graph ((( −∞ , n B ) × R ) ∪ ( R × [ − n B , ∞ ))) ∂X = Graph (( { n B } × ( −∞ , − n B ]) ∪ ([ n B , ∞ ) × {− n B } )) X n = Graph (([ − n, n B ) × R ) ∪ ([ − n, n ] × [ − n B , ∞ ))) ∂ X n = Graph ( {− n } × R ∪ { n } × [ − n B , ∞ )) ∂ X n = Graph ( { n B } × ( −∞ , − n B ] ∪ [ n B , n ] × {− n B } )Since the sequence of events (cid:16) A X n ←−→ ∂ X n (cid:17) n>n B + R S is decreasing, we have14 X n BA ∂ X n ∂ X n n n B Figure 4: Planar pictures corresponding to X , X n , ∂ X n and ∂ X n lim n →∞ P p h A X n ←−→ ∂ X n i = P p " \ n>n B + R S (cid:16) A X n ←−→ ∂ X n (cid:17) ≤ P p h A X ←→ ∞ i = P p h(cid:16) A X ←→ ∞ (cid:17) ∩ (cid:16) A X ←→ ∂X (cid:17)i . (12)(The last equality results from the fact that the infinite set V \ X intersects theinfinite cluster almost surely.)The sequence (cid:16) A X n ←−→ ∂ X n (cid:17) n>n B + R S is increasing, hence we havelim n →∞ P p h A X n ←−→ ∂ X n i = P p " [ n>n B + R S (cid:16) A X n ←−→ ∂ X n (cid:17) = P p [ A ←→ ∂X ] . (13)Since p ∈ (0 ,
1) and A is finite, the probability that A is connected to ∂X butintersects only finite clusters is positive. Thus the following strict inequalityholds P p h(cid:16) A X ←→ ∞ (cid:17) ∩ (cid:16) A X ←→ ∂X (cid:17)i < P p [ A ←→ ∂X ] . (14)From (12), (13) and (14), we can pick n > n B + R S large enough such that,for all n ≥ n , P p h A X n ←−→ ∂ X n i < P p h A X n ←−→ ∂ X n i . Fix n ≥ n and h ≥
0, then define ‘ = ‘ B ( n, h ). For these parameters, we have15 ⊂ C ( n, h, ‘ ) ⊂ X n and LR ( n, h, ‘ ) ⊂ ∂ X n , which gives P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) ≤ P p h A X n ←−→ ∂ X n i < P p h A X n ←−→ ∂ X n i ≤ P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) . The last inequality follows from the observation that each path connecting A to ∂ X n inside X n has to cross UD ( n, h, ‘ ).The computation above shows that the following srict inequality holds for n ≥ n , h ≥
0, and ‘ = ‘ B ( n, h ) P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) < P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) . (15)In the same way, we find n such that for all n ≥ n and h ≤
0, equation (15)holds for ‘ = ‘ B ( n, h ). Taking n = max( n , n ) ends the proof of the fact.In the rest of the proof, we fix n as in the previous fact. For h ∈ R , define ‘ eq ( h ) = sup n ‘ ≥ ‘ B ( n, h ) − P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) ≥ P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) o . Fact 2.
For all h ∈ R , the quantity ‘ eq ( h ) is finite.Proof of fact 2. We fix h ∈ R and use the same technique as developed in theproof of the fact 1. Define Y = Graph ([ − n, n ] × R ) ∂Y = Graph ( {− n, n } × R )In the same way we proved equations (12) and (13), we have herelim ‘ →∞ P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) = P p h A Y ←→ ∞ i lim ‘ →∞ P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) = P p [ A ←→ ∂Y ]Thus, we can find a finite ‘ large enough such that P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) < P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) . The quantity ‘ eq plays a central role in our proof, linking geometric andprobabilistic estimates. We can apply Lemma 3.2 with the two events appearing16n inequality (11), to obtain the following alternative:If ‘ < ‘ eq ( h ), then P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) > − ε . (16a)If ‘ > ‘ eq ( h ), then P p (cid:20) A C ( n,h,‘ ) ←−−−−→ LR ( n, h, ‘ ) (cid:21) > − ε . (16b)Fix ( h opt , ‘ ) ∈ R × R + such that ‘ eq ( h opt ) < ‘ < inf h ∈ R ( ‘ eq ( h )) + . (17)With such notation, we derive from (16b) P p (cid:20) A C ( n,h opt ,‘ ) ←−−−−−−−→ LR ( n, h opt , ‘ ) (cid:21) > − ε . Another application of Lemma 3.2 ensures then the existence of a real number h of the form h = h opt + σ‘ / σ ∈ {− , , +2 } ) such that P p (cid:20) A C ( n,h opt ,‘ ) ←−−−−−−−→ LR ( n, h , ‘ / (cid:21) > − ε . Recall that LR ( n, h , ‘ /
3) = L ( a , b ) ∪ L ( − a , − b ) with a = ( n, h − ‘ /
3) and b = ( n, h + ‘ / A is connected inside C ( n, h , ‘ /
3) to L ( a , b ) and to L ( − a , − b ) with equal probabilities. Applying again Lemma 3.2gives P p (cid:20) A C ( n,h opt ,‘ ) ←−−−−−−−→ L ( a , b ) (cid:21) > − ε . Then, use Lemma 3.3 to split L ( a , b ) into two parts that both have a goodprobability to be connected to A : we can pick u = ( n, h ) ∈ [ a , b ] such thatboth P p (cid:20) A C ( n,h opt ,‘ ) ←−−−−−−−→ L ( a , u ) (cid:21) and P p (cid:20) A C ( n,h opt ,‘ ) ←−−−−−−−→ L ( u, b ) (cid:21) exceed 1 − ε . Finally, pick ‘ such that ‘ eq ( h ) − < ‘ < ‘ eq ( h ). Define a = u + (0 , − ‘ ) and b = u + (0 , ‘ ). In particular, we have u = a + b . Our choice of ‘ (see equation (17)) implies that ‘ > ‘ − ≥ ‘ , and the following inclusionshold: L ( a , u ) ⊂ L ( a, u ) L ( u, b ) ⊂ L ( u, b ) C ( n, h opt , ‘ ) ⊂ R ( a, b )These three inclusions together with the estimates above conclude the point (ii)of Lemma 3.5 for Z = L ( a, u ) and Z = L ( u, b ).Now, let us construct a suitable vector v ∈ [ − a, b ] such that the point (ii) ofLemma 3.5 is verified for Z = L ( − a, v ) and Z = L ( v, b ). Equation (16a) impliesthat P p (cid:20) A C ( n,h,‘ ) ←−−−−→ UD ( n, h, ‘ ) (cid:21) > − ε .
17s above, using UD ( n, h, ‘ ) = L ( − a, b ) ∪ L ( − b, a ), symmetries and Lemma 3.2,we obtain P p (cid:20) A C ( n,h,‘ ) ←−−−−→ L ( − a, b ) (cid:21) > − ε . By Lemma 3.3, we can pick v ∈ [ − a, b ] such that the following estimate holdsfor Z = L ( − a, v ) , L ( v, b ): P p (cid:20) A C ( n,h,‘ ) ←−−−−→ Z (cid:21) > − ε ≥ − ε. It remains to verify the point (i). For Z = L ( a, u ) , L ( u, b ), it follows from n > n B and the definition of n B , see equation (10). For Z = L ( − a, v ) , L ( v, b ),it follows from ‘ > ‘ B ( n, h ) − ‘ B ( n, h ). In this section, we will define a finite block together with a local event that“characterize” supercritical percolation — in the sense that the event happeningon this block with high probability will guarantee supercriticality. This blockwill be used in section 4 for a coarse graining argument.In Grimmett and Marstrand’s proof of Theorem 1.2, the coarse grainingargument uses “seeds” (big balls, all the edges of which are open) in order topropagate an infinite cluster from local connections. More precisely, they definean exploration process of the infinite cluster: at each step, the exploration issuccesful if it creates a new seed in a suitable place, from which the processcan iterate. If the probability of success at each step is large enough, then, withpositive probability, the exploration process does not stop and an infinite clusteris created.In their proof, the seeds grow in the unexplored region. Since we cannotcontrol this region, we use the explored region to produce seeds instead. For-mally, long finite self-avoiding paths will play the role of the seeds in the proofof Grimmett and Marstrand. The idea is the following: if a point is reached atsome step of the exploration process, it must be connected to a long self-avoidingpath, which is enough to iterate the process.
Lemma 3.6.
For all ε > , there exists m ∈ N such that, for any fixed self-avoiding path γ of length m , P p [ γ ←→ ∞ ] > − ε. Proof.
By translation invariance we can restrict ourselves to self-avoiding pathsstarting at the origin 0. Fix ε >
0. For all k ∈ N we consider one self-avoidingpath γ ( k ) starting at the origin that minimizes the probability to intersect theinfinite cluster among all the self-avoiding paths of length k : P p h γ ( k ) ←→ ∞ i = min γ : length( γ )= k P p [ γ ←→ ∞ ] . By diagonal extraction, we can consider an infinite self-avoiding path γ ( ∞ ) suchthat, for any k ∈ N , (cid:16) γ ( ∞ )0 , γ ( ∞ )1 , . . . , γ ( ∞ ) k (cid:17) is the beginning of infinitely many18 ( k ) ’s. By Lemma 3.4, γ ( ∞ ) intersects almost surely the infinite cluster of a p -percolation. Thus, there exists an integer k such that P p hn γ ( ∞ )0 , γ ( ∞ )1 , . . . , γ ( ∞ ) k o ←→ ∞ i > − ε. Finally, there exists m such that γ m begins with the sequence( γ ( ∞ )0 , γ ( ∞ )1 , . . . , γ ( ∞ ) k ) , thus it intersects the infinite cluster of a p -percolation with probability exceeding1 − ε . By choice of γ ( m ) , it holds for any other self-avoiding path γ of length m that P p [ γ ←→ ∞ ] > − ε. We will focus on paths that start close to the origin. Let us define S ( m ) tobe the set of self-avoiding paths of length m that start in B (1). Lemma 3.7.
For any η > , there exist two integers m, N ∈ N and a goodquadruple ( a, b, u, v ) such that ∀ γ ∈ S ( m ) , ∀ Z ∈ Z ( a, b, u, v ) P p (cid:20) γ R ( a,b ) ∩ B ( N ) ←−−−−−−−→ Z ∩ B ( N ) (cid:21) > − η. Proof.
By Lemma 3.6, we can pick m such that any self-avoiding path γ ∈ S ( m )verifies P p [ γ ←→ ∞ ] > − η. Pick k ≥ m + 1 such that P p [ B ( k ) ←→ ∞ ] > − η . The number of disjoint clusters (for the configuration restricted to B ( n + 1))connecting B ( k ) to B ( n ) c converges when n tends to infinity to the numberof infinite clusters intersecting B ( k ). The infinite cluster being unique, we canpick n such that P p (cid:20) B ( k ) ! B ( n +1)! ←−−−−−→ B ( n ) c (cid:21) > − η. (18)Applying Lemma 3.5 with A = B ( k ) and B = B ( n + 1) provides a goodquadruple ( a, b, u, v ) such that the following two properties hold for any Z ∈Z ( a, b, u, v ):(i) B ( n + 1) ∩ Z = ∅ ,(ii) P p (cid:20) B ( k ) R ( a,b ) ←−−−→ Z (cid:21) > − η .Note that condition (i) implies in particular that B ( n + 1) is a subset of R ( a, b ).Equation (18) provides with high probability a “uniqueness zone” between B ( k )and B ( n ) c : any pair of open paths crossing this region must be connected inside B ( n + 1). In particular, when γ is connected to infinity, and B ( k ) is connected19o Z inside R ( a, b ), this “uniqueness zone” ensures that γ is connected to Z byan open path lying inside R ( a, b ): P p (cid:20) γ R ( a,b ) ←−−−→ Z (cid:21) ≥ P p (cid:20) { γ ←→ ∞} ∩ (cid:26) B ( k ) ! B ( n +1)! ←−−−−−→ B ( n ) c (cid:27) ∩ (cid:26) B ( k ) R ( a,b ) ←−−−→ Z (cid:27)(cid:21) > − η. The identity P p (cid:20) γ R ( a,b ) ←−−−→ Z (cid:21) = lim N →∞ P p (cid:20) γ R ( a,b ) ∩ B ( N ) ←−−−−−−−→ Z ∩ B ( N ) (cid:21) concludes the proof of Lemma 3.7. In this section, we give a precise definition of the finite-size criterion FC ( p, N, η )used in lemmas 2.3 and 2.4. Its construction is based on Lemma 3.7.Recall that, up to now, we worked with a fixed orthonormal basis e , whichwas hidden in the definition of Graph = Graph e , see equation (5). In order toperform the coarse graining argument in any marked group G • / Λ close to G • ,we will need to have the conclusion of Lemma 3.7 for all the orthonormal bases.Denote by B the set of the orthonormal basis of R r . It is a compact subsetof R r × r . If we fix X ⊂ R , a positive integer N and e ∈ B then the followinginclusion holds for any orthonormal basis f close enough to e in B : Graph e ( X ) ∩ B ( N ) ⊂ ( Graph f ( X ) + B (1)) ∩ B ( N ) . (19)We define N ( e , N ) ⊂ B to be the neighbourhood of e formed by the orthonor-mal bases f for which the inclusion above holds. A slight modification of theorthonormal basis in Lemma 3.7 keeps its conclusion with the same integer N and the same vectors a, b, u, v , but with • Z + B (1) in place of Z • and R ( a, b ) + B (1) instead of R ( a, b ).In order to state this result properly, let us define: Z N, e ( a, b, u, v ) = { ( Z + B (1)) ∩ B ( N ) : Z ∈ Z e ( a, b, u, v ) } ; R N, e ( a, b ) = ( R ( a, b ) + B (1)) ∩ B ( N ) . Note that we add the subscript e here to insist on the dependence in the basis e . This dependence was implicit for the sets Z and R ( a, b ) which were definedvia the function Graph .We are ready to define the finite size criterion FC ( p, N, η ) that appears inlemmas 2.3 and 2.4. Definition of the finite-size criterion.
Let N ≥ η >
0. We say thatthe finite size criterion FC ( p, N, η ) is satisfied if for any e ∈ B , there exist m ≥ a, b, u, v ) such that: ∀ γ ∈ S ( m ) , ∀ Z ∈ Z N, e ( a, b, u, v ) , P p (cid:20) γ R N, e ( a,b ) ←−−−−−→ Z (cid:21) > − η. (20)20 roof of Lemma 2.3. Let η >
0. Given e an orthonormal basis, Lemma 3.7provides m e , N e ∈ N , and a good quadruple ( a e , b e , u e , v e ) such that the follow-ing holds (we omit the subscript for the parameters m, a, b, u, v ): ∀ γ ∈ S ( m ) , ∀ Z ∈ Z e ( a, b, u, v ) , P p (cid:20) γ R e ( a,b ) ∩ B ( N e ) ←−−−−−−−−−→ Z ∩ B ( N e ) (cid:21) > − η. For any f ∈ N ( e , N e ), we can use inclusion (19) to derive from the estimateabove that for all γ ∈ S ( m ) and Z ∈ Z f ( a, b, u, v ), P p (cid:20) γ ( R f ( a,b )+ B (1)) ∩ B ( N e ) ←−−−−−−−−−−−−−−→ ( Z + B (1)) ∩ B ( N e ) (cid:21) > − η. By compactness of B , we can find a finite subset F ⊂ B of bases such that B = [ e ∈F N ( e , N e ) . For N := max e ∈B f N e , the finite-size criterion FC ( p, N, η ) is satisfied. Through the entire section, we fix:- G • ∈ ˜ G a marked abelian group of rank greater than two,- p ∈ (p c • ( G • ) , δ > G = ( V, E ) denote the Cayley graph associated to G • . Let us start by an observation that follows from the definition of good quadrupleat the beginning of section 3.2: there exists an absolute constant κ such thatfor any good quadruple ( a, b, u, v ) and any w ∈ R ,Card (cid:8) z ∈ Z : w + z u + z v ∈ [5 a, b, − a, − b ] (cid:9) ≤ κ. We fix κ as above and choose η > p := sup t ∈ N (cid:8) − (1 − δ/κ ) t + η (1 − p ) − t (cid:9) > p csite ( Z ) . (21)We will prove that this choice of η provides the conclusion of Lemma 2.4. Weassume that G • satisfies FC ( p, N, η ) for some positive integer N (which willbe fixed throughout this section). Let us consider a marked abelian group H • = G • / Λ of rank at least 2 and such thatΛ ∩ B (2 N + 1) = { } . (Notice that such H • ’s form a neighbourhood of G • in ˜ G by Proposition 2.1.)Under these hypotheses, we will prove that p c ( H • ) < p + δ , providing theconclusion of Lemma 2.4.The Cayley graph of H • = G • / Λ is denoted by G = ( V , E ). For x ∈ V , wewrite ¯ x for the image of x by the quotient map G → G/ Λ. This quotient mapnaturally extends to subsets of V and we write A for the image of a set A ⊂ V .21 .2 Sketch of proof Under the hypotheses above, we show that percolation occurs in G at parameter p + δ . The proof goes as follows. Step 1: Geometric construction.
We will construct a renormalized graph,that is a family of big boxes (living in G ) arranged as a square lattice. Inparticular, there will be a notion of neighbour boxes. The occurence ofthe finite-size criterion FC ( p, N, η ) will imply good connection probabilitiesbetween neighbouring boxes. This is the object of Lemma 4.2. Step 2: Construction of an infinite cluster.
The renormalized graph builtin the first step will allow us to couple a ( p + δ )-percolation on G witha percolation on Z in such a way that the existence of an infinite com-ponent in Z would imply an infinite component in G . This event willhappen with positive probability. The introduction of the parameter δ will allow us to apply a “sprinkling” technique in the coupling argumentdeveloped in the proof of Lemma 4.4. Since Λ has corank at least 2, we can fix an orthonormal basis e ∈ B such thatΛ ⊂ Ker ( π e ) × T. (22)Condition (22) ensures that sets defined in G via the function Graph e have asuitable image in the quotient G . More precisely, for any x ∈ V and any planarset X ⊂ R , we have x ∈ Graph e ( X ) ⇐⇒ x ∈ Graph e ( X ) . (23)According to FC ( p, N, η ), there exists m < N and a good quadruple ( a, b, u, v )such that ∀ γ ∈ S ( m ) , ∀ Z ∈ Z N, e ( a, b, u, v ) , P p (cid:20) γ R N, e ( a,b ) ←−−−−−→ Z (cid:21) > − η. We introduce here some subsets of G , that will play the role of vertices andedges in the renormalized graph. Box . For z in Z , define B z := Graph ( z u + z v + [ a, b, − a, − b ]) . When z and z are neigbours in Z for the standard graph structure, we write z ∼ z . In this case, we say that the two boxes B z and B z are neighbours . Corridor . For z in Z , define C z := Graph ( z u + z v + [4 a, b, − a, − b ]) . We will explore the cluster of the origin in G . If the cluster reaches a box B z ,we will try to spread it to the neighbouring boxes ( B z for z ∼ z ) by creatingpaths that lie in their respective corridors C z . For this strategy to work, weneed the boxes to have good connection probabilities and the corridors to be“sufficiently disjoint”: if the exploration is guaranted to visit each corridor atmost κ + 1 times, then we do need more than κ “sprinkling operations”. Thesetwo properties are formalized by the following two lemmas.22 emma 4.1. For all ¯ x ∈ V , Card { z ∈ Z / ¯ x ∈ C z } ≤ κ. (24) Proof.
By choice of the basis, equivalence (23) holds and implies, for any z =( z , z ) ∈ Z ,¯ x ∈ C z ⇐⇒ x ∈ Graph e ( z u + z v + [4 a, b, − a, − b ]) } By the last condition defining a good quadruple,¯ x ∈ C z = ⇒ π ( x ) ∈ z u + z v + [5 a, b, − a, − b ]The choice of κ at the beginning of the section (see equation (24)) concludesthe proof. Lemma 4.2.
For any couple of neighbouring boxes ( B z , B z ) , ∀ ¯ x ∈ B z , ∀ γ ∈ S ( m ) P p h ¯ x + γ C z ←−→ B z + B (1) i > − η. (25) Proof.
We assume that z = z + (0 , z + (1 , z + (0 , −
1) and z + ( − ,
0) are treated the same way.The assumption Λ ∩ B (2 N + 1) = { } implies that R N, e ( a, b ) is isomorphic(as a graph) to R N, e ( a, b ). It allows us to derive from estimate (20) that P p (cid:20) γ R N, e ( a,b ) ←−−−−−→ Z (cid:21) > − η. (26)Now let B z and B z be two neighbouring boxes. Let ¯ x be any vertex of B z . Bytranslation invariance, we get from (26) that P p (cid:20) x + γ ¯ x + R N. e ( a,b ) ←−−−−−−−→ ¯ x + Z (cid:21) > − η. Here comes the key geometric observation: there exists Z ∈ Z N, e ( a, b, u, v ) suchthat ¯ x + Z ⊂ B z + B (1) . This is illustrated on Figures 5 and 6 when z = (0 ,
0) and z = (0 , x + R N ( a, b ) ⊂ C z . Hence, by monotonicity, we obtain that P p h ¯ x + γ C z ←−→ B z + B (1) i > − η. Let ω be Bernoulli percolation of parameter p on G . In order to apply a“sprinkling argument”, we define for every z ∈ Z a sequence ( ξ z ( e )) e edges in C z of independent Bernoulli variables of parameter δκ . In other words, ξ z is a δκ -percolation on C z . We assume that ω and all the ξ z ’s are independent.Lemma 4.1 implies that at most κ + 1 Bernoulli variables are associated to agiven edge e : ω ( e ) and the ξ z ( e )’s for z such that e ⊂ C z .To state lemma 4.3, we also need the notion of edge-boundary. The edge-boundary of a set A of vertices is the set of the edges of G with exactly oneendpoint in A . It is denoted by ∆ A . 23 av ¯ xB z B z ¯ x + R ( a, b ) ¯ x + L ( v, b ) Figure 5: If ¯ x is on the left of the box B z , then ¯ x + L ( v, b ) ⊂ B z . bav ¯ xB z B z ¯ x + R ( a, b )¯ x + L ( − a, v ) Figure 6: If ¯ x is on the right of the box B z , then ¯ x + L ( − a, v ) ⊂ B z .24 emma 4.3. Let B z and B z be two neighbouring boxes. Let H be a subset of V . Let ( ω ( e )) e ∈ E be a family of independent Bernoulli variables of parameter P [ ω ( e ) = 1] ∈ [ p, independent of ξ z . If there exists ¯ x ∈ B z and γ ∈ S ( m ) such that ¯ x + γ ⊂ H , then P (cid:20) H C z ←−−→ ω ∨ ξ z B z + B (1) (cid:12)(cid:12)(cid:12)(cid:12) ∀ e ∈ ∆ H, ω ( e ) = 0 (cid:21) ≥ p . Proof.
In all this proof, the marginals of ω are assumed to be Bernoulli randomvariables of parameter p . The more general statement of Lemma 4.3 follows bya stochastic domination argument. The case H ∩ ( B z + B (1)) = ∅ being trivial,we assume that H ∩ ( B z + B (1)) = ∅ .Let W ⊂ ∆ H be the (random) set of edges { ¯ x, ¯ y } ⊂ C z such that(i) ¯ x ∈ H , ¯ y ∈ C z \ H and(ii) there is an ω -open path joining ¯ y to B z + B (1), lying in C z , but using noedge with an endpoint in H .In a first step, we want to say that | W | cannot be too small. The inclusions¯ x + γ ⊂ H ⊂ ( B z + B (1)) c imply that any ω -open path from ¯ x + γ to B z + B (1)must contain at least one edge of W . Thus, there is no ω -open path connecting¯ x + γ to B z + B (1) in C z when all the edges of W are ω -closed. Consequently,for any t ∈ N , we have P (cid:20)(cid:16) ¯ x + γ C z ←−→ ω B z + B (1) (cid:17) c (cid:21) ≥ P [all edges in W are ω -closed] ≥ (1 − p ) t P [ | W | ≤ t ] . To get the last inequality above, remark that the random set W is independentfrom the ω -state of the edges in ∆ H . Using estimate (25), it can be rewrittenas P [ | W | ≤ t ] ≤ η (1 − p ) − t . (27)We distinguish two cases. Either W is small, which has a probability estimatedby equation (27) above; or W is large, and we use in that case that B z + B (1) isconnected to H as soon as one edge of W is ξ z -open. The following computationmakes it quantitative: P (cid:20) H C z ←−−→ ω ∨ ξ z B z + B (1) (cid:12)(cid:12) ∀ e ∈ ∆ H, ω ( e ) = 0 (cid:21) ≥ P h at least one edge of W is ξ z -open (cid:12)(cid:12) ∀ e ∈ ∆ H, ω ( e ) = 0 i = P h at least one edge of W is ξ z -open i ≥ P h at least one edge of W is ξ z -open and | W | > t i ≥ − P h all the edges of W are ξ z -open (cid:12)(cid:12) | W | > t i − P (cid:2) | W | ≤ t (cid:3) . Using equation (27), we conclude that, for any t , P (cid:20) H C z ←−−→ ω ∨ ξ z A (cid:12)(cid:12) ∀ e ∈ ∆ H, ξ z ( e ) = 0 (cid:21) ≥ − (1 − δ/κ ) t − η (1 − p ) − t . (28)Our choice of η in (21) make the right hand side of (28) larger than p .25 emma 4.4. With positive probability, the origin is connected to infinity in theconfiguration ω total := ω ∨ _ z ∈ Z ξ z . Lemma 4.4 concludes the proof of Lemma 2.4 because ω total is stochasti-cally dominated by a ( p + δ )-percolation. Indeed, ( ω total ( e )) e is an independentsequence of Bernoulli variables such that, for any edge e , P [ ω total ( e ) = 1] ≥ − (1 − p )(1 − δ/κ ) κ ≥ p + δ. Proof of Lemma 4.4.
The strategy of the proof is similar to the one describedin the original paper of Grimmett and Marstrand: we explore the Bernoullivariables one after the other in an order prescribed by the algorithm hereafter.During the exploration, we define simultaneously random variables on the graph G and on the square lattice Z . Algorithm (0) Set z (0) = (0 , ∈ Z . Explore the connected component H ofthe origin in G in the configuration ω . Notice that only the edgesof H ∪ ∆ H have been explored in order to determine H . – If H contains a path of S ( m ), set X ((0 , U , V ) = ( { } , ∅ ) and move to ( t = 1). – Else, set X ((0 , U , V ) = ( ∅ , { } ) and move to( t = 1).( t ) Call unexplored the vertices in Z \ ( U t ∪ V t ). Examine the setof unexplored vertices neighbouring an element of U t . If this setis empty, define ( U t +1 , V t +1 ) = ( U t , V t ) and move to ( t + 1). Oth-erwise, choose such an unexplored vertex z t . In the configuration ω t +1 := ω t ∨ ξ z t , explore the connected component H t +1 of theorigin. – If H t +1 ∩ B z t = ∅ , which means in particular that B z t isconnected to 0 by an ω t +1 -open path, then set X ( z t ) = 1and ( U t +1 , V t +1 ) = ( U t ∪ { z t } , V t ) and move to ( t + 1). – Else set X ( z t ) = 0 and ( U t +1 , V t +1 ) = ( U t , V t ∪ { z t } ) andmove to ( t + 1).This algorithm defines in particular: • a random process growing in the lattice Z , S = ( U , V ) , S = ( U , V ) , . . . • a random sequence ( X ( z t )) t ≥ .Lemma 4.3 ensures that for all t ≥
1, whenever z t is defined, P [ X ( z t ) = 1 | S , S , . . . S t − ] ≥ p > p csite ( Z ) . (29)26stimate (29) states that each time we explore a new site z t , whatever the pastof the exploration is, we have a sufficiently high probability of success: togetherwith Lemma 1 of [10], it ensures that P [ | U | = ∞ ] > , where U := S t ≥ U t is the set of z t ’s such that X ( z t ) equals 1. For such z t ’s,we know that B z t is connected to the origin of G by an ω t +1 -open path. Hence,when U is infinite, there must exist an infinite open connected component inthe configuration ω ∨ _ t ≥ ξ z t , which is a subconfiguration of ω total , and Lemma 4.4 is established. Acknowledgements
We are grateful to Vincent Beffara for valuable discussions, helpful comments onthe first versions of this paper, and more generally for precious advice all alongthe project. We also thank Itai Benjamini for useful dicussions and commentson the paper, Hugo Duminil-Copin for having initiated this project and Mickaëlde la Salle for pointing out the compactness argument of lemma 2.1.
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UMPA, Ens de LyonLyon, France [email protected]@[email protected]@ens-lyon.fr