Limit theorems for the painting of graphs by clusters

URL:http://www.emath.fr/ps/

LIMIT THEOREMS FOR THE PAINTING OF GRAPHS BY CLUSTERS

Olivier Garet

Abstract. We consider a generalization of the so-calleddivide and color modelrecently introduced by H¨aggstr¨om. We investigate the behavior of the magnetization in large boxes of the lattice^Zdand its fluctuations. Thus, Laws of Large Numbers and Central Limit Theorems are proved, both quenched and annealed. We show that the properties of the underlying percolation process deeply influence the behavior of the coloring model. In the subcritical case, the limit magnetization is deterministic and the Central Limit Theorem admits a Gaussian limit. Conversely, the limit magnetization is not deterministic in the supercritical case and the limit of the Central Limit Theorem is not Gaussian, except in the particular model with exactly two colors which are equally probable. We also prove a Central Limit Theorem for the size of the intersection of the infinite cluster with large boxes in supercritical bond percolation.

Mathematics Subject Classification. 60K35, 82B20, 82B43.

Received May 15, 2001. Revised July 2 and October 3, 2001.

Introduction

The aim of this paper is to give some results concerning a pretty and natural model for the dependent coloring of vertices of a graph. This model has recently been introduced by H¨aggstr¨om [7], who presented the first results, especially concerning the presence (or absence) of percolation and the quasilocality properties. The model is easily described: choose a graph at random according to bond percolation, and then paint randomly and independently the different clusters, each cluster being monochromatic. There are several motivations for the study of such a model, the most relevant being its links with Ising or Potts models. We refer to the examples of the present article and to the introduction of H¨aggstr¨om’s paper for detailed motivations.

In H¨aggstr¨om’s model, the panel consisted of a finite number of colors, which were chosen according to a measure ν onRwith finite support. For our purpose, the natural assumptions will only be the existence of a first or a second moment forν.

We will study the mean magnetization of the random field (X(x))_x∈Zd in large boxes Λ_n={−n, . . . , n}^d: we will identify the limit

M = lim

n→∞

1

|Λ_n| X

x∈Λ_n

X(x),

Keywords and phrases:Percolation, coloring model, Law of Large Number, Central Limit Theorem.

1Laboratoire de Math´ematiques, Applications et Physique Math´ematique d’Orl´eans, UMR 6628, Universit´e d’Orl´eans, BP. 6759, 45067 Orl´eans Cedex 2, France; e-mail:Olivier.Garet@labomath.univ-orleans.fr

c EDP Sciences, SMAI 2001

and determinate its variations: we will prove Central Limit Theorems for quantities such as 1

(|Λn|)^1/2

X

x∈Λ_n

X(x)

!

− |Λn|M

! .

There are several natural questions: when is M deterministic? What is the influence of the underlying bond percolation? When is there convergence to a normal law in the Central Limit Theorem?

These questions can be asked in two different approaches. Here we shall use the vocabulary usually used in the theory of random media.

• The quenched point of view: Limit Theorems are formulated once the graph has been (randomly) fixed.

• The annealed point of view: Limit Theorems are formulated under the randomization of the graph. We average over the possible issues of the bond percolation process.

Indeed, we will show that the properties of the underlying percolation process deeply influence the behavior of the coloring model. In the subcritical case, the limit magnetization is deterministic and the Central Limit Theorem admits a Gaussian limit. Conversely, the limit magnetization is not deterministic in the supercritical case and the limit of the Central Limit Theorem is not Gaussian, except in the particular model with exactly two colors which are equally probable. As examples, we will study the case whereν is “+/−” valued and the case whereν is a Gaussian measure.

1. Notations

We will deal here with stochastic processes indexed byZ^d. Their definition will be related to some subgraphs of thed-dimensional cubic latticeL^d, which is defined byL^d= (Z^d,E^d), whereE^d={{x, y} ⊂Z^d;Pd

i=1|xi−yi|

= 1}. In the following, the expression “subgraph of L^d” will always be employed for each graph of the form G= (Z^d, E) where E is a subset ofE^d. We denote byS(L^d) the set of all subgraphs ofL^d. We say that two vertices x, y ∈ Z^d are adjacent in G if{x, y} ∈ E. Two vertices x, y ∈ Z^d are said to be connected inG if one can find a sequence of vertices containingxandy such that each element of the sequence is adjacent inG with the next one. A subset C of Z^d is said to be connected if each pair of vertices inC is connected. The maximal connected sets are called the connected components. They partitionZ^d. The connected component of xis denoted by C(x). The connected components are also called clusters. A subset D ofZ^d is said to be independent if no pair inD is constituted by adjacent vertices.

Here we will consider subgraphs ofL^d which are generated by Bernoulli bond percolation on L^d. Thus, we will denote byµp the image measure of ({0,1}^Ed,B({0,1}^Ed),((1−p)δ0+pδ1)^⊗Ed) by

x7→(Z^d,{e∈E^d;x_e= 1}), where p∈(0,1).

Let us choose a graphGat random under µ_p and recall the definition of some basic objects in percolation theory (for more details, see for example Grimmett [4]).

• The probability that 0 belongs to an infinite cluster:

θ(p) =µ_p(|C(0)|= +∞).

• The critical probability:

p_c= inf{p∈(0,1);θ(p)>0}.

• The mean size of a finite cluster:

χ^f(p) =P+∞

k=1kµp(|C(0)|=k).

The following results will be frequently used:

• µpis translation-invariant. As it is isomorphic to ({0,1}^Ed,B({0,1}^Ed),((1−p)δ0+pδ1)^⊗Ed), its tailσ-field is trivial and the ergodic theorem can be employed with full power.

• Ifp∈(0, pc), thenGcontains no infinite cluster.

• Ifp∈(p_c,1), then Gcontainsµ_p-almost surely one unique infinite cluster.

• Ifp6=pc, thenχ^f(p)<+∞.

If Gis a subgraph of L^d and ifν is a probability measure onR, we will define the color-probabilityP^G,ν as follows: P^G,ν is the unique measure on (R^Zd,B(R^Zd)) under which the canonical projections X_i – defined, as usual byX_i(ω) =ω_i – satisfy

• for eachi∈Z^d, the law ofXi isν;

• for each independent set S⊂Z^d, the variables (Xi)i∈S are independent;

• for each connected set S⊂Z^d, the variables (Xi)i∈S are identical.

The randomized color-measure is defined by P^p,ν=

Z

S(L^d)

P^G,ν dµp(G).

We also note Λ_n ={−n, . . . , n}^dand define the tailσ-fieldT by T = ∩

n≥1 FZ^d\Λn, where F^S is theσ-field generated by the (Xi)i∈S.

2. Laws of large numbers

Theorem 2.1. Let ν be a probability measure onR with a first moment. We putm=R

Rxdν(x).

Then, for p∈(0,1)\{pc}: lim

n→+∞

1

|Λn| X

x∈Λ_n

X(x) = (1−θ(p))m+θ(p)Z P^p,ν-almost surely, whereZ is the value taken byX() along the infinite component if it exists, and 0 else.

Proof. We will build here P^p,νas the image of a product measure by an appropriate transform.

Let Ω ={0,1}^Ed×R^Zd and consider the probability measureP = ((1−p)δ0+pδ1)^⊗Ed⊗ν^⊗Zdon (Ω,B(Ω)).

We define G((η, ω)) as the subgraph of L^d which is such that for each i ∈ E^d, the vertex i is open if and only if ηi = 1. For x ∈ Z^d, C(x)((η, ω)) is the cluster of x in this graph. We also define the random set I ={x∈Z^d;|C(x)|= +∞}. For each subset A⊂Z^d, we define infA as the smallest edge ofA in the lexical ordering.

We can now define the random field (X(k))_k∈Zd by X(k)((η, ω)) = ωinfC(k)(η,ω). It is easy to see that PX=P^p,ν. Z is defined byZ((η, ω)) = 0 ifI(η, ω) =∅and Z(η, ω) =ωinfI(η,ω)else.

Let us define

Qn= 1

|Λn| X

x∈Λn

X(x).

SinceP^p,νis translation-invariant, the ergodic Theorem ensures thatQnwouldP^p,ν-almost surely and in L¹(P^p,ν) converge. Our goal is now to identify this limit.

SinceGhas almost surely at most one infinite cluster, we have Q_n=D_n+|I∩Λn|

|Λn| Za.s.,

where

D_n= 1

|Λn| X

x∈Λ_n

X⁰(x) withX⁰(x) =X(x)11_|C(x)|<+∞.

Since|Λ_n∩I|= P

x∈Λn

11_{{|C(x)|=+∞}}, it follows from the ergodic theorem that

lim

n→+∞

1

|Λ_n||Λ_n∩I|=E¹1_{{|C(0)|=+∞}}=θ(p) a.s. and in L¹. (1) It remains to identify the limit of (Dn)n≥1.

For each finite set Λ⊂Z^d, let us denote byFΛ⁰ theσ-field generated by the independent variables indexed by edges and vertices which stay in the outside of Λ. Precisely,FΛ⁰ =σ((η, ω)7→(ηe, ωk);e∩Λ =∅ andk /∈Λ).

For each (η, ω), we denote by (˜ηΛ, ω⁰_Λ) the configuration obtained from (η, ω) as follows: one removes the vertices which intersect Λ and replaces the previous color of the edges inside Λ by the color 0. The colors in the outside of Λ remain unchanged. Precisely, (ω_Λ⁰)k =ωk 11k /∈Λ for eachk∈Z^d and (˜ηΛ)e =ηe 11e∩Λ=∅ for each e∈E^d.

Since X⁰((ω_Λ⁰,η˜Λ)) only differs from X⁰((ω, η)) in the color of a finite number of edges, it follows that lim

n→+∞ Dn((ω, η)) = lim

n→+∞ Dn((ω_Λ⁰,η˜Λ)). But for each n, (ω, η)7→ Dn((ω_Λ⁰,η˜Λ)) is FΛ⁰-measurable. Then (ω, η)7→ lim

n→+∞ D_n((ω, η)) isFΛ⁰-measurable. Since this holds for each finite Λ, (ω, η)7→ lim

n→+∞ D_n((ω, η)) is measurable with respect to the tailσ-field. Then, it is constant by Kolmogorov’s 0−1 law.

SinceEDn = (1−θ(p))mfor eachn, this constant is necessarily (1−θ(p))m.

This concludes the proof.

We will now formulate an easy, but important corollary:

Corollary 2.2. • Z isT-measurable.

• Forp > p_c,T is not trivial underP^p,ν as soon asν is not a Dirac measure.

Proof. The first point is a consequence of the formula given in Theorem 2.1 and the second point is a consequence of the first one, because Z is non constant as soon asν is not a Dirac measure.

The fact thatZisT-measurable is important for the formulation of annealed results, because the environment is forgotten once we have randomized under µp. Indeed, despite the infinite component can not always be recovered, the value ofX() along this component can.

2.2. Quenched law of large numbers

Theorem 2.3. Let ν be a probability measure on R with a first moment. We put m = R

Rx dν(x). Let p∈(0,1)\{pc}.

Forµp-almost allG, the following holds:

lim

n→+∞

1

|Λn| X

x∈Λn

X(x) = (1−θ(p))m+θ(p)Z P^G,ν-almost surely, (2)

whereZ is the value taken byX() along the infinite component if it exists, and 0 else.

Proof. LetC=

lim

n→+∞ 1

|Λn|

P

x∈ΛnX(x) = (1−θ(p))m+θ(p)Z

. We have

0 = 1−P^p,ν(C) = Z

S(L^d)

(1−P^G,ν(C)) dµp(G).

Since (1−P^G,ν(C))≥0, it follows that 1−P^G,ν(C) = 0 forµp-almost allG.

2.3. Examples

2.3.1. “+/−” valued spin system

It is the simplest model that we can study, since it only takes two values: “+1” and “−1”, with probability αand 1−α. In the terminology of H¨aggstr¨om [5], it is denoted as ther+s-state fractional fuzzy Potts model at inverse temperature −¹2ln(1−p), withr=αand s= 1−α. This name alludes to the fact that the fuzzy Potts model can be realized using random clusters by an analogous painting procedure. For more details, see H¨aggstr¨om [5].

We point out that forµp-almost allG, we have P^{G,(1−α)δ−1+αδ+1}=α lim

β→+∞ IG,β,h⁺ + (1−α) lim

β→+∞ IG,β,h⁻ ,

where lim

β→+∞ IG,β,h⁺

resp. lim

β→+∞ IG,β,h⁻

is the Ising Gibbs measure on G at inverse temperature β with the external field h= ¹₂ln(α/(1−α)) which is maximal (resp. minimal) for the stochastic domination. Thus,

P^{p,(1−α)δ−1+αδ+1}= lim

β→+∞

Z

S(L^d)

αIG,β,h⁺ + (1−α)IG,β,h⁻ dµ_p(G).

In this sense, we can say that P^{p,(1−α)δ−1+αδ+1} arises at the zero temperature limit of an Ising model on a randomly diluted lattice. For precise definitions and results relative to Ising ferromagnets on random subgraphs generated by bond percolation, see Georgii [3] and as well the recent article of H¨aggstr¨omet al.[6].

If we choose ν= (1−α)δ₋1+αδ1 withα∈(0,1), it follows that the magnetization is M= lim

n→+∞

1

|Λn| X

x∈Λn

X(x) = (

2α(1−θ(p)) + 2θ(p)−1 with probabilityα

2α(1−θ(p))−1 with probability 1−α. (3) When p ∈ (0, pc), the magnetization is deterministic. Moreover, it follows from (3) that the sign of the magnetization is deterministic if and only if

max(α,1−α)(1−θ(p))≥ 1 2· Note that ifθ(p)≥ ¹2 the sign of the magnetization can not be deterministic.

In the case α= ¹₂, the annealed law of the magnetization has been identified by H¨aggstr¨om ([7], Prop. 2.1) using a spin-flip argument.

2.3.2. A quenched Gaussian system

Here we chooseν=N(0,1). For eachG,P^G,ν is a Gaussian measure. Here, we have M= lim

n→+∞

1

|Λ_n| X

x∈Λ_n

X(x) =θ(p)Z. (4)

In other words,M is almost surely null when p < pcand M∼ N(0, θ(p)²) whenp > pc.

We emphasize that these Large Numbers Theorems are valid both quenched and annealed. This will no longer be so simple for Central Limit Theorems.

3. Central Limit Theorems

In the following, we write X_n =⇒ µ to denote the weak convergence of a sequence of random variables (Xn)n≥1 to a probability measureµ.

3.1. Quenched Central Limit Theorem

Theorem 3.1. Let ν be a probability measure on R with a second moment. We put m = R

Rx dν(x) and σ²=R

R(x−m)² dν(x).

Forµp-almost allG, the following holds:

• The subcritical case If p∈(0, p_c), then

1

|Λ_n|^1/2 X

x∈Λn

(X(x)−m)

!

=⇒ N(0, χ^f(p)σ²).

• The supercritical case If p∈(p_c,1), then

1

|Λ_n|^1/2



 X

x∈Λn\I

(X(x)−m)



=⇒ N(0, χ^f(p)σ²)

where I is the infinite component ofG.

The following lemma will be very useful:

Lemma 3.2. For each subgraphGofL^d, let us denote by(Ai)i∈I the partition ofGinto connected components.

Then, ifp6=pc, we have forµp-almost allG:

lim

n→∞

1

|Λn|

X

i∈I;|Ai|<+∞

|A_i∩Λ_n|²=χ^f(p),

where

χ^f(p) =

+∞

X

k=1

kP(|C(0)|=k).

Proof. Let us defineC⁰(x) by

C⁰(x) =

(C(x) if|C(x)|<+∞

∅ otherwise andC_n⁰(x) =C⁰(x)∩Λn.

It is easy to see that

X

i∈I;|A_i|<+∞

|A_i∩Λ_n|²= X

x∈Λn

|C_n⁰(x)|. (5)

We have|C_n⁰(x)| ≤ |C(x)|, and the equality holds if and only ifC⁰(x)⊂Λn.

The quantity residing in connected components intersecting the boundary of Λn can be controlled using well-known results about the distribution of the size of finite clusters. In both subcritical case and supercritical case, we can foundK >0 andβ >0 such that

P(+∞>|C(x)| ≥n)≤exp(−Kn^β).

(We can takeβ= 1 whenp < pcandβ = (d−1)/difp > pc. See for example the reference book of Grimmett [4]

for a detailed historical bibliography.) It follows from a standard Borel–Cantelli argument that forµp-almost allG, there exists a (random)N such that

∀n≥N max

x∈Λ_n|C⁰(x)| ≤(lnn)^2/β.

If follows that for eachx∈Λ_n−(lnn)2/β,C⁰(x) is completely inside Λ_n, and thereforeC⁰(x) =C_n⁰(x). Then, X

x∈Λ_{n−(lnn)2/β}

|C⁰(x)| ≤ X

x∈Λn

|C_n⁰(x)| ≤ X

x∈Λn

|C⁰(x)| 1

|Λ_n|

X

x∈Λ

n−(lnn)2/β

|C⁰(x)| ≤ 1

|Λ_n| X

x∈Λ_n

|C_n⁰(x)| ≤ 1

|Λ_n| X

x∈Λ_n

|C⁰(x)|.

By the ergodic Theorem, we haveµ_p-almost surely

n→lim+∞

1

|Λn| X

x∈Λn

|C⁰(x)|=E|C⁰(0)|=χ^f(p).

Since limn→+∞|Λ

n−(lnn)2/β|

|Λ_n| = 1, the result follows:

Remark. If we forget technical controls, the key point of this proof is the identity (5). It is interesting to note that Grimmett [4] used an analogous trick to prove that lim

n→+∞ k(n)/|Λ_n|=κ(p)-almost surely, wherek(n) is the number of open clusters in Λn andκ(p) =E|C(0)|⁻¹.

We can now proof Theorem 3.1. For simplicity, we will give the proof in the supercritical case – which contains the proof of the subcritical case.

Proof.

X

x∈Λ_n\I

(X(x)−m) =

+P∞

i=1 |C_n⁰(ai)|(X(ai)−m).

Then

1

|Λ_n|^1/2 X

x∈Λ_n\I

(X(x)−m) = s²_n

|Λn| ^1/2 1

sn +P∞

i=1 |C_n⁰(ai)|(X(ai)−m), with

s²_n=

+P∞

i=1 |C_n⁰(ai)|². By Lemma 3.2, we have forµ^p-almost allGlimn→+∞ s²_n

|Λn| =χ^f(p).

Now, it remains to prove that 1 sn

+P∞

i=1 |C_n⁰(ai)|(X(ai)−m) =⇒ N(0, σ²). (6) Therefore, we will prove that forµ^p-almost allG, the sequenceYn,k=|C_n⁰(ai)|(X(ai)−m) satisfies the Lindeberg condition. For eachε >0, we have

+∞

X

k=1

1 s²_n

Z

|Yn,k|≥εsn

Y_n,k² dP^G,ν=

+∞

X

k=1

|C_n⁰(a_k)|² s²_n

Z

|C_n⁰(ak)||x|≥εsn

(x−m)²dν(x)

≤Z

|x|≥_ηn^ε

(x−m)² dν(x),

withηn= ^supk≥1_s^|C0n(ak)|

n . Thus, the Lindeberg condition is fulfilled if limηn= 0. But we have already seen that sn∼(χ^f(p)|Λn|)^1/2, whereas sup_k≥1|C_n⁰(ak)|=O((lnn)^2/β). This concludes the proof.

3.2. Annealed Central Limit Theorem

Theorem 3.3. Let ν be a probability measure on R with a second moment. We put m = R

Rx dν(x) and σ²=R

R(x−m)² dν(x). Letp∈(0,1)\{pc}. We emphasize thatGis randomized under µp.

• The subcritical case If p∈(0, p_c), then

1

|Λ_n|^1/2 X

x∈Λ_n

(X(x)−m)

!

=⇒ N(0, χ^f(p)σ²).

• The supercritical case If p∈(pc,1), then

1

|Λn|^1/2 X

x∈Λ_n

X(x)−((1−θ(p))m+θ(p)Z)|Λ_n|)

!

=⇒γ

where γis the image ofN(0, χ^f(p)σ²)× N(0, σ²_p)×ν by(x, y, z)7→x+y(z−m), with σ_p²= P

k∈Z^d (P(0∈Iand k∈I)−θ(p)²), where I is the infinite component ofG.

In the subcritical case, the Annealed Central Limit Theorem is a simple consequence of the Quenched Central Limit Theorem.

In order to prove this result in the supercritical case, we will need a Central Limit Theorem related to the variations of the size of the intersection of the infinite cluster with large boxes.

Yu Zhang had recently proved the following [12]:

|Λ_n∩I| −θ(p)|Λ_n|

rn1/2 =⇒ N(0,1),

withrn = Var|Λn∩I|. Our result is a little bit more precise, in the following sense that it gives the asymptotic behavior of (rn)n≥1: rn∼σ²_p|Λn|. Thus, independently of [12], we will prove:

Proposition 3.4. Under µp, we have

|Λ_n∩I| −θ(p)|Λ_n|

|Λn|^1/2 =⇒ N(0, σ²_p), whereI is the infinite component of G.

Proof.

|Λn∩I(ω)| −θ(p)|Λn|= X

k∈Λ_n

f(T^kω),

where T^k is the translation operator defined by T^k(ω) = (ωn+k)_n∈Zd and f =11_{{|C(0)|=+∞}}−θ(p). Moreover, f is an increasing function andµ^p satisfies the F.K.G. inequalities. Then (f(T^kω))_k∈Zd) is a stationary random field of square integrable variables satisfying to the F.K.G. inequalities. Therefore, according to Newman [9], the Central Limit Theorem is true if we prove that the quantity

X

k∈Z^d

Cov(f, f◦T^k) (7)

is finite.

This way of proving Central Limit Theorems for the density of infinite clusters in percolation models satisfying to the F.K.G. inequalities is not new, because it has already been pointed out by Newman and Schulman [10,11].

The new result is that it applies to classical independent bond percolation,i.e. that the series in (7) is convergent.

Next, Cov(f, f◦T^k) = Cov(Y0, Yk), withYk =11_{|C(k)|<+∞}. Now,

Yk =

+∞

X

n=0

Fn,k,withFn,k=11_{|C(k)|=n}.

Now

Cov(Y₀, Y_k) =

+∞

X

n=0 +∞

X

m=0

Cov(F_n,0, F_m,k)

=

+∞

X

n=0

Cov(F_n,0, F_n,k) + 2

n−1

X

m=0

Cov(F_n,0, F_m,k)

!

=

+∞

X

n>kkk/2−1

Cov(Fn,0, Fn,k) + 2

nX−1 m=0

Cov(Fn,0, Fm,k)

!

=

+∞

X

n>kkk/2−1

Cov Fn,0, Fn,k+ 2

nX−1 m=0

Fm,k

! ,

becauseFn,0 andFm,k are independent as soon askkk ≥m+n+ 2.

SinceF_n,0≥0 and 0≤F_n,k+ 2

nP−1 m=0

F_m,k ≤2, we have

Cov Fn,0, Fn,k+ 2

nX−1 m=0

Fm,k

!≤2EFn,0= 2P(|C(0)|=n).

Then, Cov(Y₀, Y_k)| ≤ P

n>kkk/2−1

2P(|C(0)|=n) and X

k∈Z^d

Cov(f, f◦T^k)≤2

+P∞

n=1 |Λ_2(n+1)|P(|C(0)|=n).

Since Kesten and Zhang [8] have proved the existence ofη(p)>0 such that

∀n∈Z⁺ P(|C(0)|=n)≤exp(−η(p)n^(d−1)/d),

it follows that the series converges. Of course, such a sharp estimate is not necessary for our purpose. Estimates derived from Chayeset al.[2], and from Chayeset al.[1] would have been sufficient.

Proof. In this proof, it will be useful to considerGas a random variable. Let Ω =S(L^d)×R^Zd and define the probabilityP onB(Ω) as a skew-product: for measurable A×B ∈ B(S(L^d))× B(R^Zd), we have P(A×B) = R

AP^G,ν dµp(G). Then, the law of the marginalsGand X areP^G=µp andP^X =P^p,ν. Rearranging the terms of the sum, we easily obtain

X

x∈Λn

X(x)−((1−θ(p))m+θ(p)Z)|Λn|)

!

= X

x∈Λn\I

(X(x)−m) + (Z−m)(|I∩Λn| − |Λn|θ(p)).

We will now put

Q_n= 1

|Λ_n|^1/2 X

x∈Λn

X(x)−((1−θ(p))m+θ(p)Z)|Λ_n|)

! ,

and define

∀t∈R φ_n(t) =E exp(iQ_n) and

∀t∈R φn,z(t) =E exp(iQn)|{Z =z} ·

As usual, it means that E (exp(iQn)|Z) = φn,z(Z). It is also important to emphasize that the following properties are fulfilled under P:

• Gis independent fromZ;

• (Xk11_{k /∈I})_k∈Zd is independent fromZ.

Thereby, we have

φ_n,z(t) =E exp



− it

|Λn|^1/2 X

x∈Λn\I

(X(x)−m) + (z−m)(|I∩Λ_n| − |Λ_n|θ(p))



.

Conditioning byσ(G) and using the fact thatI isσ(G)-measurable, we getφn,z(t) =Efn(t, .)gn((z−m)t, .), with

fn(t, ω) =E exp



− it

|Λ_n|^1/2 X

x∈Λn\I

(X(x)−m)|σ(G)





= Z

exp



− it

|Λ_n|^1/2 X

x∈Λn\I(ω)

(X(x)−m) dP^G(ω),ν





and

gn(t, ω) = exp

− it

|Λn|^1/2(|I(ω)∩Λn| − |Λn|θ(p))

.

By Theorem 3.1 we have for eacht∈RandP^p,ν-almost allω: lim

n→+∞ fn(t, ω) = exp(−^t2²χ^f(p)σ²). Then, by dominated convergence

lim

n→+∞E

f_n(t, .)−exp

−t²

2χ^f(p)σ²

g_n((z−m)t, .) = 0.

Next

n→lim+∞Efn(t, .)gn((z−m)t, .) = lim

n→+∞exp

−t²

2χ^f(p)σ²

Egn((z−m)t, .)

= exp

−t²

2χ^f(p)σ²

exp

−t²

2(z−m)²σ²_p

where the last equality follows from Proposition 3.4. We have just proved that lim

n→∞ φ_n,z(t) = exp

−t²

2(χ^f(p)σ²+ (z−m)²σ²_p)

.

Sinceφn(t) =R

φn,z(t) dν(z), we get lim

n→∞ φ_n(t) = Z

exp

−t²

2(χ^f(p)σ²+ (z−m)²σ²_p)

dν(z)

= Z

exp(itx)dγ(x).

By the theorem of Levy, it follows thatQn=⇒γ.

3.3. Examples

3.3.1. “+/−” valued spin system

If we choose ν= (1−α)δ₋1+αδ1 withα∈(0,1), it follows that:

• in the subcritical case p∈(0, pc), then the limit in the Central Limit Theorem is αN(0,4α(1−α)χ^f(p));

• in the supercritical casep∈(pc,1), then the limit in the Central Limit Theorem is αN(0,4α(1−α)χ^f(p) + 4(1−α)²σ_p²) + (1−α)N(0,4α(1−α)χ^f(p) + 4α²σ²_p).

Remarks.

1. For the “+/−” valued spin system in the subcritical case, the annealed Central Limit Theorem can be simply proved without using the quenched one: since R

ωk dP^G,ν =m for eachk and eachG, it follows that the covariance ofX0 andXk under P^p,ν is

Cov(X0, Xk) = Z

S(L^d)

Z

(ω0−m)(ωk−m) dP^G,ν

dµp(G)

= Z

σ²11_{k∈C(0)}dµ_p(G)

=σ²P(k∈C(0)).

Then,

P

k∈Z^d Cov(X0, Xk) = P

k∈Z^d

Z

S(L^d)

σ²11_{k∈C(0)}dµp(G)

= σ² Z

S(L^d)

P

k∈Z^d 11_{k∈C(0)}dµ_p(G)

= σ² Z

S(L^d)

|C(0)|dµp(G)

= σ²χ(p), withχ(p) =E|C(0)|=χ^f(p) +θ(p)(+∞). P

k∈Z^d Cov(X0, Xk) is a convergent series whenp < pc and a divergent one else.

In the subcritical case, the theorem of Newman [9] ensures that the Central Limit is valid as soon as the translation-invariant measureP^p,ν satisfy to the F.K.G. inequalities. Since H¨aggstr¨om and Schramm [7]

have proved the F.K.G. inequalities for the “+/−” valued spin system, we get a simple proof for the annealed Central Limit Theorem in this case.

2. Whenα= ¹₂, γ is a Gaussian measure as well in the subcritical case (the limit isN(0, χ^f(p))) as in the supercritical case (the limit isN(0, χ^f(p) +σ²_p)). It provides an example where there is a classical Central Limit Theorem whereas the “susceptibility” P

k∈Z^d Cov(X0, Xk) is infinite.

It is the “only” case with a Gaussian limit in the supercritical case, as you see from the following remark.

3. Ifp∈(p_c,1), then γis Gaussian if and only if there exista, b∈Rsuch thatν =¹₂(δ_a+δ_b).

Proof. Using the characteristic function, it is easy to see that γ is Gaussian if and only if γ⁰ = R N(0, (z−m)²σ_p²)(t)dν(z) is. Let us define, fork∈Z⁺: mk =R

z^2k dγ(z). We have mk =

Z

N(0,(z−m)²σ²_p)(x7→x^2k)dν(z)

=

Z (2k)!

k!2^k(z−m)^2kdν(z)

= (2k)!

k!2^k Z

(z−m)^2kdν(z).

By definition ofγ⁰,γ⁰ is a symmetric measure. So ifγ⁰ if Gaussian, it is centered and we have

∀k∈Z⁺ mk =(2k)!

k!2^km^k₁.

Then, we have

∀k∈Z⁺ Z

(z−m)^2kdν(z) =m^k₁.

If we denote byν⁰ the image ofν byz7→(z−m)², we have

∀k∈Z⁺ Z

R+

z^kdν⁰(z) =m^k₁. Then, we have

supp ess ν⁰= lim

k→+∞

Z

R+

z^kdν⁰(z)

!1/k

=m1= Z

R+

zdν⁰(z).

It follows that forν⁰-almost allz,z= supp essν⁰: ν⁰is a Dirac measure.

Therefore, suppν ⊂ {m− √m1, m+√m1}. Since m = R

z dν(z), we necessary have ν(m− √m1)

=ν(m+√m1) =¹₂ and thenν =¹₂(δm−√m₁+δm+√m₁).

3.3.2. The quenched Gaussian system

In the caseν =N(0,1), Theorem 3.3 takes the following form:

• The subcritical case Ifp∈(0, pc), then

1

|Λ_n|^1/2 X

x∈Λn

X(x)

!

=⇒ N(0, χ^f(p)σ²).

• The supercritical case Ifp∈(pc,1), then

1

|Λn|^1/2 X

x∈Λ_n

X(x)−θ(p)Z|Λn|

!

=⇒γ

whereγ is the image ofN(0, I_R3) by (x, y, z)7→(χ^f(p))^1/2x+σ_pyz,with σ_p²= P

k∈Z^d (P(0∈Iandk∈I)−θ(p)²), whereI is the infinite component ofG.

In this case the limit is a Gaussian measure whenp < pc whereas it is a Gaussian chaos of order 2 forp > pc. I would like to thank the referees for their carefully reading, specially for suggesting me a simplification in the proof of Theorem 2.1.

References

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