L
2+
ε
-Esimates on Exponential Decay of
Correlations in Equilibrium States of Classical
Continuous Systems of Point Particles
Gennadiy Shchepanyuk
Institute of Mathematics, National Academy of Sciences of Ukraine, Kiev, Ukraine
Copyright c⃝2015 Horizon Research Publishing All rights reserved.
Abstract
We present and prove L2+ε-estimates onexponential decay of correlations in equilibrium states of classical continuous systems of point particles interacting via an exponentially decaying pair potential of interaction, where ε is arbitrary small and positive real number. The obtained estimates exhibit not only the explicit dependence on the distance between the areas of the equilibrium classical systems between which the correlations are estimated but also on the volume of these areas, which can be used in the future for the investigation of the corresponding non-equilibrium and dynamic systems.
Keywords
Equilibrium States, Classical Continuous Systems, Point Particles, Cluster Expansions, Decay of Correlations1
Introduction
In the present work, we provide and proveL2+ε-estimates
on exponential decay of correlations in equilibrium states of classical continuous systems of point particles interacting via regular stable pair potential that exponentially decays with distance between particles. The parameterεcan be any (ar-bitrary small) positive real number. In our estimates, we ob-tain an explicit dependence of correlations not only on the distance between the regions of an equilibrium classical sys-tem of point particles but also on their volumes, that, in the future, can be used for the investigation of the corresponding non-equilibrium and dynamic systems.
The contents of the present work is as follows. In Sec-tion 2, we briefly remind the definiSec-tion and basic properties of the Lebesgue-Poisson measure over the space of locally-finite configurations and provide some formulae needed for the further consideration. In Section 3, we consider continu-ous systems of point particles interacting via a pair potential, formulate conditions that this pair potential should satisfy, provide a definition of a limiting Gibbs measure correspond-ing to equilibrium states of continuous systems of point par-ticles interacting via the pair potential, and construct clus-ter expansions for the density of the corresponding limiting
Gibbs measure with respect to the Lebesgue-Poisson measure over the configuration space. In Section 4, we formulate the result on the existence of the limiting Gibbs measure corre-sponding to equilibrium states of continuous systems of point particles at a sufficiently high temperature and the sufficiently low activity of the system and prove it by showing the con-vergence of the corresponding cluster expansions constructed for the density of the limiting Gibbs measure with respect to the Lebesgue-Poisson measure over the corresponding con-figuration space. Finally, Section 5 is devoted to formulation and proof of the main result of the present work concerning L2+ε-estimates on the exponential decay of correlations in
hight temperature dilute equilibrium states of continuous sys-tems of point particles interacting via a regular stable pair po-tential that exponentially decays with distance between par-ticles.
To prove the main result of the present paper, that is formu-lated in Section 5, we use the method of cluster expansions proposed and further developed by M. Duneau, B. Souillard, D. Iagolnitzer, V. Malyshev, and R. Minlos in [1, 2, 3, 4, 7].
In application to equilibrium systems of classical statisti-cal mechanics, the method of cluster expansions consists in representing macro- and microscopic observables of an infi-nite system of interacting particles in a form of the conver-gent series, each term of which takes into account interac-tion between particles only withing some their groups or, in other words, clusters. Such presentation allows to investi-gate various properties of the said macro- and microscopic observables of the equilibrium states of the classical system for those values of the thermodynamic parameters of the clas-sical system for which the series of cluster expansions con-verge.
In particular, the method of cluster expansions is very con-venient for investigation of decay of correlations in equilib-rium systems of classical statistical mechanics as it allows to reduce the problem of investigation of correlations in equilib-rium systems of infinitely many interacting particles to a set of more simple problems of investigating the correlations in analogous systems of particles interacting only withing some groups of particles, or clusters.
the distance between the particles allows to show that some terms in expressions that describe decay of correlations in such systems mutually cancel and, from the remaining terms, one can extract the common factor that exponentially decays with distance. That leads to the final estimates on decay of correlations in such systems of interacting particles.
All this technique is detailed in the proof of Lemma 4, the statement of which constitutes the most important estimate leading to the main result of the present paper. As to the main result of this paper, it is formulated in Theorem 2.
It should be noted that similar results concerning decay of correlations for truncated correlation functions of lattice systems was obtained in the classical paper by M. Duneau, B. Souillard, D. Iagolnitzer [3].
2
A Poisson Measure on
the Configuration Space
LetB(Rd)denote a Borelσ-algebra onRdand letL(Λ)
denote the system of all bounded Borel subsets of the Borel setΛ∈B(Rd).
Definition 1. For any Borel setΛ ∈B(Rd), the
configura-tion spaceΓΛis defined as the set of all locally finite subsets
of the setΛ, that is,
ΓΛ:=
{
γ⊂Λ|#(γ∩Λ′)<∞, ∀Λ′∈ L(Λ)}, (1)
where#γdenotes the number of elements of the setγ.
Next, for any Borel setsΛandΛ′ fromB(Rd)such that
Λ′ ⊂Λ,we denote byBΛ′(ΓΛ)theσ-algebra onΓΛ
gener-ated by all possible mappings of the form
γ7→#(γ∩Λ′′)∈Z+:={0,1,2, . . .},
whereΛ′′ ∈ L(Λ′)and configurationsγare taken from the spaceΓΛ.
Remark 1. In what follows, we are going to systematically omit the subindices atΓandBin case they are equal toRd,
that is, we are going to writeΓ and Binstead ofΓRd and BRd.
For any Borel setΛ∈B(Rd), we will also denote byγ
Λ
the canonical projection of the configurationγonto the space
ΓΛ, that is, by definition,γΛ:=γ∩Λ.
Definition 2. For any Borel setΛ ∈B(Rd), the Lebesgue-Poisson measureλzΛoverB(ΓΛ)with intensityz >0is
de-fined in such a way that for anyBΛ(ΓΛ)-measurable
func-tionF
∫
ΓΛ
F(γ)λzΛ(dγ) =
∞
∑
n=0
zn
n!
∫
Λn F
(
{xi}ni=1
)
(dx)1n, (2)
where{xi}ni=1:={x1, . . . xn}and(dx)1n :=dx1. . . dxn.
Lemma 1. For any Borel setsΛ,Λ′∈B(Rd)such thatΛ⊂
Λ′ and|Λ\Λ′| < ∞, and for any summable functionF ∈ L1
(
ΓΛ,BΛ′(ΓΛ), λzΛ
)
, the following equality holds:
∫
ΓΛ
F[γΛ′]λzΛ(dγΛ) = exp
[
z|Λ\Λ′|] ∫
ΓΛ′
F[γΛ′]λzΛ′(dγΛ′).
Lemma 2. For any Borel sets Λ1,Λ2 ∈ B(Rd)such that Λ1 ∩ Λ2 = ∅ and for any square-integrable functions
F1, F2 ∈ L2
(
ΓΛ1∪Λ2,BΛ1∪Λ2(ΓΛ1∪Λ2), λ
z
Λ1∪Λ2
)
, the fol-lowing equality holds:
∫
ΓΛ1∪Λ2
F1[γΛ1]F2[γΛ2]λ
z
Λ1∪Λ2(dγΛ1∪Λ2)
=
∫
ΓΛ1
F1[γΛ1]λ
z
Λ1(dγΛ1)
∫
ΓΛ2
F2[γΛ2]λ
z
Λ2(dγΛ2).
Lemmas 1 and 2 easily follow from Definition 2.
For further consideration, we will also need the following lemma, proof of which can be obtained directly from Defini-tion 2. (See also [7, 8].)
Lemma 3. For any Borel set Λ ∈ B(Rd) and arbitrary summable functionsF1, . . . , Fp∈L1(ΓΛ,BΛ(ΓΛ), λzΛ), the
following equalities hold:
∫
ΓΛ
λzΛ(dγ) ∑ (γ1,...,γp)
γ1∪···∪γp=γ
γi∩γj=∅, i̸=j
F1(γ1)· · ·Fp(γp)
=
p
∏
i=1
∫
ΓΛ
Fi(γi)λzΛ(dγ
i) (3)
and
∫
ΓΛ\{∅}
λzΛ(dγ) ∑ (γ1,...,γp)
γ1∪···∪γp=γ γi∩γj=∅, i̸=j γi̸=∅, i=1,...,p
F1(γ1)· · ·Fp(γp)
=
p
∏
i=1
∫
ΓΛ\{∅}
Fi(γi)λzΛ(dγ
i). (4)
The summation in the left-hand side of equality (3) is under-taken over all ordered partitions of the configurationγ∈ΓΛ
onto subsets that are mutually non-intersecting and, in gen-eral, can be empty. As to the summation in the left-hand side of equality (4), it is undertaken over all ordered partitions of the configurationγ ∈ ΓΛ onto thenon-emptysubsets that
are mutually non-intersecting.
3
Limiting Gibbs measures
corresponding to continuous
systems of point particles
In what follows, we consider classical continuous systems of point particles interacting via a pair potentialv that is as-sumed to be
1) symmetric, that is, such that
v(x, x′) =v(x′, x)
for allx, x′∈Rd,
2) translationaly invariant, that is, such that v(x, x′) = ˆv(x−x′)
3) stable, that is, such that satisfies the following stability condition:
∃B >0 : ∀n≥2, ∀x1, . . . , xn∈Rd
∑
1≤i<j≤n
v(xi, xj)≥ −Bn, (5)
4) and regular, that is, such that C(β) :=
∫
Rd
|e−βvˆ(x)−1|dx <∞.
Remark 2. From the physical point of view, it is natural to assume even more, namely, that the pair potential of inter-actionvis not only symmetric and translationally invariant but also rotationally invariant, that is, such thatv(x, x′) = ˜
v(|x−x′|).Having in mind exactly such potential of pair in-teraction of point particles,we, nevertheless, will not use the assumption of rotational invariance of the potential of pair interaction of point particles.
Definition 3. In a finite volumeΛ ∈ L(Rd),we define the
equilibrium Gibbs states Gz,βΛ (·) of continuous systems of point particles interacting via a pair potentialvat an inverse temperatureβ and an activityz as probability measures on the configuration space(ΓΛ,B(ΓΛ))such that for any
con-figurationγΛ∈ΓΛ
Gz,βΛ (dγΛ)
λz
Λ(dγΛ)
= 1
Ξz,βΛ exp
[
−βU(γΛ)
]
, (6)
where
U(γΛ) :=
∑
{x, x′}⊂γΛ
v(x, x′) (7)
and
Ξz,βΛ :=
∫
ΓΛ
exp
[
−βU(γΛ)
]
λzΛ(dγΛ). (8)
The summation in the right-hand side of (7) is undertaken over all two-element subsets of the configurationγΛ.
Definition 4. The limiting Gibbs states corresponding to the equilibrium states of continuous systems of point particles interacting via a pair potentialvat an inverse temperatureβ
and activityzare defined as the limits
Gz,β = lim Λ↗RdG
z,β
Λ
that is understood in the following sense:
∫
Γ
FΛ0(γ)dG
z,β(γ) = lim
Λ↗Rd
∫
ΓΛ
FΛ0(γΛ)dG
z,β
Λ (γ), (9)
whereFΛ0is an arbitrary boundedΛ0-local function,that is, measurable with respect to theσ-algebraBΛ0.
Remark 3. The stability condition imposed on the pair po-tential of interaction ensures existence of the Gibbs state
Gz,βΛ (·)in the finite volumeΛ∈ L(Rd).
It is far not trivial question whether the set of all limiting Gibbs states of a classical continuous system of point parti-cles interacting via a pair potentialvis not empty at least for some values of activityzand inverse temperatureβ.
One of the possible ways to prove that the question above has a positive answer is to construct a cluster expansion for the quantity
pΛ(γΛ) :=
Gz,βΛ (dγΛ)
λz
Λ(dγΛ)
(10)
= 1
Ξz,βΛ exp
[
−βU(γΛ)
]
, (11)
Λ∈ L(Rd), and to prove its convergence atΛ↗Rd. Let ∑(γ)denote the sum over all subdivisions of an ar-bitrary configurationγonto non-empty subsets, that is,
∑(γ) [. . .] =
#γ
∑
n=1
∑
{γ1,...,γn} γ1∪···∪γn=γ γi∩γj=∅, i̸=j γi̸=∅, i=1,...,n
[...]
= #γ
∑
n=1 1 n!
∑
(γ1,...,γn)
γ1∪···∪γn=γ
γi∩γj=∅, i̸=j γi̸=∅, i=1,...,n
[...]. (12)
We denote the sets of all graphs, all connected graphs, and all trees1whose vertices coincide with the elements of the set γ∈Γ(or, in other words, with the points of the configuration γ∈Γ) byG(γ),Gc(γ), andT(γ), respectively. Let also
δ∅(γ) =
{
1, γ=∅,
0, γ̸=∅.
In the following, we are going to employ the cluster ex-pansions suggested and further developed in [1, 2, 3, 4, 7]. Namely, for an arbitrary bounded Borel setΛ ∈ L(Rd), we
present the quantitypΛ(γΛ)as
pΛ(γΛ) = 1
Ξz,βΛ
{
δ∅(γΛ) +
∑(γΛ)
k(γΛ1). . . k(γΛn)
}
, (13)
wherek(·)is a function defined onΓ\ {∅}and such that
|k(γ)| ≤(e2βB)#γ ∑
T ∈T(γ)
∏
{x,x′}∈T
e−βv(x,x′)−1. (14)
Under most general assumptions on stability and regularity of the pair potential of interactionv, (13) and (14) yield the existence of the limiting Gibbs state that corresponds to a classical continuous system of point particles in equilibrium interacting via the pair potential v at the sufficiently small inverse temperatureβand activityz. See, for example, The-orem 1 below.
Remark 4. For the consideration to follow, it is also conve-nient to setk(∅) := 0.
Remark 5. It follows from estimate (14) on the absolute value of the function k, definition of the Lebesgue-Poisson measure(2), inequality
nn−2≤enn!, (15)
the fact[10],that
∑
T ∈T({1,...,n})
1 =nn−2, (16)
and Remark4that
k∈L1(ΓΛ,B(ΓΛ), λzΛ)
for anyΛ∈ L(Rd).
Statement 1. If the formula (13)for the cluster expansion of the quantitypΛ(γΛ),Λ∈ L(Rd), given by(10)holds and
k∈L1(ΓΛ, λzΛ), then
Ξz,βΛ =
∫
ΓΛ
λzΛ(dγΛ)
{
δ∅(γΛ) +
∑(γΛ)
k(γΛ1). . . k(γΛn)
}
= exp
[ ∫
ΓΛ
k(γΛ)λzΛ(dγΛ)
]
. (17)
Proof.Formulae (10), (8), and (13) yeild that
1 =
∫
ΓΛ
pΛ(γΛ)λzΛ(dγΛ) =
(
Ξz,βΛ )−1×
×
∫
ΓΛ
λzΛ(dγΛ)
{
δ∅(γΛ) +
∑(γΛ)
k(γΛ1). . . k(γnΛ)
}
.
From this, we get the first equality of formula (17), whereas the second equality of formula (17) is an obvious corollary of Lemma 3, Remark 4, definition (12), and the fact that λz
Λ(∅) = 1for allΛ⊆Rd.
To construct a cluster expansion (13), it suffices to show that for any non-empty finite configurationγ
exp
[
−β ∑
{x,x′}⊂γ
v(x, x′)
]
= ∑(γ)k(γ1). . . k(γn). (18)
However,
exp
[
−β ∑
{x,x′}⊂γ
v(w, x′)
]
= ∑
G∈G(γ)
∏
{x,x′}∈G
(
exp[−βv(x, x′)]−1),
and therefore equality (18) holds with k(γ) := ∑
Gc∈Gc(γ)
∏
{x,x′}∈Gc
(
exp[−βv(x, x′)]−1
)
(19)
for anyγ∈Γ\ {∅}.
The proof of estimate (14) on the absolute value of the functionk(γ)defined by formula (19) can be found in [7] (Lemma 2 from Subsection 4 of Section 4).
4
Convergence of the Cluster
Expansions and Existence of the
Limiting Gibbs Measure
Estimate (14) yields (see, for example, [7]) the following the-orem of existance of a limiting Gibbs state.
Theorem 1. For a stable regular potential of pair interac-tionv, sufficiently small inverse temperatureβ,and activity
zsuch that
2ze2βB+1C(β)<1, (20)
there exists a limiting Gibbs stateGz,βcorresponding to the
limit (9)of Gibbs statesGz,βΛ . Moreover,the density of its restriction ontoBΛ′(Γ),Λ′ ∈ L(Rd),
pΛ′(γΛ′) :=
Gz,β BΛ′(Γ)(dγΛ′)
λz
Λ′(dγΛ′)
(21)
can be represented by the formula
pΛ′(γΛ′) =f(Λ′)peΛ
′
(γΛ′), (22)
where
e
pΛ′(γΛ′) =
{
δ∅(γΛ′) +
∑(γΛ′)
rΛ′(γΛ1′). . . rΛ ′
(γΛn′)
}
,
rΛ′(γΛ′) :=
∫
ΓΛ′c
k(γΛ′∪γΛ′c)λzΛ′c(dγΛ′c) (23)
and
f(Λ′) = exp
[ ∫
ΓΛ′c
k(γΛ′c)λΛz′c(dγΛ′c)−
∫
Γ
k(γ)λz(dγ)
]
.
(24)
Proof. It is easy to see that for any Λ′ ∈ B(Λ), Λ ∈
L(Rd),the density
pΛΛ′(γΛ′) :=
Gz,βΛ BΛ′(ΓΛ)(dγΛ′)
λz
Λ′(dγΛ′)
of the restriction of the Gibbs stateGz,βΛ in finite volumeΛ
ontoBΛ′(ΓΛ)is given by the formula
pΛΛ′(γΛ′) =fΛ(Λ′)
{
δ∅(γΛ′)+
∑(γΛ′)
rΛΛ′(γΛ1′). . . rΛ ′
Λ(γ
n
Λ′)
}
, (25) where
rΛΛ′(γΛ′) :=
∫
ΓΛ\Λ′
k(γΛ′∪γΛ\Λ′)λzΛ\Λ′(dγΛ\Λ′) (26)
and
fΛ(Λ′) := Ξz,βΛ\Λ′
Ξz,βΛ
= exp
[ ∫
ΓΛ\Λ′
k(γΛ\Λ′)λΛz\Λ′(dγΛ\Λ′)
−
∫
ΓΛ
k(γΛ)λzΛ(dγΛ)
]
. (27)
Then, as it follows from the cluster property of the Lebesgue-Poisson measure,
∫
ΓΛ
k(γΛ)λzΛ(dγΛ)−
∫
ΓΛ\Λ′
k(γΛ\Λ′)λΛz\Λ′(dγΛ\Λ′)
=
∫
ΓΛ′\{∅}
λzΛ′(dγΛ′)
∫
ΓΛ\Λ′
and thus, the expression forfΛ(Λ′)can also be presented as
follows:
fΛ(Λ′) = exp
[
−
∫
ΓΛ′\{∅}
λzΛ′(dγΛ′)×
×
∫
ΓΛ\Λ′
λzΛ\Λ′(dγΛ\Λ′)k(γΛ′∪γΛ\Λ′)
]
. (27′)
Next, estimate (14) on the absolute value of the func-tionk, definition of the Lebesgue-Poisson measure (2), for-mula (16), and inequality (15) yield that
∫
ΓΛ′\{∅}
λzΛ′(dγΛ′)
∫
ΓΛ\Λ′
λzΛ\Λ′(dγΛ\Λ′)k(γΛ′∪γΛ\Λ′)
≤α|Λ′|, (28) whereαis a function of the parametersz,β,δandκ. This means that the quantity
∫
ΓΛ\Λ′
λzΛ\Λ′(dγΛ\Λ′)k(γΛ′∪γΛ\Λ′)
is uniformly bounded (with respect to Λ ⊆ Rd) for
λzΛ′-almost allγΛ′ ∈ΓΛ′ \ {∅}.
It is easy to see that, for an arbitrary configurationγ ∈Γ, there exists the limit of the quantity
χΓΛ\Λ′(γ)k(γΛ′∪γ)
asΛ↗Rd.
Moreover, for an arbitrary configurationγ∈Γ lim
Λ↗RdχΓΛ\Λ′(γ)k(γΛ′∪γ) =χΓ(Λ′)c(γ)k(γΛ′∪γ).
Therefore, for λz
Λ′-almost all γΛ′ ∈ ΓΛ′ \ {∅}, there exist limits of the quantitiesfΛ(Λ′),rΛ
′
Λ(γΛ′), andpΛ
′
Λ(γΛ′)
as Λ ↗ Rd and, moreover, these limits equal to f(Λ′), rΛ′(γΛ′),andpΛ
′
(γΛ′),respectively.
This proves the existance of the limiting Gibbs stateGz,β and the correctness of formulae (22)–(24).
Remark 6. Formulae(27′)and(28)also yield that
f(Λ′)≤eα|Λ′|. (29)
5
Decay of Correlations
Under an additional assumption that the potential of pair interaction v exponentially decays with distance, from the convergence of the cluster expansions (13), one can get some Lp-estimates showing that correlations in limiting Gibbs
measures corresponding to equilibrium states of classical continuous systems of point particles interacting via the pair potential v also exponentially decay with distance. In this section, we formulate and prove the corresponding L2+ε
-estimates, whereεis an arbitrary small but positive real num-ber.
Definition 5. We will say that a symmetric translation-invariant potential of pair interactionvexponentially decays
with distance if there exist such positive constantsv0, r0, and
κthat
|ˆv(x)| ≤v0e−κ|x| ∀x:|x| ≥r0, (30)
where,as before,v(x, x′) = ˆv(x−x′).
For further consideration, we need the following two propositions:
Statement 2. Suppose that the functionf : R+ 7→R
satis-fies the conditionf(x)≤Ce−x,then
|ef(x)−1| ≤ψ(C)e−x, x∈R+,
whereψ(C) :=CeC.
Statement 3. If the potential of pair interaction v
exponentially decays with distance,then
|e−βˆv(x)−1|< K(β)e−κ|x|, (31)
where
K(β) = max{(e2βB+ 1)eκr0, ψ(βv
0)
}
.
Remark 7. If the potential of pair interactionvexponentially decays with distance,then
C(β) =
∫
Rd
|e−βˆv(x)−1|dx
<K(β)
∫
Rd
e−κ|x|dx
=Ω(d)K(β)(d−1)!
κd , (32)
whereΩ(d)denotes the volume ofd-dimensional sphere with radius 1.
Theorem 2. Suppose that the potential of pair interaction
v is stable, regular, and exponentially decays with distance in the sense of Definition5. Then, for any positiveεandδ, sufficiently small inverse temperatureβ, and activityzsuch that
2ze2βB+1Ω(d)K(β)(d−1)! κd
(
1 +1 δ
)d
<1, (33)
arbitrary bounded Borel sets Λ1,Λ2 ∈ L(Rd) such
that Λ1 ∩ Λ2 = ∅, and arbitrary functions Fi ∈
L2+ε
(
Γ,BΛi(Γ), G
z,β), i= 1,2,
⟨F1F2⟩ − ⟨F1⟩⟨F2⟩
≤Dexp
[
− κε
(1 +δ)(2 +ε)dist(Λ1,Λ2)
]
, (34)
where
D= 10
(
α1|Λ1| |Λ2|eα2|Λ1∪Λ2|
) ε
2+ε
∥F1∥L2+ε∥F2∥L2+ε, (35) α1, α2are some functions of the parametersz, β, δ, κ, and
⟨ · ⟩denotes expectation with respect to the probability mea-sureGz,β constructed in Theorem1.
Remark 8. As it has already been noted in Remark7, for-mula (33)yields formula (20),and so,according to Theo-rem1, the limiting Gibbs measure Gz,β mentioned in
With the standard technique (see, for example, the proof of Theorem 17.2.2 from [5]), one can get the statement of Theorem 2 from the followingL∞-estimate on exponential decay of correlations in the limiting Gibbs stateGz,β.
Lemma 4. Suppose that the potential of pair interaction
v is stable and exponentially decays with distance. Then for any positive δ,arbitrary bounded Borel sets Λ1,Λ2 ∈
L(Rd) such that Λ
1 ∩Λ2 = ∅, and any functions Fi ∈
L∞(Γ,BΛi(Γ), G
z,β), i= 1,2,
⟨F1F2⟩ − ⟨F1⟩⟨F2⟩≤D′∥F1∥L∞∥F2∥L∞×
×exp
[
− κ
1 +δdist(Λ1,Λ2)
]
, (36)
for sufficiently small inverse temperatureβand activityzthat satisfy estimate(33),where
D′ =α1|Λ1| |Λ2|eα2|Λ1∪Λ2| (37)
andα1andα2are some fuctions from the parametersz, β, δ
andκ.
Proof.Let us denote byI12the following integral:
∫
ΓΛ1∪Λ2
F1(γΛ1)F2(γΛ2)pe
Λ1∪Λ2(γ
Λ1∪Λ2)λ
z
Λ1∪Λ2(dγΛ1∪Λ2)
Then, using formula (21), the first inequality from (22), and the fact that
f(Λ1∪Λ2)I12≤ ∥F1∥L∞∥F2∥L∞,
we can write that
⟨F1F2⟩ − ⟨F1⟩⟨F2⟩
≤1−f(Λ1)f(Λ2) f(Λ1∪Λ2)
∥F1∥L∞∥F2∥L∞+
+f(Λ1)f(Λ2)I12−I1I2, (38)
where, fori= 1,2, we denoted Ii:=
∫
ΓΛi
Fi(γΛi)pe
Λi(γ
Λi)λ
z
Λi(dγΛi).
Next, using formula (24), one can get that
f(Λ1)f(Λ2)
f(Λ1∪Λ2) − 1 = exp [ − ∫
{γ∈Γ|γ∩Λ1̸=∅,γ∩Λ2̸=∅}
k(γ)λz(dγ)
]
−1
. (39)
An attentive consideration of the integral in the right-hand side of equality (39) leads to the inequality
∫
{γ∈Γ|γ∩Λ1̸=∅,γ∩Λ2̸=∅}
k(γ)λz(dγ)
≤D0exp
[
− κ
1 +δdist(Λ1,Λ2)
]
, (40) whereD0 =α3|Λ1| |Λ2|andα3is a function of the inverse
temperatureβ,activityz, and the parametersδandκ.
Thus, as it follows from Statement 2, the first term in the right-hand side of formula (38) satisfies the following esti-mate:
1−f(Λ1)f(Λ2) f(Λ1∪Λ2)
∥F1∥L∞∥F2∥L∞ ≤ψ(D0)×
×exp
[
− κ
1 +δdist(Λ1,Λ2)
]
∥F1∥L∞∥F2∥L∞. (41)
Next, using the second equality from (22), formula (23), and the cluster property of the Lebesgue-Poisson measure, one can rewrite the second term from the right-hand side of formula (38) in the form
f(Λ1)f(Λ2)
∫
ΓΛ1∪Λ2
λzΛ1∪Λ2(dγΛ1∪Λ2)F1(γΛ1)F2(γΛ2)×
×
[
∑(γΛ1,γΛ2)∏n
i=1
J12i − ∑(γΛ1)∑(γΛ2)×
× ∑′
(ΘΛ1,ΘΛ2)
n1
∏
i1=1
Ji1
1
n2
∏
i2=1
Ji2
2
]
, (42)
where, forj= 1,2,
Jij
j :=
∫
Γ(Λ1∪Λ2)c
λz(Λ
1∪Λ2)c
(
dγij
(Λ1∪Λ2)c
)
×
×
∫
ΘijΛ
j λzΛ
2(dγ
ij
Λj)k
(
γij
Λ1∪γ
ij
Λ2∪γ
ij
(Λ1∪Λ2)c
)
,
J12i :=
∫
Γ(Λ1∪Λ2)c
k(γΛi1∪Λ2∪γ
i
(Λ1∪Λ2)c
)
λz(Λ1∪Λ2)c
(
dγ(Λi 1∪Λ2)c
)
,
∑(γΛ1,γΛ2)
denotes the sum over all partitions of the config-urationγΛ1∪γΛ2onto the non-empty subsets in such a way
that at least one of these subsets contains points (or elements) of both configurationγΛ1andγΛ2, that is,
∑(γ1,γ2)
[. . .] =
#(γ∑1∪γ2)
n=1
∑
{γ1,...,γn} γ1∪···∪γn=γ1∪γ2
γi∩γj=∅, i̸=j γi̸=∅, i=1,...,n ∃k∈{1,...,n}:γi∩γk̸=∅, i=1,2
[. . .], (43)
and ∑′
(ΘΛ1,ΘΛ2)
denotes the sum over all ordered collections
(ΘΛ1,ΘΛ2) := (Θ
1
Λ1, . . . ,Θ
n1
Λ1,Θ
1
Λ2, . . . ,Θ
n2
Λ2),
each element of which is either a set of configurations that contains only the empty configuration or a set of configura-tions that contains all the configuration space with exception of the empty configuration, moreover, each of such ordered collections should contain at least one element that is not an empty set, that is,
∑′
(ΘΛ1,ΘΛ2)
[. . .] = ∑
(Θ1Λ1,...,Θn1 Λ1,Θ
1 Λ2,...,Θ
n2 Λ2)
Θi1
Λ1∈{{∅},ΓΛ1\{∅}}, i1=1,...,n1
Θi2
Λ2∈{{∅},ΓΛ2\{∅}}, i2=1,...,n2
Θ1Λ1∪···∪Θn1
Λ1∪Θ
1 Λ2∪···∪Θ
n2 Λ2̸={∅}
The already obtained estimates onf(Λ)together with the attentive consideration of the integrals that contain the ex-pression (42) allow us to estimate it by the quantity
D1exp
[
− κ
1 +δdist(Λ1,Λ2)
]
∥F1∥L∞∥F2∥L∞,
whereD1=α4|Λ1| |Λ2|eα5|Λ1∪Λ2|andα4andα5are some
functions from the inverse temperatureβ,activityz, and the parametersδandκ, that completes the proof of Lemma 4.
Acknowledgements
The present research has been partially supported by the Alexander-von-Humboldt Foundation.
REFERENCES
[1] M. Duneau, D. Iagolnitzer, and B. Souillard. Decrease Proper-ties of Truncated Correlation Functions and Analyticity Prop-erties for Classical Lattices and Continuous Systems, Commu-nications in Mathematical Physics,31, (1973), pp. 191–208. [2] M. Duneau, D. Iagolnitzer, and B. Souillard. Strong Cluster
Properties for Classical Systems with Finite Range Interac-tion, Communications in Mathematical Physics, 35 (1974), pp. 307–320.
[3] M. Duneau, B. Souillard, and D. Iagolnitzer. Decay of Corre-lations for Infinite-Range Interactions,Journal of Mathemati-cal Physics,16(1975), pp. 1662–1666.
[4] M. Duneau and B. Souillard. Cluster properties of lattice and continuous systems, Communications in Mathematical Physics,47(1976), pp. 155–166.
[5] I. A. Ibragimov and Yu. V. LinnikIndependent and Stationary Sequences of Random Variables. Wolters-Noordhoff Publish-ing, Groningen, The Netherlands, 1971.
[6] Yu. G. Kondratiev, R. A. Minlos, M. R¨ockner, and G. V. Shchepanyuk. Exponential Mixing for Classical Continuous Systems. In: Canadian Mathematical Society Conference Proceedings, Vol. 28: “Stochastic Processes, Physics and Geometry: New Interplays. I.” Proceedings of the International Conference on Infinite Dimensional (Stochastic) Analysis and Quantum Physics, 18–22 January 1999, Leipzig. Fritz Gesztesy, Helge Holden, J¨urgen Jost, Sylvie Paycha, Michael R¨ockner and Sergio Scarlatti Editors. Published by the American Mathematical Society for the Canadian Mathematical Society, Providence, Rhode Island, 2000, pp. 243–254.
[7] V. A. Malyshev and R. A. Minlos. Gibbs Random Fields. Cluster expansions. Kluwer Academic Publishers, Dordrecht–Boston–London, 1991.
[8] R. A. Minlos. Lectures on Statistical Physics, Uspekhi Matematicheskikh Nauk, no. 1 (1968), p. 137–196.
[9] R. A. Minlos. Limiting Gibbs Distribution, Funktsionalnyj Analiz i ego Prilozhenija,1, no. 2 (1967), p. 60–73.
