Gibbs Delaunay tessellations with geometric
hardcore conditions
David Dereudre
Université de Valenciennes et du Hainaut-Cambrésis
Le Mont Houy
59313 Valenciennes Cedex 09, France
e-mail : [email protected]
November 14, 2006
Abstract
We prove the existence of infinite Gibbs Delaunay tessellations on R2 . The interaction
depends only on the local geometry of the tessellation. We introduce a geometric hardcore
condition on small and large cells. Through this, we can construct more regular infinite
random Delaunay tessellations.
AMS Classifications: 60D05 - 60K35.
KEY-WORDS: Stochastic Geometry, Gibbs measure, Delaunay tessellation.
1
1
Introduction
In [4], [5] and [6], Bertin, Billiot and Drouilhet deal with the Gibbs point process where
the interaction depends on the structure of the Delaunay tessellation. More precisely, they
consider the Delaunay triangulation in the space where the vertices are given by a Poisson
point process and the interaction is built thanks to the triangles of this tessellation. It is
a functional from the local geometry of the Delaunay tessellation. That produces, for the
classical point of view, an infinite multibody interaction (see [8], [16] and [17] for a description of the classical statistical mecanics point of view). In the first paper, they study
the finite volume case. For example, they give some algorithms to simulate this Gibbs
tessellation and they give applications for the modelling of the cells of the prostatic tissue.
In the second and third papers, they prove the existence of some infinite Gibbs states. The
results are proved in the classical context of Gibbs point process. They use the classical
results from Preston; see [14]. Our paper is the continuation of this results but in a little
different point of view that is more geometric and uses different methods.
Before precising this consideration, let us introduce the paper of H. zessin. In [20], the
Author propose a purely geometric approach of Gibbs simplicial complexes. The interaction potential is a functional from the local geometry. There is no point process behind
the simplicial complexes; the interaction is purely geometric. Moreover, in this paper, he
proposes to find an intrinsic description of Gibbs geometric states. How to define a geometric Gibbs structure on the space of simplicial complexes? The objective is the problem
of the pure quantum gravity; see the paper by Zessin for many references. He does not give
answers at this question and not deal with the existence of Gibbs structure. The author
proves some results about the specific energy of random stationary simplicial complexes.
We are interested in the development of a purely geometric description of the random
simplicial complexes, built on the Delaunay tessellation. Why do we choose the Delaunay
tessellation? Because it is the simplest and the most natural triangulation in the space.
So, we want to develop a natural concept of interacting Gibbs Delaunay tessellation in
the spirit of H. Zessin. We take his point of view for the definition of the Energy and we
propose a new and elegant description of the local density. Indeed, the local density is
built completely from the intrinsic geometry of the tessellation. In the papers [4], [5] and
[6], the description of the energy and the local density is in the context of classical Gibbs
point process. In some cases, this description and our description are equivalent. It is the
case, for example, when the energy is completely finite; we explain it more precisely in
the section 3. But in the case where we introduce some geometric hardcore interactions,
our approach is different from the classical case. Let us give a simple example to show the
difference.
Let us suppose that we want to build some tessellations whose the cells are not too small
or too large; it will be the case in the section 4. So, we fix the energy of a triangle equals to
infinity if this triangle is small or large (for example, using the radius of the circumscribed
ball). Kowever, this kind of energy is not authorized in the classical point of view. Indeed,
Let us imagine a tessellation containing a large triangle (the energy is equal to infinity)
and let us add a point inside this triangle. Then, it is possible that the energy becomes
finite. This kind of situation is completely forbidden in the classical case (see assumption
(I4) in [17]). In this example, we can not calculate, in general, the energy of a point in a
configuration. It’s not defined.
So, we develop some techniques to build the Gibbs Delaunay tessellation without the clas2
sical tools of Gibbs point processes. We would like to thank T. Schreiber for his help. We
use some of his ideas from [18] to control the convergence of the finite volume Gibbs tessellation. Thanks to this methods, we relax an assumption of quasilocality for the interaction
in the paper [5]. Indeed, in the classical context of Gibbs point processes, the interaction
of a configuration goes to zero when the diameter of this configuration goes to infinity. In
our case, we do not need this assumption. Thanks to our geometric approach, the locality
of the interaction is given directly by the locality of the tessellation. The interaction is
not null when the size of the cell goes to infinity and sometimes the energy is even not
bounded (or equal to plus infinity). This phenomena is not anecdotal, because in many
natural geometric interactions, the energy is not bounded when the cells are large. For
example, in the pure quantum gravity, evoked by Hans zessin, the energy is equal to the
curvature minus the local volume which is completely unbounded, when the cells become
large.
2
Definitions and notations
2.1 State spaces
If X denotes a Polish space endowed with the Borel σ-algebra σ(X), then B(X) is the
set of bounded sets of X. M(X) is the set of the integer-valued measures Γ on X such that,
for every Λ ∈ B(X), Γ(Λ) ∈ N. M(X) is endowed with the σ-algebra σ(M(X)) generated
by the sets {Γ ∈ M(X), Γ(Λ) = n}, n ∈ N∗ , Λ ∈ B(X). Any measure Γ ∈ M(X) has the
following representation
X
Γ=
δXi ,
i∈I
where I is a subset of N, (Xi )i ∈ I are elements of X and δX is the Dirac measure at X.
We write X ∈ Γ if Γ({X}) > 0. Γ is said simple if for every X ∈ Γ, Γ({X}) = 1
In our case, X denotes sometime R2 endowed with the Euclidian metric, sometimes the
space E defined below. So, we note M(R2 ) the space of integer-valued measures in R2
and γ denote a typical element of M(R2 ). We are interesting with a strictly smaller space
than M(R2 ) containing only the integer valued measures associated to the Delaunay
tessellation. More precisely, we note
- four points of γ are not on a same circle
2
2
MD (R ) = γ ∈ M(R ) such that
.
- any half plane in R2 contains some points of γ
We note E the space
E=
- 1 ≤ Card(A) ≤ 3
A ⊂ R2 such that
.
- A is affinely independent
Three points are affinely independent if they are not on a same line. In the others cases,
it is always true. We note E (1) := {A ∈ E, card(A) = 1} the space of vertices in R2 ,
E (2) := {A ∈ E, card(A) = 2} the space of edges in R2 and E (3) := {A ∈ E, card(A) = 3}
the space of triangles in R2 . The space E is endowed with the distance dE defined below.
For X, Y ∈ E we note X ⊂ Y if X is a subset of Y in R2 .
3
For X, Y ∈ E, we define N (X) =Card(X) and
(
+∞
if N (X) 6= N (Y )
dE (X, Y ) =
max supx∈X inf y∈Y |x − y|, supy∈Y inf x∈X |x − y|
if N (X) = N (Y )
(1)
where |.| denotes the Euclidian norm in R2 . It is easy to verify that dE is an distance on
E. (E, dE ) is a seperable metric space but it is not completed. So we note E¯ a completion
¯ We define M(E) the space of
of E for the metric dE and dĒ the extended metric on E.
integer-valued measures in E and Γ denotes a typical element of M(E). For Γ ∈ M(E)
and k = 1, 2 or 3 we note
X
δX .
Γ(k) =
X∈Γ
N (X)=k
Let us define now the space of Delaunay tessellation. For every X ∈ E (3) , we note B(X)
the open circumscribed ball of X and B̄(X) the closure of B(X). Γ ∈ M(E) is a Delaunay
tessellation if
i) Γ(1) is in MD (R2 )
ii) for every X ∈ Γ for every Y ⊂ X then Y ∈ G
(3)
iii) for every Y ∈ Γ, there exists X
∈ Γ such that Y ∈ X
(3)
(1)
iv) for every X ∈ Γ , Γ B(X) = 0.
We note MD (E) the space of Delaunay tessellations. The conditions ii), iii) give to Γ the
structure of simplicial complex (for more details, see [19]). The property v) is a typical
property for the Delaunay tessellation (see [12] ).
Let us remark that a Delaunay tessellation is uniquely and completely determined by Γ(1)
(see for example [12]). So, for every γ ∈ MD (R2 ) we note Γ := γ̄ the unique Delaunay
tessellation in MD (E) such that Γ(1) = γ.
For every X ∈SE, we note < X > the convex hull of X in R2 and for every Γ ∈ M(E) we
note < Γ >= X∈Γ < X >. Let us remark that for every Γ ∈ MD (E), < Γ >= R2 . Let
Γ, Γ′ are in M(E), we note Γ′ ≺ Γ if the measure Γ′ is absolutely continuous with respect
to Γ. In fact, Γ′ ≺ Γ if for every X in Γ′ then X is in Γ.
4
Fig 1) an example of Delaunay tessellation
2.2 Interaction
In this section, we define the energy of finite connected configurations. For the moment,
we don’t give the assumptions needed in the following theorems to prove the existence of
Gibbs Delaunay tessellation. Let p be a fixed integer until the end of the paper. p is the
maximum number of cells for a typical interacting configuration. More precisely, we Note
−Γ(E) ≤ p
Sp (E) = Γ ∈ M(E) .
− < Γ >:= ∪X∈Γ < X > is a connected set
Let us remark that an element Γ in Sp (E) does not satisfy in general the assumptions ii),
iii),iv). An interaction potential V is a measurable function
V :
Sp (E)
−→
R ∪ {+∞}.
Now, we can define the local energy of an infinite Delaunay tessellation. Let Λ be a bounded
set in B(R2 ) and Γ a Delaunay tessellation in M(E), we define the energy of Γ inside Λ
with the following definition
X
EΛ (Γ) =
V (Γ′ ).
(2)
Γ′ ≺ Γ
Γ′ ∈ Sp (E)
< Γ′ > ∩Λ 6= ∅
Let us remark that the sum above is finite thanks to the locality of the Delaunay tessellation. Since Γ is determined by γ, we can define the energy of γ inside Λ by the simple
formula EΛ (γ) := EΛ (γ̄); we use the same notation for the energy of Γ in Λ and the
2
∞
2
energy of γ in Λ. We note M∞
D (R ) (respectively MD (E)) the subset of MD (R ) (respectively MD (E)) such that the configurations γ (respectively Γ) have a finite local energy
EΛ (γ) < +∞ (respectively EΛ (Γ) < +∞) for all Λ ∈ B(R2 ).
This definition of the energy is similar at this one given in [19].
5
2.3 The Reference measure and the local specifications
We note λ the lebesgue measure on R2 , π denotes the Poisson Process on R2 with
intensity λ. It is a probability measure on M(R2 ). For every Λ ∈ B(R2 )), πΛ denotes
the Poisson process on Λ with intensity 1IΛ λ. We note π̄ the natural Poisson Delaunay
tessellation which is the image of π to MD (E) by the application γ → γ̄. π̄ is a Probability
measure on MD (E) and it is the reference process for the gibbs tessellations. We can see
it like the random Delaunay tessellation without interaction. In fact, we want to construct
Gibbs Delaunay tessellation which are absolutely continuous with respect to π̄. The local
density is given by local specifications that we are now defining.
conditioning First, let us explain how the refence measure π̄ satisfies the local conditioning.
In the classical continuous case, the reference measure is π and so the local conditioning
is trivial. For every Λ ∈ B(R2 ), the process γ under π has two independent parts γΛ (the
projection of γ inside Λ) and γΛc (the projection of γ outside Λ). In our case, it is more
complicate, because the inside and outside of Γ under π̄ are not independent. And what
are exactly the inside and outside of Λ? There are many possibilities to define them. In
[4],[5] and [6] , they consider only the points of Γ(1) and so the inside and outside are
clearly and easily defined but they lose the structure of the tessellation near the boundary
of Λ. In our case, we prefer to keep completely the structure of the Delaunay tessellation
outside of Λ. More precisely, we have the following definition.
Let Λ be in B(R2 ) and Γ ∈ MD (E), we note ΓΛc the process
X
Γ Λc =
δX .
(3)
X∈Γ
∃Y ∈Γ(3) , X⊂Y
<Y >∩Λc 6=∅
In fact, ΓΛc corresponds to the part of the Delaunay tessellation which intersects the outside of Λ. Let us remark that every X ∈ ΓΛc does not necessary satisfy < X > ∩Λc 6= ∅
but it is contained in a triangle Y which satisfies it. We have in the Figure 2 an example
of configuration ΓΛc .
Fig 2) an example of configuration ΓΛc
6
Now, in the following proposition we identify the law of Γ under π̄ given ΓΛc . Let us
define before the set SΛ (Γ) in R2
[
2
SΛ (Γ) = R \
B̄(X)
(4)
(3)
X∈ΓΛc
and the function Υ
Υ : MD (E) × M(R2 ) × B(R2 ) −→ MD (E)
(1)
(Γ, γ, Λ)
−→
ΓSΛ (Γ)c + γSΛ (Γ) .
(1)
With this definition, Υ(Γ, γ, Λ) is not correctly defined if ΓSΛ (Γ)c +γSΛ (Γ) is not in MD (R2 )
(1)
( it is possible if ΓSΛ (Γ)c + γSΛ (Γ) has four points on the same circle). In this case, we put
Υ(Γ, γ, Λ) = Γ.
The function Υ extends the partial Delaunay tessellation ΓΛc inserting the points of γ
inside SΛ (Γ) and completing the Delaunay structure.
Fig 3) an example of set SΛ (Γ) represented as the dark area
Proposition 1 let Λ be an bounded set in R2 then for all bounded measurable functions
f and g we have
Z
Z
hZ
i
c
c
g(ΓΛ )
f Υ(Γ, γ, Λ) πSΛ (Γ) (dγ) π̄(dΓ).
g(ΓΛ )f (Γ)π̄(dΓ) =
MD (E)
M(R2 )
MD (E)
Proof :
We have for all Λ in R2 and all f, g measurable bounded functions on M(E)
Z
Z
g(γ̄Λc )f (γ̄)π(dγ)
g(ΓΛc )f (Γ)π̄(dΓ) =
M(R2 )
MD (E)
Z
hZ
i
′
g(γ̄Λc )
f Υ(γ̄, γ, Λ π(dγ|γ̄Λc = γ̄Λ′ c ) π(dγ ′ ).
=
M(R2 )
M(R2 )
7
Since Υ(Γ, γ, Λ) depends only on ΓΛc and γSΛ (Γ) , we have
Z
Z
hZ
i
g(ΓΛc )
f Υ(Γ, γ, Λ π(dγ|γ̄Λc = ΓΛc ) π̄(dΓ)
g(ΓΛc )f (Γ)π̄(dΓ) =
M(R2 )
M(R2 )
MD (E)
Z
i
hZ
(1)
f Υ(Γ, γ, Λ π(dγ|γSΛ (Γ)c = ΓSΛ (Γ)c ) π̄(dΓ)
g(ΓΛc )
=
M(R2 )
M(R2 )
Z
hZ
i
g(ΓΛc )
f Υ(Γ, γ, Λ) πSΛ (Γ) (dγ) π̄(dΓ).
=
M(R2 )
MD (E)
The proposition is proved.
This proposition gives the conditional law under π̄ of Γ knowing ΓΛc . It is the first step to
define the D.L.R. equations. This description of π̄ given ΓΛc gives us a natural description
of the local law of π̄. We base our Gibbs structure on this result. It is not really new because
it is in the spirit of the geometric gibbsian structure for the Arak processes (see [1], [2]
and [3]) and it is suggested by H. Zessin in [19] to obtain a purely geometric description of
kernels for Gibbs simplicial complexes. Now, let us give the local specifications with inter∞
action. For every Λ ∈ B(R2 ), we define the following kernel ΞΛ on M∞
D (E) × P(MD (E)).
∞
So, for every Γ ∈ M∞
D (E) and every bounded continuous function f from MD (E) to R we
define
Z
1
−EΛ Υ(Γ,γ,Λ)
e
πSΛ (Γ) (dγ),
(5)
f Υ(Γ, γ, Λ)
ΞΛ (Γ, f ) =
ZΛ (Γ)
M(R2 )
where ZΛ (Γ) is the normalization constant
Z
e−EΛ
ZΛ (Γ) =
M(R2 )
Υ(Γ,γ,Λ)
πSΛ (Γ) (dγ).
We have to justify that the constant ZΛ (Γ) is finite and not null for all Γ ∈ M∞
D (E) .
That depends on the interaction V . We’ll prove later when we’ll introduce precisely the
assumptions on V . In fact that comes from the stability of V and a hardcore property.
We have to extend the definition of ΞΛ (Γ, .) in the case where SΛ (Γ) = ∅. In this case, we
put ΞΛ (Γ, f ) = f (Γ) and the probability measure ΞΛ (Γ, .) is in fact δΓ .
Now, the family of kernel (ΞΛ )Λ∈B(R2 ) is a specification if the kernels are compatible; that
means
∀Λ, Λ′ ∈ B(R2 ) such that Λ ⊂ Λ′ , then ΞΛ′ ΞΛ = ΞΛ′ .
In general this fact is obvious in statistical mechanics; that comes from the additivity of
the energy. But in our case the space state and the interaction are a little different, we
prefer to give a short proof of this result.
Proposition 2 the kernels (ΞΛ )Λ∈B(R2 ) are compatible.
Proof:
Let Λ, Λ′ ∈ B(R2 ), Γ ∈ MD (E) and f a bounded measurable function on MD (E). We
8
suppose Λ ⊂ Λ′ ;so, we have
Z
Z
′′ ′′
′
′′
′
′
f (Γ )ΞΛ (Γ, dΓ |ΓΛ = ΓΛ ) ΞΛ′ (Γ, dΓ′ )
ΞΛ (Γ, f ) =
MD (E)
M (E)
Z D
Z
1
′′
′′
′′
f (Γ′′ )
=
e−EΛ (Γ )− EΛ′ (Γ )−EΛ (Γ ) Ξ0Λ′
ZΛ′ (Γ)
MD (E)
MD (E)
(Γ, dΓ′′ |Γ′′Λ = Γ′Λ ) ΞΛ′ (Γ, dΓ′ ),
where (Ξ0Λ )Λ∈B(R2 ) are the kernels without interaction (V = 0) and the convention EΛ′ (Γ′′ )−
EΛ (Γ′′ ) = 0 if EΛ′ (Γ′′ ) = +∞ and EΛ (Γ′′ ) = +∞ . Thanks to the proposition 1
Ξ0Λ′ (Γ, dΓ′′ |Γ′′Λ = Γ′Λ ) = Ξ0Λ (Γ′ , dΓ′′ ).
So, we have
ΞΛ′ (Γ, f ) =
Z
Z
MD (E)
′′
1
f (Γ )
e−EΛ (Γ )−
′
ZΛ (Γ)
MD (E)
Z
MD (E)
′′
′′
1
e−EΛ (Γ )−
ZΛ′ (Γ)
EΛ′ (Γ′′ )−EΛ (Γ′′ )
EΛ′ (Γ′′ )−EΛ (Γ′′ )
Ξ0Λ (Γ′ , dΓ′′ )
−1
Ξ0Λ (Γ′ , dΓ′′ )
ΞΛ′ (Γ, dΓ′ ).
Since EΛ′ (Γ) − EΛ (Γ) and ZΛ′ (Γ) depend only on ΓΛc , we have
Z
Z
1
′′
−EΛ (Γ′′ ) 0
′
′′
f (Γ )
e
ΞΛ (Γ , dΓ ) ΞΛ′ (Γ, dΓ′ )
ΞΛ′ (Γ, f ) =
′)
Z
(Γ
Λ
M (E)
MD (E)
Z D
Z
′′
′
′′
f (Γ )ΞΛ (Γ , dΓ ) ΞΛ′ (Γ, dΓ′ ).
=
MD (E)
MD (E)
the proposition is proved.
Now we can define the Gibbs Delaunay tessellation.
Definition 1 A probability measure µ on MD (E) is a Gibbs Delaunay tessellation for the
interaction potential V if for every bounded set Λ in B(R2 ) and every bounded measurable
function f from MD (E) to R we have
Z
Z
Z
f (Γ′ )ΞΛ (Γ, dΓ′ )µ(dΓ).
(6)
f (Γ)µ(dΓ) =
MD (E)
MD (E)
MD (E)
The equations (6) are called DLR equations (Dobrushin,Landford,Ruelle). It is equivalent
to: for µ almost every Γ, every bounded set Λ in B(R2 ) and every bounded measurable
function f from MD (E) to R
µ(f |ΓΛc ) = ΞΛ (Γ, f ).
3
Existence of Gibbs tessellation without hardcore condition
In this section, our result is exactly a direct extension of a result in [5]. In this paper, the
authors prove the existence of Gibbs Delaunay tessellation for an interaction on the finite
9
configurations, which is null if the configuration is included in a triangle having minimal
angle smaller than a fixed angle α0 > 0. Moreover, they assumed the quasilocality of the
interaction; it means that the interaction of configuration goes to zero when the diameter
of the configuration goes to infinity. With these two assumptions, they can use the Preston
theorem (see [14]) to prove the existence of Gibbs Delaunay tessellation (see page 727 in
[5] and page 899 in [6]).
Our methods are completely different. Firstly, the Gibbs Delaunay tessellation concept is
different. Nevertheless, in the case without hardcore interaction we can prove that their
concept, which is purely a point process concept, and our concept of Gibbs Delaunay
tessellation, which is more geometric, are equivalent. It will not be the case in the next
section. Secondly, we don’t use the Preston Theorems. These theorems don’t use the
geometric structure of Delaunay tessellation to prove the locality of the interaction. In
fact, we use the locality of the Delaunay tessellation and not the locality of the interaction
to prove the existence of Gibbs tessellations.
Let us define the assumption for V in this case without hardcore. We suppose until the
end of this section that V satisfies the following assumptions.
(TI): The translation invariant condition
∀h ∈ R2 , ∀Γ ∈ Sp (E)
V (τh Γ) = V (Γ),
where τh is the shift operator for the vector h ∈ R2 . Now, let us give the assumption which
assures that V has not an hardcore part
(F): The finite energy condition
∀Γ ∈ Sp (E),
V (Γ) < +∞.
Let X be a triangle in E (3) , we note α(X) the smallest angle in X. Now let us define a
condition about the support of V
(S): The support condition
there exists α0 > 0 such that for every Γ in Sp (E).
V (Γ) 6= 0 =⇒ ∀X ∈ Γ ∃Y ∈ Γ(3) such that X ⊂ Y and α(Y ) > α0
(S) means that V charges only the configurations Γ not having too flat triangles. The
interaction potential is bounded if
(B): The bounded condition
∃M > 0, ∀Γ ∈ Sp (E), then V (Γ) < +∞ =⇒ |V (Γ)| ≤ M .
There is a last assumption about the regularity of V .
(R): The regular condition
V is π a.s. continuous. That means there exists Ω ⊂ MD (E) with π(Ω) = 1 such that for
every Γ ∈ Ω, every Γ′ ≺ Γ in Sp (E), V is continuous at Γ′ .
Now, let us give some remarks and consequences of these assumptions.
(TI) is needed because we study only invariant translation Gibbs Delaunay tessellations.
Moreover, we need this invariant translation to construct the Gibbs tessellation and to
control the convergence. The conditions (S) and (B) are the same as in [5]. For the
theoretical point of view, they are strong. But for the practical point of view, it is very
weak because we can choose α0 very small and M very large. The condition (R) is a little
technical but it is satisfied for every natural interaction. This condition is not needed in
10
[5].
If the interaction V satisfies (F), we can define the energy of point x in a configuration γ.
Let x ∈ R2 and γ ∈ MD (R2 ) such that γ + δx is in MD (R2 ); then, we define the energy
of the point x in γ by the following formula
X
X
V (Γ′ ) −
V (Γ′ )
(7)
E(x, γ) =
Γ′ ≺γ
Γ′ ⊀γ+δx
Γ′ ≺γ+δx
Γ′ ⊀γ
Thanks to this definition we have
EΛ (γ) = EΛ (γΛc ) +
k
X
E(xi , γΛc + δx1 + . . . δxi−1 ),
(8)
i=1
where γΛ = δx1 + . . . δxk .
Now, let us remark that the definition of kernels (Ξ)Λ⊂R2 in (5) would be exactly the
same (in the case where V satisfies (F)) if we substitute EΛ (γ) by the cumulated sum
of the energy (i.e. the second part in the right term in equality (8)) because EΛ (γΛc ) is
a constant. That is the reason for which the techniques, that we use in this section, are
based partially on classical tools for the point process. For instead, we will use the reduced
Campbell measure on M(R2 ) that we remind below.
Let P be a probability measure on M(R2 ), we define the reduced Campbell measure CP!
on M(R2 ) × R2 by
Z Z
CP! (f ) =
f (x, γ − δx )γ(dx)P (dγ),
where f is a bounded measurable function from M(R2 ) × R2 to R. Now let us give the
first Theorem.
Theorem 1 For any α0 > 0, there exists a Gibbs Delaunay tessellation for a potential V
which satisfies (TI),(F),(S),(B) and (R).
Proof :
First of all, we have to prove that the kernels (ΞΛ ) (see (5)) are well defined. Let us show
that 0 < ZL (Γ) < +∞ for every Γ ∈ M∞
D (E). We remind the definition of ZΛ (Γ) (we
suppose that SΛ (Γ) 6= ∅)
Z
−EΛ Υ(Γ,γ,Λ)
πSΛ (Γ) (dγ).
e
ZΛ (Γ) =
M(R2 )
Since there is no hardcore, it is obvious that ZΛ (Γ) > 0. Now, we are proving the stability.
Let x be in Γ(1) and Γ′ ≺ Γ be in Sp (E) such that V (Γ) 6= 0. Thanks to the assumption
(S), Γ′ is included, at the maximum, in p connected triangles having each angle bigger
than α0 . The number of such p connected triangles configurations in Γ with large angles
containing x is bounded by C := (p α2π0 )p . That permits to control the first sum of the
energy E(x, Γ(1) ) defined in (7) since |V (.)| ≤ M (M coming from the assumption (B)).
For the second part, let us remark that the number of triangle X ∈ Γ(1) − δx such that
α(X) > α0 and X ∈
/ Γ is bounded by α2π0 (see proposition 3 in [6]). So the number of
configurations Γ ∈ Sp (E) included in Γ(1) − δx but not included in Γ such that V (Γ) 6= 0
11
is also bounded by the constant C. We deduce that |E(x, γ)| is bounded by 2M C and the
interaction V is stable. ZΛ (Γ) is finite and non null.
Now, the proof of the theorem can be sketched into two parts. The first is to find and
construct a probability measure µ in MD (E) that will be a good candidate to be a Gibbs
measure.So we define the bounded periodic specifications associated to the potential V
and we prove the convergence of these measures. Afterward, in a second part we show that
µ satisfies the DLR equations characterizing the Gibbs structure. In this second part, we
have to prove essentially that the kernels (ΞΛ )Λ∈B(R2 ) are continuous for µ almost every
Γ with respect to the Topology that we use.
So let us define the stationary periodic specifications.
Let Λn be the bounded set [−n, n]2 ⊂ R2 . For every γ ∈ M(Λn ) we note pn (γ) the periodic
configuration
X X
pn (γ) =
δx+h .
h∈2nZd x∈γ
We define the stationary periodic specifications µn for every n ∈ N∗ by the following way:
for every positive bounded function f from MD (E) to R
Z
Z
−E
p
(γ)
n
1
Λn
f pn (γ) e
πΛn (dγ),
(9)
f (Γ)µn (dΓ) =
Z
M(Λn ) n
MD (E)
R
with Zn = M(Λn ) e−EΛn (pn (γ)) πΛn (dγ). If γ = 0 (respectively γ = δx ) then we define
P
pn (0) = 0 (respectively pn (δx ) = n∈2nZd δ{x} ) and EΛn pn (γ) = 0 in the both cases.
Let us remark that Zn is finite for all n ∈ N∗ because V is stable; see the calculus above.
First step, convergence of (µn ):
Let us look at precisely the convergence concept. The converge of the probability
measures µn is for the weak topology on P(M(E)) where the topology on M(E) is induced
by the metric dM(E) . dM(E) is the associated metric for the vague topology on M(E), E
being endowed by the metric dE defined in (1). see for example [7] page 607 for an appendix
about the topology of the convergence of measures and [11] page 108 for the convergence
of integer valued measure.
Let us begin to analyze the metric dE and to remind that the set E is not a completed
space. So, for the moment we have to define the convergence in E¯ for which we have the
extended metric dĒ . dĒ (X) is defined by the limit of dE (Xn ), when n goes to infinity, and
∗
for any set (Xn ) which tends to X. So, a set (Xn )n∈N∗ ∈ Ē N converges to an element
X ∈ Ē if and only if there exists n0 in N∗ such that ∀n ≥ n0 , N (Xn ) = N (X) and we can
write Xn like {xn } or {xn , yn } or {xn , yn , zn } (idem for X, {x} or {x, y} or {x, y, z}) with
limn→∞ xn = x, limn→∞ yn = y, limn→∞ yn = y.
Next, Let us look at the metric dM(E¯) (for a precise definition, we can see [11] page 108∗
¯ if and only if for every
109). A set of measure (Γn ) ∈ M(Ē)N converges to Γ ∈ M(E)
∗
¯ there exists ε > 0 and n0 ∈ N such that for all n ≥ n0
open bounded set Λ ⊂ E,
ε
ε
Γn (Λ ) = Γ(Λ ) = Γ(Λ) with
n
o
Λε = X ∈ E¯ such that B(X, ε) ⊂ Λ .
PΓ(Λ)
PΓ(Λ)
Moreover, we can write for every 1 ≤ k ≤ Γ(Λ) (Γn )Λε = k=1 δXkn , ΓΛε = k=1 δXk ,
and we have limn→∞ Xkn = Xk for the metric dĒ . The set Λε is introduced to control the
12
boundary effects which can appear if the configurations Γn have points near the boundary
of Λ. The convergence of Γn to Γ for the metric dM(Ē ) is equivalent at the convergence
¯ it means that for every bounded positive continuous
for the vague topology on M(E);
function f on Ē, which is identically null outside of suitable bounded set of Ē, we have
Z
Z
f (X)Γ(dX).
f (X)Γn (dX) =
lim
n→∞ Ē
Ē
Proposition 3 The probability measures (µn )n∈N∗ on MD (E) convergence for a sub set
to a shift invariant probability measure µ on M(E). Moreover µ-almost every Γ ∈ M(E)
satisfies the condition i’), ii), iii), iv), where i’) is
i’) Γ(1) is in MD (R2 ) or Γ = 0 .
Moreover,
µ Γ(1) ∈ MD (R2 ) > 0.
Proof :
To prove the convergence of µn to a probability measure µ we need to control the first
moments of µn . So, we have the following lemma.
Lemma 1 There exists a constant C1 such that for every n in N∗ and every Λ ∈ B(R2 )
we have
Eµn Γ(1) (Λ) ≤ C1 λ(Λ).
Proof :
µn is obviously shift invariant. So it is sufficient to prove that µn has a σ-finite moment
uniformly bounded for n ∈ N∗ . For that, we can use the reduced Campbell measure of
µn to identify the first moment. Since µn is a finite-volume Gibbs measure and since the
energy can be written like in (8) we have
Cµ!
n(1)
(dx, dγ) = e−En (x,γ) λ ⊗ µn(1)
(10)
where En (x, γ) is the periodic energy of x in γ
En (x, γ) = EΛn (pn (γ + δx )) − EΛn (pn (γ)).
(11)
In fact, we have
EΛn (pn (γ)) =
k
X
En (xi , δx1 + . . . δxi−1 )
for every γ ∈ M(Λn )
(12)
i=1
where γ = δx1 + . . . + δxn .
For the classical techniques concerning the calculus about the Campbell measure, we
can see for example [13]. In particulary, we can find the equation (10) for the Campbell
measure of a Gibbs state. Like above, it is easy to show that |En (x, γ)| ≤ 2CM . So, the
first moments of measures (µn )n∈N∗ are uniformely bounded by e2CM λ. The lemma is
prooved for the constant C1 = e2CM .
In [7], Corollary A2.6.V page 607 gives a characterization for the Relative compacity of
¯ is relatively
a set of radon measures on a locally compact space. So, a set (Γn ) in M(E)
compact if
∀Λ ∈ B(E), ∃C > 0, ∀n ∈ N∗ ,
Γn (Λ) ≤ C.
13
In our case, thanks to the lemma 1 we have
∀Λ ∈ B(E), ∃C > 0, ∀n ∈ N∗ ,
Eµn (Γ(Λ)) ≤ C.
So, the family of probability measures (µn ) is uniformly tight in M(E). We can extract a
sub set (µϕ(n) ) converging weakly in M(E) to a probability measure µ. The probability
measure is clearly shift invariant.
Now, we have to prove some properties of this measure µ. We not are absolutely sure that
µ(MD (E)) = 1.
¯ such
There exists a probability space (Ω̃, F̃ , P̃ ) and processes (Γ̃n )n∈N∗ , Γ̃ from Ω̃ to M(E)
∗
that for every n ∈ N the law of Γ̃n (respectively Γ̃) is µn (respectively µ) and Γ̃n converge
P̃ -almost surely to Γ̃ for the metric dM(Ē) (we skip until the end of the proof the notation
µϕ(n) ; we just write µn ) .
For the moment, the only one Delaunay property of Γ̃n , which is kept for the limit Γ̃, is
ii). Let us begin the precise analyze of Γ̃ with the following lemma.
Lemma 2 P̃ almost surely (Γ̃n ) satisfies
∀R > 0, ∃A > 0, ∀n ∈ N∗ , ∀X = {x, y, z} ∈ Γ̃(3)
n
|x| ≤ R ⇒ max(|y|, |z|) ≤ A.
(13)
Proof:
Let us suppose that (13) is false. It means that, with a strictly positive probability,
∃R > 0, ∀A > 0, ∃n ∈ N∗ , ∃X = {x, y, z} ∈ Γ̃(3)
n
|x| ≤ R and max(|y|, |z|) ≥ A.
In other words, there exists R > 0 and a subset of triangles (Xϕ(n) )n∈N∗ such that for
all n ∈ N∗ Xϕ(n) = {xϕ(n) , xϕ(n) , xϕ(n) } ∈ Γ̃ϕ(n) , xϕ(n) ≤ R and the radius rϕ(n) of the
circumscribed ball of Xϕ(n) goes to infinity. So, it proves the existence of an half plane in
R2 without points for the limit process Γ̃. But Γ̃(B(0, R)) ≥ 1 since Γ̃ϕn B(0, R) ≥ 1 for
every n ∈ N∗ . There is a contradiction because a stationary point process, having an half
plane without points, is necessary the process null. The lemma is proved. Now thanks to this lemma, we have that Γ̃ satisfies the assumption iii) P̃ -almost surely.
Moreover, P̃ -almost surely, we can say that every edge in Γ̃ is contained in two triangles.
So the convex hull of Γ̃, < Γ̃ > has no boundary. Therefore < Γ̃ > is equal to R2 or the
empty set. It implies Γ(1) is equal to 0 or has not half plane without points. To prove
i), it stays to show that Γ̃ has not almost surely four points on a same circle. It is a
direct consequence of the following lemma which claims that the probability measure µ(1)
is locally absolutely continuous with respect to π.
Lemma 3 µ(1) is locally absolutely continuous with respect to π.
Proof:
If we consider the equation (10) and the domination of En (x, γ) by the constant 2CM
we have for every bounded continuous function f , Cµ! (1) (f ) ≤ e2CM λ ⊗ µn(1) (f ). Let us
n
n goes to infinity we have Cµ! (1) (f ) ≤ eCM λ ⊗ µ(1) (f ). So, Cµ(1) is absolutely continuous
with respect λ ⊗ µ. We deduce thanks to a Theorem by Glotzl (Theorem 1 [10]) that µ(1)
is a Gibbs point process for some potential. That implies the local absolute continuity of
µ(1) with respect to π.
14
Thanks to this lemma we have many consequences for the process Γ̃. First of all, we finish
the proof that Γ̃ satisfies almost surely the property i). Secondly, we show that every
¯ Moreover, the property
X ∈ Γ̃ is in E because, in principle, we just know that X is in E.
iv) is also satisfied by Γ̃. Concerning the last property in the proposition 3, it is a direct
consequence of this following lemma.
Lemma 4 There exists a constant C2 > 0 such that for every n in N∗ and every Λ ∈ B(R2 )
we have
Eµn Γ(1) (Λ) ≥ C2 λ(Λ).
Proof:
It is exactly the same proof in the lemma 1.
The proof of the proposition 3 is finished. Let us Remark that we won’t be sure that µ Γ(1) ∈ MD (R2 ) = 1 until the end of
the paper, but the definition of Gibbs Delaunay tessellation implies this condition. In
fact, the good candidateto be the Gibbs Delaunay tessellation is the probability measure
µ := µ .Γ(1) ∈ MD (R2 ) . We will see it more precisely later.
Second step, the DLR equations:
The DLR equations are essentially based on the continuity of the kernels (ΞΛ ) . But,
it is easy to see that ΞΛ (., f ) is not continuous everywhere because the potential V is not
continuous and the Delaunay structure is not stable near a configuration having almost
four points on a same circle. To solve this problem, we prove only the continuity of the
kernels for µ-almost every Γ. The lemma 3 is crucial.
Let us give a precise definition of the set for which we have exactly the continuity of the
kernels. Let (∆m )m∈N∗ a set of bounded sets in R2 growing to R2 and having smooth
boundaries (∂∆m )n∈N∗ ; for example we can take ∆m = B(0, m). We define


a) Γ = 0 or Γ ∈ M∞


D (E)




b) ∀m ∈ N∗ , Γ(1) (∂∆m ) = 0
.
M = Γ ∈ M(E) such that

c) ∀m ∈ N∗ , any X ∈ Γ(2) is not tangential to ∂∆m 




d) V is continuous for every subconfiguration Γ′ in Γ
(14)
We note M∗ = M − {0}. Thanks to the proposition 3, the lemma 3 and the assumption
R) we claim
µ(M) = 1.
It is easy to see that for every m ∈ N∗ the function from MD (E) to M(E): Γ −→ Γ∆cm is
continuous at each Γ ∈ M∗ because they are no boundary problems thanks to the points
b) and c) . Let us remind that the topology on M(E) is induced by the metric dM(E) .
We deduce that the function from MD (E) × M(R2 ) to MD (E): (Γ, γ) −→ Υ(Γ, γ, ∆m )
is continuous for every m ∈ N∗ , every Γ ∈ M∗ and for π-almost every γ ∈ M(R2 ). Now,
thanks to the assumption d) in the definition of M, we affirm that, for every m ∈ N∗ , the
function from MD (E) × M(R2 ) to MD (E): (Γ, γ) −→ E∆m (Υ(Γ, γ, ∆m )) is continuous at
every Γ ∈ M∗ and π-almost every γ ∈ M(R2 ).
Now, we can see precisely the continuity of the kernels (Ξ∆m )n∈N∗ . Let us remind the
15
definition: for every Γ ∈ M∞
D (E) and every test function f from M(E) to R we define
Z
1
f Υ(Γ, γ, Λ)
ΞΛ (Γ, f ) =
e−EΛ Υ(Γ,γ,Λ) πSΛ (Γ) (dγ),
ZΛ (Γ)
M(R2 )
where ZΛ (Γ) is
ZΛ (Γ) =
Z
−EΛ Υ(Γ,γ,Λ)
e
M(R2 )
πSΛ (Γ) (dγ)
if SΛ (Γ) 6= ∅, and else
ΞΛ (Γ, f ) = f (Γ).
So to define the kernels on M, we have to extend the definition at 0. We put ΞΛ (0, .) = δ0
where 0 is the measure null on M(E). Now, let us give the principal lemma.
Lemma 5 For every m ∈ N∗ , for every bounded continuous function f from M(E) to R,
the function from MD (E) ∪ {O} to R: Γ −→ Ξ∆m (Γ, f ) is continuous at every Γ ∈ M.
Proof:
Let m be in N∗ and f a bounded continuous function from M(E) to R.
Let us begin by the the trivial case where Γ = 0. Let us (Γn ) a set of elements in MD (E)
(1)
such that Γn goes to 0 when n goes to infinity. Then, for n large enough, Γn has not points
in Λm . So, S∆m (Γn ) is empty. Therefore Ξ∆m (Γn , .) = δΓn and limn→∞ Ξ∆m (Γn , f ) =
Ξ∆m (0, f ) = f (0).
Now, let us suppose Γ ∈ MD (E) ∩ M∗ . Similarly, let us Γn a set of elements in MD (E)
such that Γn goes to Γ when n goes to infinity. If S∆m = ∅ the proof is the same like
above. Else, thanks to the dominated convergence Theorem by Lebesgue , we have
Z
−E∆m Υ(Γn ,γ,∆m )
e
πS∆m (Γn ) (dγ)
lim Z∆m (Γn ) = lim
n→∞
n→∞ M(R2 )
Z
e−E∆m Υ(Γ,γ,Λm ) πS∆m (Γ) (dγ)
=
M(R2 )
= Z∆m (Γ).
The uniform domination comes from uniform stability constants for the potential V (see
the beginning of the proof in the proposition 3). The simple convergence of the function
inside the integral comes from the continuity of the function E∆m (., ., ∆m ). With the same
argument, it is easy to see that
lim Ξ∆m (Γn , f ) = Ξ∆m (Γ, f ).
n→∞
The lemma is proved.
Now, let us give an equilibrium equation for µn . Since the measures (µn ) are the periodic
Gibbs Delaunay tessellations on Λn , we introduce the periodic equilibrium equations. We
note for every Γ ∈ MD (E), every Λ ∈ B(R2 )
ΓnΛc =
X
X∈Γ
∃Y ∈Γ(3) , X⊂Y
<Y >∩pn (Λ)c 6=∅
16
δX .
(15)
where pn (Λ) is the set ∪x∈Λ pn (x). Let us define the set SΛn (Γ) in R2
[
n
2
B̄(X)
SΛ (Γ) = R \
(16)
X∈(Γn
)(3)
Λc
and the function Υn
Υn : MD (E) × M(R2 ) × B(R2 ) −→ MD (E)
(1)
(Γ, γ, Λ)
−→
ΓS n (Γ)c + γSΛn (Γ) .
Λ
(1)
If ΓS n (Γ)c + γSΛn (Γ) is not in MD (R2 ), we put Υn (Γ, γ, Λ) = Γ.
Λ
∞
∞
Now we define the following kernels ΞnΛ on M∞
D (E) × P(MD (E)). For every Γ ∈ MD (E)
∞
and every test function f from MD (E) to R we define
Z
1
n
e−EΛ Υ (Γ,pn (γ),Λ) πSΛ (Γ)∩Λn (dγ),(17)
f Υn (Γ, pn (γ), Λ) n
ΞnΛ (Γ, f ) =
ZΛ (Γ)
M(R2 )
where ZΛn (Γ) is the normalization constant
Z
e−EΛ
ZΛn (Γ) =
Υn (Γ,pn (γ),Λ)
M(R2 )
πSΛ (Γ) (dγ).
The kernels (ΞnΛ )Λ∈B(R2 ) are just the periodic version of the kernels (ΞΛ )Λ∈B(R2 ) . Thanks to
these kernels, we can write the following equilibrium equation which is just the same result
that the compatibility of the kernels (ΞnΛ )Λ∈B(R2 ) (see proposition 2). For every bounded
set Λ in B(R2 ) and every bounded continuous function f from MD (E) to R we have
Z
Z
Z
f (Γ′ )ΞnΛ (Γ, dΓ′ )µn (dΓ).
(18)
f (Γ)µn (dΓ) =
MD (E)
MD (E)
MD (E)
So, let us prove the DLR equations (6) for µ with Λ = ∆m , m ∈ N∗
Z
Z
f (Γ)µn (dΓ)
f (Γ)µ(dΓ) = lim
n→∞ M (E)
MD (E)
D
Z
Z
f (Γ′ )Ξn∆m (Γ, dΓ′ )µn (dΓ)
= lim
n→∞ M (E) M (E)
D
Z
Z D
ΞnΛm (Γ, f ) − Ξ∆m (Γ, f ) µn (dΓ)
Ξ∆m (Γ, f )µn (dΓ) + lim
= lim
n→∞ M (E)
n→∞ M (E)
D
D
Z Z
n
Ξ∆m (Γ̃n , f ) − Ξ∆m (Γ̃n , f ) dP̃ (dΓ).
(19)
Ξ∆m (Γ, f )µ(dΓ) + lim
=
n→∞
MD (E)
In the last equality, the first part comes from the lemma 5 and the fact µ(M) = 1. The
second part comes from the representation of the weak convergence introduced just before
the lemma 2.
To prove that the second part of the last equality is null let us show that the function
Ξn∆m (Γ̃n , f )−Ξδm (Γ̃n , f ) goes to 0 almost surely, when n goes to infinity, and the dominated
convergence Theorem completes the proof. If Γ̃n goes to 0, the both kernels testing on f
goes to f (0); the difference goes to 0. Now if Γ̃n goes to Γ̃ 6= 0. Since f is continuous for
17
the metric dM(E) , f is the uniform limit of a set of local continuous functions. So, it is
enough to prove the convergence for a local function f . Since Γ̃n goes to Γ̃ 6= 0 when n
goes to infinity, then for n large enough the kernels Ξn∆m (Γ̃n , f ) and Ξ∆m (Γ̃n , f ) are equal;
the difference is null. So, the equality (19) becomes
Z
Z
Ξ∆m (Γ, f )µ(dΓ).
(20)
f (Γ)µ(dΓ) =
MD (E)
MD (E)
It is exactly the equation DLR (6) for Λ = ∆m . To have the equilibrium equations (6) for
any Λ, it is just a consequence of the DLR equation for some ∆m containing Λ and the
compatibility of kernels. Now let us remark that µ is not a Gibbs Delaunay tessellation
because µ(MD (E)) = µ(Γ 6= 0) may be not equal to 1. Thanks to the proposition 3,
we know that µ(Γ 6= 0) > 0. So, the Gibbs Delaunay tessellation is in fact the measure
µ = µ(.|Γ 6= 0). Let us verify that µ satisfies the DLR equations (6): from (20)
Z
f (Γ) µ(dΓ|Γ 6= 0)µ(Γ 6= 0) + µ(dΓ|Γ = 0)µ(Γ = 0)
=
MD (E)
Z
ΞΛ (Γ, f ) µ(dΓ|Γ 6= 0)µ(Γ 6= 0) + µ(dΓ|Γ = 0)µ(Γ = 0) .
MD (E)
If we expand, that becomes
µ(Γ 6= 0)
Z
µ(Γ 6= 0)
Z
µ(dΓ) + f (0)µ(Γ = 0) =
f (Γ)µ
MD (E)
µ(dΓ) + f (0)µ(Γ = 0)
ΞΛ (Γ, f )µ
MD (E)
and so
Z
µ(dΓ) =
f (Γ)µ
MD (E)
Z
µ(dΓ).
ΞΛ (Γ, f )µ
MD (E)
The theorem 1 is proved.
The proof of the theorem 1 is based on classical techniques for Gibbs point process. For
example, we use general results by Glotzl about the Campbell measure. Nevertheless, we
make a general proof of this theorem 1. Indeed, it bases on three fundamental lemmas
1, 3 and 4. In the case where V satisfies the assumptions (TI),(F),(S),(B) and (R)
(Theorem 1), it is possible to prove these lemmas with the classical techniques for points
processes. But if we change the assumptions on V , we have just to prove that these
lemmas are still true. It is what we will do in the next section but we will use completely
different techniques. They are not based on the classical point process tools which are not
usable. For example, the Campbell measure is inefficient in the case of hardcore geometric
interactions because we have not the decomposition (7) for the energy.
4
Existence of Gibbs tessellation with hardcore geometric
conditions
In this part we propose to introduce a geometric hardcore condition on the potential V . It
is a hardcore condition for small and large cells. In fact, we put the energy of a cell equals
18
to plus infinity if this cell is small or large. Let us make some remarks about the nature of
this interaction. The hardcore condition about the small triangle is classical because it is
almost equivalent to the hardcore condition between the points (hardball condition). But
the condition about the large cells is absolutely not a classical point hardcore condition.
It depends completely on the geometry of the tesselation. Moreover, it is not quasilocal
because the energy goes to infinity when the size of the cell goes to infinity.
So, let us now introduce precisely the hardcore interaction for the potential V .
(HC): The hardcore condition
There exists strictly positive constants r and R such that for all X in Sp (E)
V (X) = +∞ ⇐⇒
X is an edge with a length smaller or equal than r
or X is a triangle with the radius of B(X) greater or equal than R.
In the following theorem we suppose that V satisfies (HC) and (R). Let us remark
there is not contradiction. Indeed, (HC) implies that V is discontinuous for every X in
E (2) , having a length equals to r, or every X in E (3) with the radius of B(X) equals to
R. In fact, it is not incompatible with (R) because these configurations do not appear
π̄-almost surely.
Now let us give the theorem with an hardcore geometric interaction.
Theorem 2 There exists a Gibbs Delaunay tessellation for a potential V which satisfies
(TI),(HC),(B) and (R) with R > r.
Proof :
The scheme of the proof is exactly the same in the Theorem 1. We have just to prove that
the kernels (ΞΛ )Λ∈B(R2 ) are well defined and that the lemmas 1, 3 and 4 are still true in
the case where V satisfies (TI),(HC),(B) and (R).
Let us begin to justify that the kernels (ΞΛ )Λ∈B(R2 ) are well defined. The problem is to
prove that 0 < ZΛ (Γ) < +∞ for every Γ ∈ M∞
D (E) such that SΛ (Γ) 6= ∅. By construction,
SΛ is an opened set in R2 . Moreover the condition (HC) implies that the set M∞
D (E) is
locally an opened set in M (E). It means that for every configuration Γ ∈ M∞
(E)
we can
D
∞
perturb locally the configuration Γ such that this perturbation is still in MD (E). So it is
clear that ZΛ (Γ) > 0.
Now let us prove the second inequality. We use a classical stability argument. Let us
′
Γ′ ∈ MD (E). Either EΛ (Γ′ ) = +∞ and so e−EΛ (Γ ) = 0, or Γ′ ∈ M∞
D (E) and, thanks
to the assumption (HC), we prove that every triangle X in Γ′(3) has a radius of B(X)
smaller than R and has sides with length greater than r. It implies the existence of a
uniform minimal angle α0 > 0 for every triangle in Γ′(3) . With a same argument, in the
proof of the boundness of ZΛ (Γ) in the Theorem 1, we conclude there exists a constant C
such that EΛ (Γ′ ) < CΓ′(1) (Λ). The potential is stable and so ZΛ (Γ) < +∞.
Now let us remark that the lemmas 1, 4 are obviously true if (HC) is satisfies. The difficulty
in this part is essentially concentrated in the following lemma which is the analogue of the
lemma 3.
Lemma 6 Under the assumptions (TI),(R),(B) and (HC) with (R > r), µ(1) is locally
absolutely continuous with respect to π.
19
Proof:
We can not use in this case the beautiful argument, based on the Campbell measure
developed in the lemma 3, because the energy of a point in a configuration is not defined.
So we can show that the Campbell measure Cµ! (1) is not absolutely continuous with respect
to the measure λ ⊗ µ(1) . We propose another method.
Let Λ be a bounded closed set in R2 , let us directly show that the projection of µ(1) on
Λ is absolutely continuous with respect to πΛ . Let us remind that the support of µ is in
principle included in Ad(M∞
D (E)), where Ad means the adherence for the metric dM(E) .
We note for Λ′ ⊃ Λ
n
o
∅
(1)
AΛ = Γ ∈ Ad (M∞
D (E)) such that Γ (Λ) = 0
′
′
∞
AΛ
Λ = Γ ∈ Ad (MD (E)) such that ΓΛ′ is (Λ, Λ )-good
(3)
where ΓΛ′ is (Λ, Λ′ )- good means for every X ∈ ΓΛ′ , every x ∈ B̄(X) then B̄(x, R)∩Λ = ∅.
We have to give an extension of the definition of B̄(X) because in principle X is in Ē (3)
(X is not necessary affinely independent). Since Γ is in Ad(M∞
D (E)), Γ is a limit of set
(3) , (Γ ) → Γ denotes a set (Γ )
∞
(Γn ) in M∞
(E).
So,
for
each
X
∈
Γ
n
n n∈N∗ in MD (E) which
D
converges to Γ and Xn the element of Γn which converges to X ∈ Γ. We have the following
definition for B̄(X),
[
B̄(X) =
lim sup B̄(Xn ).
(Γn )→Γ
′
If Γ is in M∞
D (E), B̄(X) is clearly the closed circumscribed ball of X and so ΓΛ′ is (Λ, Λ )c
good if for every x ∈ SΛ′ (Γ) then B̄(x, R) ∩ Λ = ∅. We remark for example that for every
′
∅
Λ′
∞
Γ ∈ AΛ
Λ ∩ MD (E) then Λ ⊂ SΛ′ (Γ). Thanks to the lemma 2, it is clear that µ(AΛ ∪ AΛ )
goes to 1 when Λ′ growths up to R2 . So, it is sufficient to prove that µ(1) is localy absolutely
′
∅
2
continuous with respect to πΛ on the projection of AΛ and AΛ
Λ on M(R ). Let us remark
∅
that µ(1) is obviously localy absolutely continuous on the projection of AΛ . It stays to
′
prove that µ(1) is localy absolutely continuous on the projection of AΛ
Λ .′ Let us fΛ a FΛ measurable bounded continuous function from MD (R2 ) to R . Since AΛ
Λ is an opened set
in M(E) and since (µn ) converges weakly to µ (see proposition 3), we have
Z
Z
(21)
fΛ Γ(1) 1IAΛ′ (Γ)µn (dΓ).
fΛ Γ(1) 1IAΛ′ (Γ)µ(dΓ) ≤ lim sup
Λ
M(Ē )
n→∞
Λ
M(E)
For each n ∈ N∗ , µn is the solution of the periodic equilibrium equation (18);
Z
Z
(1)
fΛ Γ′(1) 1IAΛ′ (Γ′ )ΞnΛ′ (Γ, dΓ′ )µn (dΓ).
fΛ Γ 1IAΛ′ (Γ)µn (dΓ) =
M(E)
Λ
Λ
M(E)
(22)
Now we want to substitute ΞnΛ′ with ΞΛ′ ; using the same techniques at the equation (19),
we find
Z
Z
(1)
fΛ Γ′(1) 1IAΛ′ (Γ′ )ΞΛ′ (Γ, dΓ′ )µn (dΓ)
fΛ Γ 1IAΛ′ (Γ)µ(dΓ) ≤ lim sup
Λ
Λ
n→∞
M(E)
M(E)
Z
(23)
≤ lim sup fΛ Γ′(1) 1IAΛ′ (Γ′ )ΞΛ′ (Γ̃n , dΓ′ )dP̃ .
n→∞
Ω̃
20
Λ
We fix P̃ -a.s. (Γ̃n ) converging to Γ̃.
Z
fΛ Γ′(1) 1IAΛ′ (Γ′ )ΞΛ′ (Γ̃n , dΓ′ )dP̃
Λ
Ω̃
=
Z
−EΛ′ Υ(Γ̃n ,γ,Λ′ )
MD (R2 )
fΛ γ)1IAΛ′ (Γ̃n )
where
ZΛ′ (Γ̃n ) =
e
ZΛ′ (Γ̃n )
Λ
Z
e−EΛ′
Υ(Γ̃n ,γ,Λ′ )
MD (R2 )
πS
πS
Λ′ (Γ̃n )
Λ′ (Γ̃n )
(dγ)
(dγ).
Now let us introduce the both following sets included in M SΛ′ (Γ̃n )
o
n
KΛ′ (Γ̃n ) = γ ∈ M SΛ′ (Γ̃n ) such that EΛ′ Υ(Γ̃n , γ, Λ′ ) 6= +∞
and for ε > 0
Z
n
ε
KΛ′ (Γ̃n ) = γ ∈ KΛ′ (Γ̃n ) such that
(24)
(25)
o
′
′
1IKΛ′ (Γ̃n ) (γΛ′ −Λ +γ )πΛ∩SΛ′ (Γ̃n ) (dγ ) ≥ ε .
M Λ∩SΛ′ (Γ̃n )
′
Let us define the set AΛ
Λ (Γ̃n ) =
following majoration
(26)
′
.
We
have
the
γ ∈ SΛ′ (Γ̃n ) s.t. Υ(Γ̃n , γ, Λ′ ) ∈ AΛ
Λ
Z
−EΛ′ Υ(Γ̃n ,γ,Λ′ )
e
f γ)1IAΛ′ (Γ̃n )
πSΛ′ (Γ̃n )(dγ)
Λ
MD (R2 ) Λ
ZΛ′ (Γ̃n )
′
Z
e−EΛ′ Υ(Γ̃n ,γ,Λ )
fΛ γ)1IAΛ′ (Γ̃n )1IKε (Γ̃n ) (γ)
−
πSΛ′ (Γ̃n )(dγ)
′
Λ
Λ
2
′
ZΛ (Γ̃n )
MD (R )
′
Z
e−EΛ′ Υ(Γ̃n ,γ,Λ ) ≤
fΛ γ)1IAΛ′ (Γ̃n )
1 − 1IKε ′ (Γ̃n ) (γ) πSΛ′ (Γ̃n )(dγ)
Λ
Λ
ZΛ′ (Γ̃n )
MD (R2 ) ′
ε (Γ̃ )
πS ′ (Γ̃n ) KΛ′ (Γ̃n ) − KΛ
supγ∈K ′ (Γ̃n ) e−EΛ′ Υ(Γ̃n ,γ,Λ )
′
n
Λ
1IAΛ′ (Γ̃n ) Λ
≤ kfΛ k∞
′
Λ
−EΛ′ Υ(Γ̃n ,γ,Λ )
′
(
Γ̃
)
(K
π
n
inf γ∈K ′ (Γ̃n ) e
Λ
SΛ′ (Γ̃n )
Λ
ε (Γ̃ )
πS ′ (Γ̃n ) KΛ′ (Γ̃n ) − KΛ
′
n
≤ e2CΛ′ kfΛ k∞ 1IAΛ′ (Γ̃n ) Λ
(27)
Λ
πS ′ (Γ̃n ) (KΛ′ (Γ̃n )
Λ
where CΛ′ is the uniform bound of the function |EΛ′ Υ(Γ̃n , ., Λ′ ) | on KΛ′ (Γ̃n ), which exists
thanks to the assumptions (HC) and (B). CΛ′ does not depend on (Γ̃n ). To control the
last term in the inequality (27), we have the following lemma
Lemma 7 there exists a constant ρ > 0 such that for every ε > 0
ε (Γ̃ )
πS ′ (Γ̃n ) KΛ′ (Γ̃n ) − KΛ
′
n
Λ
≤ ρε.
1IAΛ′ (Γ̃n )
Λ
πS ′ (Γ̃n ) (KΛ′ (Γ̃n )
Λ
21
proof:
′
ε
is
Let us Γ̃n in AΛ
Λ and γ in KΛ′ (Γ̃n ) − KΛ′ (Γ̃n ). The problem of a such
configuration
′
′
that the points of γ inside Λ are almost frozen in the sens where πΛ γΛ γΛ′ −Λ + γΛ ∈
KΛ′ (Γ̃n ) ≤ ε. We want to construct a new configuration which is equal to γΛ′ −Λ plus a
ε (Γ̃ ). Let
second part γ̂ = δx̂1 + . . . + δx̂l , such that this configuration γΛ′ −Λ + γ̂ is in KΛ
′
n
us define a function from KΛ′ (Γ̃n ) to M(SΛ′ (Γ̃n ): γ −→ γ̂ = ŷ1 + . . . ŷl such that for all
(hi )1≤i≤l in B(0, 3δ )l we have
γΛ′ −Λ + δŷ1 +h1 + . . . + δŷl +hl is in KΛ′ (Γ̃n )
(28)
ε
′
where δ is equal to R−r
2 . Let us remark that, if (28) is satisfied, γΛ −Λ + γ̂ is in KΛ′ (Γ̃n )
for ε small enough. How to construct this configuration γ̂?
In fact, we construct γ̂ recursively. We fix γ̂ = 0 and we test if γ̂ satisfies the following
assumption H)
∀x ∈ SΛ′ (Γ̃n ), Υ Γ̃n , γΛ′ −Λ + γ̂, Λ′ (B(x, r + δ)) > 0.
If it is the case, we stop the recursive process. If it is not the case: we choose arbitrary a
point x̂ in SΛ′ (Γ̃n ), such that Υ Γ̃n , γΛ′ −Λ + γ̂, Λ′ (B(x, r +δ)) = 0, and we put γ̂ := γ̂ +δx̂ .
We test again the assumption H) for this new γ̂. if H) is true, we stop here else we choose
another point x̂′ like above and we put γ̂ := γ̂ + δx̂′ . We go on like this until the process
stops which is always the case since SΛ′ (Γ̃n ) is bounded and since the minimal distance
between two points in γ̂ is larger than r. To have that the application γ → γ̂ is measurable,
it is needed to chose the points x̂ with a measurable function. We can use, for example,
the lexicographic order in R2 .
Now let us prove that (γ, γ̂) satisfies (28). Let us write γ̂ = δŷ1 + . . . + δŷl and for all
(hi )1≤i≤l in B(0, α3 )l , we note γ ′ = γΛ′ −Λ + δŷ1 +h1 + . . . + δŷl +hl . For any distinct points x, y
in Υ(Γ̃n , γ ′ , Λ′ )(1) , the norm |x − y| > r because γ is in KΛ′ (Γ̃n ) and H). Moreover, since γ̂
satisfies H), any ball B(x, R) where x is in SΛ′ (Γ̃n ) contains some points of Υ(Γ̃n , γ ′ , Λ′ ). It
′
is the same if x is not in SΛ′ (Γ̃n ) because Γ̃n is in AΛ
Λ and γ is in KΛ′ (Γ̃n ). We deduce that
any triangle X in Υ(Γ̃n , γ, Λ′ ) has the radius of B(X) strictly lower than R. So Υ(Γ̃n , γ, Λ′ )
is in KΛ′ (Γ̃n ).
ε (Γ̃ ) .
Now we can begin the domination of πS ′ (Γ̃n ) KΛ′ (Γ̃n ) − KΛ
′
n
Λ
ε
πS ′ (Γ̃n ) KΛ′ (Γ̃n ) − KΛ
′ (Γ̃n )
Λ
−λ SΛ′ (Γ̃n )−Λ
= e
X
k0
k=0
Z
M(Λ)
1IK
1
l!
Z
ε
Λ′ (Γ̃n )−KΛ′ (Γ̃n )
...
Z
(SΛ′ (Γ̃n )−Λ)k
(δx1 + . . . + δxk
+ γ )πΛ (dγ ) dx1 . . . dxn ,
′
′
(29)
where k0 is a bound for the number of points of elements in KΛ′ (Γ̃n ). For (xi )1≤i≤k in
SΛ′ (Γ̃n ) − Λ and γ ′ in M(Λ) such that η = δx1 + . . . + δxk + γ ′ in KΛ′ (Γ̃n ), we note
η̂ = δŷ1 + . . . + δŷl . Let us remark that (ŷi )1≤i≤l depends only on x1 , . . . , xk . So, we note
K the set of elements γ in M(SΛ′ (Γ̃n ) − Λ) such that there exists γ ′ ∈ M(Λ) for which
22
ε (Γ̃ ) the following domination
γ + γ ′ is in KΛ′ (Γ̃n ). We have from the definition of KΛ
′
n
Z
1IK ′ (Γ̃n )−Kε (Γ̃n ) (δx1 + . . . + δxk + γ ′ )πΛ (dγ ′ )
M(Λ)
Λ
Λ′
≤ ε1IK (δx1 + . . . + δxk )1IK ′ (Γ̃n ) (δx1 + . . . + δxk + δŷ1 + . . . + δŷl ),
Λ
Z
α −l Z
2
≤ ε1IK (δx1 + . . . + δxk ) π
...
3
B(ŷ1 , α )
B(ŷl , α )
3
3
1IK ′ (Γ̃n ) (δx1 + . . . + δxk + δy1 + . . . + δyl )dy1 . . . dyl
Λ
Z
Z
k0
α −k0 X
2
≤ ε π
1IK ′ (Γ̃n ) (δx1 + . . . + δxk + δy1 + . . . + δyl )dy1 . . . dyl .
...
Λ
3
(SΛ′ (Γ̃n ))l
l=0
(30)
Compiling (29) and (30), we find
ε
πS ′ (Γ̃n ) KΛ′ (Γ̃n ) − KΛ
′ (Γ̃n )
Λ
α −k0
2
≤ ε π
e−λ
3
1IK
SΛ′ (Γ̃n )−Λ
X
k0 X
k0
l=0 k=0
1
k!
Z
...
Z
(SΛ′ (Γ̃n ))k+l
(δx1 + . . . + δxk + δy1 + . . . + δyl )dx1 . . . dxk dy1 . . . dyl
≤ ρεπS ′ (Γ̃n ) (KΛ′ (Γ̃n ) ,
Λ′ (Γ̃n )
Λ
(31)
where ρ is a strictly positive constant. The lemma 7 is proved.
Thanks to the lemma 7 and the inequality (27), the equality (24) becomes
Z
fΛ Γ′(1) 1IAΛ′ (Γ′ )dP ΞΛ′ (Γ̃n , dΓ′ )
Λ
Ω̃
′
Z
e−EΛ′ Υ(Γ̃n ,γ,Λ )
πSΛ′ (Γ̃n )(dγ) + ρe2CΛ′ kfΛ k∞ ε.
fΛ γ)1IAΛ′ (Γ̃n )1IKε (Γ̃n ) (γ)
≤
′
Λ
Λ
2
′
(
Γ̃
)
Z
MD (R )
n
Λ
(32)
′
For every Γ̃n ∈ AΛ
Λ , we have Λ ⊂ SΛ′ (Γ̃n ) and πSΛ′ (Γ̃n ) = πΛ πSΛ′ (Γ̃n ). So
Z
fΛ Γ′(1) 1IAΛ′ (Γ′ )ΞΛ′ (Γ̃n , dΓ′ )dP̃
Λ
ZΩ̃
(1)
fΛ γ)1IAΛ′ (Γ̃n )1IKε ′ (Γ) (γΛ + ΓS ′ (Γ)−Λ )
≤
Λ
Λ
Λ
2
MD (R )
(1)
−E ′ Υ(Γ,γΛ +ΓS (Γ)−Λ ,Λ′ )
Λ′
e Λ
πΛ (dγΛ )ΞΛ′ (Γ̃n , dΓ) + ρe2CΛ′ kfΛ k∞ ε.
(1)
′
R
−EΛ′ Υ(Γ,γΛ +ΓS (Γ)−Λ ,Λ′ )
Λ′
πΛ (dγΛ′ )
MD (R2 ) e
When
(1)
γΛ +ΓS ′ (Γ)−Λ
Λ
(1)
is in
′ +Γ
R
,Λ′ )
−EΛ′ Υ(Γ,γΛ
S ′ (Γ)−Λ
ε (Γ) then
Λ
KΛ
′
MD (R2 ) e
23
πΛ (dγ ′ ) ≥ εe−CΛ′ .
So,
Z
≤
Z
Ω̃
fΛ Γ′(1) 1IAΛ′ (Γ̃n )ΞΛ′ (Γ̃n , dΓ′ )dP̃
MD (R2 )
Λ
fΛ γ)
e2CΛ′
πΛ (dγ) + ρe2CΛ′ kfΛ k∞ ε
ε
(33)
From (33) and (23), we deduce
Z
M(E)
fΛ Γ(1) 1IAΛ′ (Γ)µ(dΓ) ≤
Λ
Z
MD (R2 )
fΛ γ)
e2CΛ′
πΛ (dγ) + ρe2CΛ′ kfΛ k∞ ε.
ε
Thank to a class monotone argument, for every A in the σ-fields of M(Λ), we have
µ(1) (A) ≤ e2CΛ′ πΛε(A) + ρε . If πΛ (A) = 0 and if ε goes to 0, we have µ(A) = 0. So, µ is
locally absolutely continuous with respect to πΛ . The lemma 6 is proved.
To finish the proof of the Theorem 2, we have just to justify that µ(M) = 1, where M is dec
fined in (14). Because in the case with hardcore, we have to show that µ(M∞
D (E) −{0}) =
0. But µ is the limit of (µn ) where the support of each µn is in M∞
D (E), so the support of
∞ (E), where ∂M∞ (E) is the boundary of M∞ (E) for the
µ is in principle M∞
(E)
∪
∂M
D
D
D
D
metric dM(E) . Thanks to (HC), it is clear that π̄(∂M∞
D (E) − {0}) = 0. The Theorem 2 is
proved.
5
Remarks and others models
The Gibbs Delaunay tessellations, which we construct in the previous section, have not
(with probability one) small and large cells. It is an interesting result for the stochastic
geometry. Indeed, in [15], the author proposes a connection between the Gibbs point
process and the Delaunay Tessellations to control the size of cells. It is exactly what we
do. We construct more regular infinite Delaunay tessellations.
Now if we want to construct an infinite random Delaunay tessellation for which each angle
is bigger than a fix angle α0 . We propose two approaches. In the first, we can consider an
interaction potential which equals to plus infinity if and only if a triangle has a minimal
angle lower than α0 . It is possible to construct a Gibbs Delaunay tessellation with this
potential but the calculus and the proof are complicated and we have only the result for
α0 small enough. In a second easier approach, we can use the Theorem 2 and fix r and R
such that any triangle, with the edges larger than r and the radius of the circumscribed
ball lower than R, has the minimal angle necessary bigger than α0 . In the Theorem 2, the
assumptions on r and R are only R > r; so, the only condition on α0 is α0 < π6 .
Now, to finish this paper, we would like to say that it is possible to extend our results for
many different models of tessellation. but it is difficult to give general results about Gibbs
tessellation because the interaction depends strongly on the nature of the tessellation and
the interaction. it is needed to adapt the results at the model and the interaction.
24
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26