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Journal of Statistical Physics (2020) 179:972–996https://doi.org/10.1007/s10955-020-02559-3
Marked Gibbs Point Processes with Unbounded Interaction:An Existence Result
Sylvie Rœlly1 · Alexander Zass1
Received: 4 December 2019 / Accepted: 27 April 2020 / Published online: 22 May 2020© The Author(s) 2020
AbstractWe construct marked Gibbs point processes in R
d under quite general assumptions. Firstly,we allow for interaction functionals thatmay be unbounded andwhose range is not assumed tobe uniformly bounded. Indeed, our typical interaction admits an a.s. finite but random range.Secondly, the random marks—attached to the locations in R
d—belong to a general normedspace S . They are not bounded, but their law should admit a super-exponential moment.The approach used here relies on the so-called entropy method and large-deviation tools inorder to prove tightness of a family of finite-volume Gibbs point processes. An applicationto infinite-dimensional interacting diffusions is also presented.
Keywords Marked Gibbs process · Infinite-dimensional interacting diffusion · Specificentropy · DLR equation
Mathematics Subject Classification 60K35 · 60H10 · 60G55 · 60G60 · 82B21 · 82C22
1 Introduction
In this paper we construct a certain class of continuous marked Gibbs point processes. Recallthat a marked point consists of a pair: a location x ∈ R
d , d ≥ 1, and a mark m belongingto a general normed spaceS . The interactions we consider here are described by an energyfunctional H , which acts both on locations and on marks. This includes, in particular, thecase of k-body potentials, but is indeed a more general framework, useful to treat examples
Communicated by Aernout van Enter.
This work has been partially funded by Deutsche Forschungsgemeinschaft(DFG)—SFB1294/1—318763901.
B Alexander [email protected]
Sylvie Rœ[email protected]
1 Institüt für Mathematik der Universität Potsdam, Karl-Liebknecht Str. 24-25, 14476 Potsdam, Germany
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Marked Gibbs Point Processes with Unbounded Interaction 973
coming from the field of stochastic geometry (as e.g. the area- or the Quermass-interactionmodel, see Example 1).
The novelty of the results presented in this paper is threefold.Firstly, we do not assume a specific form of the interaction—like pairwise or k-body—but only make assumptions (in Sect. 2.3) on the resulting energy functional H itself. Inparticular, we do not assume superstability of the interaction, but only rely on the stabilityassumptions (Hst ) and (Hloc.st ). In the field of stochastic geometry, in particular, many quitenatural energy functionals are stable but not superstable, like the Quermass-interactionmodelpresented in Example 1.Secondly, theGibbsian energy functional we consider has an unbounded range: it is finite, butrandom and not uniformly bounded—as opposed to models treated for example in [1] whichdeal with a bounded-range interaction; see Assumption (Hr ). For a very recent existenceproof in the case of infinite-range interaction (without marks) see [8]. Moreover, unlike thehyper-edge interactions presented in [7], we treat the case of interactions which are highlynon local: the range of the conditional energy on a bounded region of an infinite configuration(see Definition 3) requires knowledge of the whole configuration and cannot be determinedonly by a local restriction of the configuration.Lastly, we work with a mark reference distribution whose support is a priori unbounded butonly fulfils a super-exponential integrability condition (see Assumption (Hm)).
Let us mention recent works on the existence of marked Gibbs point processes for partic-ular models. In [5] D. Dereudre proves the existence of the Quermass-interaction processas a planar germ-grain model; we draw inspiration from his approach, presenting here anexistence result for more general processes, under weaker assumptions. In [3] and [1] theauthors treat the case of unbounded marks in R
d with finite-range energy functional whichis induced by a pairwise interaction.
The main thread of our approach is the reduction of the general marked point processto a germ-grain model, where two marked points (x1,m1), (x2,m2) ∈ R
d × S do notinteract as soon as the balls with centre xi and radius ‖mi‖, i = 1, 2, do not intersect. Theframework we work in requires the introduction of a notion of tempered configurations (seeSect. 2.2) in order to better control the support of the Gibbs measure we construct. In thisway, the size growth of the marks of far away points is bounded. In Sect. 3.3 we see thatthis procedure is justified by the fact that the constructed infinite-volume Gibbs measure isactually concentrated on tempered configurations.
The originality of our method to construct an infinite-volume measure consists in the useof the specific entropy as a tightness tool. This relies on the fact that the level sets of thespecific-entropy functional are relatively compact in the local convergence topology; seeSect. 3.2. This powerful topological property was first shown in the setting of marked pointprocesses by H.-O. Georgii and H. Zessin in [12]. Indeed, we prove in Proposition 1, usinglarge-deviation tools, that the entropy of some sequence of finite-volume Gibbs measuresis uniformly bounded. This sequence is therefore tight, and admits an accumulation point.Let us remark that the entropy tool relies mainly on stability assumptions of the energyH , without the need for superstability. The usual approach (see e.g. [21]), in fact, uses thesuperstability condition to precisely control the local density of points; in our framework,we do this thanks to an equi-integrability property, which holds on the entropy level sets (seeLemma 6). Furthermore, the stability notion we use here is weaker than the classic one ofRuelle, as it includes a term depending on the marks of the configuration. For more detailsand examples, see Sect. 2.3.
The last step of the proof consists in showing that this accumulationpoint satisfies theGibb-sian property. Since the interaction is not local and not bounded, this property is not inherited
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974 S. Rœlly, A. Zass
automatically from the finite-volume approximations, but instead requires an accurate anal-ysis, which is done in Sect. 3.4. In Sect. 4 we propose an application to infinite-dimensionalinteracting diffusions.
1.1 Point-Measure Formalism
The point configurations considered here live in the product state space E ..= Rd ×S , d ≥ 1,
where(S , ‖·‖) is a general normed space: each point location in R
d has an associated markbelonging to S . The location space R
d is endowed with the Euclidean norm |·|, and theassociated Borel σ -algebra B(Rd ); we denote by Bb(R
d) ⊂ B(Rd) the set of boundedBorel subsets of R
d . A set Λ belonging to Bb(Rd) will often be called a finite volume. We
denote by B(S ) the Borel σ -algebra on S .The set of point measures on E is denoted byM ; it consists of the integer-valued, σ -finite
measures γ on E :
M ..= {γ =
∑
i
δxi : xi = (xi ,mi ) ∈ Rd × S
}.
We endow M with the canonical σ -algebra generated by the family of local counting func-tions on M ,
γ =∑
i
δ(xi ,mi ) �→ Card({i : xi ∈ Λ, mi ∈ A}), Λ ∈ Bb(Rd), A ∈ B(S ).
We denote by o the zero point measure whose support is the empty set. Since, in theframework developed in this paper, we only consider simple point measures, we identifythem with the subset of their atoms:
γ ≡ {x1, . . . , xn, . . .
} ⊂ E .
For a point configuration γ ∈ M and a fixed set Λ ⊂ Rd , we denote by γΛ the restriction of
the point measure γ to the set Λ × S :
γΛ..= γ ∩ (Λ × S ) =
∑
{i : xi∈Λ}δ(xi ,mi ).
A functional is a measurable R ∪ {+∞}-valued map defined on M . We introduce specificnotations for some of them: the mass of a point measure γ is denoted by |γ |. It correspondsto the number of its atoms if γ is simple.We also denote by m the supremum of the size of the marks of a configuration:
m(γ ) ..= sup(x,m)∈γ
‖m‖, γ ∈ M .
The integral of a fixed function f : E → R under the measure γ ∈ M—when it exists—isdenoted by
〈γ, f 〉 ..=∫
f dγ =∑
x∈γ
f (x).
For a finite volume Δ, we call local or more precisely Δ-local, any functional F satisfying
F(γ ) = F(γΔ), γ ∈ M .
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Marked Gibbs Point Processes with Unbounded Interaction 975
We also define the set of finite point measures on E :
M f..= {
γ ∈ M : |γ | < +∞}.
Moreover, for any bounded subset Λ ⊂ Rd ,MΛ is the subset ofM f consisting of the point
measures whose support is included in Λ × S :
MΛ..= {
γ ∈ M : γ = γΛ
} ⊂ M f .
Let P(M ) denote the set of probability measures on M .We write N
∗ for the set of non-zero natural numbers N \ {0}. The open ball in Rd centred in
y ∈ Rd with radius r ∈ R+ is denoted by B(y, r).
2 Gibbsian Setting
2.1 Mark Reference Distribution
The mark associated to any point of a configuration is random. We assume that the referencemark distribution R on S is such that its image under the map m �→ ‖m‖ is a probabilitymeasure ρ on R+ that admits a super-exponential moment, in the following sense:
(Hm) There exits δ > 0 such that∫
R+e�d+2δ
ρ(d�) < +∞. (1)
Throughout Sects. 2 and 3 of the paper, the parameter δ is fixed.
Remark The probability measure ρ is the density of a positive random variable X such that
X2d +ε is subgaussian for some ε > 0 (see e.g. [13,16]).
2.2 Tempered Configurations
We introduce the concept of tempered configuration. For such a configuration γ , the numberof its points in any finite volume Λ, |γΛ|, should grow sublinearly w.r.t. the volume, while itsmarks should grow as a fraction of it. More precisely, we define the spaceM temp of temperedconfigurations as the following increasing union
M temp ..=⋃
t∈N
M t,
where
M t = {γ ∈ M : ∀l ∈ N
∗, 〈γB(0,l), f 〉 ≤ t ld for f (x,m) ..= 1 + ‖m‖d+δ}. (2)
We now prove some properties satisfied by tempered configurations.
Lemma 1 The mark associated to a point in a tempered configuration is asymptoticallynegligible with respect to the norm of the said point: any tempered configuration γ ∈ M temp
satisfies
liml→+∞
1
lm(γB(0,l)) = 0.
123
976 S. Rœlly, A. Zass
Fig. 1 For (x,m) ∈ γ ∈ M t,t ≥ 1, such that |x | ≥ 2 l(t) + 1,B(x, ‖m‖) does not intersectB
(0, l(t)
)
Proof Let γ ∈ M t, t ≥ 1. From (2), recalling that m(γ ) = sup(x,m)∈γ
‖m‖, we get that, for alll ≥ 1,
m(γB(0,l)) ≤ (tld
)1/d + δ = (tld)1/d + δ
ll.
Define, for any η ∈ (0, 1),
l1(t, η) ..=( t
ηd+δ
)1/δ
. (3)
Then, if l ≥ l1(t, η),
m(γB(0,l))
l≤ (tld)1/d + δ
l≤ η ∈ (0, 1), (4)
and the Lemma is proved. ��Lemma 2 Let γ ∈ M t, t ≥ 1, and define l(t) ..= 1
2 l1(t,12 ), where l1 is defined by (3). Then,
for all l ≥ l(t), the following implication holds:
x = (x,m) ∈ γB(0,2l+1)c �⇒ B(x, ‖m‖) ∩ B(0, l) = ∅.
Proof Let γ ∈ M t and (x,m) ∈ γ such that |x | ≥ 2l + 1.
By definition of l1(t, 12 ), since (x,m) ∈ γB(0,�x�),
|x | − ‖m‖ (4)≥ |x | − 12�|x |� ≥ 1
2 |x | − 12 ≥ l.
��The assertion of Lemma 2 is illustrated in Fig. 1. Define the germ-grain set à of a
configuration γ as usual by
Γ ..=⋃
(x,m)∈γ
B(x, ‖m‖) ⊂ Rd ,
where the point x is the germ and the ball B(0, ‖m‖) is the grain. Lemma 2 then impliesthat, for tempered configurations, only a finite number of balls of their germ-grain set canintersect a fixed bounded subset of R
d . This remark will be very useful when defining therange of the interaction in (9).
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Marked Gibbs Point Processes with Unbounded Interaction 977
2.3 Energy Functionals and Finite-Volume Gibbs Measures
For a fixed finite volume Λ ⊂ Rd , we consider, as reference marked point process, the
Poisson point process π zΛ on E with intensity measure zdxΛ ⊗ R(dm). The coefficient z
is a positive real number, dxΛ is the Lebesgue measure on Λ, and the probability measureR on S was introduced in Sect. 2.1. In this toy model, since the spatial component of theintensity measure is diffuse, the configurations are a.s. simple. Moreover, the random marksof different points of the configuration are independent random variables.
To model and quantify a possible interaction between the point locations and the marksof a configuration, one introduces the general notion of energy functional.
Definition 1 An energy functional H is a translation-invariant measurable functional on thespace of finite configurations
H : M f → R ∪ {+∞}.We use the convention H(o) = 0.
Configurations with infinite energy will be negligible with respect to Gibbs measures.
Definition 2 ForΛ ∈ Bb(Rd), the finite-volumeGibbsmeasurewith free boundary condition
is the probability measure PΛ on M defined by
PΛ(dγ ) ..= 1
ZΛ
e−H(γΛ) π zΛ(dγ ). (5)
The normalisation constant ZΛ is called partition function. We will see in Lemma 4 why thisquantity is well defined under the assumptions we work with.
Notice how π zΛ—and therefore PΛ—is actually concentrated on MΛ, the finite point
configurations with atoms in Λ.The measure π z
Λ extends naturally to an infinite-volume measure π z ; the question we
explore in this work is how to do the same for PΛ. The first step in order to define aninfinite-volume Gibbs measure is to be able to consider the energy of configurations withinfinitely many points. In order to do this, we approximate any (tempered) configuration γ
by a sequence of finite ones (γΛn )n . Using a terminology that goes back to Föllmer [9], weintroduce the following
Definition 3 For Λ ∈ Bb(Rd), the conditional energy of γ on Λ given its environment is
the functional HΛ defined, on the tempered configurations, as the following limit:
HΛ(γ ) = limn→∞
(H(γΛn ) − H(γΛn\Λ)
), γ ∈ M temp, (6)
whereΛn..= [−n, n)d is an increasing sequence of centred cubes of volume (2n)d , converg-
ing to Rd .
Remarks i. Notice that the conditional energy of finite configurations confined in Λ
coincides with their energy: HΛ(γΛ) ≡ H(γΛ). In general, however, the conditionalenergy HΛ(γ ) of an infinite configuration γ does not reduce to H(γΛ) because of thepossible interaction between (external) points of γΛc and (internal) points of γΛ. In otherwords, the conditional energy is possibly not a local functional. In this paper, we areinterested in this general framework.
123
978 S. Rœlly, A. Zass
ii. Indeed, we will work with energy functionals H for which the limit in (6) is stationary,i.e. reached for a finite n (that depends on γ ). Assumption (Hr ) below ensures thisproperty.iii. Since π z
Λ only charges configurations in Λ, PΛ can be equivalently defined as
PΛ(dγ ) = 1
ZΛ
e−HΛ(γ )π zΛ(dγ ).
The key property of such conditional energy functionals is the following additivity; the proofof this lemma is analogous to the one in [5], Lemma 2.4, that works in the more specificsetting of Quermass-interaction processes.
Lemma 3 The family of conditional energy functionals is additive, i.e. for any Λ ⊂ Δ ∈Bb(R
d), there exists a measurable function φΛ,Δ : M temp → R such that
HΔ(γ ) = HΛ(γ ) + φΛ,Δ(γΛc ), γ ∈ M temp. (7)
Let us nowdescribe the framework of our study, by considering for the energy functional Ha global stability assumption (Hst ), a range assumption (Hr ) and a locally-uniform stabilityassumption (Hloc.st ):
(Hst ) There exists a constant c ≥ 0 such that the following stability inequality holds
H(γ ) ≥ −c 〈γ, 1 + ‖m‖d+δ〉, γ ∈ M f . (8)
(Hr ) Fix Λ ∈ Bb(Rd). For any γ ∈ M t, t ≥ 1, there exists a positive finite number
r = r(γ,Λ) such that
HΛ(γ ) = H(γΛ⊕B(0,r)
) − H(γΛ⊕B(0,r))\Λ
), (9)
where Λ ⊕ B(0, r) ..= {x ∈ R
d : ∃y ∈ Λ, |y − x | ≤ r}. Equivalently, the limit in (6)
is already attained at the smallest n ≥ 1 such that Λn ⊃ Λ ⊕ B(0, r). Indeed, one canchoose
r(γ,Λ) = 2 l(t) + 2m(γΛ) + 1.
(Hloc.st ) Fix Λ ∈ Bb(Rd). For any t ≥ 1 there exists a constant c′ = c′(Λ, t) ≥ 0 such
that the following stability of the conditional energy holds, uniformly for all ξ ∈ M t:
HΛ(γΛξΛc ) ≥ −c′〈γΛ, 1 + ‖m‖d+δ〉, γΛ ∈ MΛ. (10)
Remarks i. Notice how the stability assumption (Hst ) is weaker than the usual Ruellestability H(γ ) ≥ −c|γ | = −c〈γ, 1〉, for the presence of the mark-dependent negative
term −c〈γ, ‖m‖d+δ〉.ii. Two points x = (x,m), y = (y, n) ∈ E of a configuration γ are not in interac-tion whenever B(x, ‖m‖) ∩ B(y, ‖n‖) = ∅, so that Assumption (9) has the followinginterpretation: there is no influence from the points of γ(Λ⊕B(0,r))c on the points of γΛ:
HΛ(γ ) = HΛ(γΛ⊕B(0,r)). Therefore, the range of the energy HΛ at the configuration
γ is smaller than r(γ,Λ), which is finite but random since it depends on γ . This rangemay not be uniformly bounded when γ varies.
Lemma 4 Under assumptions (Hst ) and (Hm) the partition function ZΛ is well defined, thatis, it is finite and positive.
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Marked Gibbs Point Processes with Unbounded Interaction 979
Proof
ZΛ ≥ π zΛ(o) = e−z|Λ| > 0;
ZΛ =∫
e−H(γΛ)π zΛ(dγ )
(8)≤∫
ec〈γΛ,1+‖m‖d+δ〉π zΛ(dγ )
≤ e−z|Λ| exp{ecz|Λ|
∫
R+ec�
d+δ
ρ(d�)} (1)
< +∞.
��
We provide here examples of energy functionals on marked configurations, which satisfythe assumptions above. Sect. 4 provides, in the context of interacting diffusions, a furtherexample of a pair interaction that acts on both locations and marks of a configuration, wherethe mark space is a path space.
Example 1 (Geometric multi-body interaction in R2)
Consider the marked-point state space E = R2 × R+, and recall that, to any finite configu-
ration γ = {(x1,m1), . . . , (xN ,mN )}, N ≥ 1, one can associate the germ-grain set
Γ =N⋃
i=1
B(xi ,mi ) ⊂ R2.
Consider, as reference mark measure, a measure R on R+ satisfying (Hm), that is, thereexists δ > 0 such that
∫
R+e�2+2δ
R(d�) < +∞.
The Quermass energy functional HQ (see [15]) is defined as any linear combination of area,perimeter, and Euler-Poincaré characteristic functionals:
HQ(γ ) = α1 Area(Γ ) + α2 Per(Γ ) + α3χ(Γ ), α1, α2, α3 ∈ R.
Notice how this interaction, depending on the values of the parameters αi , can be attractiveor repulsive. It is difficult (and not useful) to decompose this multi-body energy functionalas the sum of several k-body interactions. The functional HQ satisfies assumptions (Hst ),(Hr ), and (Hloc.st ). Indeed, it even satisfies the following stronger conditions:
• There exists a constant c such that, for any finite configuration γ ,
|HQ(γ )| ≤ c〈γ, 1 + ‖m‖2〉; (two-sided stability)
• For anyΛ ∈ Bb(R2) and t ≥ 1, there exists c′(Λ, t) such that, for any γ ∈ M , ξ ∈ M t,
|HQΛ (γΛξΛc )| ≤ c′(Λ, t)〈γΛ, 1 + ‖m‖2〉. (two-sided loc. stability)
Under these stronger conditions than ours, the existence for the Quermass-interaction modelwas proved in [5]; notice that HQ is not superstable.
For more examples of geometric interactions, see [6].
123
980 S. Rœlly, A. Zass
Example 2 (Two-body interactions)
(i) Interacting hard spheres of random radii On E = Rd × R+, consider a model of
hard balls centred at points xi , of random radii mi distributed according to a measure Rsatisfying Assumption (Hm). The hard-core energy functional of a finite configurationγ = {(x1,m1), . . . (xN ,mN )}, N ≥ 1, is given by
H(γ ) =∑
1≤i< j≤N
(+∞) 1{B(xi ,mi )∩B(x j ,m j )�=∅},
with the convention +∞ · 0 = 0.(ii) Non-negative pair interaction On E = R
d × S , consider any energy functional Hof the form H(γ ) = ∑
1≤i< j≤N Φ(xi , x j ), where
Φ(xi , x j ) = φ(|xi − x j |) 1{|xi−x j |≤‖mi‖+‖m j‖},
where φ is non-negative and null at 0.
In both cases, since H is a non-negative functional, it satisfies (Hst ) and (Hloc.st ). It is alsoeasy to see that, by construction, the range assumption (Hr ) also holds.
2.4 Local Topology
We endow the space of point measures with the topology of local convergence (see [11],[12]), defined as the weak* topology induced by a class of functionals onM which we nowintroduce.
Definition 4 A functional F is called tame if there exists a constant c > 0 such that
|F(γ )| ≤ c(1 + 〈γ, 1 + ‖m‖d+δ〉), γ ∈ M .
We denote by L the set of all tame and local functionals. The topology τL of localconvergence onP(M ) is then defined as the weak* topology induced byL , i.e. the smallesttopology on P(M ) under which all the mappings P �→ ∫
F dP , F ∈ L , are continuous.
3 Construction of an Infinite-Volume GibbsMeasure
Let us first precise the terminology (see [10]).
Definition 5 Let H be an energy functional satisfying the three assumptions (Hst ), (Hr ), and(Hloc.st ). We say that a probability measure P on M is an infinite-volume Gibbs measurewith energy functional H if, for every finite volume Λ ⊂ R
d and for any measurable,bounded and local functional F : M → R, the following identity (calledDLR equation afterDobrushin–Lanford–Ruelle) holds under P:
∫
MF(γ ) P(dγ ) =
∫
M
∫
MΛ
F(γΛξΛc ) ΞΛ(ξ, dγ ) P(dξ), (DLR)Λ
where ΞΛ, called the Gibbsian probability kernel associated to H , is defined on MΛ by
ΞΛ(ξ, dγ ) ..= e−HΛ(γΛξΛc )
ZΛ(ξ)π z
Λ(dγ ), (11)
where ZΛ(ξ) ..= ∫MΛ
e−HΛ(γΛξΛc )π zΛ(dγ ).
123
Marked Gibbs Point Processes with Unbounded Interaction 981
Remarks i. The probability kernel ΞΛ(ξ, ·) is not necessarily well-defined for any ξ ∈ M .In Lemma 7, we will show that this is the case when we restrict it to the subspaceM temp.
ii. The map ξ �→ ΞΛ(ξ, dγ ) is a priori not local since ξ �→ HΛ(γΛξΛc ) may depend onthe full configuration ξΛc .
iii. The renormalisation factor ZΛ(ξ)–when it exists—only depends on the external config-uration ξΛc . Therefore ΞΛ(ξ, ·) ≡ ΞΛ(ξΛc , ·).
We can now state the main result of this paper:
Theorem 1 Under assumptions (Hm), (Hst ), (Hr ), and (Hloc.st ), there exists at least oneinfinite-volume Gibbs measure with energy functional H.
This section will have the following structure.
3.1 We define a sequence of stationarised finite-volume Gibbs measures (P̄n)n .3.2 We use uniform bounds on the entropy to show the convergence, up to a subsequence,
to an infinite-volume measure P̄ .3.3 We prove, using an ergodic property, that P̄ carries only the space of tempered config-
urations.3.4 Noticing that, for any fixed Λ ∈ Bb(R
d), P̄n does not satisfy (DLR)Λ, we introducea new sequence (P̂n)n asymptotically equivalent to (P̄n)n but satisfying (DLR)Λ. We
use appropriate approximations, by localising the interaction, to show that also P̄ satis-fies (DLR)Λ.
3.1 A Stationarised Sequence
In this subsection,we extend each finite-volumemeasure Pn ..= PΛn ,Λn = [−n, n)d , definedonMΛn to a probability measure P̄n on the full spaceM , invariant under lattice-translations.
We start with the following
Lemma 5 There exists a constant a1 such that
∀n ≥ 1, Jn ..=∫
M〈γ, 1 + ‖m‖d+δ 〉Pn(dγ ) ≤ a1|Λn |. (12)
Proof We partition the space of configurations MΛn in three sets:
M(1)Λn
..= {γ ∈ MΛn : 〈γ, 1 + ‖m‖d+δ〉 ≤ a11|Λn |},M
(2)Λn
..= {γ ∈ MΛn : 〈γ, 1 + ‖m‖d+δ〉 > a11|Λn |, |γ | > a12|Λn |},M
(3)Λn
..= {γ ∈ MΛn : 〈γ, 1 + ‖m‖d+δ〉 > a11|Λn |, |γ | ≤ a12|Λn |},for some constants a11, a12 which will be fixed later. Therefore, the integral Jn can be writtenas the sum of three integrals, J (1)
n , J (2)n , J (3)
n , resp. over each of these sets.The first term is straightforward:
J (1)n
..=∫
M(1)Λn
〈γΛn , 1 + ‖m‖d+δ〉Pn(dγ ) ≤ a11|Λn |.
123
982 S. Rœlly, A. Zass
For the second term,
J (2)n
..=∫
M(2)Λn
〈γΛn , 1 + ‖m‖d+δ〉Pn(dγ )
(5)≤∫
MΛn
1{|γΛn |>a12|Λn |}〈γ, 1 + ‖m‖d+δ〉 1
ZΛn
e−H(γΛn )π zΛn
(dγ )
(8)≤ e−z|Λn |
ZΛn
+∞∑
k=a12|Λn |
(z|Λn |)kk!
∫
S kec
∑ki=1(1+‖mi‖d+δ)
k∑
j=1
(1 + ‖m j‖d+δ)R(dm1) . . . R(dmk)
≤+∞∑
k=a12|Λn |
(z|Λn |)kk! k
(∫(1 + �d+δ)ec(1+�d+δ)ρ(d�)
) (∫ec(1+�d+δ)ρ(d�)
)k−1
.
Using (1), we are able to find a constant b1 such that∫
(1 + �d+δ) ec(1+�d+δ)ρ(d�) ≤ b1.
We then get
J (2)n ≤
+∞∑
k=a12|Λn |
(zb1|Λn |)kk! k ≤
+∞∑
k=a12|Λn |
(2zb1|Λn |)kk! ≤ e2zb1|Λn |P
(S|Λn | ≥ a12|Λn |
),
for a sequence (Sm)m≥1 of Poisson random variables with parameter 2zb1m.Recalling the Cramér-Chernoff inequality (cf. [2])
P( 1m Sm ≥ a12) ≤ e−mL∗(a12),
where L∗(x) = 2zb1 + x log x − x(1 + log(2zb1)) is the Legendre transform associatedto the Poisson random variable of parameter 2zb1, we can choose a12 large enough, so thatlog a12 ≥ 1 + log(2zb1). Thus L∗(a12) ≥ 2zb1, and we get that J (2)
n ≤ 1.For the third term,
J (3)n
(5)=∫
MΛn
1{〈γ,1+‖m‖d+δ〉>a11|Λn |, |γ |≤a12|Λn |}〈γ, 1 + ‖m‖d+δ〉 1
ZΛn
e−H(γΛn )π zΛn
(dγ )
≤ e−z|Λn |
ZΛn
a12|Λn |∑
k=0
(z|Λn |)kk!
∫
S k1{∑k
i=1(1+‖mi‖d+δ)>a11|Λn |}
ec∑k
i=1(1+‖mi‖d+δ)k∑
j=1
(1 + ‖m j‖d+δ)R(dm1) . . . R(dmk)
≤a12|Λn |∑
k=0
(z|Λn |)kk!
∫
Rk+1{∑k
i=1(1+�d+δi )>a11|Λn |}
ec∑k
i=1(1+�d+δi )
k∑
j=1
(1 + �d+δj )ρ(d�1) . . . ρ(d�k).
123
Marked Gibbs Point Processes with Unbounded Interaction 983
Applying the Cauchy-Schwarz inequality, we find:
J (3)n ≤
a12|Λn |∑
k=0
(z|Λn |)kk!
√√√√ρ⊗k
( k∑
i=1
(1 + �d+δ
i
)> a11|Λn |
)
√√√√
∫
Rk+e2c
∑ki=1(1+�d+δ
i )( k∑
j=1
(1 + �d+δj )
)2ρ(d�1) . . . ρ(d�k)
≤√√√√ρ⊗a12|Λn |
( a12|Λn |∑
i=1
(1 + �d+δ
i
)> a11|Λn |
)
a12|Λn |∑
k=0
(z|Λn |)kk!
√
k2∫
(1 + �d+δ)2e2c(1+�d+δ)ρ(d�)
(∫e2c(1+�d )ρ(d�)
)k−1
.
Using (1), there exists a positive constant b2 such that∫
(1 + �d+δ)2e2c(1+�d+δ)ρ(d�) ≤ b2.
Thus
J (3)n ≤
√√√√ρ⊗a12|Λn |( a12|Λn |∑
i=1
(1 + �d+δ
i
)> a11|Λn |
) a11|Λn |∑
k=0
(z√b2|Λn |)kk! k
≤√√√√ρ⊗a12|Λn |
( a12|Λn |∑
i=1
(1 + �d+δ
i
)> a11|Λn |
)e2z
√b2|Λn |.
Using again theCramér-Chernoff inequality,we can choosea11 large enough such that L̄∗, theLegendre transform of the image measure of ρ by � �→ 1+ �d+δ , satisfies L̄∗(a11) ≥ 4z
√b2
(since it is stricly increasing on the positive half-line). Thus
ρ⊗a12|Λn |( a12|Λn |∑
i=1
(1 + �d+δ
i
)> a11|Λn |
)≤ e−4z
√b2|Λn |,
which yields J (3)n ≤ 1.
Putting it all together, the claim of Lemma 5 follows with a1 ..= a11 + 2. ��We start by considering the probability measure P̃n onM , under which the configurations
in the disjoint blocks Λκn
..= Λn + 2nκ , κ ∈ Zd , are independent, with identical distribution
Pn . We then build the empirical field associated to the probability measure P̃n , i.e. thesequence of lattice-stationarised probability measures
P̄n = 1
(2n)d
∑
κ∈Λn∩Zd
P̃n ◦ ϑ−1κ , (13)
where ϑκ is the translation on Rd by the vector κ ∈ Z
d .
Remarks i. As usual we identify the translation ϑκ onRd with the image of a point measure
under such translation.
123
984 S. Rœlly, A. Zass
ii. So constructed, the probability measure P̄n is invariant under (ϑκ)κ∈Zd .iii. An upper bound similar to (12) holds also under P̄n :
∃a2 > 0,∀n ≥ 1,∫
M〈γΛn , 1 + ‖m‖d+δ 〉P̄n(dγ ) ≤ a2 |Λn |.
Moreover, using stationarity and the fact that the covering Λn = ⋃κ Λκ
1 contains nd
terms,∫
M〈γΛ1 , 1 + ‖m‖d+δ 〉P̄n(dγ ) =
∫
M
1
nd∑
κ
〈γΛκ1, 1 + ‖m‖d+δ 〉P̄n(dγ )
= 1
nd
∫
M〈γΛn , 1 + ‖m‖d+δ〉P̄n(dγ )
≤ 1
nd(2n)da2 = 2da2.
(14)
As we will see in the following subsection, in order to prove that (P̄n)n admits an accu-mulation point, it is enough to prove that all elements of the sequence belong to the sameentropy level set.
3.2 Entropy Bounds
Let us now introduce the main tool of our study, the specific entropy of a (stationary) proba-bility measure on M .
Definition 6 Given two probability measures Q and Q′ on M , and any finite-volume Λ ⊂Rd , the relative entropy of Q′ with respect to Q on Λ is defined as
IΛ(Q|Q′) ..={∫
log f dQΛ if QΛ � Q′Λ with f ..= dQΛ
dQ′Λ
,
+∞ otherwise,
where QΛ (resp. Q′Λ) is the image of Q (resp. Q′) under the mapping γ �→ γΛ.
As usual,
Definition 7 The specific entropy of Q with respect to Q′ is defined by
I (Q|Q′) = limn→+∞
1
|Λn | IΛn (Q|Q′).
From now on, the reference measure Q′ will be the marked Poisson point process π z withintensity measure z dx ⊗ R(dm). In this case, the specific entropy of a probability measureQ with respect to π z is always well defined if Q is stationary under the lattice translations(ϑκ)κ∈Zd . Moreover, recall that for any a > 0, the a-entropy level set
P(M )≤a..=
{Q ∈ P(M ), stationary under (ϑκ)κ∈Zd : I (Q|π z) ≤ a
}
is relatively compact for the topology τL , as proved in [12].
Proposition 1 There exists a constant a3 > 0 such that,
∀n ≥ 1, I (P̄n |π z) ≤ a3
where P̄n ∈ P(M ) is the empirical field defined by (13).
123
Marked Gibbs Point Processes with Unbounded Interaction 985
Proof Since the map Q �→ I (Q|π z) is affine, it holds
I (P̄n |π z) = 1
(2n)d
∑
κ∈Zd∩Λn
I (P̃n ◦ ϑ−1κ |π z)
= I (P̃n |π z) = limm→+∞
1
|2mΛn | I2mΛn (P̃n |π z)
= limm→+∞
1
(2m)d |Λn | (2m)d IΛn (Pn |π z) = 1
|Λn | IΛn (Pn |π z).
Using the stability of the energy functional, we find
IΛn (Pn |π z) = −∫
H(γ )Pn(dγ ) − log(ZΛn )(8)≤ c
∫〈γ, 1 + ‖m‖d+δ〉Pn(dγ ) + z|Λn |.
From Lemma 5, we know that inequality (12) holds. Defining a3 ..= ca1 + z, we concludethat, uniformly in n ≥ 1, I (P̄n |π z) ≤ a3. ��
From the above proposition we deduce that the sequence (P̄n)n≥1 belongs to the relativelycompact set P(M )≤a3 . It then admits at least one converging subsequence which we willstill denote by (P̄n)n≥1 for simplicity. The limit measure, here denoted by P̄ , is stationaryunder the translations (ϑκ)κ∈Zd . We will prove in what follows that P̄ is the infinite-volumeGibbs measure we are looking for.
3.3 Support of the Infinite-Volume Limit Measure
We now justify the introduction of a set of tempered configurations as the right support ofeach of the probability measures P̄n , n ≥ 1, as well as of the constructed limit probabilitymeasure P̄ .
Proposition 2 The measures P̄n, n ≥ 1, and the limit measure P̄ are all supported on thetempered configurations, i.e.
∀n ≥ 1, P̄n(M temp) = P̄
(M temp) = 1.
Proof Let us show that, for P̄ (resp. P̄n)-a.e. γ ∈ M , there exists t = t(γ ) ≥ 1 such that
supl∈N∗
1
ld〈γB(0,l), 1 + ‖m‖d+δ〉 ≤ t. (15)
From (14), we know that
∀n ≥ 1,∫
〈γ[−1,1)d , 1 + ‖m‖d+δ〉P̄n(dγ ) ≤ 2da2. (16)
Since the integrand is a tame local function, the same inequality remains true when passingto the limit:
∫〈γ[−1,1)d , 1 + ‖m‖d+δ〉P̄(dγ ) ≤ 2da2.
The integrability of 〈γ[−1,1)d , 1 + ‖m‖d+δ〉 under P̄ (resp. P̄n) is precisely what we need inorder to apply the ergodic theorem in [18]. Doing so yields the following spatial asymptotics,
123
986 S. Rœlly, A. Zass
where we have P̄ (resp. P̄n)-a.s. convergence to the conditional expectation under P̄ (resp.P̄n) with respect to the σ -field J of (ϑκ)κ∈Zd - invariant sets:
liml→+∞
1
|B(0, l)| 〈γB(0,l), 1 + ‖m‖d+δ〉 = 1
2dEP̄
[〈γ[−1,1)d , 1 + ‖m‖d+δ〉 | J
]
(resp. EP̄n ). This implies that, P̄ (resp. P̄n)-a.s.,
liml→+∞
1
|B(0, l)| 〈γB(0,l), 1 + ‖m‖d+δ〉 < +∞
so that, a fortiori, (15) holds, and the proposition is proved. ��In Subsect. 3.4, in order to prove Gibbsianity of the limit measure, we need more: a
uniform estimate of the support of the measures P̄n , n ≥ 1. For this reason, we introduce theincreasing family (M l)l∈N∗ of subsets of M temp, defined by
M l ..= {γ ∈ M temp : ∀k ∈ N
∗, k ≥ l, ∀(x,m) ∈ γB(0,2k+1)c , B(x, ‖m‖) ∩ B(0, k) = ∅}.
Notice that, thanks to Lemma 2, for any t ≥ 1, M t ⊂ M l(t) (see Fig. 1).
Proposition 3 For any ε > 0, there exists l ≥ 1 such that
∀n ≥ 1, P̄n(Ml) ≥ 1 − ε.
Proof We want to find l ≥ 1 such that
P̄n(supk≥l
sup(x,m)∈γB(0,2k+1)c
‖m‖|x | ≥ 1
2
)≤ ε. (17)
For any κ = (κ1, . . . , κd) ∈ Zd , let Dκ = [κ1, κ1 + 1) × · · · × [κd , κd + 1) ⊂ R
d .We list all the elements of Z
d by a sequence (κi )i∈N ⊂ Zd that forms a spiral, starting at
κ0 = 0; in particular, there exist constants a, b > 0 (depending on the dimension d), suchthat ia ≤ |κi |d ≤ ib. We can then compute, for any l ≥ 1,
∑
κ∈Zd :|κ|≥2l
P̄n(m(γDκ ) ≥ 1
2 |κ|) =∑
i≥1:|κi |≥2l
P̄n(m(γDκi
) ≥ 12 |κi |
)
≤∑
i≥(2l)d/b
P̄n(m(γDκi
) ≥ 12 |κi |
)
≤∑
i≥(2l)d/b
P̄n(m(γDκi
)d ≥ a
2di)
≤∑
i≥(2l)d/b
P̄n( 2d
a
∑
(x,m)∈γD0
(1 + ‖m‖d)︸ ︷︷ ︸
=.. F(γ )
≥ i)
≤ EP̄n [1{F(γ )≥(2l)d/b}F(γ )]
= 2d
aEP̄n
[1{∑(x,m)∈γD0
(1+‖m‖d )≥ ab l
d }∑
(x,m)∈γD0
(1 + ‖m‖d)].
To control this expression, recall the following result (due to H.-O. Georgii and H. Zessin),which proves that point configurations in R
d with marks in a complete, separable metric
123
Marked Gibbs Point Processes with Unbounded Interaction 987
space, satisfy a local equi-integrability property on entropy level sets, with respect to themarks:
Lemma 6 ([12], Lemma 5.2) For any measurable non-negative function f : S → R+ andfor every a > 0 and Δ ∈ Bb(R
d),
limN→∞ sup
P∈P (M )≤a
EP
[1{〈γΔ, f 〉≥N }〈γΔ, f 〉
]= 0.
Applying this result to the sequence (P̄n)n , with f (x,m) = 1+‖m‖d and Δ = D0, we havethat, for any ε > 0, there exists l ≥ 1 large enough, such that
∀n ≥ 1, P̄n(
supκ∈Zd , |κ|≥2l
1
|κ|m(γDκ ) ≥ 1
2
)≤ ε.
For any (x,m) ∈ γB(0,2k+1)c , with k ≥ l, there exists κ ∈ Zd with |κ| ≥ 2k, such that
(x,m) ∈ γDκ ; since then‖m‖|x | ≤ ‖m‖
|κ| , we find that (17) holds, and the claim follows. ��Remark One could have thought that such a uniform estimate held in M t, for some t ≥ 1,but this is not the case; we thank one of the referees for pointing how this would not work. Inorder to have the uniform estimate, we had then to enlarge the set of tempered configurationsby introducing M l instead.
3.4 The Limit Measure is Gibbsian
We are now ready to prove that the infinite-volume P̄ measure we have constructed satisfiesthe Gibbsian property.
Lemma 7 Consider the Gibbsian kernel ΞΛ defined by (11). It satisfies:
(i) For any ξ ∈ M temp, ΞΛ(ξ, dγ ) is well defined: ZΛ(ξ) < +∞.(ii) For any Λ-local tame functional F on M , the map ξ �→ ∫
MΛF(γ )ΞΛ(ξ, dγ ) defined
on M temp is measurable.(iii) The family (ΞΛ)Λ∈Bb(R
d ) satisfies a finite-volume compatibility condition, in the sensethat, for any ordered finite-volumes Λ ⊂ Δ,
∫
MΔ\ΛΞΛ(ζΔ\ΛξΔc , dγΛ)ΞΔ(ξΔc , dζΔ\Λ) = ΞΔ(ξΔc , d(γΛζΔ\Λ)). (18)
Proof (i) We have to show that, for any ξ ∈ M temp, 0 < ZΛ(ξ) < +∞. Lemma 4 dealtwith the free boundary condition case, so this followed from the stability assumption (8).Since HΛ(γΛξΛc ) �= H(γΛ), this now follows in the same way from (10).
(ii) Themeasurability of themap ξ �→ ∫MΛ
F(γ )ΞΛ(ξ, dγ ) follows from themeasurabilityof ξ �→ H(γΛξΛc ) and ξ �→ ZΛ(ξ).
(iii) The compatibility of the family (ΞΛ)Λ follows, as in [19], from the additivity (7) of theconditional energy functional.
��We now state the main result of this subsection:
Proposition 4 The probability measure P̄ is an infinite-volume Gibbs measure with energyfunctional H.
123
988 S. Rœlly, A. Zass
Proof Since P̄ is concentrated on the tempered configurations, we have to check that, forany finite-volume Λ, the following DLR equation is satisfied under P̄:
∫
M tempF(γ ) P̄(dγ ) =
∫
M temp
∫
MΛ
F(γ )ΞΛ(ξ, dγ ) P̄(dξ),
where F is a measurable, bounded and Λ-local functional.Fix Λ ∈ Bb(R
d). We would like to use the fact that its finite-volume approximations(P̄n)n satisfy (DLR)Λ; but since they are lattice-stationary and periodic, this is not true. Toovercome this difficulty, we use some approximation techniques, articulated in the followingthree steps:
(i) An equivalent sequenceWe introduce a new sequence (P̂n)n and show it is asymptot-ically equivalent to (P̄n)n .(ii) A cut-off kernel We introduce a cut off of the Gibbsian kernel by a local functional.(iii) Gibbsianity of the limit measure We use estimations via the cut-off kernel to provethat P̄ satisfies (DLR)Λ.
(i) An equivalent sequenceWe introduce a modified sequence of measures (P̂n)n satisfy-ing (DLR)Λ and having the same asymptotic behaviour as (P̄n)n : for every n ≥ 1, consider
P̂n = 1
|Λn |∑
κ∈Λn∩Zd :
ϑκ (Λn)⊃Λ
Pn ◦ ϑ−1κ .
Since the above sum is not taken over all κ ∈ Λn ∩ Zd , P̂n is not a probability measure.
Moreover, P̂n is bounded from above by P̄n , in the sense that, for any measurable A ⊂ M ,P̂n(A) ≤ P̄n(A).
We introduce the index i0 ∈ N as the smallest n ≥ 1 such that Λ is contained in the boxΛn . Using the compatibility of the kernels (18), sinceΛ ⊂ Λn , for every n ≥ i0, the measureP̂n satisfies (DLR)Λ.
The sequences (P̂n)n and (P̄n)n are locally asymptotically equivalent, in the sense that,for every tame Λ-local functional G in L ,
limn→∞
∣∣∣∣
∫G(γ )P̂n(dγ ) −
∫G(γ )P̄n(dγ )
∣∣∣∣ = 0.
In particular, asymptotically P̂n is a probability measure, i.e. for any ε > 0, we can findn0 such that
∀n ≥ n0, P̂n(M ) ≥ 1 − ε. (19)
Indeed: let G be a tame Λ-local functional in L as in Definition 4, and set
δ1..=
∣∣∣∣
∫
M tempG(γ ) P̂n(dγ ) −
∫
M tempG(γ ) P̄n(dγ )
∣∣∣∣
=∣∣∣∣
1
(2n)d
∑
κ∈Λn∩Zd :
ϑκ (Λn)⊃Λ
∫G(γ )Pn ◦ ϑ−1
κ (dγ )
− 1
(2n)d
∑
κ∈Λn∩Zd
∫G(γ )P̃n ◦ ϑ−1
κ (dγ )
∣∣∣.
123
Marked Gibbs Point Processes with Unbounded Interaction 989
We then have
δ1 ≤ 1
(2n)d
∑
κ∈Λn∩Zd :
ϑκ (Λn)�Λ
∣∣∣∫
G(γ )P̃n ◦ ϑ−1κ (dγ )
∣∣∣
(loc.+tame)≤ c
(2n)d
∑
κ∈Λn∩Zd :
ϑκ (Λn)�Λ
∫ (1 + 〈γΛ, 1 + ‖m‖d+δ〉
)P̃n ◦ ϑ−1
κ (dγ ).
As Λ ⊂ Λi0 , the number of κ ∈ Λn ∩ Zd such that ϑκ(Λn) � Λ is
Card{κ ∈ Λn ∩ Z
d : ϑκ(Λn) � Λ} ≤ i02d(2n − 1)d−1,
since: for Λ to be moved out of Λn , one of the components of κ should be larger than n − i0(i0 options for this); there are 2d directions Λ can be moved through Λn ; and the other d − 1components of κ ∈ Λn ∩ Z
d are left free (2n − 1 options).Calling c′ ..= i0dc, we find
δ1 ≤ c′
n+ c
(2n)d
∑
κ∈Λn∩Zd :
ϑκ (Λn)�Λ
∫〈γΛ, 1 + ‖m‖d+δ〉P̃n ◦ ϑ−1
κ (dγ ).
Now, for any a4 > 0 (which will be fixed later), we split the above integral over the set{ ∑(x,m)∈γΛ
(1 + ‖m‖d+δ) ≥ a4}and its complement. We obtain
δ1 ≤ c′
n+ a4c′
n
+ c
(2n)d
∑
κ∈Λn∩Zd :
ϑκ (Λn)�Λ
∫
{〈γΛ,1+‖m‖d+δ〉≥a4}〈γΛ, 1 + ‖m‖d+δ〉P̃n ◦ ϑ−1
κ (dγ )
≤ (1 + a4)c′
n︸ ︷︷ ︸first term
+ c∫
{〈γΛ,1+‖m‖d+δ〉≥a4}〈γΛ, 1 + ‖m‖d+δ〉P̄n(dγ )
︸ ︷︷ ︸second term
.
Fix ε > 0; for n ≥ 2(1+a4)c′ε
, the first term is smaller than ε/2.To control the second term, we apply Lemma 6 to the sequence (P̄n)n , and the function
f (x,m) = 1 + ‖m‖d+δ; we find a4 > 0 such that the second term is smaller than ε/2,uniformly in n, and conclude the proof of this step.
(ii) A cut-off kernel We know that P̂n satisfies (DLR)Λ, i.e. for any Λ-local and boundedfunctional F
∫F(γ ) P̂n(dγ ) =
∫ ∫
MΛ
F(γ )ΞΛ(ξ, dγ ) P̂n(dξ).
If ξ �→ ∫MΛ
F(γ )ΞΛ(ξ, dγ ) were a local functional, we would be able to conclude
simply by taking the limit in n on both sides of the above expression, since P̄ = limn P̂n forthe topology of local convergence. But this is not the case because of the unboundedness ofthe range of the interaction. We are then obliged to consider some approximation tools.
To that aim, we introduce a (Δ,m0)-cut off of the Gibbsian kernels ΞΛ(ξ, dγ ), whichtakes into account only the points of ξ belonging to a finite volume Δ and having markssmaller than m0.
123
990 S. Rœlly, A. Zass
Definition 8 Let Δ ∈ Bb(Rd) with Δ ⊃ Λ. The (Δ,m0)-cut off Ξ
Δ,m0Λ of the Gibbsian
kernel ΞΛ is defined as follows:for every measurable, Λ-local and bounded functional G : MΛ → R,
∫
MΛ
G(γ )ΞΔ,m0Λ (ξ, dγ )
..= 1
ZΔ,m0Λ (ξΔ\Λ)
∫
MΛ
1{m(γ )≤m0}G(γ )e−HΛ(γΛξΔ\Λ)π zΛ(dγ ),
where ZΔ,m0Λ (ξΔ\Λ) is the normalisation constant.
Remarks i. ΞΔ,m0Λ is well defined since the normalisation constant Z
Δ,m0Λ is positive and
finite:
0 < e−z|Λ| ≤ ZΔ,m0Λ (ξΔ\Λ) < +∞.
ii. The functional
ξ �→∫
MΛ
G(γ )ΞΔ,m0Λ (ξ, dγ )
is now local and bounded, since the supremum norm of G is bounded.
We now show that ΞΔ,m0Λ is a uniform local approximation of the Gibbsian kernel ΞΛ,
i.e.For any ε > 0, t ≥ 1, for any measurable, Λ-local and bounded functional F , there exist
m0 > 0 and Δ ⊃ Λ such that, for any m0 ≥ m0 and Δ ⊃ Δ, we have
supξ∈M t
∣∣∣∣
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ ) −
∫
MΛ
F(γ )ΞΛ(ξ, dγ )
∣∣∣∣ ≤ ε. (20)
Indeed, let ξ ∈ M t. First notice that, since HΛ(γΛξΔ\Λ) = HΛ(γΛξΛc ) as soon asΔ ⊇ Λ ⊕ B
(0, 2l(t) + 2m(γΛ) + 1
)then e−HΛ(γΛξΔ\Λ) − e−HΛ(γΛξΛc ) = 0 on the set of
configurations {γ : m(γΛ) ≤ m0 and Λ ⊕ B(0, 2l(t) + 2m(γΛ) + 1
) ⊂ Δ}.Considering the difference between both partition functions, we obtain
∣∣∣ZΔ,m0Λ (ξΔ\Λ) − ZΛ(ξΛc )
∣∣∣
=∣∣∣∣
∫ (1{m(γΛ)≤m0}e−HΛ(γΛξΔ\Λ) − e−HΛ(γΛξΛc )
)π z
Λ(dγ )
∣∣∣∣
≤∫
1{m(γΛ)>m0}∪{Λ⊕B(0,2l(t)+2m(γΛ)+1
)�Δ}
(e−HΛ(γΛξΔ\Λ) + e−HΛ(γΛξΛc )
)π z
Λ(dγ )
(10)≤∫
1{m(γΛ)>m0}∪{Λ⊕B(0,2l(t)+2m(γΛ)+1
)�Δ}2e
c′〈γΛ,1+‖m‖d+δ〉π zΛ(dγ ).
Notice that this upper bound does not depend on ξ anymore. Thanks to the integrabilityassumption (1), by dominated convergence this implies that the map
ξ �→ ZΔ,m0Λ (ξΔ\Λ) − ZΛ(ξΛc )
converges to 0 as m0 ↑ ∞ and Δ ↑ Rd uniformly in ξ ∈ M t.
123
Marked Gibbs Point Processes with Unbounded Interaction 991
Similarly,
ξ �→∫
MΛ
1{m(γΛ)≤m0}F(γΛ)e−HΛ(γΛξΔ\Λ)π zΛ(dγ )
−∫
MΛ
F(γΛ)e−HΛ(γΛξΛc )π zΛ(dγ )
converges to 0 as m0 ↑ ∞ and Δ ↑ Rd , uniformly in ξ ∈ M t. This concludes the proof of
this step: we can find m0 = m0(ε, t) and Δ = Δ(ε, t) such that (20) holds for any m0 ≥ m0and Δ ⊃ Δ.
(iii)Gibbsianity of the limit measure To prove the Gibbsianity of P̄ we have to check that
δ2..=
∣∣∣∣
∫
M temp
∫
MΛ
F(γ )ΞΛ(ξ, dγ ) P̄(dξ) −∫
M tempF(γ ) P̄(dγ )
∣∣∣∣
vanishes.We first show that for large enough t ≥ 1, the sets M t,M l(t) are close to the support of
the measures P̄ , P̄n , and P̂n : let m0 and Δ be large enough in the above sense, and satisfyΔ ⊃ Λ ⊕ B
(0, 2l(t) + 2m0 + 1
). Thanks to the results on the supports of P̄ (Proposition 2)
and P̄n (Proposition 3), we can find a5 > 0, independent of n, such that, for any m0 and t
larger than a5, and all n ≥ 1,
P̄(M t) ≥ 1 − ε, P̄n(Ml(t)) ≥ 1 − ε, P̄n
({γ ∈ M : m(γΛ) ≤ m0
}) (4)≥ 1 − ε. (21)
Since, by construction, P̄n dominates P̂n , using (19) yields, for n ≥ n0,
P̂n(Ml(t)) ≥ 1 − 2ε, P̂n
({γ ∈ M : m(γΛ) ≤ m0
}) ≥ 1 − 2ε. (22)
The following steps deal with the estimation of δ2: using (21), we have that (w.l.o.g.‖F‖∞ ≤ 1)
δ2 ≤ ‖F‖∞ P̄((M t)c
)
︸ ︷︷ ︸≤ε
+∣∣∣∣
∫
M t
∫
MΛ
F(γ )ΞΛ(ξ, dγ )P̄(dξ) −∫
M tF(γ )P̄(dγ )
∣∣∣∣︸ ︷︷ ︸
=:δ21
.
Using the estimates of (20), we have
δ21(20)≤ ε +
∣∣∣∣
∫
M t
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ )P̄(dξ) −
∫
M tF(γ )P̄(dγ )
∣∣∣∣
(21)≤ 2ε +∣∣∣∣
∫
M temp
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ )P̄(dξ) −
∫
M tempF(γ )P̄(dγ )
∣∣∣∣︸ ︷︷ ︸
=:δ22
.
By construction, the functional ξ �→ ∫MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ ) is local; thus the local
convergence of (P̂n)n to P̄ implies that there exists n1 ∈ N∗ such that, for n ≥ n1, both
estimates hold:∣∣∣∣
∫
M temp
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ )P̄(dξ) −
∫
M temp
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ )P̂n(dξ)
∣∣∣∣ ≤ ε,
∣∣∣∣
∫
M tempF(γ )P̄(dγ ) −
∫
M tempF(γ )P̂n(dγ )
∣∣∣∣ ≤ ε.
123
992 S. Rœlly, A. Zass
Therefore,
δ22 ≤ 2ε +∣∣∣∣
∫
M temp
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ )P̂n(dξ) −
∫
M tempF(γ )P̂n(dγ )
∣∣∣∣
︸ ︷︷ ︸=:δ23
.
Now δ23 can be further decomposed:
δ23(22)≤ 2ε
+∣∣∣∣
∫
M l(t)
∫
MΛ
F(γ )ΞΔ,m0Λ (ξ, dγ )P̂n(dγ ) −
∫
M l(t)
∫
MΛ
F(γ )ΞΛ(ξ, dγ )P̂n(dξ)
∣∣∣∣
︸ ︷︷ ︸=:δ24
+∣∣∣∣
∫
M l(t)
∫
MΛ
F(γ )ΞΛ(ξ, dγ ) P̂n(dξ) −∫
M tempF(γ )P̂n(dγ )
∣∣∣∣ .
We can estimate δ4 by conditioning
δ24 =∣∣∣∣
∫
M l(t)
∫
MΛ
F(γ )(Ξ
Δ,m0Λ (ξ, dγ ) − ΞΛ(ξ, dγ )
)P̂n(dγ )
∣∣∣∣
=∣∣∣∣
∫
M l(t)
∫
MΛ
F(γ )(Ξ
Δ,m0Λ (ξ, dγ )P̂n(dξ)
− ΞΛ
(ξ, dγ | {γ : m(γ ) ≤ m0}
)(1 − ΞΛ
(ξ, {γ ′ : m(γ ′) > m0}
))
+ ΞΛ
(ξ, dγ | {γ : m(γ ) > m0}
)ΞΛ
(ξ, {γ ′ : m(γ ′) > m0}
))P̂n(dξ)
≤∣∣∣∣
∫
M l(t)
∫
MΛ
F(γ )(Ξ
Δ,m0Λ (ξ, dγ ) − ΞΛ
(ξ, dγ | {γ : m(γ ) ≤ m0}
))P̂n(dξ)
∣∣∣∣
+ 2∫
M l(t)ΞΛ
(ξ, {γ ′ : m(γ ′) > m0}
)P̂n(dξ).
The first term in the above inequality vanishes if the two kernels coincide onM l(t), which
is the case since Λ ⊕ B(0, 2l(t) + 2m(γΛ) + 1
) ⊂ Δ. Since P̂n satisfies (DLR)Λ, for thesecond term we have
∫
M l(t)ΞΛ
(ξ, {γ ′ : m(γ ′) > m0}
)P̂n(dξ) ≤ P̂n
({γ ′ : m(γ ′Λ) > m0}
) (22)≤ 2ε.
We then have δ24 ≤ 4ε. Putting it all together,
δ2 ≤ 11 ε +∣∣∣∣
∫
M l(t)
∫
MΛ
F(γ )ΞΛ(ξ, dγ ) P̂n(dξ) −∫
M tempF(γ )P̂n(dγ )
∣∣∣∣
(22)≤ 13 ε +∣∣∣∣
∫
M temp
∫
MΛ
F(γ )ΞΛ(ξ, dγ ) P̂n(dξ) −∫
M tempF(γ )P̂n(dγ )
∣∣∣∣ = 13 ε,
since P̂n satisfies (DLR)Λ. Thanks to the arbitrariness of ε > 0, we can conclude that alsoP̄ satisfies (DLR)Λ.
In conclusion, P̄ satisfies (DLR)Λ for any finite volume Λ, so Proposition 4—and conse-
quently Theorem1—is proved: P̄ is an infinite-volumeGibbsmeasurewith energy functionalH . ��
123
Marked Gibbs Point Processes with Unbounded Interaction 993
4 Application to Infinite-Dimensional Interacting Diffusions
We consider the case where the space of marks S is C0([0, 1], R
2), the set of R
2-valuedcontinuous paths on [0, 1] starting at 0, endowed with the supremum norm ‖m(·)‖ :=maxs∈[0,1]|m(s)|.In other words, a marked point x = (x,m(·)) ∈ R
2 × C0([0, 1], R
2)is identified with the
continuous path (x + m(s), s ∈ [0, 1]) starting in x .The random evolution of a reference path follows a gradient dynamics which solves thefollowing Langevin stochastic differential equation:
dXs = −1
2∇V (Xs)ds + dBs, s ∈ [0, 1], (23)
where B is an R2-valued Brownian motion.
Since we are looking for a random mark whose norm admits a reference law with asuper-exponential moment (Assumption (Hm) in Sect. 2.1), we shall restrict our attention topotentials V which force the gradient dynamics to be strongly confined. Let us thus assumethat the potential V is smooth (i.e. of class C 2) and that it satisfies the following boundsoutside some compact set of R
2:
∃δ′, a1, a2 > 0, V (x) ≥ a1|x |2+δ′and ΔV (x) − 1
2|∇V (x)|2 ≤ −a2|x |2+2δ′
. (24)
Indeed, the bounds in (24) imply the conditions of [20] for existence anduniqueness of a strongsolution of the Langevin equation (23); moreover, they also ensure the ultracontractivity ofthis diffusion with respect to its invariant measure μ(dy) = e−V (y)dy (see Example 3.5 in[14]). In particular, this implies (see Theorem 4.7.1 in [4]) that there exists a constant a > 0such that, for any δ < δ′/2,
sups∈[0,1]
E
[e|Xs |2+2δ |X0 = 0
]≤ a
∫
R2e|y|2+2δ
μ(dy) = a
∫
R2e|y|2+2δ
e−V (y)dy < +∞.
We denote by ρ the law on R+ of the supremum norm of the Langevin diffusion starting in 0;since the process Yt ..= sups∈[0,t]|Xs |2+δ is a submartingale, we can apply Doob’s inequalityto get
∫
R+e�2+2δ
ρ(d�) = E
[esups∈[0,1]|Xs |2+2δ |X0 = 0
]< +∞.
Remark The previous reasoning can be generalised by considering the evolution of theLangevin dynamics in R
d for any d > 2. Assumption (24) should then be reinforced byreplacing the (2+ δ′)-exponent with a (d + δ′) exponent, in order to obtain the finiteness ofthe super-exponential moment (Hm).
Let us now describe the kind of interaction we consider between the marked points of aconfiguration. It is not necessarily superstable, and consists of a pair interaction, concerningseparately the (starting) points and their attached diffusion paths, and a self interaction (Fig.2).
The energy of a finite configuration γ = {x1, . . . , xN } is taken of the form
H(γ ) =N∑
i=1
Ψ (xi ) +N∑
i=1
∑
j<i
Φ(xi , x j ), (25)
where:
123
994 S. Rœlly, A. Zass
Fig. 2 Two paths of a Langevindiffusion in R
2. Each circle iscentred in the starting point; theradius of the coloured circlescorrespond to their maximumdisplacement in the time interval[0, 1]; the dotted circles representthe security distance a0/2
Fig. 3 An example of radial andstable pair potential φ is theLennard-Jones potential
φ(u) = 16(( 3/2
u)12 − ( 3/2
u)6); it
is always negative after a0 = 32
• The self-potential term Ψ satisfies inf x∈R2 Ψ (x + m) ≥ −c1(1 + ‖m‖2+δ), for someconstant c1 > 0;
• The two-body potential Φ is defined by
Φ(x1, x2) =(φ(|x1 − x2|) +
∫ 1
0φ̃(|m1(s) − m2(s)|)ds
)1{|x1−x2|≤a0+‖m1‖+‖m2‖},
where φ : R+ → R ∪ {+∞} is a stable pair potential, i.e. there exists a constant cφ > 0such that, for any {x1, . . . , xN } ⊂ R
2, N ≥ 1,
N∑
i=1
∑
j<i
φ(|xi − x j |) ≥ −cφN ,
see [21] and [17]; moreover, we assume that φ(|x |) ≤ 0 for any |x | ≥ a0 (see, e.g.,Fig. 3). Moreover, we assume that φ is bounded from below by some constant −c2. Thepair potential φ̃ : R
2 → R+ ∪ {+∞} is a non-negative function.
123
Marked Gibbs Point Processes with Unbounded Interaction 995
Under these assumptions, the potentialΦ is stablewith constant cΦ = cφ in the followingsense:
∑
{x,y}⊂γ
Φ(x, y) ≥ −cΦ |γ |, ∀γ ∈ M f .
It is then straightforward to prove that such an energy functional H satisfies the stabilityassumption (Hst ) with constant c = c1 ∨ cΦ , and the range assumption (Hr ). Moreover, thelocal uniform-stability assumption (Hloc.st ) also holds:
Let γ ∈ M and ξ ∈ M t, t ≥ 1, and denote Δ = Λ ⊕ B(0, r). We have
HΛ(γΛξΔ\Λ) =∑
x∈γΛ
Ψ (x) +∑
{x,y}⊂γΛ
Φ(x, y)
︸ ︷︷ ︸(H st )≥ −(c1∨cΦ)
∑x∈γΛ
(1+‖m‖2+δ)
+∑
x∈γΛy∈ξΔ\Λ
Φ(x, y).
Since ξ ∈ M t, there exists c4(t) such that |ξΔ| ≤ c4(t). Therefore,
∑
x∈γΛy∈ξΔ\Λ
Φ(x, y) ≥∑
x∈γΛy∈ξΔ\Λ
φ(|x − y|) +∑
x∈γΛy∈ξΔ\Λ
∫ 1
0φ̃(|mx (s) − my(s)|
)ds
≥∑
x∈γΛy∈ξΔ\Λ
φ(|x − y|) ≥ −c2 |γΛ||ξΔ\Λ|
(ξ∈M t)≥ −c̄(t)|γΛ|,where c̄(t) ..= c2c4(t), so that (Hloc.st ) holds with c′(t) ..= c1 ∨ c̄(t).
Remark The above is an example of a pair potential with finite but not uniformly boundedrange.
Example 3 A concrete example of functions satisfying (24)–(25) is as follows:
• For the Langevin dynamics, consider V (x) = |x |4; then the diffusion is ultracontractivewith δ′ = 2. The invariant measure μ(dx) = e|x |4dx is a Subbotin measure (see [22]);
• For the interaction, let ψ(z) = −1 − z5/2, and φ be a Lennard-Jones pair potential, i.e.
there exist constants a, b > 0 such that
φ(z) = az−12 + bz−6.
Acknowledgements Open Access funding provided by Projekt DEAL. The authors would like to warmlythank David Dereudre for the fruitful discussions they had together on the topic. We also thank the tworeferees for the accurate reading and the many comments and remarks, which notably improved the earlierversions of this paper.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, whichpermits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you giveappropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence,and indicate if changes were made. The images or other third party material in this article are included in thearticle’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material isnot included in the article’s Creative Commons licence and your intended use is not permitted by statutoryregulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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