Measures on two-component configuration spaces

We study measures on the configuration spaces of two type particles. Gibbs measures on the such spaces are described. Main properties of corresponding relative energies densities and correlation functions are considered. In particular, we show that a support set for the such Gibbs measure is the set of pairs of non-intersected configurations.

In this work we study these equations for the simplest case of the space of marks: {+, −}. We extend approach proposed in [2] for this marked (two-component) system. We concentrate our attention on the properties of the Gibbs type measures without studying existence and uniqueness problems. One may study this using Ruelle technique in the same way as in [2], which we represent in the forthcoming paper. Another approach for proving existence and non-uniqueness was proposed in [4].
Let us describe the content of the work in more detail.
Preliminary constructions for the one-component case are presented in Section 2. In Section 3 we consider main properties of a measure on the two-component configuration spaces which is locally absolutely continuous with respect to (w.r.t.) product of two Poisson measures. Note that it is natural that these Poisson measures have the same intensities since they should not be orthogonal. This is impossible for different constant intensities but for non-constant ones we need some additional conditions (see, e.g., [15]). Hence, for simplicity we consider the same Poisson measures. One of the main results of this section is connection between correlation functions of a measure and of their marginal distribution. In Section 4 we describe the Gibbs measures in terms of the so-called relative energies densities, which characterized the energy between particle of one type and configurations of the both types. Main properties of these densities allow us to show that the corresponding Gibbs measure is locally absolutely continuous w.r.t. product of Poisson measures. As a result, we may study such measure only on the subspace of the two-component configuration space which includes only pairs of configurations which are not intersect. This plays important role for studying different dynamics on the two-component configuration spaces, namely, we have useful support set for a big class of measures (see, e.g., [3], [1]). At we end we show an example of the pair-potentials Gibbs measure which coincides with studying in [4].
We don't construct in this work specifications of the Gibbs measure and corresponding DLR approach. This may be considered analogously to [2] as well as it possible to show the equivalence between such two approaches (that goes back to [13]). All our considerations may be extended on the case of the product of finite number of the configuration spaces over different C ∞ manifolds.

Preliminaries
Let X be a connected oriented C ∞ manifold. The configuration space Γ := Γ X over X is defined as the set of all locally finite subsets of X, where |·| denotes the cardinality of a set and γ Λ := γ ∩ Λ. As usual we identify each γ ∈ Γ with the non-negative Radon measure x∈γ δ x ∈ M(X), where δ x is the Dirac measure with unit mass at x, x∈∅ δ x is, by definition, the zero measure, and M(X) denotes the space of all non-negative Radon measures on the Borel σ-algebra B(X). This identification allows to endow Γ with the topology induced by the vague topology on M(X), i.e., the weakest topology on Γ with respect to which all mappings are continuous. Here C 0 (X) denotes the set of all continuous functions on X with compact support. We denote by B(Γ) the corresponding Borel σ-algebra on Γ.
Let us now consider the space of finite configurations where Γ (n) := Γ (n) X := {γ ∈ Γ : |γ| = n} for n ∈ N and Γ (0) := {∅}. For n ∈ N, there is a natural bijection between the space Γ (n) and the symmetrization X n S n of the set X n := {(x 1 , ..., x n ) ∈ X n : x i = x j if i = j} under the permutation group S n over {1, ..., n} acting on X n by permuting the coordinate indexes. This bijection induces a metrizable topology on Γ (n) , and we endow Γ 0 with the topology of disjoint union of topological spaces. By B(Γ (n) ) and B(Γ 0 ) we denote the corresponding Borel σ-algebras on Γ (n) and Γ 0 , respectively.
We suppose from the beginning that there exists a sequence {Λ m } m∈N ⊂ B c (X) such that m∈N Λ m = X.
Since µ Λm is absolutely continuous w.r.t. λ σ × λ σ it is enough to prove that But if we denote for any fixed The remark that is fulfilled the proof.
be a locally absolutely continuous measure w.r.t. π σ × π σ and let A be a B(X)-measurable set such that σ(A) = 0. Then the following set has zero µ-measure.
Proof. Using the same trick as in the previous Proposition one can show that it is enough to prove that for any m ∈ N But the left hand side is equal to The statement is proven.
We define the marginal distribution of µ in a usual way, namely, Hence, for example, µ + is a probability measure on Γ + , B(Γ + ) . Then one can consider projection of µ + on Γ + Λ : On the other hand we may consider marginal distribution of µ Λ whose we denote by (µ Λ ) + .
It's easy to see that Indeed, let F : Γ 2 → R be a measurable function such that there a exist measurable function On the other hand Remark 3.4. Using (3.3) it is clear that if µ is locally absolutely continuous w.r.t. π σ × π σ then µ ± are locally absolutely continuous w.r.t. π σ .
Definition 2. We will say that locally absolutely continuous w.r.t. π σ × π σ probability measure µ is satisfied local Ruelle bound if for any For the measure µ from Definition 2 one can define a correlation function k µ , namely, for It follows from infinitely-divisible property of λ σ that r.h.s. of (3.5) doesn't depend on Λ ± . Also, from definition of λ σ and (3.4) one has that Correlation function of the marginal distribution µ + we will denote k + µ and define as 4 Two-component Gibbs measures such that for all non-negative measurable functions We denote class of such measures G(r + , r − , σ).
We will call the functions r ± partial relative energy densities of the measure µ. With necessity these function have the following properties.

4)
as well as the balance identity holds and, analogously, Comparing right hand sides of these equalities we obtain (4.3). J := on the other hand, Comparing right hand sides of these equalities we obtain (4.5).
In the same way we obtain that As a result, using (4.5), (4.9), (4.10), one has that proves the statement.
Cocycle and balance identities allow us to construct more complicate objects which characterized energies between finite and infinite configurations.
Next theorem present Ruelle-type identity for Gibbs measure µ which also called "infinitely divisible property". Theorem 4.8. Let µ ∈ G(r + , r − , σ). Then for any non-negative measurable function F : Γ 2 → [0; +∞) and for any Λ ± ∈ B c (X) Proof. Set for x ∈ X, n ∈ N, A − ∈ B(Γ − ) and for measurable non-negative measurable F then using (4.1) we obtain for any non-negative measurable F . Apply this formula for functioñ with fixed x, y. Then Repeating this procedure we obtain, as a result, Then Analogously, for any A + ∈ B(Γ + ) Hence, the statement is followed from (4.17).
Proof. Using (3.5), (4.21), Lemma 4.6 and (4.18) we obtain The second formula one can obtain in the same way or just putting η − = ∅ in the previous one and using (4.17), (4.13).
At the end of article we consider examples of partial relative energies densities r ± which satisfied (4.3)-(4.5).
The simplest examples of r 1,2 are also pair potential densities: let φ ± : X 2 → R ∪ {∞} be symmetric functions and Then µ 1,2 are classical pair-potential Gibbs measures and µ is a measure of type which is considered in [4]. As a result, in this case and, therefore,