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Loss network representation of Peierls contours

We present a probabilistic approach for the study of systems with exclusions, in the regime traditionally studied via cluster-expansion methods. In this paper we focus on its application for the gases of Peierls contours found in the study of the Ising model at low temperatures, but most of the results are general. We realize the equilibrium measure as the invariant measure of a loss-network process whose existence is ensured by a subcriticality condition of a dominant branching process. In this regime, the approach yields, besides existence and uniqueness of the measure, properties such as exponential space convergence and mixing, and a central limit theorem. The loss network converges exponentially fast to the equilibrium measure, without metastable traps. This convergence is faster at low temperatures, where it leads to the proof of an asymptotic Poisson distribution of contours. Our results on the mixing properties of the measure are comparable to those obtained with ``duplicated-variables expansion'', used to treat systems with disorder and coupled map lattices. It works in a larger region of validity than usual cluster-expansion formalisms, and it is not tied to the analyticity of the pressure. In fact, it does not lead to any kind of expansion for the latter, and the properties of the equilibrium measure are obtained without resorting to combinatorial or complex analysis techniques.


Introduction
In this paper we develop a probabilistic approach to the study of the equilibrium measure of systems with exclusions -like hard-core gases, contours, polymers or animals-in the lowdensity or extreme-temperature regime. This regime has traditionally been studied via clusterexpansion methods, which relied either on sophisticated combinatorial estimations [Malyshev (1980), Seiler (1982), Brydges (1984)] or on astute inductive hypotheses plus complex analysis [Kotecký and Preiss (1986), Dobrushin (1996Dobrushin ( , 1996a]. In contrast, we realize the equilibrium measure as the invariant measure of a loss network process that can be studied using standards tools and notions from probabilistic models and processes. Loss networks, first introduced by Erlang in 1917, encompass a rather general family of processes as discussed in Kelly (1991) and references therein. Technically we work with the so-called fixed-routing loss networks. We build on ideas of Ferrari and Garcia (1998) to show (Section 3.4) that the existence of the loss network can be related to the absence of percolation in an oriented percolation process. This condition also yields other properties of the process and its invariant measure, like uniqueness, convergence through sequences of finite volumes and mixing properties (Theorem 4.6). More precise results can be obtained by resorting to a dominant multitype branching process (Section 5). Roughly speaking, the mean number of branches of this process becomes the driving parameter: Subcriticality is a sufficient condition for the construction to work. Time and space rates of convergence and mixing rates are explicitly obtained in terms of this parameter (Theorem 2.10).
The approach of this paper was already exploited in Fernández, Ferrari and Garcia (1998).
Here we refine and complete the theory presented there, extending their region of validity and including proofs of exponential mixing and convergence to a Poisson process.
The subcriticality condition of the corresponding branching process is sup γ 1 |γ| θ:θ ∼γ |θ| w(θ) < 1 , (1.1) role in some applications of cluster expansions [see eg. Zahradníck (1984), Borgs and Imbrie (1989)]. In fact, the approach is designed so to bypass expansions of this type. It is a probabilistic approach designed to answer probabilistic questions -existence of expectations, properties of correlation functions-in a direct way, without combinatorial or complex-analysis techniques.
It is not an alternative to cluster expansions: it has a different regime of validity and different aims.
While the ideas behind our results are natural and simple, their formalization requires many intermediate technical results that may obscure the development of our theory. Let us, therefore, present a sort of "road map" of the paper to guide the reader. The main results are presented, in a self-contained manner, in Section 2. The actual construction of the loss network is the subject of Section 3. We start with a reference "free" process of Poissonian births and exponentially distributed deaths, with respect to which the loss network is absolutely continuous. The novelty of our approach resides in the fact that, rather than independently generating birth and deathtimes, the lifetimes are associated to each birth-time as a mark. Hence, unlike comparable constructions, each death-time has an associated birth-time and defines an space-time "cylinder" representing the presence of a loss event (contour). This permits to comb the process either backwards or forward in time with equal ease. The "time backwards" point of view leads to the notion of "backwards oriented percolation" which will be our main conceptual and practical tool. The idea is to construct the loss network by erasing from the free process those cylinders that conflict with pre-existing ones. This can only be done if the set of preexisting cylinders (the "clan of ancestors") is finite with probability one. This is precisely the condition of absence of backwards oriented percolation. The main technical result of this Section 3 is the proof that this construction yields precisely the loss network. [This is contained in the proof of Theorem 3.20].
It is apparent that the same construction works for any other process absolutely continuous with respect to a Poisson birth and death process, for instance point or Boolean processes (Baddeley and van Lieshout, 1995;Kendall, 1997Kendall, , 1998. Once the process is so constructed, the course of action is clear. First, we relate time and space mixing properties with the time and space size of the percolation clan. This is done in Section (4) [ Theorem 4.6]. The only slightly involved part of this section is related to the proof of the mixing properties. Indeed, we resort to a -standard but unusual-coupling between clans, together with a continuous-time construction (Section 4.5), to improve a little over results proved previously by the method of "duplicated variables" (von Dreifus, Klein and Perez, 1995; Bricmont and Kupiainen, 1996).
In order to present quantitative estimates in terms of the parameters of the problem, we follow the known technique (used, for instance by Hall, 1985)  Our work was motivated in part by the posthumous review of Roland Dobrushin (1996a), where he complained that while "perturbation methods are intensively used by mathematical physicists, they are not so popular as correlation inequalities among the probabilists". He called for a "systematical exposition oriented to the mathematicians". In this paper we follow his call: we concentrate on probabilistic issues and exploit probabilistic arguments.

The contour model
We consider the d-dimensional lattice Z d and call plaquettes the (d − 1)-dimensional unit cubes centered at points of Z d . We identify each plaquette with its center. A set of plaquettes is called surface. Two plaquettes are adjacent if they share a (d − 2)-dimensional face. This defines a notion of connection: a set γ of plaquettes forms a connected (hyper) surface if for every two plaquettes x,y there is a sequence of pairwise adjacent plaquettes starting at x and ending at y.
A surface is closed if every (d − 2)-dimensional face is shared by an even number of plaquettes in the surface. A contour γ is a connected and closed family of plaquettes. Two contours γ and θ are incompatible if they share some (d − 2)-dimensional face. In this case we denote γ ∼ θ.
Denote by G Λ the set of all contours in the volume Λ. The subset X Λ ⊂ N G Λ of compatible configurations is defined as that is, a configuration of contours is compatible if it does not contain two incompatible contours.
We denote X = X Z d and G = G Z d . As usual, we endowed X with the product topology.
For each fixed β ∈ R + , a parameter usually called the inverse temperature, and for each finite Λ define the measure µ Λ on X Λ by where |γ| is the number of plaquettes in γ and Z Λ is a renormalization constant making µ Λ a probability.

The loss network
We introduce a Markov process called (fixed routing) loss network in the set of compatible contours. This process was introduced by Erlang in 1917 (see Brockmeyer, Halstrom and Jensen (1948), p. 139). An account of its properties can be found in Kelly (1991). In the traditional interpretation a contour γ represents the route taken up by a call. The plaquettes encompasing γ are the circuits held by the call. For a finite or infinite set Λ ⊂ Z d and f real continuous function on X Λ , the generator of the process is defined by: where δ γ (θ) = 1{θ = γ} and the sum of configurations is defined pointwisely in N G Λ : (η+ξ)(γ) = η(γ) + ξ(γ). In words, each contour γ attempts to appear at rate e −β|γ| but it does so only if it is compatible with all present contours. Present contours disappear at rate 1. Our first result is the following sufficient condition for the existence of a process with generator A Λ for any (infinite) Λ.
then for any (infinite) Λ the Markov process with generator A Λ exists and admits at least one invariant measure.
This theorem is proven in Section 6. We denote η Λ,ζ t the corresponding process in Λ with initial configuration ζ. We omit the volume superindex when Λ = Z d :

Results on the invariant measure
We say that f has support in Υ ⊂ Z d if f depends only on contours intersecting Υ (not necessarily contained in Υ

Exponential mixing. For measurable functions f and g depending on contours contained
in an arbitrary set Λ ⊂ Z d : where M 2 ad M 3 are the same of (2.12). 5. Central limit theorem. Let f be a measurable function on X with finite support such that µf = 0 and µ(|f | 2+δ ) < ∞ for some δ > 0. Let τ x be the translation by x and assume where the double arrow means convergence in distribution.
This theorem strengthens the results R1-R5 of Fernández, Ferrari and Garcia (1998). In that paper β * was replaced by a value β M defined as the solution of α 0 (β M ) = 1/(d − 1). This value is strictly bigger than β * . Item 5 generalizes (the central limit) Theorem 7.4 of Dobrushin (1996a), where only functions depending on a finite number of contours are considered. The above theorem will be proven in Section 7.
Finally, we prove a Poisson approximation. Consider the equivalence relation induced by the translation of contours. Let G j be a the set formed by one representative containing the origin from each class of equivalence of contours with length j. For each Borel set V ⊂ R d and Fix a contour length j > 0. For each γ ∈ G j , let M γ,β be defined by where η β is distributed according to the invariant measure for the given β. Let (M γ,∞ , γ ∈ G j ) be a family of independent unit Poisson processes in R d .

Theorem 2.17
For each contour length j, it is possible to jointly construct the d-dimensional processes {M γ,β : β * < β ≤ ∞, γ ∈ G j } in such a way that for all regions V equal to a product of intervals, where c(|γ|, V ) is a computable constant. As a consequence {M γ,β : γ ∈ G j } converges in distribution to a family consisting of | G j | independent Poisson processes with mean e −βj .
Theorem 2.17 is proven in Section 8.

Graphical representation of loss networks
We construct the loss network as a function of stationary marked Poisson processes (a la Harris) each of which indicates the attempted birth-times of a contour. A lifetime is associated to each attempted birth. The triple (contour , attempted birth , lifetime) is called a cylinder. The loss network is constructed by erasing cylinders which at birth violate the exclusion condition. The crucial point in this construction is the association of the lifetime to the birth-time. This allows us to study the process backwards in time by studying a Markovian oriented percolation process of cylinders. In contrast, the standard construction uses independent Poisson processes for the birth-times and death-times respectively. In this case the backwards construction looks hard.

Marked Poisson processes
To each contour γ ∈ G we associate an independent (of everything) marked Poisson process N γ with rate e −β|γ| . We call T k (γ), γ ∈ G, the ordered time-events of N γ with the convention that T 0 (γ) < 0 < T 1 (γ). For each occurrence time T i (γ) of the process N γ we choose an independent mark S i (γ) exponentially distributed with mean 1. At the Poisson time-event T i (γ) a contour γ appears and it lasts S i (γ) time units.
The random family C = {{(γ, T i (γ), S i (γ)) : i ∈ Z} : γ ∈ G} consists of independent marked Poisson processes. A marked point (γ, T k (γ), S k (γ)) ∈ C is identified with γ × [T k (γ), T k (γ) + S k (γ)], the cylinder with basis γ, birth-time T k (γ) and lifetime S k (γ). The life of the cylinder is the time interval [T k (γ), T k (γ) + S k (γ)]. For a generic cylinder C = (γ, t, s), we use the notation We define incompatibility between cylinders C and C ′ by otherwise C ′ ∼ C (compatible). We say that two sets of cylinders A and A ′ are incompatible if there is a cylinder in A incompatible with a cylinder in A ′ : A ∼ A ′ if and only if C ∼ C ′ for some C ∈ A and C ′ ∈ A ′ . Let S := (S 0 i (θ) : θ ∈ G, i ≥ 1) be a (countable) family of iid exponential times of mean 1 independent of C. These are the lifetimes that, when necessary, will be associated to the contours of the initial configuration. Indeed, we identify S with the set of cylinders {(θ, 0, S 0 i (θ)) : the family of cylinders associated to the initial configuration ξ, all with birth-time 0. Notice that ξ may have more than one cylinder with the same basis. For s < t define Remark. In this paper we will work with the probability space given by the product of the spaces generated by C and S. We call it (Ω, F, P). We write E for the respective expectation.
In section 4.5.2 we use the direct product of this space by itself, while in Section 5 we need to consider countable products of this space. We use the same notation P and E for the corresponding probability and expectation in these enlarged spaces.

The free network
The above process, called the free network, is a product of independent birth-and-death processes on N G with initial configuration ξ whose generator is given by The invariant (and reversible) measure for this process is the product measure µ 0 on N G with Poisson marginals In terms of loss networks, ξ ξ t is the process for which all the calls are accepted, that is ξ ξ t (γ) is the number of calls on route γ at time t when there is no restriction on the number of calls a circuit can accept and the initial configuration of calls is ξ.

Finite-volume construction of a loss network
In the construction of a loss network in a finite volume Λ with an initial condition ζ ∈ X Λ we use only the finite set of Poisson processes (N γ : γ ⊂ Λ) and the finite family of initial lifetimes , defined as in (3.4), be such that all its cylinders are mutually compatible. We realize the dynamics η Λ,ζ t as a (deterministic) function of C Λ and S(ζ).

We construct inductively
Then, we move forward in time and consider the first Poisson mark: The corresponding cylinder is erased if it is incompatible with any of the cylinders already in This procedure is successively performed mark by mark until all cylinders born before t are (3.11) because in the free network ξ ξ t all cylinders are kept.
Since Λ is finite, there exists a sequence of random times We can, in particular, consider t i as the entrance times of ξ t in the set {ξ : ξ(γ) = 0, for all γ ∈ Λ}. Since this process has a unique invariant measure which gives positive probability to this set, (t i ) is a stationary renewal process with inter-renewal time with finite mean. We extend the construction of a set of kept cylinders to t ∈ R forgetting the set S(ζ), by doing the above procedure in each time ]. This can be done because no cylinder intersects {t i : i ∈ Z}. Let us denote K Λ the resulting set of kept cylinders and η Λ t its projection in the sense of (3.9). By construction K Λ has a time-translation invariant distribution. The process η Λ t has generator A Λ and distribution independent of t, hence given by µ Λ . This implies that for any f : Since η Λ t (γ) ≤ ξ t (γ) for all γ ∈ Λ and ξ t (γ) has Poisson distribution with mean e −β|γ| , we have, (3.13)

Infinite-volume construction. Backwards oriented percolation
If we try to perform an analogous construction in infinite volume we are confronted with the problem that there is no first mark. To overcome this we follow the original approach of Harris (1972) [see also Durrett (1995)] and introduce the notion of percolation. The goal is to partition the set of cylinders in finite subsets to which the previous mark-by-mark construction can be applied.
We come back to the infinite-volume construction of Section 3.1. For an arbitrary space-time point (x, t) define the set of cylinders containing the point (x, t) by For any cylinder C define the set of ancestors of C as the set of cylinders born before C that are incompatible with C: The definition of ancestor of C does not depend on the lifetime of C. Recursively for n ≥ 2, the nth generation of ancestors of (x, t) is defined as and for a given cylinder C, We say that there is backward oriented percolation in C if there exists a space-time point (x, t) such that A x,t n = ∅ for all n, that is, there exists a point with infinitely many generations of ancestors. Let the clan of the space-time point (x, t) be the union of its ancestors: In the next theorem we give a sufficient condition for the existence of the infinite-volume process in any finite time interval in terms of backwards percolation.
is finite for any x ∈ Z d and t ≥ 0, then for any (possibly infinite) Λ ⊂ Z d , the process with generator A Λ is well defined for any initial configuration ζ ∈ X Λ and has at least one invariant measure µ Λ .
Proof. We construct the process for Λ = Z d ; the construction for other regions Λ is analogous.
The initial configuration is denoted ζ ∈ X and the initial cylinders are given by S(ζ), defined in (3.4). Note that that all the cylinders of S(ζ) are mutually compatible. We then partition into a set of kept cylinders, denoted by K, and a set of erased cylinders denoted by D.
The construction is as follows. First, all cylinders in S(ζ) are kept. Second, for each x ∈ Λ the percolation clan of ( , is partitioned in kept and deleted cylinders as in the finite-volume case. To do so we order the cylinders of A x,t ∩ C[0, t] by birth-time. This can be done because by hypothesis A x,t ∩ C[0, t] has a finite number of cylinders. Then we successively classify each cylinder as kept if it is compatible with all cylinders already classified as kept (including those in S(ζ)); if not, we classify it as erased. We denote the resulting sets we have that The process is now defined as in (3.9) by The reader can check that for finite Λ the above construction is equivalent to that of Section To show that η ζ t has generator A, denote η t = η ζ t and K = K ζ and write + {other things}, (3.24) where {other things} refer to events with more than one Poisson mark in the time interval [t, t + h] for the contours in the (finite) support of f . Since the total rate of the Poisson marks in this set is finite, the event {other things} has a probability of order h 2 . Now, denoting To compute the second term of (3.24), observe that Life (C) is independent of Birth (C) and both the event {C ∈ K[0, t]} and η t are F t -measurable. Here F t is the σ-algebra generated by the births and deaths occurred before t. Hence Since Life (C) is exponentially distributed with mean 1, Taking the expectation of (3.24) and substituting (3.26)- (3.29) we get which dividing by h and taking limit gives The existence of an invariant measure follows by compactness as our process is defined in the compact space X . See Chapter 1 of Liggett (1985).
We show in the next theorem that under stronger hypothesis the process can be constructed for times in the whole real line. Since the construction is time-translation invariant, the distribution of η t will be invariant. Proof. The lack of percolation allows us to construct a set K ⊂ C as K ζ [0, t] was constructed from C[0, t] ∪ S(ζ) in the proof of the previous theorem. We just proceed clan by clan and simply ignore the cylinders of S. Note that K is both space and time translation-invariant by construction. Analogously to the previous theorem we define η t as the section of K at time t: By construction, the distribution of η t does not depend on t, hence its distribution is an invariant measure for the process.
Let us denote µ the distribution of η t , in anticipation of the fact that this is precisely the measure of Theorem 2.10.
As in the finite case, for all γ ∈ G. This implies that the distribution µ inherits property (3.13): be the clan of Υ ⊂ Z d (at time 0).
Remarks. 1. It follows from (3.33) that for any t ∈ R and continuous f , The presence/absence of a contour γ at time t depends only on the clan of ancestors of (x, t) for any x ∈ γ through a certain function. More generally, for each f there exists a function Φ such that For instance, η t (γ) = 1 if and only if A x,t 1 contains a cylinder in K with basis γ whose life contains t. This depends only on the set A x,t . In particular, with the notation (3.36), Analogous statements are true for the process starting with a fixed configuration at time zero: is the set of cylinders in C[0, t]∪S(ζ) which determines the value of f (η ζ t ) when f has support Υ.

Percolation, space-time convergence and mixing
In this section we exploit the relation between the loss-network process and the absence of percolation to prove a more precise version of Theorem 2.10. In the proof of the mixing properties we shall need a continuous-time construction of the backwards percolation clan.

The key theorem
The precise statement of next theorem requires the notion of non-oriented percolation in a time interval. For any time interval (s, t) and any space-time point (x, t ′ ) define and Notice that in the definition of G n there is no exigency that the birth time of C ′ be previous to the birth-time of C or that the lifetimes intersect. Let We say that there is no (non-oriented) percolation in [s, t] if for any space-time point (x, t ′ ), contains a finite number of cylinders. We will show later that the condition α < ∞ is sufficient for the existence of an h such that the probability that there is no non-oriented percolation in [0, h] is one.
In addition we need the following definitions.
• The time-length and the space-width of the family of cylinders A x,t are respectively Basis (C) . t and the first birth in the family of ancestors of (x, t).
• A Λ (Supp (f )) is the set of ancestors of Supp (f ) constructed from C Λ as A(Supp (f )) was constructed from C. [Notice that this is not the same as A(Supp (f )) ∩ C Λ ].
In item 4 of next theorem we enlarge our probability space to the direct product of our working space with itself: (Ω, F, P) × (Ω, F, P). As warned before, we continue to use P and E for the probability and expectation of this space. Furthermore, for any b ∈ (0, 1).
3. Space convergence. As Λ → Z d , µ Λ converges weakly to µ. More precisely, if f is a function depending on contours contained in a finite set Λ, then where τ x is translation by x. More precisely, where A(Supp (g)) has the same distribution as A(Supp (g)) but is independent of Existence of µ has been proven in Theorem 3.32. In the rest of the section we prove the other properties.

Time convergence and uniqueness
We use the same Poisson marks to construct simultaneously the stationary process η t and a process starting at time zero with an arbitrary initial configuration ζ. The second process is (4.13) Since we are using C to construct η t and C[0, t] ∪ S(ζ) to construct η ζ t , it follows from (3.38) and (3.40) that To see this notice that . Display (4.14) shows (4.8).
To prove the weak convergence (4.7) we fix b ∈ [0, 1] and bound the indicator function in the right hand side of (4.14) by The expected value of the first term in (4.15) goes to zero because A x,t has a finite number of cylinders with probability one. The second term in (4.15) is bounded above by Since S and A x,t are independent and S 0 i are iid exponentially distributed random variables of mean 1, Since ζ is a configuration of compatible contours, it contains at most one contour per site, i.e.
|{γ ∋ x : ζ(γ) = 1}| ≤ 1 for all x ∈ Z d . This implies that at most SW (A x,t ) cylinders of C(ζ) can be incompatible with cylinders in A x,t . Hence, (4.17) is bounded by The expectation of (4.18) is given by because the distribution of A x,t does not depend on t. Our hypothesis of no backwards oriented percolation implies that A x,0 contains a finite number of (finite) contours. Hence To prove (4.9) we start from the expectation of (4.15) and use (4.19) to bound the expected value of the second term by The above arguments prove that the process converges, uniformly in the initial configuration, to the invariant measure µ. An immediate consequence is that µ is the unique invariant measure.
This concludes the proof of (1) and (2) of Theorem 4.6.
Since the spatial projections of the set of ancestors of Supp (f ) are finite, the right hand side of (4.10) goes to zero, proving, in particular, the weak convergence of µ Λ to µ.

Mixing. Its relation with a coupling construction
The proof of (4) of Theorem 4.6 is very similar in spirit to the above proof but it requires a somewhat more delicate argument based on the coupling of two continuous-time versions of the backwards percolation process. We first notice that (4.11) is a straightforward consequence of (4.12), because in the absence of backwards percolation, the spatial projections of the set of ancestors of Supp (f ) and Supp (g) are finite. This implies that the right hand side of (4.12) goes to zero.
To prove (4.12) we use (3.37) and (3.38) to get where Φ is the function referred to in (3.38) and ( A(Supp (f )), A(Supp (g))) has the same marginal distributions as (A(Supp (f )), A(Supp (g))) but its marginals are independent.
Identity (4.23) shows that to obtain (4.11) it is enought to construct a coupling (joint construction) of the four processes such that
In the remaining of the section we discuss the construction of the coupling with properties (4.24)-(4.25). The construction is natural and straightforward, but unavoidably technical. As an alternative we mention the approach based on "duplicated variables" (von Dreifus, Klein and Perez, 1995; Bricmont and Kupiainen, 1996), which is probabilistically simpler but requires some combinatorial input.

Construction of a four-clan coupling
We need to couple two clans in the same random set of cylinders with two independent copies with the same marginal distribution. Moreover, to strengthen our results we need to ensure that the marginal realizations remain the same as much as possible. The coupling (Section 4.5.2), is based on a construction of backwards percolation clans as non-homogeneous continuous time Markov processes (Section 4.5.1). The hypothesis on the absence of non-oriented percolation for some time interval (0, h) is needed for the infinite-volume construction of the coupling. [−t − h, −t] whose corresponding cylinder is incompatible with some C ′ ∈ A t (Υ). That is, if C is a cylinder with Basis (C) ∼ Basis (C ′ ) for some C ′ ∈ A t ,

A continuous-time construction of the backwards percolation clan
(4.27) We have denoted  From (4.29), as in the computation of the forward Kolmogorov equations, we get where the sum is over the set {γ ∈ G : γ ∼ Basis (C ′ ) for some C ′ ∈ A t }. This equation characterizes the law of the process A t (Υ) as a non-homogeneous Markov process. If there is a Poisson birth mark at time −t whose corresponding cylinder is called C ′′ , then

We now construct
where the incompatibility between cylinders was defined in (3.2).
Infinite volume case. In infinite volume the construction can be performed using a percolation argument as in Section 3.

A coupling between two interacting and two independent clans
We take two independent marked Poisson process whose marks and cylinders we respectively call blue and red. We enlarge our probability space and continue using P and E for the probability and expectation with respect to the space generated by the product of the blue and red Poisson processes. Using these marks we construct simultaneously the processes in the following way.
1. The processes A t (Λ 1 ) and A t (Λ 2 ) are constructed using only the blue marks, as described in Subsection 4.5.1, and ignoring the red marks. Hence, they are the clans of Λ 1 and Λ 2 respectively.
2. The process A t (Λ 1 ) is also constructed only with the blue marks, hence it coincides with 3. The process A t (Λ 2 ) is constructed with a precise combination of blue and red marks in such a way that (a) it coincides with A t (Λ 2 ) for a time interval that is as long as possible; (b) it is independent of A t (Λ 1 ), and (c) it has the same marginal distribution as A t (Λ 2 ).
Property (3) is achieved in the following way.
Finite-volume case. If both Λ 1 and Λ 2 are finite sets, we order the marks by appearance and introduce a flag variable, Flag(t) ∈ {0, 1}, which indicates if some cylinder of A t (Λ 1 ) is incompatible with some cylinder of A t (Λ 2 ): We now proceed as follows, mark by mark backwards in time. First, we set Flag(0) = 0. The construction guarantees that Flag(t) = 0 implies A t (Λ i ) = A t (Λ i ) for i = 1, 2.
• If at time −t a (blue or red) mark is present and Flag(t−) = 0, then -If the mark is blue and the corresponding cylinder can be included in A t− (Λ 1 ) but not in A t (Λ 2 ), then include it in A t (Λ 1 ) and A t (Λ 1 ). Analogously, if it can be included in A t− (Λ 2 ) but not in A t (Λ 1 ), then include it in A t (Λ 2 ) and A t (Λ 2 ). Keep the Flag = 0.
-If the mark is blue and the corresponding cylinder can be included in both A t− (Λ 1 ) and A t− (Λ 2 ), then include it in both A t (Λ 1 ) and A t (Λ 2 ) but include it only in A t (Λ 1 ). Set the Flag = 1.
-If the mark is red and the corresponding cylinder can be included in both A t− (Λ 1 ) and A t− (Λ 2 ), then include it only in A t (Λ 2 ). Set the Flag = 1.
-If the mark is red and the corresponding cylinder can be included in either A t− (Λ 1 ) or A t− (Λ 2 ) but not in both of them, then ignore the mark. Keep the Flag = 0.
-If the mark is red or blue but the corresponding cylinder can be included in neither A t− (Λ 1 ) nor A t− (Λ 2 ) then ignore the mark. Keep the Flag = 0.
• If at time −t a (blue or red) mark appears and Flag(t−) = 1, then use blue marks for , and A t (Λ 1 ) and red marks for A t (Λ 2 ). Keep the Flag = 1.
To verify that the above coupling has the right marginals, it suffices to notice that the rate of inclusion of cylinders in each one of the marginals is precisely given by (4.30).
Infinite-volume case. In the case in which at least one of Λ 1 and Λ 2 is infinite, we consider families H k analogous to the H k given in (4.32) but defined using the time interval [−h/2, 0] and both red and blue marks. Therefore in the combined set of cylinders there is no non-oriented percolation and we can construct the coupling working in a finite set of H k 's at a time. We then continue working in time intervals of lenght h/2 to reach arbitrary times t.
By construction, the coupling satisfies (4.24). It also satisfies (4.25) because the flag changes from 0 to 1 (and remains 1 forever) the first time a cylinder of A t (Supp (g)) is incompatible with a cylinder of A t (Supp (f )). This implies (4.34)

Branching processes. Time length and space width
In this section we estimate the time-length and space-width of the families of ancestors A x,t . We follow the well known approach of introducing a branching process that dominates the backward percolation process (see eg. Hall, 1985), though we must consider multitype branching. The main result of this section is Theorem 5.27 which shows that the hypotheses of Theorem 2.10 lead to exponential upper bounds of both TL (A x,t ) and SW (A x,t ).

Multitype branching processes
We introduce a multitype branching process B n , in the set of cylinders, which dominates A n .
To do this we look "backwards in time" and let "ancestors" play the role of "branches". In particular, births in the original marked Poisson process correspond to dissapearance of branches.
We reserve the words "birth" and "death" for the original forward-time Poisson process.
We start by enlarging our probability space and defining, for any given set {C 1 , . . . , C k }, independent random sets B C i 1 with the same marginal distribution as A C i 1 . The important point here is that The proof of this fact relies in fixing a way to distribute common ancestors. For example, consider the total order ≺ in the set of cylinders induced by the birth-times. That is C ≺ C ′ if and only if Birth (C) ≤ Birth (C ′ ). For any finite set of cylinders {C 1 , . . . , C k } such that This ensures that A C j 1 are independent sets and On the other hand, for any C, The procedure defined by B 1 naturally induces a multitype branching process in the space of cylinders. We define the n-th generation of the branching process by where for all C ′ , B C ′ 1 has the same distribution as A C ′ 1 and are independent random sets depending only on C ′ . Inductively, Indeed, where in the definition of A C ′ 1 we use {C 1 , . . . , C k } = ∪ n−1 i=0 A C i . Hence, the inductive hypothesis . . , n − 1, yields (5.5).
In consistence with our previous notation, we denote the branching clans of C, (x, t) and Υ (at time 0) respectively. By (5.5), Defining the time length and space width of this clan as in (4.4) and (4.5), we get and similarly for the respective space widths.
The (multitype) branching process B n induces naturally a multitype branching process in the set of contours. For a cylinder C with basis γ and birth-time 0, define b γ n ∈ N G as the number of cylinders in the nth generation of ancestors of C with basis θ: This process will be useful in estimating the space properties of the clans of ancestors. We have the following relationship: The process b n is a multitype branching process whose offspring distributions are Poisson with means To see this, notice that the cylinders C ′ with basis θ that are potential ancestors of C (with basis γ) form a Poisson process of rate 1{γ ∼ θ} e −β|θ| . Each of those cylinders is an ancestor of C if its lifetime is bigger than the difference between the birth-time of C and C ′ . The lifetimes of different cylinders are independent exponentially distributed random variables of rate 1. The probability that the lifetime of any given cylinder is bigger than t is given by e −t . Hence, the birth-times of the ancestors of C with basis θ form a (non homogeneous) Poisson process of rate depending on t given by 1{γ ∼ θ} e −β|θ| e −t . The mean number of births is therefore given by (5.12).
Proof. This lemma shows, in particular, that the branching process b n is subcritical if α < 1.

Continuous-time branching process
Let C be a cylinder with basis γ and birth-time 0. Combing backwards continuously in time the branching clan B C we define a continuous-time multitype branching process ψ γ t (θ) = number of contours of type θ present at time t (of this process) whose initial configuration is δ γ . Each C ′ ∈ B C is a branch, that is, belongs to the first generation of ancestors of a unique cylinder U (C ′ ) in B C . In the branching process ψ t all the branches (ancestors) of U (C ′ ) appear simultaneously at the birth of U (C ′ ), that is when U (C ′ ) dissapears if we look backwards in time. Therefore the part of C ′ in the interval [Birth (U (C ′ )), Death (C ′ )] is ignored. Formally, In the process ψ t , each contour γ lives a mean-one exponential time after which it dies and gives birth to k θ contours θ, θ ∈ G, with probability θ e m(γ,θ) m(γ, θ) k θ k θ ! (5.17) for k θ ≥ 0. These are independent Poisson distributions of mean m(γ, θ). The infinitesimal generator of the process is given by The branching process ψ γ t allows us to estimate the time-length of a clan, due to the obvious fact: Let M t (γ, θ) be the mean number of contours of type θ in ψ t and R t (γ) its sum over θ: The bound we need is given in the next lemma.

Lemma 5.21
The mean number of branches, R t (γ) satisfies Proof. The first inequality is immediate because θ ψ γ t (θ) assumes non-negative integer values and R t (γ) is its mean value.
To show the second inequality we first use the generator given by (5.18) to get the Kolmogorov backwards equations for R t (γ): where m is the matrix with entries m(γ, γ ′ ) and I is the identity matrix. This can be rewritten t n n! |γ| α n , (5.26) where the last bound is just the leftmost inequality in (5.14).

Time length and space width
We are now ready to provide bounds for the time length and space width of the percolation clan.
Proof (i) We follow an idea of Hall (1985). For each C ∈ C we use the domination (5.5) and the identity (5.11). Therefore, to prove that there is no backward oriented percolation it is enough to prove that, for fixed γ P θ b γ n (θ) = 0 for infinitely many n = 0. Hence, Substituting this in (5.38) and using that we get (v) It suffices to use (iv) and the exponential Chevichev inequality and to notice that a must be less than β − β * to avoid a zero in the denominator of (5.30).
Remarks. 1. Part (ii) can in fact be proven by a more elementary argument not requiring the continuous-time construction of Section 5.2. The argument gives the same rate of decay as in (5.28) but a worse leading constant. Let us sketch it.
where S i are independent mean one exponentially distributed random variables and independent of γ i . The time S i represents the period between the birth of γ i and γ i+1 . As the sum of independent exponentials is a gamma distribution with parameters k and 1, (5.43) Therefore (5.42) is bounded by 2. In Fernández, Ferrari and Garcia (1998) we offered an alternative proof of part (iii), based on a the computation of the exponential moment of the total population of a subcritical single-type branching process, which dominates the space width. However this proof works in a smaller range of β.
6 Proof of Theorem 2.4 The following theorem shows that the condition α < ∞ implies the hypothesis of Theorem 3.20.
This proves Theorem 2.4. Proof. Let C be a cylinder with basis γ and birth-time 0. Recall the definition of U (C ′ ) just before display (5.16) and define The process ψ γ t signals all contours born in [0, t] in the process ψ γ t . Notice that for x ∈ γ, We prove that this is finite with probability one by showing it has a finite mean. Indeed, reasoning as in the previous section, Exponential space convergence In view of (4.21) it suffices to bound By (5.31), this is bounded by Exponential mixing We shall use part (4) of Theorem 4.6. We first show, in the next lemma, that α < ∞ implies the existence of an h such that there is non-oriented percolation in the interval (0, h).

Lemma 7.3 For all h > 0 such that
the probability that there is no (non-oriented) percolation in (0, h) is one.
Proof. Analogously to Section 5.1, we dominate the construction of the set ancestors of the non oriented percolation process by a multitype branching process. In this branching process, the mean number of ancestors θ of a contour γ is m(γ, θ) = h 1{θ ∼ γ} e −β|θ| (7.5) As in Lemma 5.13, this branching process is subcritical if α = α h < 1.
We prove now (2.13). From (3.37) and (3.12): By (4.12) it is enough to bound where A(Supp (g)) has the same distribution as A(Supp (g)) but is independent of A(Supp (f )).

This is bounded by
Using the following inequality which is valid for independent random variables S 1 and S 2 , and the exponential decay of (5.31) we get the decay stated in (2.13).

Proof of Theorem 2.17. Poisson approximation
We define first a common probability space where all processes {η β : β * < β ≤ ∞} can simultaneously be constructed. For each γ ∈ G = ∪ j G j let N γ be a marked Poisson process on R d+2 of rate 1. The event points of this process are denoted (u, t, r, s), where u ∈ R d , t, r ∈ R and s ∈ R + . The coordinate t is interpreted as time while the coordinate r is later used to tune the rate of the projected process (u, t). The coordinate s -the mark -is an exponential random variable with mean one independent of everything (used later to determine the lifetime of the corresponding point/cylinder). Denote by P the product measure generated by (N γ : γ ∈ G), and by E the corresponding expectation. We identify the random counting measure N γ with the corresponding discrete random subset of R d+1 × R + .
Fix a contour length j and an inverse temperature β. By counting only those points in N θ whose r coordinate is in [0, e −β(|θ|−j) ] we generate the (d + 1)-dimensional marked process N θ,β = marked Poisson process of rate e −β(|θ|−j) . The life of each point (u, t, s) ∈ N θ,β × R + is the interval [t, t + s].
Define a family of marked point processes indexed by β and τ x θ, x ∈ Z d , θ ∈ G for Borel sets I ⊂ R by Now we use the processes N τxθ,β to perform the graphical construction of Section 3.1. We call C β be the family of cylinders so obtained. Let ξ β t be the free network of Section 3.2. and η β t the loss networks of (3.6). As in (3.8) and (3.37) these processes have invariant distributions µ 0 β and µ β respectively.
The process M γ,∞ is a Poisson process in R d of rate one. This is because the lifetimes are independent exponentials of mean one and for |γ| = j, N γ,∞ is a Poisson process of rate one.