Continuum Percolation for Quermass Model

0.29 MB·English

by  David Coupier

Continuum Percolation for Quermass Model
David Coupier
Laboratoire Paul Painlevé U.M.R. CNRS 8524
Université Lille 1, France
e-mail : [email protected]
David Dereudre
2 Laboratoire Paul Painlevé U.M.R. CNRS 8524
1
Université Lille 1, France
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2 e-mail : [email protected]

n
January 12, 2012
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J

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Abstract

] The continuum percolation for Markov (or Gibbs) germ-grain models is investigated.
R
The grains are assumed circular with random radii on a compact support. The morpho-
P
logical interaction is the so-called Quermass interaction defined by a linear combination
.
h
of the classical Minkowski functionals (area, perimeter and Euler-Poincaré characteristic).
t
a
We show thatthe percolation occurs forany coefficient of this linear combination and fora
m
large enough activity parameter. An application to the phase transition of the multi-type
[
Quermass model is given.

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KEY-WORDS: Stochastic geometry, Gibbs point process, germ-grain model, Quermass
4
4 interaction, percolation, phase transition.
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1 Introduction
The germ-grain model is built by unifying random convex sets– the grains –centered at
the points– the germs –of a spatial point process. It is used for modelling random surfaces
and interfaces, geometrical structures growing from germs, etc. For such models, the con-
tinuum percolation refers mainly to the existence of an unbounded connected component.
This phenomenon expressessomemacroscopic properties ofmaterials as permeability, con-
ductivity, etc. Moreover, it turns out to be an efficient tool to exhibit phase transition
in Statistical Mechanics [2, 4]. For theses reasons, the continuum percolation has been
abundantly studied since the eighties and the pioneer paper of Hall [8].
When the grains are independent and identically distributed, and the germs are given
by the locations of a Poisson point process (PPP), the germ-grain model is known as the
Boolean model. In this context, the continuum percolation is well understood; see the
book of Meester and Roy [13] for a very complete reference. One of the first results is the
existenceof apercolation threshold z fortheintensity parameter z of thestationary PPP:
∗
provided the mean volume of the grain is finite, percolation occurs for z > z and not for
∗
z <z .
∗
Because of the independence properties of the PPP, the Boolean model is sometimes
caricaturalfortheapplications inBiologyorPhysics. Meckeanditscoauthors[11,12]have
mentioned the need of developing, via Markov or Gibbs process, an interacting germ-grain
model in which the interaction would locally depend on the geometry of the set. For this
purpose, let us cite the Widom-Rowlinson model [16], the area interaction process [1] and
the morphological model [12]. Thus, Kendall, Van Lieshout and Baddeley suggested in
[9] a generalization of the previous models, called the Quermass Interaction Process. In
this model, the formal Hamiltonian is a linear combination of the fundamental Minkowski
functionals,namelyinR2 thearea ,theperimeter andtheEuler-Poincaré characteristic
A L
χ:
H = θ +θ +θ χ .
1 2 3
A L
The existence of infinite volume Gibbs point processes for the Hamiltonian H has been
recently proved in [3]. This paper focuses on the continuum percolation for such processes.
The existence of a percolation threshold z for the Boolean model relies on a basic
∗
(but essential) monotonicity argument: see [13], Chapter 2.2. This argument fails in the
case of Gibbs point processes with Hamiltonian H. So, no percolation threshold can be
expected in our context. However, other stochastic arguments as stochastic domination or
FKG lead to percolation results. In [2], Chayes et al prove that percolation occurs for z
large enough and θ = θ = 0. To our knowledge, the percolation phenomenon for other
2 3
values of parameters θ ,θ ,θ has not been investigated yet.
1 2 3
Our main result (Theorem 1) states that, for any θ ,θ ,θ (positive or negative), per-
1 2 3
colation occurs with probability 1 for z large enough. The only assumption bears on the
random radii of the circular grains: they have to belong to a compact set not containing
0. The proof of this theorem is relatively easy in the case θ = 0. Indeed, the local energy
3
h((x,R),ω)– theenergy variationwhen thegrain B¯(x,R)is addedtotheconfiguration ω –
isuniformlybounded(seeLemma4.12)andbyclassicalstochasticcomparisonwithrespect
to the Poisson Process the result follows. So the main challenge of the present paper is the
proofofTheorem1whenθ = 0. Inthissetting, thelocalenergybecomesunbounded from
3
6
2
above and below and classical stochastic comparison arguments for point processes fail. In
interpreting the percolation in our model via a site percolation model (see the beginning
of Section 4), we prove the result thanks to a stochastic domination result for graphs due
to Liggett et al (Theorem 1.3 in [10] or Lemma 4.2 below). An arduous control of the
hole number variation, when a new grain is added, is the main technical issue. We prove
essentially that this variation is moderate for a large enough set of admissible locations of
grains. Let us mention that our proof is inspired by the one of Proposition 3.1 in [4].
Following [2,4], weuseourpercolation result(Theorem 1)toexhibitaphasetransition
phenomenon for Quermass model with several type of particles (Theorem 2).
Our paper is organized as follows. In Section 2, the Quermass model and the main
notations are introduced. The local energy h((x,R),ω) is defined in (2.3). Section 3
contains the results of the paper. Section 3.2 is devoted to the case θ = 0 and Section 3.3
3
to the phase transition result. The proof of Theorem 1 is developed in Section 4.
2 Quermass Model
2.1 Notations
We denote by (R2) thesetof bounded Borel sets in R2 with a positiveLebesgue measure.
B
For any Λ and ∆ in (R2), Λ ∆ stands for the Minkoswki sum of these sets. Let
B ⊕
0 R R be some positive reals and be the product space R2 [R ,R ] endowed
0 1 0 1
≤ ≤ E ×
with its natural Euclidean Borel σ-algebra σ( ). For any Λ (R2), denotes the space
Λ
E ∈ B E
Λ [R ,R ]. A configuration ω is a subset of which is locally finite with respect to its
0 1
× E
firstcoordinate: #(ω )isfinite forany Λin (R2). The configuration setΩ is endowed
Λ
∩E B
with the σ-algebra generated by the functions ω #(ω A) for any A in σ( ).
F 7→ ∩ E
We will merely denote by ω instead of ω the restriction of the configuration ω (with
Λ Λ
∩E
respect to its first coordinate) to Λ. Moreover, for any (x,R) in , we will write ω (x,R)
E ∪
instead of ω (x,R) .
∪{ }
A configuration ω Ω can be interpreted as a marked configuration on R2 with marks
∈
in [R ,R ]. To each (x,R) ω is associated the closed ball B¯(x,R) (the grain) centered
0 1
∈
at x (the germ) with radius R. The germ-grain surface ω¯ is defined as
ω¯ = B¯(x,R) .
(x,R) ω
[∈
2.2 Quermass interaction
LetusdefinetheQuermassinteractionasinKendalletal. [9]. Theenergy(orHamiltonian)
of a finite configuration ω in Ω is defined by
H(ω) =θ (ω¯)+θ (ω¯)+θ χ(ω¯) , (2.1)
1 2 3
A L
where θ , θ and θ are three real numbers, and , and χ are the three fundamen-
1 2 3
A L
tal Minkowski functionals, respectively area, perimeter and Euler-Poincaré characteristic.
This last one is the difference between the number of connected components and the num-
ber of holes. Recall that a hole of ω¯ is a bounded connected component of ω¯c. Hadwiger’s
Theorem ensures that any functional F defined on the space of finite unions of convex
3
compact sets, which is continuous for the Hausdorff topology, invariant under isometric
transformations and additive (i.e. F(A B) = F(A) + F(B) F(A B)) can be de-
∪ − ∩
composed as in (2.1). This universal representation justifies the choice of the Quermass
interaction for modelling mesoscopic random surfaces [11, 12].
The energy inside Λ (R2) of any given configuration ω in Ω (finite or not) is defined
∈ B
by
H (ω) = H(ω ) H(ω ) , (2.2)
Λ ∆ ∆ Λ
− \
where ∆ is any subset of R2 containing Λ B(0,2R ). By additivity of functionals ,
1
⊕ A L
and χ, the difference H (ω) does not depend on the chosen set ∆.
Λ
Let us end with defining the local energy h((x,R),ω) of the marked point (x,R)
∈ E
(or of the associated ball B¯(x,R)) with respect to the configuration ω:
h((x,R),ω) = H (ω (x,R)) H (ω) , (2.3)
Λ Λ
∪ −
for any Λ (R2) containing x. Remark this definition does not depend on the choice
∈ B
of the set Λ. The local energy h((x,R),ω) represents the energy variation when the ball
B¯(x,R) is added to the configuration ω.
2.3 The Gibbs property
Let Q be a reference probability measure on [R ,R ]. Without loss of generality, R and
0 1 0
R can be chosen such that, for every ε > 0,
1
Q([R +ε,R ]) < 1 and Q([R ,R ε]) < 1 . (2.4)
0 1 0 1
−
Let z > 0. Let us denote by λ the Lebesgue measure on R2 and by πz the PPP on
with intensity measure zλ Q. Under πz, the law of the random surface ω¯ is the
E ⊗
stationary boolean model with intensity z > 0 and distribution of radius Q. Finally, for
any Λ (R2), let us denote by πz the PPP on with intensity measure zλ Q, where
∈ B Λ EΛ Λ⊗
λ is the restriction of the Lebesgue measure λ to Λ.
Λ
Definition 2.1. A probability measure P on Ω is a Quermass Process for the intensity
z >0 and the parameters θ ,θ ,θ if P is stationary and if for every Λ in (R2), for every
1 2 3
B
bounded positive measurable function f from Ω to R,
1
f(ω)P(dω) = f(ωΛ′ ∪ωΛc)ZΛ(ωΛc)e−HΛ(ωΛ′ ∪ωΛc)πΛz(dωΛ′ )P(dω) , (2.5)
Z Z Z
where ZΛ(ωΛc) is the partition function
ZΛ(ωΛc)= e−HΛ(ωΛ′ ∪ωΛc)πΛz(dωΛ′ ) .
Z Z
The equations (2.5)– for all Λ (R2) –are called DLR for Dobrushin, Landford and
∈ B
Ruelle. They are equivalent to: for any Λ (R2), the law of ωΛ under P given ωΛc is
∈ B
absolutely continuous with respect to the Poisson Process πz with the local density
Λ
1
gΛ(ωΛ′ |ωΛc)= ZΛ(ωΛc)e−HΛ(ωΛ′ ∪ωΛc) . (2.6)
4
See [15] for a general presentation of Gibbs measures and DLR equations.
The existence, the uniqueness or non-uniqueness (phase transition) of Quermass pro-
cesses are difficult problems in statistical mechanics. The existence has been proved re-
cently in [3], Theorem 2.1 for any parameters z > 0 and θ ,θ ,θ in R . A phase transition
1 2 3
result is proved in [2, 6, 16] for R = R , θ = θ = 0 and for θ = z large enough.
0 1 2 3 1
3 Results
3.1 Percolation occurs
We say that percolation occurs for a given configuration ω Ω if the subset ω¯ of R2 con-
∈
tains at least one unbounded connected component. The set of configurations such that
percolation occurs is a translation invariant event. Its probability, called the percolation
probability, equals to 0 or 1 for any ergodic Quermass process. However, the Quermass
processesarenotnecessarilyergodic(theyareonlystationary)andtheirpercolationproba-
bilitiesmay bedifferentfrom0and1. Besides,in[2],Chayes etalhavebuilttwoQuermass
processes, both corresponding to θ = θ = 0 and θ = z large enough, whose percolation
2 3 1
probabilities respectively equal to 0 and 1. Since any mixture of these two processes is still
a Quermass process, the authors obtain Quermass processes whose percolation probabili-
ties equal to any value between 0 and 1.
Our main result states that percolation occurs with probability 1 for any (ergodic or not)
Quermass process whenever the intensity z is large enough.
Theorem 1. Let R > 0 and θ ,θ ,θ R. There exists z > 0 such that for any
0 1 2 3 ∗
∈
Quermass process P associated to the parameters θ ,θ ,θ and z > z , percolation occurs
1 2 3 ∗
P-almost surely.
The proof of Theorem 1 is based on a discretization argument which allows to reduce
thepercolation problemfromthe(continuum) Quermass modeltoasitepercolationmodel
on the lattice Z2 (up to a scale factor). This proof is rather long and technical so it is
addressed in Section 4.
Let us point out here that our theorem does not claim z is a percolation threshold. In
∗
other words, for z < z , the percolation may be lost and recovered on different successive
∗
ranges.
Another natural question involves the number of unbounded connected components. Fol-
lowing the classical arguments for continuum percolation, we prove that this number is
almost surely equal to zero or one.
Proposition 3.1. For any Quermass process P the number of unbounded connected com-
ponent is a random variable in 0,1 .
{ }
Proof. It is well-known that any Gibbs measure is a mixture of extremal ergodic Gibbs
measures. For each ergodic Quermass process P, the number of connected component is
almost surely a constant in N + . For any Λ (R2), thanks to the DLR equations
∪{ ∞} ∈ B
(2.5), it is easy to prove that the law of ω under P is equivalent to πz. Therefore, in
Λ Λ
following the general scheme of the proof of Theorem 2.1 in [13], we show that the number
of connected components is necessary 0 or 1.
5
3.2 Percolation when θ = 0
3
In the particular case θ = 0, Theorem 1 can be completed and proved in a simple way.
3
First, let us recall the definitions involving the stochastic domination for point pro-
cesses. We follow the notations given in [5]. An event A in is called increasing if for
F
every ω A and any ω Ω containing ω then ω A too. Let P and P be two proba-
′ ′ ′
∈ ∈ ∈
bility measures on Ω. We say that P is dominated by P , denoted by P P , if for every
′ ′
(cid:22)
increasing event A , P(A) P (A). In this section, we focus our attention on the
′
∈ F ≤
increasing event "there exists an unbounded connected component".
Let P be any Quermass process and assume θ = 0 and R > 0. Thanks to Lemma
3 0
4.12, the local energy can be uniformly bound: there exist constants C and C such that
0 1
for any (x,R) and ω Ω,
∈ E ∈
C h((x,R),ω) C . (3.1)
0 1
≤ ≤
Combining (3.1) and Theorem 1.1 in [5], we get the following stochastic dominations:
πze−C1 P πze−C0 .
(cid:22) (cid:22)
Now, the (stationary) Boolean models corresponding to πze−C1 and πze−C0 admit positive
and finite percolation thresholds (see [14], Chapter 3). It follows :
Proposition 3.2. Let R > 0. For every θ ,θ in R, there exist constants z ,z such that
0 1 2 0 1
for any Quermas Process P associated to parameters z,θ ,θ and θ = 0, the percolation
1 2 3
occurs P-almost surely if z > z and does not occur P-almost surely if z < z .
1 0
Proposition 3.2 improves Theorem 1 in the case θ = 0 since it ensures the existence
3
of a subcritical regime.
It is worth pointing out here that the uniform bounds in (3.1) do not hold whenever
θ = 0. Precisely, this is the hole number variation which cannot be uniformly bounded.
3
6
3.3 Phase transition for multi-type Quermass Process
In this section, the multi-type Quermass model is introduced and a phase transition is
exhibited, i.e. the existence of several Gibbs processes for the same parameters is proved.
Let K be a positive integer. The K-type Quermass model is defined on the space Ω
K
of configurations in = R2 [R ,R ] 1,2,...,K . Each disc is now marked by a
K 0 1
E × × { }
number specifying its type. We don’t give the natural extension of the notations involving
the sigma-field and so on.
The following Quermass energy function is defined such that all discs of a connected com-
ponent have the same number. This is a non-overlapping multi-type germ-grain model.
Precisely the energy of a finite configuration ω is now given by
H(ω) = θ (ω¯)+θ (ω¯)+θ χ(ω¯)+ φ(x y R R) , (3.2)
1 2 3 ′
A L | − |− −
(x,R,i),(y,R′,j) ω
Xi=j ∈
6
where φ is an hardcore potential equals to infinity on ] ,0] and zero on ]0,+ ]. The
−∞ ∞
energy inside Λ (R2) of any finite or infinite configuration ω is defined as in (2.2) with
∈ B
6
the convention + = + . The definition of the K-type Quermass process via the
∞−∞ ∞
DLR equations follows as in Definition 2.1.
The proof of the existence of such processes is similar to the one of the existence of
Quermass process. See Theorem 2.1 of [3] for more details. Here is our phase transition
result:
Theorem 2. Let R > 0. For any θ , θ and θ in R, there exists z > 0 such that, for
0 1 2 3 0
any z > z , there exist several K-type Quermass Processes. The phase transition occurs.
0
Theproofessentially followstheschemeoftheoneofTheorem2.2of[2]orTheorem1.1
of[4]. Itisbasedonarandom-clusterrepresentation(orGrayRepresentation)analogousto
the Fortuin-Kasteleyn representation of the Potts model. The existence of an unbounded
connected component allows toprove the existenceof a K-typeQuermass processin which
the density of particles of a given type is larger than the ones of the other types. By
symmetry of the types, we prove the existence of at least K different K-type Quermass
processes.
4 Proof of Theorem 1
4.1 General scheme
In the following, P denotes a stationary Quermass process on Ω associated to the intensity
z >0 and the parameters θ ,θ ,θ R.
1 2 3
∈
Let ℓ be a real number such that ℓ > 2R +2R . Let us define the diamond box ∆ as
1 0
the interior of the convex hull of the eight points (3ℓ,0), (6ℓ,0), (9ℓ,3ℓ), (9ℓ,6ℓ), (6ℓ,9ℓ),
(3ℓ,9ℓ), (0,6ℓ) and (0,3ℓ). This large octagon contains four smaller boxes BN, BS, BE
and BW with side length ℓ; precisely BN = (4ℓ,7ℓ) + [0,ℓ]2, BS = (4ℓ,ℓ) + [0,ℓ]2, BE =
(7ℓ,4ℓ)+[0,ℓ]2 and BW = (ℓ,4ℓ)+[0,ℓ]2. The subscripts N, S, E and W refer to the cardinal
directions. See Figure 1. Thus, let us introduce the indicator function ξ defined on Ω and
equal to 1 if and only if the two following conditions are satisfied:
(C1) Each box BN, BS, BE and BW, contains at least one point of ω∆;
(C2) The number N∆(ω)of connected components of ω¯ having at leastoneball centered
cc ∆
in one of the boxes BN, BS, BE or BW, is equal to 1.
In other words, ξ(ω) = 1 means the boxes BN, BS, BE and BW are connected through ω¯∆.
Foranyx (6ℓZ)2,letτ bethetranslationoperatorontheconfigurationset defined
x
∈ E
by (y,R) τ ω if and only if (y+x,R) ω. Hence, we can define the translated indicator
x
∈ ∈
function ξ of ξ on the translated box ∆ = x+∆ by ξ (ω) =ξ(τ ω). Let us remark that
x x x x
ξ (ω) only depends on the restriction of the configuration ω to the box ∆ . Moreover,
x x
thanks to the stationary character of the Quermass process P, the random variables ξ ,
x
x (6ℓZ)2, are identically distributed. They are dependent too.
∈
Let us consider x,y (6ℓZ)2 such that y = (6ℓ,0)+x. The boxes ∆ and ∆ have in
x y
∈
common a cardinal box, i.e. x+BE = y+BW. So, the condition ξx(ω) = ξy(ω) = 1 ensures
that the cardinal boxes of ∆ and ∆ are connected together through the restriction of ω¯
x y
to ∆ ∆ . The same is true when y = (0,6ℓ)+x. This induces a graph structure on the
x y
∪
vertex set V = (6ℓZ)2: for any x,y V, x,y belongs to the edge set E if and only if
∈ { }
y x (6ℓ,0), (0,6ℓ) .
− ∈ {± ± }
7
9ℓ
6ℓ
3ℓ
3ℓ 6ℓ 9ℓ
Figure 1: Hereisthe diamondbox∆. Thelightgraysetrepresentsthe configurationω restricted
to ∆. The dark gray squares are the fourth cardinal boxes BN, BS, BE and BW with side length ℓ.
On this picture, conditions (C1) and (C2) are fulfilled, i.e. ξ(ω)=1.
The graph (V,E) is merely the square lattice Z2 with the scale factor 6ℓ. The family
ξ ,x V provides a site percolation process on the graph (V,E). It has been built so
x
{ ∈ }
as to satisfy the following statement.
Lemma 4.1. Let ω Ω such that percolation occurs in the site percolation process ξ ,x
x
∈ { ∈
V . Then, so does for ω.
}
Let Π be the Bernoulli (with parameter p) product measure on 0,1 V. A stochastic
p
{ }
domination resultofLiggett etal[10](Theorem 1.3)allows tocompare thesitepercolation
processes induced by the family ξ ,x V and Π . Here is an adaptated version to our
x p
{ ∈ }
context. Basicdefinitionsaboutstochasticdominationforlatticestatespacesarenotrecall
here. They are similar to the ones presented in Section 3.2 for point processes. See also
[7].
Lemma 4.2. Let p [0,1]. Assume that, for any vertex x V,
∈ ∈
P (ξ = 1 ξ : x,y / E) p a.s. (4.1)
x y
| { } ∈ ≥
Then the distribution of the family ξ ,x V stochastically dominates the probability
x
{ ∈ }
measure Π , where f : [0,1] [0,1] is a deterministic function such that f(p) tends to
f(p)
→
1 as p tends to 1.
Actually, Theorem 1 straight derives from Lemmas 4.1 and 4.2. Let us first recall that
inthesitepercolationmodelonthegraph(V,E), thereexistsathresholdvaluep < 1such
∗
that percolation occurs with Π -probability 1 whenever p > p . See the book of Grimmett
p ∗
[7], p. 25. So, let p be a real number in [0,1] such that
f(p)> p . (4.2)
∗
8
Whenever the Quermass process P satisfies (4.1) for that p, then combining Lemmas 4.1
and 4.2 percolation occurs P-a.s. Therefore it remains to show that for any p > 0, hypoth-
esis (4.1) holds for z large enough.
The next result claims that each Borel set of R2, sufficiently thick in some sense,
contains at least one element of the configuration ω with a probability tending to 1 as the
intensity z tends to infinity. It will be proved at the end of this section.
Lemma 4.3. Let V R2 such that there exist U (R2) with positive Lebesgue measure
⊂ ∈B
and ε > 0 satisfying U B¯(0,R +R +ε) V. Then there exists a constant C > 0,
1 0
⊕ ⊂
depending on λ(U) and ε, such that for any configuration ω Ω and for any z > 0,
∈
P (ωV = ωVc) Cz−1 .
∅| ≤
SincetheQuermassprocessP isstationary, itissufficienttoprove(4.1)withx = (0,0).
So, we focus our attention on the diamond box ∆ = ∆ and use Lemma 4.3 to check
(0,0)
that condition (C1) is fulfilled in this box. Since BN, BS, BE and BW are sufficiently thick
(with side length ℓ > 2R +2R ), it follows
1 0
P (ωBi = ∅|ω∆c)= P P ωBi = ∅|ωBic |ω∆c ≤ Cz−1 ,
for any i N,S,E,W . So the conditio(cid:0)nal(cid:0)probability th(cid:1)at ω s(cid:1)atisfies (C1) is larger than
∈ { }
1 4Cz 1.
−
−
The equation Nc∆c(ω) = 0 forces the box BN (for instance) to be empty of points of the
configuration ω. Hence,
P Nc∆c(ω)= 0 ω∆c Cz−1 .
| ≤
Checkingthatcondition (C2)is(cid:0)fulfilledinthediam(cid:1) ond box∆needs whatwecalltheCon-
nection Lemma (Lemma 4.4). This result states the conditional probability that N∆(ω)
cc
is larger than 2 converges to 0 uniformly on the configuration outside ∆. This is the heart
of the proof of Theorem 1. Its technical proof is given in Section 4.2.
Lemma 4.4 (The Connection Lemma). There exists a constant C > 0 such that for any
′
configuration ω Ω and for any z > 0,
∈
P Nc∆c(ω) 2 ω∆c C′z−1 . (4.3)
≥ | ≤
The above inequalities and(cid:0)the Connection L(cid:1)emma imply that conditions (C1) and
(C2) are fulfilled in ∆ with a probability tending to 1 as z tends to :
∞
P ξ(0,0)(ω) = 1 ω∆c 1 (5C +C′)z−1 .
| ≥ −
The hypothesis (4.1) then(cid:0) follows. Let x b(cid:1)e a vertex of the graph (V,E) which is not a
neighbor of (0,0). By construction, the box ∆ is included in ∆c = ∆c (since ∆ is an
x (0,0)
open set). This means the random variable ξ is measurable with respect to the σ-algebra
x
induced by the configurations restricted to ∆c . So,
(0,0)
P ξ = 1 ξ : (0,0),x / E 1 (5C +C )z 1 ,
(0,0) x ′ −
| { } ∈ ≥ −
and the hypothesis(cid:0)(4.1) holds with x = (0,0) and(cid:1) any p [0,1[, provided the intensity z
∈
is large enough. This ends the proof of Theorem 1.
9
Proof. (Lemma4.3)LetU (R2)beaboundedBorelsetwithpositiveLebesguemeasure
and V U B¯(0,R +R∈+Bε). First, let us write:
1 0
⊃ ⊕
1
P (ωV = ∅|ωVc) = ZV(ωVc)ZΩV 1IωV=∅e−HV(ωV∪ωVc)πVz(dωV)
e zλ(V)
−
= , (4.4)
ZV(ωVc)
since the empty configuration has a null energy, i.e. HV(ωVc) = 0. A configuration ω
whose restriction to V satisfies
#ω = 1 and ω =
U×[R0,R0+ε] V\U ∅
is reduced to aball B¯(x,R) centered at ax in U andwith a radius R < R < R +ε. Since
0 0
the ball B¯(x,R) does not overlap ω¯Vc, its energy HV((x,R) ωVc) is easy to compute;
∪
HV((x,R) ωVc)= θ12πR+θ2πR2+θ3
∪
(itisnotworthusinginequalities ofLemma4.12here). So, HV((x,R) ωVc)isboundedby
∪
a positive constant K only depending on parameters θ , θ , θ and radius R . Henceforth,
1 2 3 1
P #ωU×[R0,R0+ε] = 1, ωV\U = ∅|ωVc
1
(cid:0) = (cid:1) e HV((x,R) ωVc)ze zλ(V)λ(dx)Q(dR)
− ∪ −
ZV(ωVc)ZU×[R0,R0+ε]
e zλ(V)
− ze Kλ(U)Q([R ,R +ε]) .
− 0 0
≥ ZV(ωVc)
Recall that Q([R ,R +ε]) is positive by (2.4). Using the identity (4.4), we finally upper-
0 0
bound the conditional probability P(ωV = ωVc) by
∅|
ze−Kλ(U)Q([R0,R0+ε]) −1P #ωU×[R0,R0+ε] = 1, ωV\U = ∅|ωVc .
This prov(cid:0)es Lemma 4.3 with C = (e(cid:1) Kλ(U(cid:0))Q([R ,R +ε])) 1. (cid:1)
− 0 0 −
4.2 Proof of the Connection Lemma
4.2.1 Outline
Let us recall that N∆(ω) denotes the number of connected components of ω¯ having at
cc ∆
least one ball centered in one of the four cardinal boxes BN, BS, BE or BW. In this section,
we assume N∆(ω) 2. Our strategy consists in exhibiting a subset B of the diamond box
cc
≥
∆ in which ω = . Moreover, for x B, if we are able to control uniformly the energy
B
∅ ∈
HB((x,R) ωBc) on ωBc, then the configuration ω should contain a point centered in B
∪
with large probability as z tends to infinity. This leads to the Connection Lemma.
For x B, let us denote by hol((x,R),ωBc) the hole number variation when the ball
∈ N
B¯(x,R) is added to the configuration ωBc. This quantity is central in our proof. Indeed,
a first upperbound for the energy HB((x,R) ωBc) is given by Lemma 4.12:
∪
HB((x,R) ωBc)= h((x,R),ωBc) K θ3 hol((x,R),ωBc) , (4.5)
∪ ≤ − N
where K is a positive constant only depending on parameters θ , θ , θ and radii R , R .
1 2 3 0 1
So, to upperbound the energy HB((x,R) ωBc) it suffices to upperbound the number of
∪
created holes (resp. deleted holes) when θ is negative (resp. positive). This is the reason
3
why the proof of the Connection Lemma differs according to the sign of the parameter θ .
3
10