Cluster Expansions for Gibbs Point Processes

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AIEJANSEN SABINE Introduction Contents 1 t ini otnu eso faconver- a of version continuum a is rion 29 0 = tra uuatdniis(correlation densities cumulant ctorial rwt ovretepnin fthe of expansions convergent with er besohsi nerl ihrespect with integrals stochastic uble tosfrtes rnhn processes, branching trees, for ctions sbsdo h Kirkwood-Salsburg the on based is f sao,Frńe n Procacci Fernández and issacot, irpsadpiso partitions, of pairs and tigraphs snaini efcnandand self-contained is esentation ai xasos(ect and (Peccati expansions matic nhn rcse,Boolean processes, anching ,te r end roughly, defined, are they e, uino ib point Gibbs of bution ehnc,sohsi ge- stochastic mechanics, l 18 15 10 39 39 34 33 31 30 3 3 1 9 8 3 2 SABINE JANSEN as modifications of Poisson point processes. The modification involves a factor exp(−H(η)) where H(η) incorporates interactions between points and the magnitude of H captures how far the Gibbs point process might be from the a priori Poisson point process. In infinite volume, Gibbs measures are defined instead by structural properties such as the GNZ equation and the DLR conditions, named after Georgii, Nguyen and Zessin, and Dobrushin, Lanford, and Ruelle, respectively, and proving the mere existence of such measures requires some work. A notorious difficulty when dealing with Gibbs measures is that many quantities cannot be computed explicitly. For example, the intensity measure ρ of a Gibbs point process is a highly non-trivial function of the intensity measure λz of the underlying Poisson point process, which leads to challenges when estimating, say, intensity parameters of the Poisson point process based on observations of the density of the Gibbs point process [BN12]. As a way out, physicists and mathematical physicists have long worked with power series expansions [Rue69, Chapter 4.3]: If interactions are sufficiently weak, then the Gibbs point process should be close to the non- interacting Poisson point process, and correction terms may be captured by convergent power series in the Poisson intensity parameter, called activity or fugacity in statistical mechanics. The expansions obtained in this way are called cluster expansions; the name stems from combinatorial expressions for the expansion coefficients in terms of connected graphs. The mathematical theory of cluster expansions is rich and well-developed [Bry86]. Cluster expansions feature prominently in mathematical physics, and have found applications in combinatorics [SS05, Far10] and random graphs [Yin12]. Their popularity in the community of point processes, however, stays somewhat limited (see, nevertheless, [NPZ13] and the references therein), in stark contrast with expansion techniques from the realm of Poisson-Malliavin calculus [PT11, LP17]. Interestingly, in their seminal article on combinatorics and stochastic integration [RW97], Rota and Wallstrom explicitly mention Feynman diagrams, “physicists aiming at the development of nonlinear quantum field theories,” and “probabilists in search of new point processes that would not turn out to be Poisson distributions in disguise,” but do not mention statistical physics or cluster expansions at all. It is our hope that the present article helps make the theory of cluster expansions accessible to a broader community of probabilists working with point processes. Our first main result is a sufficient criterion for the uniqueness of Gibbs point processes with non-negative pair interactions (Theorem 2.3), accompanied by convergent expansions for the log- Laplace functional of the Gibbs point process as well as its factorial moment and factorial cumulant densities (also known as correlation functions and truncated correlation functions), see Theorems 2.4, 2.5, and 2.6. The existence of such expansions, and their convergence in some non-empty domain, is well-known [Rue69, PU09, NPZ13], however our generalized convergence criterion was previously proven only for discrete polymer systems [FP07] and hard spheres [FPS07]. The results in the present article cover non-negative pairwise interactions only. For pairwise interactions with possibly negative values, many convergence conditions are available in the literature, see e.g. [PU09, PY17] and Section 2.4. Cluster expansions for multi-body interactions are much less developed, some results can be found in [Mor76, PS00, Reb05]. Convergence conditions in the theory of cluster expansions often mirror fixed point equations for generating functions of trees [Far10, FP07, Jan15]. The equations are reminiscent of an equation satisfied by extinction probabilities in branching processes. In Section 2.5 we discuss these relations in more depth and show that if a convergence condition due to Ueltschi [Uel04] holds true, then there is extinction in an associated multi-type branching process (Proposition 2.12). This observation is interesting because in turn, extinction of the branching process implies absence of percolation in a related random connection model [MR96]. In Section 2.6 we discuss these relations in the context of disagreement percolation, an expansion-free method for proving uniqueness of Gibbs measures and exponential mixing [GHM01]. In the future these consideration may help systematize relations between Gibbs point processes and the random connection model, perhaps starting from the duality between hard spheres and Boolean percolation model discussed by Torquato [Tor12], or expansion methods and integral equations for connectedness functions [CDAF77, GS90, LZ17]. CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 3

Our second main result consists in two propositions on moments and cumulants of double stochastic integrals with respect to a random Poisson measure (not compensated). Proposition 2.16 involves pairs of partitions, it is the analogue of similar formulas for multiple stochastic integrals with respect to compensated Poisson measures [PT11]. Proposition 2.14 replaces pairs of partitions with multigraphs, i.e., graphs with multiple edges. The propositions elucidate the relation between cluster expansions and diagrammatic expansions of cumulants used in stochastic geometry [PT11, LPST14]. As this relation provides a modern point of contact between point processes and cluster expansions, let us try to convey the main idea; details are given in Sections 2.7 and 8. Let ηz be a homogeneous Poisson point process in a bounded region Λ ⊂ Rd with intensity parameter z, and v : Rd × Rd → R a symmetric function (v(x, y) = v(y, x)). Consider the random variable (2) (2) Xz := Λ2 vdηz , with ηz the second factorial measure of ηz. The cumulants of Xz, if they exist, are the coeﬃcients κ (X ) in the Taylor expansion R m z βm log E eβXz = κ (X ) (β → 0). m! m z h i mX≥1 The left-hand side is closely related to the so-called grand-canonical partition function from statistical mechanics, given by ∞ zn ΞΛ(β,z) := 1 + exp −β v(xi, xj ) dx n! n n=1 Λ ! X Z 1≤Xi

z|Λ| −βXz/2 ΞΛ(β,z)=e E e . The theory of cluster expansions yields an expansionh of the logarithi m of the partition function in powers of z, of the form ∞ n z −βv(xi,xj ) log ΞΛ(β,z)= z|Λ| + e − 1 dx n! Λn n=2 Z G∈Cn {i,j}∈E(G) X X Y with Cn the connected graphs with vertices 1,...,n, and E(G) the set of edges of G. In order to arrive at an expansion in powers of β rather than z, we need to expand exp(−βv(xi, xj )). This results in a power series in two variables, β and z. The coeﬃcient of znβm is associated with a multigraph with m edges on n vertices, and the relation between cluster expansions and diagrammatic expansions of cumulants is this: Cluster expansions group multigraphs according to their number of vertices, cumulants group multigraphs according to their number of edges. The article is organized as follows: Section 2 provides the setting and formulates the main results. Section 3 is devoted to the proof of the uniqueness theorem, based on the Kirkwood- Salsburg equations. Section 4 addresses the expansions of correlation functions (factorial moment densities), Section 5 deduces expansions for the truncated correlation functions (factorial cumulant densities) and for the log-Laplace functional. The special case of bounded volumes Λ is dealt with in Section 6. Section 7 explains the relation between Kirkwood-Salsburg relations on the one hand and trees and forests on the other hand. In Section 8 we prove the representations for stochastic integrals that we sketched above. Appendix A shows how to go from the GNZ equation to the Kirkwood-Salsburg equation, Appendix B gives a self-contained proof of existence of Gibbs measures adapted to our setup. Appendix C provides some mathematical details for the random connection model discussed informally in Section 2.6.

2. Main results

2.1. Setting. Let (X, dist) be a complete separable metric space, X the Borel σ-algebra, and Xb the collection of bounded Borel sets. Let λ be a reference measure on X that is locally finite, i.e., λ(B) < ∞ for all B ∈ Xb. A locally finite counting measure is a measure η on (X, X ) with η(B) ∈ N0 for all B ∈ Xb. Let N and Nf be the spaces of locally finite and finite counting κ measures, respectively. Each non-zero η ∈ N can be written as η = j=1 δxj with κ ∈ N ∪ {∞} P 4 SABINE JANSEN and x1, x2,... ∈ X. We define equip N with the σ-algebra N generated by the integer-valued κ b random variables η 7→ η(B), B ∈ X . The n-th factorial measure of η = j=1 δxj ∈ N is denoted η(n). It is the unique measure on Xn such that for all measurable non-negative g : Xn → R , we P + have 6= (n) g dη = g(xi1 ,...,xin ) Xn Z i1,...,iXn≤κ with summation over pairwise distinct indices i1,...,in. Fix a non-negative pair potential v, i.e., a measurable function v : X × X → R+ ∪ {∞} that is symmetric, v(x, y)= v(y, x). Set H0 = 0 and

Hn(x1,...,xn)= v(xi, xj ) (x1,...,xn ∈ X) (2.1) 1≤Xi

W (x; η)= v(x, y)dη(y) (x ∈ X, η ∈ N ). X Z Cartesian products such as Xn or X × N are equipped with product σ-algebras, e.g., X ⊗n, X ⊗ N . We adopt the conventions log 0 = −∞, 0 log 0 = 0, 00 = 1, exp(−∞) = 0, and exp(∞)= ∞.

b Deﬁnition 2.1. Let z : X → R+ be measurable map with B zdλ < ∞ for all B ∈ X . A probability measure P on (N , N) is a Gibbs measure with pair interaction v(x, y) and activity z if R −W (x;η) E F (x, η)dη(x) = E F (x, η + δx)e z(x)dλ(x) (GNZ) X X hZ i Z h i for all measurable F : X × N → [0, ∞). The set of Gibbs measures is denoted G (z). The GNZ equation named after Georgii, Nguyen and Zessin is equivalent to the DLR conditions (named after Dobrushin, Lanford, Ruelle) more familiar to mathematical physicists, see [Geo76, NZ79]. It is a probabilistic cousin of the Kirkwood-Salsburg equations [Rue69, Chapter 4.2] for correlation functions recalled in Lemma 3.1 below. The GNZ equation is also intimately related to the detailed balance equation for the Markovian birth and death process with formal generator

−W (x;η) (Lg)(η)= g(η − δx) − g(η) dη(x)+ z(x)e g(η + δx) − g(η) dλ(x), (2.2) X X Z Z sometimes called continuum Glauber dynamics, see [Pre75, KL05] for the construction of such processes and [FFG01, BCC02, KKO13, SS14, FGS16] for some applications. When the interaction vanishes (v ≡ 0), the GNZ equation reduces to the Mecke formula for the Poisson point process with intensity measure λz and, in the context of Poisson-Malliavin calculus the operator L and the generated semi-group are called Ornstein-Uhlenbeck operator and Mehler semi-group [Las16]. The n-th factorial moment density ρn(x1,...,xn), called n-point correlation function in statis- dαn tical mechanics, is the Radon-Nikod´ym derivative ρn = dλn of the n-th factorial moment measure αn of P [DVJ08, LP17]. Thus

(n) n E g dη = gρndλ Xn Xn hZ i Z for all non-negative measurable g. The correlation functions are symmetric, i.e., invariant with respect to permutation of the arguments. For Gibbs measures, they admit the following concrete representation.

Lemma 2.2. Let P ∈ G (z). Then the n-point functions ρn exist and satisfy

n −Hn(x1,...,xn) − P W (xi;η) ρn(x1,...,xn)= z(x1) ··· z(xn)e e i=1 dP(η) (2.3) ZN n n for all n ∈ N and λ -almost all (x1,...,xn) ∈ X . CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 5

The lemma allows us to adopt Eq. (2.3) as the definition of the correlation functions, thus removing indeterminacies on null sets from the definition as a Radon-Nikodým derivative. The lemma is well-known, for the reader’s convenience we present a proof in Appendix A. A simple consequence that we use repeatedly is the estimate

ρn(x1,...,xn) ≤ z(x1) ··· z(xn). (2.4) T The factorial cumulant densities ρn(x1,...,xn), called truncated correlation functions in statistical mechanics, are uniquely defined by the requirement that they are symmetric and satisfy, for all n ∈ N, n r T ρn(x1,...,xn)= ρ#Vk (xi)i∈Vk (2.5) r=1 {V1,...,Vr }∈Pn k=1 X X Y T where Pn is the collection of set partitions of {1,...,n}. For example, ρ1 (x1) = ρ1(x1) and T ρ2 (x1, x2)= ρ2(x1, x2) − ρ1(x1)ρ1(x2). Example (Mixture of hard spheres and Poisson exclusion process with random radii). Let X = d R × R+, equipped with the Euclidean distance, and λ the Lebesgue measure. Consider the interaction v((x, r), (y, R)) := ∞ 1l{|x−y|≤R+r} so that −v((x,r),(y,R)) e = 1l{Bd(x,r)∩Bd(y,R)=∅} d with Bd(x, r) ⊂ R the closed ball of radius r centered at x. Let P ∈ G (z). Any η ∈ N can κ P be written as η = j=1 δ(xj ,rj ), and we have Bd(xi, ri) ∩ Bd(xj , rj ) = ∅ for all i 6= j, -almost surely. The measure describes a mixture of spheres of different radii, distinct spheres are not P d allowed to overlap. P is the distribution of a marked point process on R , with marks in R+. ∞ A priori, the points are Poisson distributed with intensity z0(x) = 0 z(x, r)dr (assuming that each z (x) is finite), and sphere at point x has a random radius with probability density function 0 R r 7→ p(x, r) given by p(x, r)= z(x, r)/z0(x). We can think of P, roughly, as the a priori distribution conditioned on non-overlap of the spheres. The example is interesting because of deep connections between mixtures of hard spheres and Boolean percolation, see [HTH19, Tor12] and Section 2.6. 2.2. Uniqueness and convergent expansions in infinite volume. Our main results are (i) a sufficient condition for the uniqueness of Gibbs measures, and (ii) expansions of the log-Laplace functional, correlation functions and truncated correlation functions (factorial moment and factorial cumulant densities) of the Gibbs measure. Our sufficient condition for the absolute convergence of the expansions generalizes a condition by Fernández and Procacci [FP07] for hard-core interactions and discrete spaces X to non-negative interactions and more general spaces X. Define Mayer’s f-function f(x, y) := e−v(x,y) − 1 (x, y ∈ X). Our first result is a sufficient criterion for uniqueness of Gibbs measures. The criterion also implies convergence of the expansions and analyticity of generating functions. The reader primarily intereted in uniqueness should thus be warned that the condition might be way too strong for his needs and other approaches may work in a bigger domain, see the discussion in [FFG01].

2 2 b Theorem 2.3. Assume f(x, y) ≤ 0 for λ -almost all (x, y) ∈ X and B zdλ< ∞ for all B ∈ X . Suppose that there exists a measurable function a : X → R and some t> 0 such that + R ∞ tk k k e a(yj ) k a(x0) |f(x0,yj )| (1 + f(yi,yj)) z(yj)e dλ (y) ≤ e − 1 (FPt) k! Xk j=1 j=1 Xk=1 Z Y 1≤i

t a(y) e |f(x0,y)|e z(y)dλ(y) ≤ a(x0) (KPUt) X Z 6 SABINE JANSEN for λ-almost all x0 ∈ X. Condition (KPUt) with t = 0 is discussed in depth in [Uel04]; it is a continuum version of the Kotecký-Preiss condition for polymer systems [KP86]. Condition (KPUt) with t > 0 is a special case of Assumption 3 in [Pog13], who considered both non-negative and attractive potentials. Condition (FPt) with t = 0 corresponds to the criterion for discrete polymer models by Fernández and Procacci [FP07] and for hard spheres by Fernández, Procacci and Scoppola [FPS07]. Example (Hard spheres / Poisson exclusion process). Choose X = Rd with the Euclidean norm | · | and the Lebesgue measure λ = Leb. Let v(x, y)= ∞1l{|x−y|≤r} with r> 0. Suppose that z Leb(B(0, r)) < 1/e ≃ 0.36787 (2.6)

Because of 1/e = maxa>0 a exp(−a), we can ﬁnd a,t > 0 such that

et |f(x, y)|eazdy =etLeb(B(0, r))ea ≤ a Rd Z and conditions (KPUt) and (FPt) holds true for some constant function a(y) = a. The criterion (2.6) is well-known [Rue69, Chapter 4]. As shown in [FPS07], for a two-dimensional gas of hard spheres, condition (FPt) allows one to improve (2.6) to z Leb(B(0, r)) < 0.5107.

The condition (FPt) is enough to ensure that generating functionals and correlation functions admit convergent expansions; for homogeneous models (constant activity z(x)= z), the expansions are power series in z. Furthermore the expansion coeﬃcients can be expressed as sums over weighted graphs. Let Gn be the collection of graphs G = (V, E) with vertex set V = {1,...,n} = [n] and edge set E ⊂ {{i, j} | i, j ∈ V, i 6= j}. We write E(G) = E for the edge set of G. We deﬁne graph weights

w(G; x1,...,xn) := f(xi, xj ) (2.7) {i,j}∈YE(G) Let Cn ⊂ Gn be the collection of connected graphs with vertex set [n] and

T ϕn(x1,...,xn) := w(G; x1,...,xn) (2.8) GX∈Cn the n-th Ursell function. Let λz be the measure on X that is absolutely continuous with respect to λ with Radon-Nikod´ym derivative z(·). For h : X → R ∪ {∞}, we introduce the integrability condition −h(x) a(x) |e − 1|e dλz(x) < ∞. (2.9) X Z Theorem 2.4. Under the conditions of Theorem 2.3, the log-Laplace functional of the unique Gibbs measure P ∈ G (z) satisﬁes

∞ 1 n T − RX hdη − Pj=1 h(xj ) n log E e = e − 1 ϕn(x1,...,xn)dλz (x) (2.10) n! Xn n=1 Z h i X for all measurable h : X → [−t, ∞) ∪ {∞} or h : X → {s ∈ C | Re s ≥ −t} that satisfy (2.9). Moreover ∞ t(n−1) e T a(x1) |ϕn(x1,...,xn)|dλz(x2) ··· dλz(xn) ≤ e − 1 (2.11) (n − 1)! Xn−1 n=2 X Z for λ-almost all x1 ∈ X. The convergence of the right-hand side of (2.10) follows from the inequality (2.11) and the estimate

n n − n h(x ) −h(x ) − j−1 h(x ) (n−1)t −h(x ) e Pj=1 j − 1 = (e j − 1)e Pi=1 i ≤ e e j − 1 . (2.12) j=1 j=1 X X

CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 7

Remark (Cumulants). Complex parameters allow us to extract bounds on cumulants via contour integrals. For example, let Λ ∈ X . Applying Theorem 2.4 to −h = sNΛ with |s|≤ t, we obtain ∞ κℓ log E esNΛ = sℓ ℓ! h i Xℓ=1 with 1 ds ℓ! E sNΛ a |κℓ| = ℓ! log e ℓ+1 ≤ ℓ e dλz. 2πi s t Λ I h i Z See [Bry93] for applications to central limit theorems.

Remark (Signed L´evy measure). Eq. (2.10) can be rewritten as

log E e− RX hdη = (e− RX hdη − 1)dΘ(η) (2.13) Nf h i Z where the signed measure Θ on Nf is defined by ∞ 1 T n gdΘ = g(δx1 + ··· + δxn )ϕn(x1,...,xn)dλz (x). (2.14) n! Xn Nf n=1 Z X Z Log-Laplace transforms such as (2.13)—but with non-negative measures Θ—appear naturally in the context of cluster point processes and infinitely divisible point processes [DVJ08, Chapter 10.2]. Following Nehring, Poghosyan and Zessin [NPZ13] we may call Θ the pseudo- or signed Lévy measure of P (compare Eq. (2.17) below). Theorem 2.5. Under the conditions of Theorem 2.3, the truncated correlation functions (factorial cumulant densities) of the unique Gibbs measure P ∈ G (z) satisfy T ρn(x1,...,xn) n ∞ T 1 T k = z(xj ) ϕn(x1,...,xn)+ ϕn+k(x1,...,xn,y1,...,yk)dλz (y) (2.15) k! Xk j=1 ! Y Xk=1 Z n n for all n ∈ N and λ -almost all (x1,...,xn) ∈ X , with ∞ 1 T (n − 1)! a(x1) |ϕn(x1,...,xn+k)|dλz(x2) ··· dλz(xn+k) ≤ t n−1 e . (2.16) k! Xn−1+k (e − 1) Xk=0 Z The theorem is proven in Section 5. The bound (2.16) proves the convergence of the expansion (2.15) in a suitable L1-norm, in agreement with the spaces used by Kuna and Tsagkaro- T giannis [KT18] and Last and Ziesche [LZ17] for the two-point function ρ2 . Pointwise estimates better suited to estimating decay of correlations and mixing properties can be found in [Uel04], see also [Han18, Corollary 5.2] and the references therein for the two-point function. Remark (Positivity of t). For the bound (2.16) the strict positivity of t is essential. Something similar happens for the factorial cumulants of random variables: Let N be an N0-valued random variable that is infinitely divisible with finite Lévy measure ν = n∈N νnδn, then ∞ −λN −λn P log E e = (e − 1)νn (λ ≥ 0). (2.17) n=1 X The factorial cumulants (αn)n∈N, if they exist, are defined as the coefficient in the asymptotic expansion ∞ α n log E e−λN = n e−λ − 1 (λ ց 0). n! n=1 n X t N If the series n νnr has a radius of convergence e > 1, then u 7→ log E[(1 + u) ] is analytic in t n |u| < e − 1, which leads to bounds on its Taylor coefficients αn. If on the other hand νnr P n diverges for r> 1, then nothing can be said about the αn’s without additional information on the P νn’s. 8 SABINE JANSEN

The correlation functions are associated with graphs subject to slightly weaker connectivity constraints. For k ≤ n, let Dk,n ⊂ Gn be the set of multi-rooted graphs G with vertices 1,...,n, roots 1,...,k, such that every non-root vertex j ∈{k +1,...,n} is connected to some root vertex i ∈{1,...,k} by a path in G. Notice Dn,n = Gn and D1,n = Cn. Set

ψk,n(x1,...,xn)= w(G; x1,...,xn). (2.18) G∈DXk,n Theorem 2.6. Under the conditions of Theorem 2.3, the correlation functions (factorial moment densities) of the unique Gibbs measure P ∈ G (z) are given by

ρn(x1,...,xn) n ∞ 1 k = z(xj ) ψn,n(x1,...,xn)+ ψn,n+k(x1,...,xn,y1,...,yk)dλz (y) (2.19) k! Xk j=1 ! Y kX=1 Z with

∞ tk e k ψn,n(x1,...,xn) + ψn,n+k(x1,...,xn,y1,...,yk) dλz (y) k! Xk k=1 Z X n Pi=1 a(xi) ≤ (1 + f(xi, xj ))e . (2.20) 1≤i

Let us briefly comment on the order in which Theorems 2.3, 2.4, 2.5 and 2.6 are proven. Theo- rem 2.3 about the uniqueness of Gibbs measures is proven first with the help of Kirkwood-Salsburg equations, in Section 3. Theorem 2.6 is proven next, in Section 4, building on the Kirkwood- Salsburg equations introduced in Section 3. The representation for one-point correlation functions is then used to prove Theorem 2.4 in Section 5, and finally Theorem 2.5 is deduced from Theo- rem 2.4, also in Section 5.

2.3. About the case t =0. The strict positivity t> 0 in condition (FPt) ensures that a certain −t linear operator Kz has operator norm ||Kz|| ≤ e < 1, leading to the uniqueness of a fixed point problem underpinning the uniqueness of infinite-volume Gibbs measure (see Section 3). If t = 0, then the contractivity of the operator is lost and our approach no longer guarantees uniqueness of the infinite-volume Gibbs measure. Nevertheless, the condition (FPt) with t = 0 stays useful. Roughly, the condition ensures that the cluster expansions for the correlation functions of Gibbs measures in finite volume Λ (with empty boundary conditions) are absolutely convergent, uniformly in the volume Λ. The uniformity in the volume guarantees the existence of the infinite-volume limit. This reasoning is fairly standard in mathematical physics. Let Λ ∈ Xb be a bounded set (thus λz(Λ) < ∞, “finite volume”). Assume λz(Λ) > 0. Let NΛ = {η ∈N |∀A ∈ X : η(A)= η(A ∩ Λ)} be the space of point configurations in Λ. The space NΛ is equipped with the trace of the σ-algebra N. It is not too difficult to see that there is a uniquely defined measure PΛ such that

−W (x,η) EΛ F (x, η)dη(x) = EΛ F (x, η + δx)e z(x)dλ(x) Λ Λ hZ i Z h i for all measurable F :Λ×N → [0, ∞). Moreover, the measure PΛ is absolutely continuous with respect to the distribution of the Poisson point process with intensity measure z1lΛ dλ, with a Radon-

Nikod´ym derivative proportional to exp(−H(η)), where H(δx1 + ···+ δxn )= Hn(x1,...,xn). The measure PΛ is usually called a ﬁnite-volume Gibbs measure with empty boundary conditions (because H(η) does not include interactions with points outside Λ). We formulate analogues of Theorems 2.4 and 2.6 before commenting on Theorem 2.3. As explained earlier, Theorem 2.5 really needs the condition t > 0 and it has no direct counterpart for t = 0 (see however [Uel04]). CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 9

2 2 b Theorem 2.7. Assume v(x, y) ≥ 0 for λ -almost all (x, y) ∈ X and B zdλ< ∞ for all B ∈ X . Suppose that there exists a measurable function a : X → R+ such that condition (FPt) holds true with t =0. Then R ∞ 1 T a(x0) |ϕn(x0,...,xn)|dλz(x1) ··· dλz(xn) ≤ e − 1 (2.21) n! Xn n=1 X Z for λ-almost all x0 ∈ X. Moreover, for every non-empty Λ ∈ Xb, and all measurable h :Λ → [0, ∞] −h a (or h complex-valued with non-negative real part) with Λ |e − 1|e dλz < ∞, we have ∞ 1 n R T − R hdη − Pj=1 h(xj ) n log EΛ e Λ = e − 1 ϕn(x1,...,xn)dλz (x). (2.22) n! n n=1 ZΛ h i X Theorem 2.8. Under the conditions of Theorem 2.7, the bound (2.20) holds true for t = 0 and n integration over X , and the n-point functions ρn,Λ of PΛ are given by (2.19) with integrals over Λn instead of Xn. Theorem 2.7 and 2.8 are proven in Section 6. Notice that the convergence estimate (2.21) really has integrals over Xn (not Λn), and the expansions for the log-Laplace transform and correlation functions depend on Λ only through the domains of integration. As a consequence, we can pass to the infinite-volume limit: Let (Λn)n∈N be a sequence of non-empty sets in Xb with Λn ր Λ. Then for all measurable h :Λ → [0, ∞] (or h complex-valued with non-negative real part) with −h a X |e − 1|e dλz < ∞, we have ∞ R − R hdη 1 − n h(x ) T n E Λn Pj=1 j lim log Λn e = e − 1 ϕn(x1,...,xn)dλz (x). (2.23) n→∞ n! Xn n=1 Z h i X A similar convergence for correlation functions ρn,Λn holds true as well, the precise statement is left to the reader. In view of (2.23), we should expect that the sequence (PΛn )n∈N converges in some suitable sense to a uniquely defined probability measure P on (N , N). The measure P should be in G (z) and its log-Laplace functional (restricted to the functions h under consideration) should be given by the right-hand side of (2.23). A rigorous statement and proof are beyond the scope of this article, but we emphasize that the uniqueness of the infinite-volume limit of Gibbs measures with empty boundary conditions replaces the uniqueness in Theorem 2.3. Mathematical physicists often work with the grand-canonical partition function ∞ 1 −Hn(x1,...,xn) n ΞΛ(z)=1+ e dλz (x). n! n n=1 Λ X Z instead of log-Laplace transforms.

b a Theorem 2.9. Under the conditions of Theorem 2.7, if Λ ∈ X with Λ e dλz < ∞, then ∞ 1 T n R log ΞΛ(z)= ϕn(x1,...,xn)dλz (x) n! Xn n=1 X Z with absolutely convergent integrals and series. a The theorem is proven in Section 6. For Λ e dλz < ∞ and measurable, non-negative h, we have − R hdη −h EΛ e Λ R= log ΞΛ(z e ) − log ΞΛ(z), and the formula (2.22) is easilyh recovered.i 2.4. Attractive pairwise interactions. The results proven in this article cover non-negative pairwise interactions only, for the reader’s convenience we summarize some known results on attractive pairwise interactions. Let v : X × X → R ∪ {∞} with v(x, y)= v(y, x) on X × X. The pair potential v is called stable [Rue69, Chapter 3.2] if for some constant B ≥ 0 and all n ∈ N and x1,...,xn ∈ X, we have v(xi, xj ) ≥−Bn. 1≤Xi

Sometimes it is more convenient to allow for x-dependent B, and replace the previous condition by n v(xi, xj ) ≥− B(xi) (2.24) i=1 1≤Xi

v(x, y)dη(y) ≥−C(x) (2.25) X Z k for some function C : X → R+ and all finite configurations η = i=1 δyi such that η + δx has finite total energy. Then for all n ∈ N and x1,...,xn ∈ X, we have P n n 1 1 v(x , x )= v(x , x ) ≥− C(x ). i j 2 i j 2 i 1≤i

−|v| B(y)+a(y) 1 − e e dλz(y) ≤ a(x) (2.26) X Z for some measurable function a : X → R+ and λ-almost all x ∈ X. Then ∞ 1 T n B(q) a(q) ϕn(q, x1,...,xn) dλz (x) ≤ e e − 1 n! Xn n=1 Z X for λ-almost all q ∈ Xn, and a suitable analogue of Theorem 2.9 (hence, Theorems 2.7 and 2.8) holds true. The proof in [PY17] is written for X = Rd, λ the Lebesgue measure, and translationally invariant pair potentials, building on a novel tree-graph inequality. The result is easily extended to the general setup by using the formulation of the tree-graph inequality from [Uel17]. For further results on attractive pairwise interactions, see [PU09] and the references therein. The condition (2.26) extends (KPUt) for t = 0; we leave as an open problem the extension of (FPt) with t = 0 to stable pair potentials. We also leave as an open question the extension of (2.26) to t > 0, i.e., whether multiplying the left-hand side of (2.26) with et for some x- independent t> 0 yields a suﬃcient condition for the uniqueness of the inﬁnite-volume Gibbs measure as in Theorem 2.3, and mention instead that related uniqueness criteria based on Kirkwood- Salsburg equations are available [Rue69, Chapter 4.2].

2.5. Trees and branching processes. One of the standard techniques to prove convergence of cluster expansions is by tree-graph estimates: sums over graphs are estimated by sums over trees with the help of tree partition schemes, see [FP07, PU09] and the references therein. Even though we follow another standard route, the method of integral equations, it is helpful to interpret the convergence conditions in the context of trees: Propositions 2.10 and 2.11 below say that the convergence conditions are in fact equivalent to the convergence of certain generating functions for trees—convergence conditions mirror ﬁxed point equations for trees. This observation is not entirely new [FP07, Far10] but usually the focus is on suﬃcient convergence conditions and the equivalence is rarely explicitly stated (see, however, [Jan15]). Trees are interesting because they are related to branching processes: the main result of this section says that the convergence condition for tree generating functions (and cluster expansions) as formulated by [Uel04] implies extinction of a related multi-type, discrete-time branching process with type space X. The branching process is of interest because extinction of the branching process implies absence of percolation in a random connection model, see Section 2.6. CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 11

2.5.1. Convergence of generating functions for trees. Let Tn be the set of trees with vertex set • ◦ [n]= {1,...,n}, Tn = Tn × [n] the rooted trees with vertex set [n], and Tn the set of trees with vertex set {0, 1,...,n}. The notation with black and white circles is borrowed from the theory of combinatorial species: black circles refer to roots or the operation of “pointing” [BLL98], white circles to “ghosts” that do not come with powers of z in the generating functions. Define a family ◦ (Tq )q∈X of generating functionals ∞ ◦ 1 Tq (z) := 1 + |w(T ; x0,...,xn)|dλz (x1) ··· dλz(xn), x0 := q (2.27) n! Xn n=1 ◦ X Z TX∈Tn and ∞ n • 1 Tq (z) := z(q)+ |w(T ; x1,...,xn)| z(xi)dλ(x1) ··· dδq(xr) ··· dλ(xn). (2.28) n! Xn n=1 • i=1 X Z (T,rX)∈Tn Y Notice • ◦ • Tq (z)= z(q)Tq (z)= z(q)exp |f(q,y)|Ty (z)dλ(y) (2.29) X Z see Faris [Far10, Section 3.1]. In Eq. (2.29), both sides are either infinite or finite. If they are • finite, the fixed point problem may have more than one solution, and Tq (z) is the smallest solution [Far10].

Proposition 2.10. Let z : X → R+ be a measurable map. The following two conditions are equivalent: ◦ (i) Tq (z) < ∞ for all q ∈ X. a(y) (ii) There exists a measurable function a : X → R+ such that X |f(q,y)|e dλz(y) ≤ a(q) for all q ∈ X. R Proof. The implication (ii) ⇒ (i) is proven by induction as in [PU09]; see also [Far10, Theorem 3.1]. The implication (i) ⇒ (ii) follows from the ﬁxed point equation (2.29), which proves that ◦ a(q) := log Tq (z) is an admissible choice. • Let (T, r) ∈ Tn . The root r induces a notion of generations and descendants. For k ∈ T , let (r) Ck ⊂ [n] be the collection of children of k. Set

w˜(T, r; x1,...,xn)= |f(xk, xi)| (1 + f(xi, xj )) . k∈T i∈C(r) i,j∈C(r) : Y Yk Yk i

12 SABINE JANSEN which can be proven as in [FP07]. 2.5.2. Extinction probabilities in branching processes. It is instructive to compare the tree generating functions with extinction probabilities of branching processes. For simplicity we treat the ◦ function Tq only. Assume that

b(q)= |f(q,y)|dλz(y) < ∞ (2.31) X Z q for all x ∈ X. Let ξ be a Poisson point process with intensity measure |f(q; y)|dλz(y), deﬁned on some probability space (Ω, F, P). It has Laplace functional

q − RX hdξ −h(x) E e = exp (e − 1)|f(q, x)|dλz(x) , X h i Z the condition (2.31) ensures that the expected number of points b(q) of ξq is finite hence in particular, ξq has only finitely many points, P-almost surely. q We can define a discrete-time, multi-type branching process (ηn)n∈N with type space X and q q q ancestor q as a Markov chain with state space N as follows: η0 := δq, η1 := ξq, and ηn+1 is q q obtained from ηn by asking, roughly, that each point x ∈ ηn, has an offspring equal in distribution x q to ξ , with the usual independence assumptions. Precisely, (ηn)n∈N0 is a Markov chain with

q − RX hdηn+1 q −h(y) q E e ηn = exp |f(x, y)|(e − 1)dλz(y) dηn(x) P-a.s. X X ! h i Z Z q for all n ∈ N0. For B ∈ Xb, the quantity ηn(B) represents the number of points in generation n with type in B. The dependence on z is suppressed from the notation of the branching process. The extinction probability q p(q) := P ∃n ∈ N : ηn =0 as a function of q, is the smallest non-negative solution of the ﬁxed point problem

p(q) = exp (p(x) − 1)|f(q, x)|dλz(x) (q ∈ X). (2.32) X Z I.e., ifp ˜(·) is another non-negative solution, then p(q) ≤ p˜(q) for all q ∈ X. For finite type spaces, this statement is proven in [Jag75, Theorem 4.2.2], the proof for infinite X is similar. The similarity of the fixed point equations (2.32) and (2.29) for extinction probabilities and trees suggests a relation between convergence and extinction. The next proposition says that convergence of tree generating functions implies extinction of the branching process.

Proposition 2.12. Assume X |f(q, x)|dλz(x) < ∞ for all q ∈ X and let p(q) be the extinction probability of the z-dependent branching process with ancestor δ . If T ◦(z) < ∞ for all q ∈ X, R q q then p(q)=1 for all q ∈ X. The proof adapts a classical result [Har63, Chapter III, Theorem 12.1]: if the principal eigenvalue of the so-called expectation operator is smaller or equal to 1 and some additional regularity conditions hold true, then the branching process goes extinct.

Proof of Proposition 2.12. Consider the kernel on X given by M(q, dx) := |f(q, x)|λz(dx) and use the same letter for the associated integral operator

∞ (Mg)(q) := |f(q, x)|g(x)dλz (x) q ∈ X, g ∈ L (X, X ) . X Z M(q,B) represents the expected number of children with type in B of a type-q individual. The expectation operator M replaces the mean number of offspring in a single-type branching process. Condition (2.31) guarantees that the expected value M(q, X) of the total number of children of a type-q individual is finite for all q ∈ X. However supq∈X M(q, X) = supq∈X b(q) can be infinite, which is why Theorem 12.1 in [Har63, Chapter III.12] is not applicable; moreover the image M of a bounded function g is not necessarily bounded. Therefore it is preferable to work with CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 13

◦ weighted supremum norms. By the convergence of Tq (z) and Proposition 2.10, there is a function a : X → R+ such that X |f(q, x)| exp(a(x))dλz (x) ≤ a(q) for all q. We deﬁne −a(q) R ||g||a := sup |g(q)|e q∈X and note that M is a contraction with respect to this norm. Indeed, if g is a function with ||g||a < ∞, then

a(x) (Mg)(q) ≤ |f(q, x)|e ||g||adλz(x) ≤ a(q)||g||a X Z and −a(q) −α 1 ||Mg||a ≤ ||g||a sup a(q)e ≤ ||g||a sup α e = ||g||a. q∈X α≥0 e The ﬁxed point equation for the extinction probability and the inequality exp(s) ≥ 1+ s yield

p(q) ≥ 1+ |f(q, x)|(p(x) − 1)dλz(x)=1+ M(p − 1) (q) X Z with 1 the constant function with value 1. Consequently 1 − p ≤ M(1 − p) pointwise on X and 1 ||1 − p|| ≤ ||M(1 − p)|| ≤ ||1 − p|| . a a e a

It follows that ||1 − p||a = 0 and p(q) = 1 for all q ∈ X. Below we provide an example for which the branching process goes extinct but the tree generating functions diverge. First we have a closer look at the relation between trees and extinction probabilities. Since every individual has only finitely many children by (2.31), the extinction probability q ∞ q p(q) is equal to the probability that the total offspring N∞ := n=0 ηn(X) is finite (to simplify formulas below, we include the ancestor in the offspring count). The probability that the total offspring consists of exactly n individuals is in turn a sum overP trees on n vertices, rooted at the ancestor. Leaves correspond to individuals without offspring. Hence

q p(q)= P(N∞ = n) (2.33) N nX∈ 0 ∞ n −z(q)b(q) 1 −z(xi)b(xi) n =e 1+ |w(T ; q, x1,...,xn)| e dλz (x) n! Xn n=1 ◦ i=1 X Z TX∈Tn Y −z(q)b(q) ◦ −bz =e Tq (ze ). (2.34) q ∞ q More generally, let η := n=0 ηn be the total progeny of δq, including the ancestor itself. Then for all h : X → R, we have P q h−bz ◦ h−bz RX hdη e Tq (ze )= E e 1l{NX(ηq )<∞} . and h i q ◦ RX bzdη Tq (z)= E e 1l{NX(ηq )<∞} . h ◦ i This equality together with Proposition 2.12 shows that Tq (z) is finite for all q ∈ X if and only if the branching process goes extinct for all possible ancestor choices and in addition the exponential moment above of the total progeny is finite. Example (Galton-Watson process and hard spheres). Suppose that z(q) = z and b(q) = b are independent of q. This is the case, for example, for a gas of hard spheres of fixed radius R in d q R , and then b = |B(0, 2R)|, compare Example 2.2. Then the generation counts Nn := NX(ηn), n ∈ N0, form a Galton-Watson process with offspring distribution Poi(zb). The distribution of the ∞ total progeny (including ancestor) N = n=0 Nn satisfies (bn)n−1 P N = n = P zn−1e−nbz (n ∈ N), n! 14 SABINE JANSEN compare the Borel distribution [Bor42]. The tree generating function becomes ∞ (bn)n−1 T ◦(z)= zn−1 = enbzP(N = n). n! n=1 N X nX∈ We have T ◦(z) < ∞ if and only if bz ≤ 1/e, a condition stronger than subcriticality bz ≤ 1. We leave open whether the analogue of Proposition 2.12 holds true for the Fernández-Procacci trees from Proposition 2.11, but sketch how some first steps might be implemented. Define n Bn(q; x1,...,xn) := |f(q; xi)| 1+ f(xi, xj ) i=1 1≤i

−1 is translationally invariant, then condition (KPUt) reads z < (e C(β)) with C(β) = Rd (1 − exp(−βv(0, x)))dx. Clearly lim C(β) = 0 hence for small β > 0, the condition z < (e C(β))−1 β→0 R is less restrictive than the condition z

2.7. Cumulants of double stochastic integrals. Here we explain how cluster expansions relate to another kind of diagrammatic expansion, namely, expansions of cumulants of multiple stochastic integrals with respect to compensated Poisson random measures, see [PT11, Chapter 7], [LP17, Chapter 12.2], and [LPST14]. To that aim we provide two expressions for the cumulants of random (2) variables of the form X2 udη where η is a Poisson point process with intensity measure λz and u : X×X → R is a function that satisfies some integrability conditions. We may view such random variables as double integralsR with respect to a Poisson random measure η (not compensated). The first formula involves edge-labelled graphs with multiple edges (Proposition 2.14), the second formula involves precisely the pairs of partitions (Proposition 2.16) that appear in the literature on multiple stochastic integrals [PT11]. To the best of our knowledge, these formulas are new, however our principal interest lies in the reasoning that allows us to go from the connected graphs of cluster expansions to cumulants and pairs of partitions, via multigraphs. Roughly, the n-th coefficient of the cluster expansion is a sum over graphs on n vertices while the m-th moment of a double stochastic integral is a sum over multigraphs with m edges. We start from the following observation. Let u : X × X → R and β ∈ R. Suppose that v = −βu satisfies the assumptions of Theorem 2.9. Then the latter theorem provides an expansion of the 1 (2) cumulant generating function of 2 β X2 udη , with η a Poisson point process of intensity λz. The expansion is not in powers of β but rather, if z(x) ≡ z is independent of x, in powers of z. It is a R sum over connected graphs, each graph comes with a product of edge weights exp(βu(xi, xj )) − 1. In order to obtain the cumulants, we need to understand the expansion in powers of β rather than z. Now, every edge weight is expanded as ∞ βmij eβu(xi,xj ) − 1= u(x , x )mij . m ! i j m =1 ij Xij The right-hand side is best interpreted as a sum over edge multiplicities, and thus graphs with multiple edges (but no self-edges i − i) naturally appear. It is convenient to label not only the vertices but also the edges.

Deﬁnition 2.13. Let A and V be two non-empty sets and E2(V )= {e ⊂ V | #e =2}.

• A labelled multigraph γ with vertex labels V and edge labels A is a map from A to E2(V ). The set of such multigraphs is denoted M(V, A). 16 SABINE JANSEN

• The multiplicity of an edge {i, j}∈E2(V ) in γ ∈M(V, A) is mij (γ) = #{a ∈ A | γ(a)= {i, j}}. • A multigraph γ is spanning if every vertex i ∈ V belongs to some edge in γ, i.e., mij (γ) ≥ 1 for some j ∈ V . The set of spanning multigraphs is denoted Ms(V, A). • A multigraph γ is connected if, for all i, j ∈ V , there exist k ∈ N and a sequence k (a1,...,ak) ∈ A such that i ∈ γ(a1), j ∈ γ(ak), and γ(ar) ∩ γ(ar+1) 6= ∅ for all r ∈{1,...,k − 1}. The set of connected multigraphs is denoted Mc(V, A).

Proposition 2.14. Let η be a Poisson point process with intensity measure λz and u : X×X → R a symmetric function. Suppose that

mij (γ) n u(xi, xj ) dλ (x) < ∞ (2.35) Xn Z 1≤i

∞ m ∞ 1 (2) β 1 2 β RX2 udη mij (γ) n log E e = u(xi, xj ) dλz (x). (2.36) m! n! Xn m=1 n=2 h i X X Z γ∈MXc([n],[m]) 1≤i

Since the cumulants are deﬁned by the relation

∞ m 1 (2) β 1 β RX2 udη (2) log E e 2 = κm udη m! 2 X2 m=1 h i X Z we may read them oﬀ from Eq. (2.36) and obtain the expression from Proposition 2.14.

Next we explain how to go from multigraphs to partition pairs so as to obtain expressions closer to [PT11]. Let γ ∈ Ms(V, A) be a spanning multigraph on V with edge labels A. We define an associated pair (π, σ) of partitions as follows: • First we define a new set S of “dedoubled” vertices: every vertex v ∈ V gives rise to as many points in S as there are edges to which it belongs. The new vertices s are labelled by the parent vertex v ∈ V and the edge label a. Precisely, we set S := {(v,a) ∈ V × A | v ∈ γ(a)}. • The partition π has the blocks Ba = {(v,a) | v ∈ γ(a)}, a ∈ A: it groups dedoubled vertices s connected by an edge. Every block Ba has cardinality exactly 2. • The partition σ lumps together new vertices s that come from a common underlying vertex v ∈ V . Put differently, the blocks of σ are the sets Tv = {(a, v) | v ∈ γ(a)}, v ∈ V . Note that the Tv’s are non-empty because the graph is spanning.

Notice that for all a ∈ A, v ∈ V , the set Ba ∩ Tv is a singleton if the vertex v belongs to the edge with label a, and empty if it does not; in particular Ba ∩ Tv is either empty or a singleton. If the multigraph γ is connected, then so is (π, σ), in the sense given below.

Definition 2.15 ([PT11]). Let S be a finite non-empty set and (π, σ) ∈ P(S) ×P(S) a pair of set partitions of S. • The pair is non-flat if for all blocks B ∈ π and T ∈ σ, the intersection B ∩ T is either empty or a singleton. • The pair is connected if for every strict subset M ( S, there is a block B of π or σ such that B intersects both M and S \ M.

Equivalently, a pair (π, σ) is non-ﬂat and connected if π ∧ σ is the partition into singletons and π ∨ σ is the partition consisting of a single block S, where ∧ and ∨ refer to the join and meet in the lattice of set partitions.

Remark (Gaussian diagrams). The pair of partitions (π, σ) is not only non-ﬂat and connected, but also has the property that every block of π has cardinality exactly 2, because we only allow for graphs with edges {i, j} (hypergraphs associated with multi-body interactions would include hyperedges consisting of 3 vertices or more). Peccati and Taqqu [PT11] call pairs where instead σ has only blocks of cardinality 2 Gaussian, and associate graphs with such pairs as well; their graphs allow for self-edges and restrict the degree of the vertices to 2, a type of graphs clearly diﬀerent from ours.

Let S be a ﬁnite set andσ ˆ = (T1,...,Tn) an ordered set partition of S with n blocks; let σ = {T1,...,Tn}∈P(S) be the underlying set partition. For s ∈ S, let i(s) ∈ {1,...,n} be the label of the block to which s belongs, i.e., s ∈ Ti(s). The ordered partitionσ ˆ induces an embedding n S n of X into X deﬁned by (x1,...,xn) 7→ (xi(s))s∈S. With any given map g : X → R we associate S a new map gσˆ : X → R by gσˆ(x1,...,xn) := g((xi(s))s∈S ). For example, if S = {1, 2, 3} and σˆ = ({1, 3}, {2}), then gσˆ(x1, x2)= g(x1, x2, x1). Changing the order of the blocks inσ ˆ permutes n the variables in gσˆ but leaves the integral Xn gσˆdλz unchanged, by a slight abuse of notation we write g dλn instead. Xn σ z R In the following proposition the m-fold tensor u ⊗···⊗ u is the function (x ,...,x ) 7→ R 1 2m u(x1, x2)u(x3, x4) ··· u(x2m−1, x2m), and ||σ|| is the number of blocks of a partition σ. 18 SABINE JANSEN

Proposition 2.16. Let Sm = {1,..., 2m}, and πm = {{1, 2}, {3, 4},..., {2m − 1, 2m}}. Under (2) the assumptions of Proposition 2.14, the m-th cumulant of X2 udη is given by

(2) R n κm udη = (u ⊗···⊗ u)σ(x1,...,x||σ||)dλz (x) X2 X||σ|| Z σ∈P(Sm): Z (πm,σ) non-ﬂat,X connected with absolutely convergent integrals. The proposition is proven in Section 8. It is deduced from Proposition 2.14 and the correspondence between connected multigraphs and non-ﬂat connected pairs of partitions. Some combinatorial subtleties arise because the correspondence is not exactly one-to-one; this is also 1 (2) (2) the reason why Proposition 2.14 looks at 2 X2 udη while Proposition 2.16 deals with X2 udη . Proposition 2.16 should be contrasted with a similar expression for the cumulants of the random variable R R

(2) I2(u)= udη − u d(η ⊗ λz) − u d(λz ⊗ η)+ ud(λz ⊗ λz). X2 X2 X2 X2 Z Z Z Z The cumulants of I2(u) are given by sums over non-ﬂat, connected diagrams (πm, σ) such that every block of σ has cardinality at least 2, see [PT11, LPST14]. This corresponds to connected multigraphs for which every vertex i ∈ V has degree j∈V mij (γ) ≥ 2. We leave open whether the known formulas for the cumulants of I (u) have a simple explanation P 2 in terms of cluster expansions. Regardless of the answer, the considerations in this section show that seemingly diﬀerent types of expansions can be put on a common footing, which might yield interesting insights in the future.

3. Kirkwood-Salsburg equation. Uniqueness Here we prove Theorem 2.3 on the uniqueness of Gibbs measures. Our proof follows a standard strategy [Rue69, Chapter 4.2]: We start from a set of integral equations satisfied by the correlation functions, reformulate these equations as a fixed point problem in some Banach space, and show that the fixed point problem involves a contraction. A minor novelty of our proof lies in our choice of norms as weighted supremum norms with weights that incorporate some information on interactions.

2 Lemma 3.1. Assume v ≥ 0, λ -almost everywhere, and X |f(x, y)|dλz(x) < ∞ for λz-almost all x ∈ X. Let P ∈ G (z). The correlation functions ρ of P satisfy the Kirkwood-Salsburg equations n R n ρn+1(x0,...,xn)= z(x0) (1 + f(x0, xi)) i=1 Y ∞ k 1 k × ρn(x1,...,xn)+ f(x0,yi)ρn+k(x1,...,xn,y1,...,yk)dλ (y) (KS)  k! Xk  j=1 kX=1 Z Y   n+1 n+1 0 for all n ∈ N0 and λ -almost all (x0,...,xn) ∈ X , with the convention ρ0 =1 and i=1(1 + f(x0, xi))=1. Q The absolute convergence of the right-hand side of (KS) is ensured by the condition (FPt) and the bound (2.4). The lemma is a consequence of the well-known equivalence of the GNZ and the Dobrushin-Lanford-Ruelle (DLR) conditions [NZ79] on the one hand, and the DLR conditions and the Kirkwood-Salsburg equation on the other hand [Rue70], see also [Kun99, KK03]. For the reader’s convenience we present a short self-contained proof of the implication (GNZ)⇒(KS) in Appendix A. The Kirkwood-Salsburg equations can be rephrased as a ﬁxed point equation in some suitable Banach space. Let E0 be the space of sequences ρ = (ρn)n∈N of real-valued measurable functions CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 19

n ρn : X → R such that n tn a(xi) |ρn(x1,...,xn)|≤ Cρe (1 + f(xi, xj )) z(xi)e i=1 1≤i

ρ = ez + Kzρ. (3.3)

Lemma 3.2. Under the assumptions of Theorem 2.3, the operator Kz is a bounded linear operator −t in E with operator norm ||Kz|| ≤ e < 1. Proof. Suppose that ρ ∈ E. Then we have, for n ∈ N, n n tn a(xi) (1 + f(x0, xi))|ρn(x1,...,xn)| ≤ ||ρ||e (1 + f(xi, xj )) z(xi)e (3.4) i=1 i=1 Y 0≤i

n k (1 + f(x0, xi)) |f(x0,yi)| |ρn+k(x1,...,xn,y1,...,yk)| i=1 i=1 Y Y k t(n+k) ≤ ||ρ||e (1 + f(xi, xj )) |f(x0,yi)| (1 + f(yi,yj )) i=1 0≤i

−t It follows that Kzρ ∈ E and ||Kzρ|| ≤ e ||ρ||. Proof of Theorem 2.3. The existence of a Gibbs measure follows from Theorem B.1 in Appendix B. For the uniqueness, we start from Lemma 2.2 and note that the sequence of n-point functions of a −t Gibbs measure P ∈ G (z) is in E. In view of ||Kz|| ≤ e < 1, the operator (id − Kz) is invertible with bounded inverse given by a Neumann series. Therefore the vector of correlation functions ρ = (ρn)n∈N, is uniquely determined by the Kirkwood-Salsburg equations and is given by ∞ −1 ℓ ρ = (id − Kz) ez = ez + Kzez. (3.8) Xℓ=1 n By Lemma 2.2, we have ρn(x1,...,xn) ≤ j=1 z(xj ) and

Q n n n E η(B) η(B) − 1 ··· η(B) − n +1 = ρndλ ≤ zdλ = λz(B) . n ZB ZB h i for all B ∈ Xb and n ∈ N0. As a consequence the correlation functions determine the measure P uniquely [LP17, Proposition 4.12] and we ﬁnd #G (z) = 1.

4. Weighted graphs. Expansion of correlation functions

In the previous section we have proven that condition (FPt) guarantees the uniqueness of the Gibbs measure P. We proceed with the proof of Theorem 2.6 on the expansion of correlation functions. As noted earlier, the vector of correlation functions ρ = (ρk)k∈N is the unique solution of a fixed point equation ρ = ez + Kzez and has the series representation (3.8). It remains to n compute the powers Kz ez, i.e., to show that they are indeed given by integrals and sums involving multirooted graphs. This is done by induction, noting that the partial sums of the series are in fact Picard iterates of the fixed point equation with initial value ez. Indeed, N+1 N n n Kz ez = ez + Kz Kz ez . n=0 n=0 X X The combinatorial counterpart to the partial sums of the Neumann series (3.8) are sums over graphs truncated at some maximum number N of vertices. Remember the functions ψk,n+k from (2.18) and the multirooted graphs Dk,n with k roots and n ≥ k vertices. Define S N (z) k(x1,...,xk) := z(x1) ··· z(xk) N 1 n−k × ψk,n(x1,...,xk,yk+1,...,yn)dλz (y) (4.1) (n − k)! Xn−k nX=k Z if 1 ≤ k ≤ N and (SN (z))k(x1,...,xk) := 0 if k ≥ N + 1. The summand for k = n is under- N n stood as ψk,k(x1,...,xk). In order to prove that SN (z) is equal to the partial sum n=1 Kz ez, we show that S1(z) = ez and that SN+1(z) is obtained from SN (z) by a Picard iteration, see Proposition 4.1 below. The proof builds on recursive properties of multirooted graphP with respect to removal of a root and as such generalizes the recursive proof with singly rooted graphs from [Uel04], see also [FV18, Chapter 5.4]. In addition, we prove that if the condition (FPt) holds true with t = 0, then the right-hand side of (4.1) converges pointwise as N → ∞, and we provide bounds. This ensures convergence of expansions even if Kz has operator norm equal to 1. In that case the limit corresponds to a fixed point of ρ = ez + Kzρ though we no longer know whether the solution is unique. This part of the proof is similar to the inductive treatment of the Kirkwood-Salsburg equation in [BFP10]. ˜ Define SN in a similar way as SN but with additional absolute values, i.e., S˜ N (z) k(x1,...,xk) := z(x1) ··· z(xk) N 1 n−k × ψk,n(x1,...,xk,yk+1,...,yn) dλz (y) (4.2) (n − k)! Xn−k n=k Z X

CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 21

˜ ˜ if 1 ≤ k ≤ N and (SN (z))k(x1,...,xk) := 0 if k ≥ N + 1. Clearly S1 = S1 = ez.

Proposition 4.1. Assume v ≥ 0, λz(B) < ∞ for all B ∈ Xb, and suppose that condition (FPt) holds true with t =0. Then we have

S1(z)= ez, SN (z)= ez + KzSN−1(z) (N ≥ 2) (4.3) with

k ˜ a(xj ) |(SN (z))k(x1,...,xk)| ≤ (SN (z))k(x1,...,xk) ≤ (1 + f(xi, xj )) z(xj )e (4.4) j=1 1≤i

J ψ I,J; (xj )j∈J := w G; (xj )j∈J (xj )j∈J ∈ X . (4.6) G∈D(I,J) X Thus Dk,n = D([k], [n]) and ψk,n(·)= ψ([k], [n]; ·). For J a ﬁnite possibly empty set, deﬁne

1, J = ∅, ψ ∅,J; (xj )j∈J := (4.7) (0, J 6= ∅. Notice that for all permutations σ of J,

ψ I,J; (xσ(j))j∈J = ψ σ(I),J; (xj )j∈J . (4.8) In particular, ψ(I,J; ·) is invariant with respect to permutations that act on J \ I only. The following set of equations is similar to [PU09, Lemma 6.2]. In fact Lemma 4.2 and [PU09, Lemma 6.2] show that our functions ψ(I,J; ·) are equal to the functions g(I,J \ I) deﬁned with Ruelle’s algebraic approach in [PU09]. We may think of D(I,J) as a collection of multi-rooted graphs with root set I and some connectivity constraints, then the lemma reﬂects a recursive combinatorial structure when removing a root.

J Lemma 4.2. Let I,J be two non-empty ﬁnite sets with I ⊂ J ⊂ N0. Fix (xj )j∈J ∈ X . Let ι(I) = min I, I′ = I \{ι(I)}, and J ′ = J \{ι(I)}. We have

′ ′ ψ I,J; (xj )j∈J = 1+ f(xι(I), xi) f(xι(I),ℓ) ψ I ∪ L,J ; (xj )j∈J′ . i∈I′ ! L⊂J\I ℓ∈L Y X Y In the sum L = ∅ is allowed. Products over empty sets are equal to 1. Lemma 4.2 is also true for attractive pairwise interactions.

Proof. For G ∈ D(I,J), let L ⊂ J \ I the set of vertices j ∈ J \ I that are adjacent to ι(I), i.e., L = {j ∈ J \ I |{ι(I), j} ∈ E(G)}. Let G′ be the graph obtained from G by removing the vertex ι(I) and all incident edges. Assume ﬁrst that #I ≥ 1 so that I′ 6= ∅. Then every vertex of G′ must be connected in G′ to at least one 22 SABINE JANSEN vertex in I′ ∪ L, thus G′ ∈ D(I′ ∪ L,J ′). The weight of G is given by

′ w G; (xj )j∈J = f(xι(I), xi) f(xι(I), xℓ)) w G ; (xj )j∈J . i∈I′: ! ℓ∈L ! {ι(I),iY}∈E(G) Y Conversely, given a possibly empty subset L ⊂ J \ I and a graph G′ ∈ D(I′ ∪ L,J ′), we construct a graph G ∈ D(I,J) by adding the vertex ι(I), adding all edges {ι(I),ℓ}, ℓ ∈ L, and adding none, some, or all of the edges {ι(I),i}, i ∈ I′. The proof for the case #I ≥ 1 is concluded by summing ﬁrst over all graphs G that correspond to a given pair (L, G′), and then summing over the pairs (L, G′). If #I = 1 and #J ≥ 2, then for a given graph G the element ι(I) has at least one adjacent vertex, i.e., L 6= ∅, and the proof is completed as before, taking into account that ψ(∅,J; ·)=0 ′ if J 6= ∅. If#I = #J = 1, then ψ(I,J; (xj )j∈J )= ψ({1}, {1}; x1) = 1 and the formula from the lemma holds true because ψ(∅, ∅; ·) = 1. The partial sums satisfy the following recursive inequality. k Lemma 4.3. For all N ≥ 2 and k ∈ N, and all (x1,...,xk) ∈ X ,

k S˜ N (z) k(x1,...,xk) ≤ z(x1) (1 + f(x1, xj )) j=2 Y N−k k+m 1 S˜ × f(x1, xℓ) N−1(z) k+m−1(x2,...,xk+m)dλz(xk+1) ··· dλz(xk+m) m! Xm m=0 Z ℓ=k+1 X Y where the term for k =1, m =0 is to be read as z(x1). Thus (4.5) holds true. S˜ Proof. Let N ≥ 2. For k ≥ N + 1 we have N (z) k(x1,...,xk) = 0 and the inequality is trivial. Consider k ∈ {2,...,N}. For n ∈ {k,...,N} and (x ,...,x ) ∈ Xn. By Lemma 4.2 applied to 1 n I = {1,...,k} and J = {1,...,n}, k ′ ′ ψk,n(x1,...,xn)= 1+ f(x1, xi) f(x1, xℓ) ψ I ∪ L,J ; (xj )j∈J′ (4.9) i=2 ! L⊂{k+1,...,n} ℓ∈L Y X Y ′ ′ with I = {2,...,k} and J = {2,...,n}. Let σ ∈ Sn be a permutation that acts on {k +1,...,n} only, i.e., σ(i)= i for all i ∈{1,...,n}. By (4.8), we have for every L ⊂{k +1,...,n},

′ ′ ′ ′ f(x1, xσ(ℓ)) ψ I ∪ L,J ; (xσ(j))j∈J′ = f(x1, xℓ) ψ I ∪ σ(L),J ; (xj )j∈J′ . (4.10) ℓ∈L ℓ∈σ(L) Y Y We apply the triangle inequality in (4.9), integrate over xk+1,...,xn, note that the contribution of each set L to the integral depends on the cardinality of L alone, and obtain

k ψk,n(x1,...,xn) dλz(xk+1) ··· dλz(xn) ≤ 1+ f(x1, xi) Xn−k i=2 ! Z Y n−k k+m−1 n − k × f(x1, xℓ) ψk+m−1,n−1(x2,...,xn) dλ(xk+1) ··· dλ(xn). m Xn−k m=0 Z ℓ=k+1 ! X Y We divide by (n − k)!, sum over n = k,...,N , and ﬁnd that N 1 ψk,n(x1,...,xn) dλz(xk+1) ··· dλz(xn) (n − k)! Xn−k n=k Z X (4.11) k N−k k+m ≤ 1+ f(x1, xi) f(x1, xℓ) ··· dλz(xk+1) ··· dλz(xk+m)) i=1 ! m=0 ℓ=k+1 ! Y X Y n o

CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 23 with N 1 ··· = ψk+m−1,n−1(x2,...,xn) dλz(xk+m+1) ··· dλz(xn). (n − k − m)! Xn−k n=k+m Z n o X Changing the summation index from n to n′ = n − 1, we see that the expression is exactly S˜ N−1(z) k+ℓ−1(x2,...,xk,yk+1,...,yk+m), except for the missing product z(x2) ··· z(xk). Hence, multiplying (4.11) with z(x ) ··· z(x ) on both sides, we obtain the required inequality when k ≥ 2. 1 k The proof for k = 1 is similar, with a careful consideration of the term m = 0 which gives rise to z(x1). Proof of Proposition 4.1. First we prove, by induction over N, that the inequality (4.4) holds true for all N, k ∈ N. For N = k = 1, we have (S˜1)1(x1)= z(x1) ≤ z(x1) exp(a(x1)) and the required inequality holds true. For N = 1 and k ≥ 2, we have (S˜1)k(x1,...,xk) = 0 and the inequality holds true as well. For the induction step, let N ≥ 2 suppose that the bound (4.4) holds true for all k ∈ N at N − 1 instead of N. We insert the bound into right-hand side of the recursive inequality from Lemma 4.3, then bound

(1 + f(xi, xj )) ≤ 1, (4.12) 1≤i≤k k+1Y≤j≤m and finally use condition (FPt) with t = 0 (compare the proof of Lemma 3.2). This yields the bound (4.4) for N and completes the induction step. Eq. (4.4) shows, in particular, that the integrals defining SN are absolutely convergent. We may now revisit the induction step, but without absolute values and triangle inequalities. The inequalities then become equalities for SN : we find S S S S N (z) k = Kz N−1 k, N (z) 1(x1)= z(x1)+ Kz N−1(z) 1(x1) (4.13) and the proof of the proposition is complete. Theorem 2.6 is a consequence of the Neumann series (3.8) and Proposition 4.1. Proof of Theorem 2.6. Passing to the limit N → ∞ in the inequality (4.4) from Proposition 4.1, we obtain the estimate (2.20) from Theorem 2.6. The equality (4.3) shows N−1 SN (z)= ez + Kzez + ··· + Kz ez. (4.14)

Thus SN (z) is a partial sum of the Neumann series (3.8). The bound (4.4) ensures that each (SN )k(x1,...,xk) converges pointwise as N → ∞. But we already know from the proof of Theorem 2.6 that the Neumann series converges in the Banach space E to the vector of correlation functions; thus ρk(x1,...,xk) = limN→∞(SN )k(x1,...,xk) and we obtain the representation of the correlation functions. The bound (2.20) with t = 0 follows from Proposition 4.1. For t > 0, t we note that ze satisﬁes (FPt) with t = 0 so we can apply the inductive bound of Proposition 4.1 to zet, and the proof is easily concluded.

5. Log-Laplace functional and truncated correlation functions Here we prove Theorems 2.4 and 2.5. Theorem 2.5 is deduced from Theorem 2.4 by exploiting that the log-Laplace functional at h is nothing else but the generating functional of the truncated correlation functions (factorial cumulant densities) at u =e−h − 1, see Eq. (5.21) below. Explicit bounds are proven with the complex contour integrals (here t> 0 is crucial). For the proof of Theorem 2.4, we first specialize Theorem 2.6, proven in the previous section, to the one-particle density (k = 1). As noted earlier, the classes of graphs D1,n and Cn are equal (if every vertex j ∈{2,...,n} connects to the vertex 1, then the graph is connected, and vice-versa). T It follows that ψ1,n = ϕn and ∞ 1 T ρ1(x1)= z(x1) ϕn(x1,...,xn)dλz(x2) ··· dλz(xn) (5.1) (n − 1)! Xn−1 n=1 X Z 24 SABINE JANSEN with absolutely convergent integrals and series, moreover the bound (2.11) stated in Theorem 2.4 is just the special case of the inequality (2.20) in Theorem 2.6. For the proof of the identity (2.10), the idea is to first prove a differentiated version of it. Formally, d E[( gdη) exp(− hdη)] − RX(h+tg)dη X X h log E e = = g(x1)ρ1 (x1)dλ(x1), dt E[exp(− hdη)] X t=0 R X R Z h i with ρh(x) the one-particle density of a tilted measure P , which we write succinctly with notation 1 R h from variational derivatives as δ − RX hdη h log E e = ρ1 (x1). (5.2) δh(x1) h i The variational derivative of the right-hand of Eq. (2.10) is ∞ 1 T n −h(x1) − Pi=2 h(xi) z(x1)e ϕn(x1,...,xn)e dλz(x2) ··· dλz(xn). (5.3) (n − 1)! Xn−1 n=1 X Z This is nothing else but the right-hand side of (5.1) with z replaced by ze−h. If the tilted mea- −h sure Ph is a Gibbs measure at tilted activity ze , we can conclude that the expressions (5.2) and (5.3) are equal, i.e., the differentiated form of Eq. (2.10) holds true and it remains to undo the differentiation. The full proof is a little technical as we need to make sure that all expressions involved are convergent and that we can exchange differentiation and integration. We start with the proof that −h the tilted measure Ph is indeed a Gibbs measure with tilted activity ze .

Lemma 5.1. Let P ∈ G (z) and h : X → R+ with X hdλz < ∞. Consider the measure Ph that is absolutely continuous with respect to P, with Radon-Nikod´ym derivative R dP exp(− hdη) h (η)= X . dP exp(− hdγ)dP(γ) N XR G −h Then Ph ∈ (ze ). R R

Proof. From h ≥ 0, Jensen’s inequality, and the bound ρ1(x) ≤ z(x) we get 0 < e− RX hdλz ≤ E e− RX hdη ≤ 1 so the normalization constant is strictly positive and ﬁnite. Let F : X × N → R+ be a measurable map. Notice X hd(η + δx)= X hdη + h(x). Then by (GNZ), we have

R − RX hdη R − RX hdη−W (x;η) −h(x) E F (x, η)e dη(x) = E[F (x, η + δx)e z(x)e dλ(x). (5.4) X X hZ i Z i − RX hdγ P P We divide on both sides by N e d (γ) and ﬁnd that h satisﬁes (GNZ) with z(x) replaced by z(x) exp(−h(x)), hence P ∈ G (ze−h). Rh Remember the function a : X → R+ from the convergence criterion (FPt).

Lemma 5.2. Let h : X → R+ be a bounded measurable function. Suppose that h is bounded and supported in some set of the form Λ ∩{x ∈ X | a(x) ≤ M} =: ΛM with λz(Λ) < ∞ and M ∈ (0, ∞). Then Eq. (2.10) holds true. Proof. Under the assumptions of the lemma, we have

a(x) h(x)e dλz(x) < ∞. (5.5) X Z −h The modiﬁed activity ze satisﬁes the bound (FPt) as well, with the same a and t as z, so we h have a representation for the one-point function ρ1 of Ph analogous to (5.1). Let us introduce functions F, G : R+ → R by ∞ 1 n T −s RX hdη −s Pi=1 h(xi) n F (s) = log E e , G(s)= (e − 1)ϕn(x1,...,xn)dλz (x). (5.6) n! Xn n=1 Z X CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 25

Clearly F (0) = G(0) = 0, we want to prove F (1) = G(1). Assuming that diﬀerentiation, integration and summations can be exchanged, we get

′ sh F (s)= − hdη dPsh(η)= h(x)ρ1 (x)dλ(x) (5.7) N X X Z Z Z T and, exploiting the symmetry of the Ursell function ϕN , ∞ n 1 n T ′ −s Pi=1 h(xi) n G (s)= − h(xi) e ϕn(x1,...,xn)dλz (x) n! Xn n=1 i=1 X Z X −sh(x1) = − h(x1)z(x1)e X Z ∞ 1 T × ϕn(x1,...,xn)dλze−sh (x2) ··· dλze−sh (xn) dλ(x1). (5.8) (n − 1)! Xn−1 n=1 ! X Z sh ′ ′ Using the analogue of (5.1) for ρ1 , we see that F (s) = G (s) for all s ≥ 0 and it follows that F (s)= G(s) for all s ≥ 0. In particular, F (1) = G(1) and Eq. (2.10) holds true for bounded h. It remains to justify (5.7) and (5.8). For s>ε> 0 we have 1 e−(s±ε)α − e−sα ≤ αe−(s−ε)α ≤ α, (5.9) ε −1 −εα and for s = 0 we have ε |e − 1|≤ α. For the difference quotients of s 7→ E[exp(−s X hdη)], we apply the inequality to α = hdη, note X R R E hdη = hρ1dλ ≤ hdλz < ∞ (5.10) X X X hZ i Z Z and conclude with dominated convergence that differentiation and expectation can be exchanged. n For the difference quotients of G(s), we apply the bound (5.9) to α = i=1 h(xi) and note that, in view of (2.11) and the inequality (2.12), we have P ∞ n 1 T n a(x) h(xj )|ϕn(x1,...,xn)|dλz (x) ≤ h(x)e dλz(x) < ∞. (5.11) n! Xn X n=1 j=1 X Z X Z and we conclude with dominated convergence that (5.8) holds true. It follows that Eq. (2.10) holds true if h satisfies (5.5).

Thus we have proven that

∞ 1 n T − RX hdη − Pj=1 h(xj ) n E e = exp (e − 1)ϕn(x)dλz (x) (5.12) n! Xn n=1 h i X Z when h is non-negative, bounded, and supported in some set ΛM =Λ∩{a ≤ M}. A straightforward argument involving monotone and dominated convergence shows that the identity extends to all non-negative h, but the extension to functions that take negative or complex values requires more work. In order to get rid of the condition h ≥ 0, we express the left- and right-hand sides of Eq. (5.12) as power series in e−h. Roughly, the idea is that if two power series of some variable s coincide and converge on s ∈ [0, 1], then their coeﬃcients and their domain of convergence must be equal and the identity extends to the whole domain of convergence. We start with the right-hand side of Eq. (5.12). Let Λ ⊂ X be such that eadλ < ∞. Set M ΛM z ∞ R 1 T k ϑm(x1,...,xm)= ϕm+k(x1,...,xn,y1,...,yk)dλz (y). k k! (X\ΛM ) Xk=0 Z 26 SABINE JANSEN

To lighten notation, we suppress the ΛM -dependence from ϑm. By (2.11), we have

∞ tm ∞ ∞ t(m+k) e m e T m+k |ϑm(x1,...,xm)|dλ (x) ≤ |ϕm+k(x)|dλz (x) m! m m!k! m c k m=1 ΛM m=1 ΛM ×(ΛM ) X Z X Xk=0 Z ∞ tn e T n = |ϕn(x1,...,xn)|1l{∃j: xj ∈ΛM }dλz (x) n! Xn n=1 X Z a M ≤ e dλz ≤ e λz(Λ) < ∞, (5.13) ZΛM m so the integrals and series in the deﬁnition of ϑm are absolutely convergent for λ -almost all m (x1,...,xm) ∈ ΛM , moreover the constant ∞ 1 m Cz(ΛM ) := ϑm(x1,...,xm)dλz (x) m! m m=1 ΛM X Z is ﬁnite.

Lemma 5.3. Let h : X → [−t, ∞) ∪ {∞} or h : X → {s ∈ C : Re s ≥ −t}. Assume that −h a X |e − 1|e dλz < ∞ and that h is supported in some set ΛM =Λ ∩{a ≤ M} with λz(Λ) < ∞. Then R ∞ 1 n T − Pj=1 h(xj ) n exp (e − 1)ϕn(x)dλz (x) n! Xn n=1 Z X ∞ n r 1 − P h(xi) −Cz(ΛM ) i∈Vk n =e 1+ e ϑ#Vk (xVk )dλz (x) =: R(h) n! n n=1 ΛM r=1 X Z X {V1,...,VXr}∈Pn kY=1 (5.14) with absolutely convergent sums and integrals.

Proof. We have already observed that the bound (2.11) holds true, therefore the convergence of the integrals and the sums on the left-hand side of (5.14) follows from the inequality (2.12) and the −h a integrability condition X |e − 1|e dλz < ∞. On the left-hand side of (5.14) the only non-zero contributions to integrals come from x with xj ∈ ΛM for some j. Straightforward computations R T together with the symmetry of the Ursell functions ϕn show that the sum inside the exponential can be rewritten as ∞ ∞ 1 n T 1 m − Pj=1 h(xj ) n − Pi=1 h(xi) m (e −1)ϕn(x)dλz (x)= (e −1)ϑm(x1,...,xm)dλz (x) n! Xn m! Λm n=1 Z m=1 Z M X X (5.15) The proof of the equation requires some exchange of order of summation, which is justiﬁed with (2.12) and estimates similar to (5.13). The right-hand side of Eq. (5.15) is actually convergent also without the factor −1. Indeed, from | exp(−h)| = exp(−Re h) ≤ exp(−t) and (5.13), we get ∞ 1 m − Pi=1 h(xi) m a e ϑm(x1,...,xm) dλz (x) ≤ e dλz < ∞. m! Λm Λ m=1 Z M Z M X Exponentiating (5.15), we ﬁnd that the left-hand side of Eq. (5.14) is given by

∞ 1 m − Pi=1 h(xi) m a exp −Cz(ΛM )+ e ϑm(x1,...,xm) dλz (x) ≤ e dλz m! Λm Λ m=1 Z M Z M ! X and the proof is concluded with a standard identity on exponential generating functions of set partitions (see Eq. (6.3)) below). CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 27

Next we turn to the left side of Eq. (5.12). Let h and ΛM be as in Lemma 5.3 and (jn,ΛM )n∈N0 such that ∞ − hdη 1 − n h(x ) n E RX Pj=1 j e = j0,ΛM + e jn,ΛM (x)dλz (x) =: L(h) (5.16) n! Λn n=1 Z M X whenever h is supported in ΛM and the right-hand side converges (with absolute values in the integrand), for example, if h ≥ 0. Thus j0,ΛM is an avoidance probability and jn,ΛM , n ≥ 1, are the Janossy densities [DVJ08, Chapter 5.3], also called density distributions [Rue70], of P in ΛM with respect to the reference measure λz. In our setup the Janossy densities exist and satisfy n ∞ k (−1) k jn,ΛM (x) z(xi)= ρn(x1,...,xn)+ ρn(x1,...,xn,y1,...,yk)dλ (y) (5.17) k! k i=1 ΛM Y Xk=1 Z with the convention ρ0 = 1. The integrals and series are absolutely convergent because of (2.4). Eq. (5.17) is the well-known inversion formula for Janossy densities and correlation functions, see [DVJ08, Chapter 5, Eq. (5.4.11)] and the ﬁrst equation after Eq. (5.33) in [Rue70]. The odd- looking product z(x1) ··· z(xn) in (5.16) appears because the ρn’s are deﬁned with the reference measure λ rather than λz.

Lemma 5.4. Let ΛM , h, and (ϑm)m∈N0 be as in Lemma 5.3. Then the Janossy densities

(jn,ΛM )n∈N0 are given by n −Cz(ΛM ) −Cz(ΛM ) j0,ΛM =e , jn,ΛM (x)=e ϑ#Vk (xVk ) (5.18) r=1 X {V1,...,VXr }∈Pn n n for all n ∈ N and λz -almost all (x1,...,xn) ∈ ΛM . Moreover Eq. (5.12) extends to all functions h supported in ΛM that take values in [−t, ∞) ∪ {∞} or in {s ∈ C | Re s ≥−t}. Proof. By Lemma 5.2 and Eq. (5.12), the functionals L(h) and R(h) from Eqs. (5.16) and 5.3 have to coincide for all nonnegative, bounded h supported in ΛM and subject to (2.9), which is enough to ensure that (5.18) holds true. It follows that the identity L(h)= R(h) extends to the functions h that satisfy the assumptions of Lemma 5.3.

Now we can complete the proof of Theorem 2.4.

Proof of Theorem 2.4. We have already observed that the bound (2.11) follows from Theorem 2.6 and the identity D1,n = Cn. Let h be a measurable function that satisﬁes the integrability condition (2.9) and take values in either [−t, ∞) ∪ {∞} or in {s ∈ C : Re s ≥ t}. For k ∈ N, set hk := h1lB(0,k)1l{a≤k}. By Lemma 5.4, we have ∞ 1 n T − RX hkdη − Pj=1 hk(xj ) n log E e = (e − 1)ϕn(x1,...,xn)dλ (x). (5.19) n! Xn n=1 h i X Z In order to pass to the limit on the left side, we write ∞ n 1 − RX hkdη −hk(xi) n E e =1+ (e − 1)ρn(x)dλ (x), n! Xn n=1 i=1 h i X Z Y a similar identity holds true with hk replaced by h. Moreover hk → h pointwise with

−hk −h −h |e − 1| = |e − 1|1lB(0,k)1l{a≤k} ≤ |e − 1| and ∞ n 1 −h(xi) n −h 1+ |e − 1|ρn(x)dλ (x) ≤ exp |e − 1|dλz < ∞. n! Xn X n=1 i=1 X Z Y Z Dominated convergence thus yields E[exp(− X hkdη)] → E[exp(− X hdη)], i.e., we can exchange limits and integration in the left-hand side of Eq. (5.19). On the right-hand side of (5.19), we can R R 28 SABINE JANSEN can exchange limits and integration too because of the inequality n n − n h (x ) (n−1)t −h (x ) (n−1)t −h(x ) |e Pi=1 k i − 1|≤ e |e k i − 1|≤ e |e i − 1| (5.20) i=1 i=1 X X (remember Eq. (2.12) !), the bound (2.11) and dominated convergence. Altogether, passing to the limit k → ∞ in (5.19), we ﬁnd that (2.10) holds true for h.

Proof of Theorem 2.5. Let h : X → R+. It follows from Corollary 9.5.VIII in [DVJ08] that

∞ ℓ 1 T − RX hdη −h(xj ) ℓ log E e = (e − 1)ρℓ (x1,...,xℓ)dλ (x) (5.21) ℓ! Xℓ j=1 h i Xℓ=1 Z Y whenever the right-hand side is absolutely convergent and h is non-negative with bounded support. In the right-hand side of Eq. (2.10) in Theorem 2.4, we express the log-Laplace functional in terms of exp(−h) − 1. We have

− n h(x ) −h(x ) e Pi=1 i − 1= (e i − 1) (5.22) I⊂[n]: i∈I IX6=∅ Y and, if |h|≤ t,

∞ n ℓ 1 n T − RX hdη −h(xi) n log E e = (e − 1)ϕn(x1,...,xn)dλz (x) (5.24) n! ℓ Xn n=1 i=1 h i X Xℓ=1 Z Y ∞ ℓ ℓ 1 −h(xj ) = (e − 1) z(xj ) ℓ! Xℓ j=1 j=1 Xℓ=1 Z Y Y ∞ 1 T k ℓ × ϕℓ+k(x1,...,xℓ,y1,...,yk)dλz (y) dλ (x) (5.25) k! Xk ! Xk=0 Z The exchange of the order of summation is justiﬁed with (5.23) and absolute convergence. Chang- ing variables to u(x)=1 − e−h(x) and comparing (5.21) and (5.24), we ﬁnd

ℓ T ℓ u(xj )ρℓ (x1,...,xℓ)dλ (x) Xℓ j=1 Z Y ℓ ∞ 1 T k ℓ = u(xj ) ϕℓ+k(x1,...,xℓ,y1,...,yk)dλz (y) dλ (x) (5.26) Xℓ k! Xk j=1 ! Z Y Xk=0 Z a for all u : X → [0, 1] that have bounded support and satisfy X ue dλz < ∞. Eq. (2.15) follows. For s ∈ C with |s|≤ et − 1, we have R ∞ ℓ ∞ |s| 1 T |ϕℓ+k+1(x0, x1,...,xℓ+k)|dλz(x1) ··· dλz(xℓ+k) ℓ! k! Xℓ+k ℓ=0 k=0 Z X ∞X 1 m T a(x0) ≤ (1 + |s|) |ϕm+1(x0, x1,...,xm)dλz(x1) ··· dλz(xm) ≤ e . m! Xm m=0 X Z CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 29

Taking contour integrals along the circle centered at the origin with radius et − 1, we deduce ∞ 1 1 T 1 ds a(x0) |ϕℓ+k+1(x0, x1,...,xn+k)|dλz(x1) ··· dλz(xℓ+k) ≤ ℓ+1 e ℓ! k! Xℓ+k 2πi s Xk=0 Z I 1 ≤ ea(x 0) (et − 1)ℓ and the bound (2.16) follows.

6. Gibbs measures in finite volume. Estimates for t =0 Here we prove Theorems 2.9 and 2.8 on the partition function and correlation functions in finite volume. The proof of Theorem 2.9 is fairly standard, once the bound (2.11) is available. The proof of Theorem 2.8 explains the appearance of the class Dk,n of multirooted graphs directly, without any reference to Kirkwood-Salsburg equations. The theorems are proven in reverse order: first Theorem 2.9, then Theorem 2.8 and finally Theorem 2.7. Proof of Theorem 2.9. First we note that the bound (2.11) with t = 0 follows from the identity T ϕn = ψ1,n and Proposition 4.1 applied to k = 1. The bound (2.21) is an immediate consequence of (2.11). The proof of Theorem 2.9 then follows standard arguments [Rue69] which we reproduce for the reader’s convenience. Let C(V ) be the collection of connected graphs with vertex set V (thus Cn = C([n])). We have

−Hn(x1,...,xn) e = 1+ f(xi, xj ) = f(xi, xj ). 1≤i

−Hn(x1,...,xn) e = w(G; x1,...,xn). (6.1) GX∈Gn As the weight of a graph is the product of the weights of its connected components, we deduce n r −Hn(x1,...,xn) e = w(Gk; xVk ) r=1 X {V1,...,VXr}∈Pn kY=1 Gk∈CX(Vk) n r T x = ϕ#Vk ( Vk ). r=1 X {V1,...,VXr}∈Pn kY=1 We integrate both sides over Λn and note that integrals factorize as well, moreover on the right- hand side they depend on the cardinality of Vi alone. Thus ∞ n 1 Ξ (z)=1+ A ··· A Λ n! #V1 #Vk n=1 r=1 X X {V1,...,VXr}∈Pn with n T n An := w(G; x1,...,xn) dλz (x)= ϕn(x1,...,xn)dλz (x). (6.2) Λn Λn Z GX∈Cn Z It is a general combinatorial fact that ∞ n ∞ 1 1 1+ A ··· A = exp A (6.3) n! #V1 #Vk n! n n=1 r=1 n=1 X X {V1,...,VXr}∈Pn X ∞ 1 whenever n=1 n! |An| < ∞. The latter condition is satisﬁed for the concrete choice (6.2) because of (2.11) with t = 0 and we obtain P ∞ 1 T n ΞΛ(z) = exp ϕn(x1,...,xn)dλz (x) . n! n n=1 Λ ! X Z 30 SABINE JANSEN

Proof of Theorem 2.8. The bound (2.20) with t = 0 follows from Proposition 4.1. In order to go from Theorem 2.9 to correlation functions we note that the generating functional of the latter is

∞ k η({x}) 1 k EΛ (1 + u(x)) =1+ u(xj )ρk,Λ(x)dλ (x). (6.4) k! k Λ j=1 hxY∈Λ i kX=1 Z Y The identity holds true for all measurable u :Λ → [−1, 0], the right-hand side of (6.4) is absolutely convergent because of the bound (2.4). By the deﬁnition of partition functions,

η({x}) ΞΛ z(1 + u) EΛ (1 + u(x)) = . (6.5) ΞΛ(z) x∈Λ h Y i In order to evaluate ΞΛ(z(1 + u)), we remember (6.1) and expand n (1 + u(xj )) w(G; x1,...,xn)= u(xj ) w(G; x1,...,xn). j=1 Y GX∈Gn JX⊂[n] jY∈J GX∈Gn Given a non-empty subset J ⊂ [n], we establish a one-to-one correspondence between graphs G and pairs (G1, G2) that consist of a graph G1 on some subset V ⊃ J subject to connectivity constraints, and a graph G2 on [n] ⊂ V , as follows. For G ∈ Gn and a non-empty subset J ⊂ [n], let V ⊂ [n] be those vertices that connect to some vertex j ∈ J by a path in G (thus J ⊂ V ⊂ [n]), and W = [n] \ V . Clearly G cannot have any edge connecting V and W . Let G|V be the graph with vertex set V and W and edge set {{i, j} | i, j ∈ V, {i, j} ∈ E(G)}, similarly for G|W . Conversely, let DJ (V ) be the graphs on V for which every vertex in V connects to some vertex j ∈ J. Given two graphs G1 ∈ DJ (V ), G2 ∈ G(W ) such that every vertex in V connects to some elements j ∈ J by a path in G1, we obtain a graph G ∈ Gn with edge set E(G) = E(G1) ∪ E(G2). In this way we have a one-to-one correspondence, and we ﬁnd

n (1 + u(xj )) w(G; x1,...,xn)= w(G; x1,...,xn) j=1 Y GX∈Gn GX∈Gn + u(xj ) w(G1; xV ) w(G2; xW ) .

J⊂[n]: {V,W }∈Pn: j∈J G1∈DJ (V ) G2∈G(W ) JX6=∅ VX⊃J Y X X Exploiting the symmetry of the graph weights, we deduce

∞ k ∞ 1 1 m ΞΛ z(1+u) = ΞΛ(z) 1+ u(xj ) ψk,k+m(x1,...,xk,y1,...,ym)dλz (y)  k! k m! m  k=1 ZΛ j=1 m=0 ZΛ X Y X  (6.6)  (think k = #J, m = #(V \ J)). The expression of the correlation functions follows from Eqs. (6.4), (6.5) and (6.6).

Proof of Theorem 2.7. The validity of (2.21) has been checked at the beginning of the proof of Theorem 2.9. The representation for the log-Laplace functional builds on Theorem 2.8 and is similar to the proof of Theorem 2.4 provided in Section 5.

7. Kirkwood-Salsburg equations for trees and forests Here we explain the connection between the Kirkwood-Salsburg equation from Lemma 3.1 and generating functions for trees. The key point is that the non-linear ﬁxed point equation (2.30) for tree generating functions translates into a linear set of equations for the generating functions of forests that is similar to the Kirkwood-Salsburg equations. As a by-product, we obtain a new proof of the implication (ii) ⇒ (i) in Proposition 2.11 that clariﬁes the relation between Kirkwood- Salsburg equations and inductive convergence proofs. CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 31

Let Bk(q; x1,...,xk), k ∈ N0, be the family of non-negative weight functions given by k B0(q) := 1, Bk(q; x1,...,xk)= |f(q; xi)| (1 + f(xi, xj )), (7.1) i=1 Y 1≤i

Fn+1(q0,...,qn; z)= z(q0) B0(q0)Fn(q1,...,qn; z) ∞ 1 ℓ + Bℓ(q0; x1,...,xℓ)Fn+ℓ(q1,...,qn, x1,...,xℓ; z)dλ (x) , ℓ! Xℓ Xℓ=1 Z (N) a variant of (KS) for forests. Let Fk (q1,...,qk; z) be the partial sums of the generating functions where we sum only over forests with vertex sets [n], n ≤ N. Proceeding as in the proof of (1) (1) Proposition 4.1, we ﬁnd F1 (q1)= z(q1), Fk = 0 for k ≥ 2, and

(N+1) (N) Fn+1 (q0,...,qn; z)= z(q0) B0(q0)Fn (q1,...,qn; z) ∞ 1 (N) ℓ + Bℓ(q0; x1,...,xℓ)Fn+ℓ(q1,...,qn, x1,...,xℓ; z)dλ (x) . ℓ! Xℓ Xℓ=1 Z An induction over N then shows that if ∞ 1 k Pi=1 a(xi) k a(xi) B0(q)+ Bk(q; x1,...,xk)e dλz (x) ≤ e , k! Xk kX=1 Z (N) n a(qi) then Fn (q1,...,qn; z) ≤ i=1 z(qi)e for all N, n, q. Passing to the limit N → ∞, we ﬁnd that the forest generating functions are convergent. In particular, the generating function for trees (i.e., 1-forests) is convergentQ as well. This proves the implication (ii)⇒(i) in Proposition 2.11.

8. Cumulants of second-order stochastic integrals Here we prove Propositions 2.14 and 2.16. κ Proof of Proposition 2.14. Pick η = i=1 δxi ∈ N and set [κ] = N if κ = ∞. Assume that u is non-negative or that |u|dη(2) < ∞. A reasoning similar to the proof of Eq. (2.36) shows X P m mij (γ) R u(xi, xj ) = u(xi, xj ) . i

m 2m (2) 1 n E udη = Fnm(x)dλz (x). X2 n! Xn n=2 hZ i X Z This proves the representation for the moments and shows that the moments of order s ≤ m of (2) X2 |u|dη are ﬁnite. For the cumulants, we use the multiplicativity of the graph weightsw ¯(γ; x)= u(x , x )mij (γ): R {i,j} i j Each γ ∈M ([n], [m]) decomposes into connected multigraphs γ ∈M (V , A ), i =1,...,r, where s i c i i Q {V1,...,Vr} and {A1,...,Ar} form set partitions of [n] and [m], and the weight of γ is the product of the weights of the connected components. Therefore m r

w¯(γ; x1,...,xn)= w¯(γℓ; xVℓ ) γ∈Ms([n],[m]) r=1 {V1,...,Vr}∈Pn ℓ=1γℓ∈Mc(Vℓ,Aℓ) X X {A1,...,AXr }∈Pm Y X and, formally, m E udη(2) X Z h 2m n i 1 n nℓ = w¯(γ; x)dλz (x) n n! N n1,...,nr X ℓ n=2 r=1 {A1,...,Ar }∈Pm n1,...,nr∈ 0: γ∈Mc([nℓ],Aℓ) Z X X X n1+···X+nr=n X

n r 2#Aℓ 1 nℓ = w¯(γ; x)dλz (x) n ! Xnℓ r=1 n =2 ℓ X {A1,...,AXr}∈Pm ℓY=1 Xℓ Z γ∈McX([nℓ],Aℓ) n

= κ#A1 ··· κ#Ar (8.1) r=1 X {A1,...,AXr}∈Pm with κs defined by the equation in Proposition 2.14. The identity is formal for now because we need to check the convergence of the integrals in κℓ. But the convergence can be checked by induction (2) s over s ∈{1,...,m}, using that the moments E[( X2 |u|dη ) ] are finite and that Eq. (8.1) holds true if we replace m by any s ≤ m. The set of equations (8.1) with s ≤ m instead of m is exactly the set of equations satisfied by the cumulants (see,R for example, [PT11]) and the solution of the set of equations is unique, so we identify κ1,...,κs as the first s cumulants. m 1 Proof of Proposition 2.16. From Proposition 2.14 and the general observation κm(X)=2 κm( 2 X) we get 2m m (2) 2 mij (γ) n κm udη = u(xi, xj ) dλz . (8.2) X2 n! Xn n=2 Z X Z γ∈MXc([n],[m]) 1≤i

γ γ 1 ≤ i < j ≤ m and si ∈ Bi , sj ∈ Bj , we have si ≺ sj. Indeed, the only degree of freedom we have is choosing the order within blocks. Since there are m blocks of cardinality 2 each, the overall number of choices is 2m. γ Given an order ≺, we define ϕ≺ : S →{1, 2,..., 2m} = Sm as the unique order-preserving bi- γ jection. Clearly ϕ≺(B1)= {1, 2}, ϕ≺(B2)= {3, 4}, etc. Put differently, ϕ≺ maps the partition π γ γ γ of S to the partition {{1, 2}, {3, 4},..., {2m−1, 2m}} = πm of Sm. Letˆσ := (ϕ≺(T1 ),...,ϕ≺(Tn )) be the ordered partition of Sm obtained from σ and the map ϕ≺. Let σ be the underlying non- ordered set partition of Sm. It is easily checked that (πm, σ) is non-flat and connected. Summarizing, consider the following two types of objects at fixed n and m: γ • pairs (γ, ≺) consisting of a multigraph γ ∈ Mc([n], [m]) and a total order γ on S that γ γ preserves the order of the blocks B1 ,...,Bm; • ordered set partitionsσ ˆ = (T1,...,Tn) of Sm with n blocks such that (πm, σ) is non-flat and connected. We have defined a systematic assignment (γ, ≺) 7→ σˆ, a careful examination shows that this is in fact a one-to-one correspondence. n n Let us come back to Eq. (8.2). Write and for sums over the types of objects (γ,≺) σˆ introduced above. As there are 2m choicesX of order ≺, weX may rewrite Eq. (8.2) as 2m n (2) 1 mij (γ) n κm udη = u(xi, xj ) dλz (x). X2 n! Xn n=2 Z X (Xγ,≺) Z 1≤i

mij (γ) n n u(xi, xj ) dλz (x)= (u ⊗···⊗ u)σˆ (x1,...,xn)dλz (x), Xn Xn Z 1≤i

Appendix A. Proof of Lemma 2.2 and Lemma 3.1 (m) κ Proof of Lemma 2.2. Let η be the m-th factorial measure of η, i.e., if η = j=1 δxj . Set m P −H(x1,...,xm|η) η({y}) e = (1 + f(xi, xj )) (1 + f(xi,y)) . i=1 X 1≤i

(m) −H(δ +···+δ |η) m E E x1 xm F (x; η)dη (x) = F (x; η+δx1 +···+δxm )e dλz (x) (MGNZ) Xm Xm Z Z h i m h i For F (x, η)= g(x) with g : X → R+, we obtain

(m) −H(x1,...,xm|η) m gdαm = E gdη = E e g(x)dλz (x). (A.1) Xm Xm Xm Z Z Z h i h i m It follows that the factorial moment measure αm is absolutely continuous with respect to λ with Radon-Nikod´ym derivative m dαm ρ (x)= (x)= z(x )E e−H(x1,...,xm|η) λm-a.e. m dλm j j=1 Y h i 34 SABINE JANSEN

Proof of Lemma 3.1. Let H be as in the proof of Lemma 2.2, We decompose

H(x0 ··· xn | η)= W (x0; x1,...,xn)+ W (x0 | η)+ H(x1,...,xn | η) and obtain from Lemma 2.2 that n −βW (x0;x1,...,xn) −β[W (x0;η)+H(x1,...,xn|η)] ρn+1(x0, x1,...,xn)= z(x0)e E z(xj )e j=1 hY i We can further expand ∞ k −W (x0;η) 1 (k) e =1+ f(x0,yj)dη (y), (A.2) k! Xk j=1 Xk=1 Z Y which is absolutely convergent for P-almost all η ∈ N because of

∞ k 1 (k) E 1+ f(x0,yj ) dη (y)  k! Xk  k=1 Z j=1 X Y   ∞ k 1 k =1+ f(x0,yj )) ρk(y)dλ (y) ≤ exp |f(x0,y)|dλz(y) < ∞. (A.3) k! Xk X k=1 Z j=1 Z X Y

In the last line we have used the integrability assumption on f and the bound ρk(y1,...,yk) ≤ k j=1 z(yj ), which follows from Lemma 2.2 and 1 + f ≤ 1. Applying (MGNZ) we get Q n k −H(x1,...,xn|η) (k) E z(xj ) f(x0,yj)e dη (y) Xk j=1 j=1 hY Z Y i k n k n+k −[H(x1,...,xn|δy +···+δy +η)+H(y1,...,yk|η)] k = E f(x0,yj)z e 1 k z(xj ) z(yj)dλ (y) Xk j=1 j=1 j=1 Z hY i Y Y k n k −H(x1,...,xn,y1,...,yk|η) k = f(x0,yj )E z(xj ) z(yj)e dλ (y) Xk j=1 j=1 j=1 Z Y hY Y i k k = f(x0,yj )ρn+k(x1,...,xn, y)dλ (y). Xk j=1 Z Y Because of (A.3), we can exchange summation and integration in (A.2) and (KS) follows.

Appendix B. Existence of Gibbs measures Remember that X is a complete metric separable space.

b Theorem B.1. Let z : X → R+ be measurable and such that λz(B)= B zdλ< ∞ for all B ∈ X . Let v : X × X → R ∪ {∞} be a measurable pair potential. Assume that: 2 2 R (i) v(x, y) ≥ 0 for λz-almost all (x, y) ∈ X . (ii) The potential v satisﬁes the integrability condition

∀x ∈ X : |f(x, y)| dλz(y) < ∞, for λ-almost all x ∈ X. (I) X Z where f(x, y)=e−v(x,y) − 1. Then G (z) 6= ∅. The theorem is a variant of results by Ruelle [Rue70] and Kondratiev and Kuna [KK03], for the reader’s convenience we include a complete proof. Ruelle deals with translationally invariant, superstable, possibly negative interactions and simple point processes in Rd. Kondratiev and Kuna work with simple point processes in more general spaces X with superstable, possibly non-negative CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 35 interactions without translational invariance, under the slightly more restrictive integrability condition ess supx∈X X |f(x, y)|dλz (y) < ∞ (essential supremum with respect to λz). The proof follows a standard scheme: R (1) Choose a topology on the space of probability measures on (N , N). The choice involves trade-offs discussed at the beginning of [Geo88, Chapter 4]. For point processes with locally finite intensity measure, a good choice is the topology τL of local convergence from [GZ93]. (2) Show that the collection of finite-volume Gibbs measures is sequentially relatively com- pact, i.e., every sequence admits a convergent subsequence. Some criteria for relative compactness are: Ruelle’s moment bounds for correlation functions [Rue70, KK03], entropy bounds [GZ93, DDG12], existence of Lyapunov functionals [KPR12, CDKP18]. Relative compactness is often easy for non-negative or locally stable interactions but can be difficult to prove for stable attractive interactions that favor accumulation of many points in small regions. (3) Show that every accumulation point of finite-volume Gibbs measures is an infinite-volume Gibbs measure. This is done by passing to the limit in one of the structural equations char- acterizing the set of Gibbs measures; usually, the DLR conditions or Kirkwood-Salsburg equations. The passage to the limit requires additional justification because the structural equations may involve functions that are not local (e.g., when the potential has infinite range). Quasi-local specifications provide conceptual framework particularly helpful for lattice systems [Geo88, Pre76]. Additional conditions on the interaction potential are needed to ensure that all functions involved can be approximated by local functions. Typ- ically this involves some decay of interactions, formulated as pointwise bounds or as an integrability condition similar to (I) [Rue69, Rue70].

We introduce some additional notation. For η ∈ N and ∆ ∈ X , let η∆ be the restriction of η to ∆, i.e.,

η∆(A)= η(A ∩ ∆) (A ∈ X ).

A function h : N → R is local if there exists ∆ ∈ Xb such that h is ∆-local, i.e., h(η)= h(η∆) for all η ∈ N . The algebra of local events is

A = {A ∈ N | 1lA is local}. Notice that the indicator of a measurable set A ∈ N is ∆-local if and only if it is in the σ-algebra N N ∆ generated by the counting variables η 7→ η(B) with measurable B ⊂ ∆, thus A = ∆∈Xb ∆. A function h : N → R is exponentially bounded if there exist ∆ ∈ Xb and C,c > 0 such that S |h(η)|≤ C exp(c η(∆)) for all η ∈ N , and tame if instead |h(η)| ≤ C(1 + η(∆)). Every exponentially bounded function is tame. Let L be the class of measurable tame local functions h : N → R and P the set of probability measures P on (N , N) with locally finite intensity, i.e., E[η(∆)] < ∞ for all ∆ ∈ Xb. The topology τL of local convergence is the smallest topology on P for which the maps eh : P 7→ L N h dP , h ∈ , are continuous. Next let (Λ ) N be an increasing sequence of non-empty bounded measurable sets such that R n n∈ every bounded set B is eventually contained in some Λn (the sequence is cofinal [Geo88, Chapter 1.2]). Write Pn for the finite-volume Gibbs measure in Λn with empty boundary conditions and activity z. Thus Pn is absolutely continuous with respect to the law of the Poisson point process with intensity measure 1lΛn λz, and the Radon-Nikodým derivative is proportional to exp(−H(η)).

Lemma B.2. The sequence (Pn)n∈N admits a subsequence (Pnr )r∈N such that for some probability measure P on (N , N) and all measurable, local, exponentially bounded h : N → R, we have

lim h dPn = h dP. r→∞ r ZN ZN

In particular, (Pnr )r∈N converges to P in the topology τL of local convergence. 36 SABINE JANSEN

Proof. Step 1: Convergence of correlation functions. For k,n ∈ N, set

(n) −H (x ,...,x ) − k W (x ,η) k 1 k E Pi=1 i ρ¯k (x1,...,xk) := e n e (x1,...,xk ∈ X). (B.1)

(n) h i By Lemma 2.2, the functionρ ¯k is the Radon-Nikod´ym derivative of the k-th factorial moment (n) ⊗k measure αk with respect to λz . Because of v ≥ 0, we have the pointwise inequality (n) k 0 ≤ ρ¯k ≤ 1 λz -a.e. (B.2)

(n) ∞ k ⊗k k for all k,n ∈ N. For ﬁxed k ∈ N, wemayview(¯ρk )n∈N as a bounded sequence in L (X , X , λz ). ∞ k ⊗k k 1 k ⊗k k Since L (X , X , λz ) is the dual of the Banach space L (X , X , λz ), the Banach-Alaoglu ∗ (nr) ∗ theorem ensures the existence of a weak -convergent subsequence (¯ρk )r∈N ⇀ ρ¯k, i.e., for all 1 k ⊗k k f ∈ L (X , X , λz ), we have

(nr) k k lim f ρ¯k dλz = f ρ¯k dλz , (B.3) r→∞ Xk Xk Z Z k k moreover 0 ≤ ρ¯k(x) ≤ 1 for λz -almost all x ∈ X . A straightforward argument with diagonal sequences yields the existence of a subsequence (Pnr )r∈N such that the convergence (B.3) holds true for all k ∈ N. Note, however, that we do not yet know that the functionsρ ¯k are factorial moment densities of some measure P.

Step 2: Convergence of Janossy densities and avoidance probabilities. For ∆ ∈ Xb, k,n ∈ N, and k (x1,...,xk) ∈ X , deﬁne

∞ m (n) (n) (−1) (n) m jk,∆(x1,...,xk) :=ρ ¯k (x1,...,xk)+ ρ¯k+m(x1,...,xk,y1,...,ym)dλz (y). (B.4) m! m m=1 ∆ X Z (n) The right-hand side is absolutely convergent because of the pointwise inequality 0 ≤ ρℓ ≤ 1 and (n) because of λz(∆) < ∞. The family (jk,∆)k∈N forms a system of Janossy densities (with respect k to λz) localized to ∆ of Pn [DVJ03, Eq.. (5.4.14)], consequently jk,∆(x) ≥ 0 for λz -almost all x ∈ ∆k. For k = 0, we deﬁne

∞ m (n) (−1) (n) m j0,∆ := 1 + ρ¯m (y1,...,ym)dλz (y) m! m m=1 ∆ X Z (n) and note that j0,∆ = Pn(η(∆) = 0) ≥ 0 is an avoidance probability [DVJ03, Example 5.4(a)]. (n) (n) Deﬁne jk,∆ by formulas analogous to jk,∆ but withρ ¯k replaced byρ ¯k. It follows from (B.3) that

(nr) (nr ) ∗ j0,∆ → j0,∆, jk,∆ ⇀ jk,∆ (k ∈ N) (B.5) as r → ∞. (Here the weak∗ convergence is with respect to the L∞ space with ∆k instead of k X .) Consequently the jk,∆’s, k ∈ N, and j0,∆ are non-negative (up to null sets), however we do not yet know that they are the Janossy densities and avoidance probabilites, respectively, of some probability measure P.

Step 3: Convergence of expectations. Let h : N → R be measurable, ∆-local, and exponentially k bounded, |h(η)|≤ C exp(c η(∆)). Then the map (x1,...,xk) 7→ h(δx1 + ··· + δxk ) from ∆ → R k is integrable against 1l∆k λz , therefore by (B.5), for all k ∈ N,

(nr) k k lim h(δx1 + ··· + δxk )jk,∆ (x) dλz (x)= h(δx1 + ··· + δxk )jk,∆(x) dλz (x). r→∞ k k Z∆ Z∆ Furthermore, by (B.4) and the bound (B.2),

1 (nr ) k λz (∆) 1 c k h(δx1 + ··· + δxk ) jk,∆ (x) dλz (x) ≤ e C e λz(∆) . k! k k! Z∆

CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 37

The right-hand side is independent of nr and the sum over k is finite, therefore we can exchange limits and summation over k and obtain ∞ 1 P (nr) (nr ) k lim hd n = lim h(0)j0,∆ + h(δx1 + ··· + δxk )jk,∆ (x)dλz (x) r→∞ r→∞ k! k ZN k=1 Z∆ ! ∞ X 1 k = h(0)j0,∆ + h(δx1 + ··· + δxk )jk,∆(x)dλz (x). k! ∆k kX=1 Z This applies in particular to the constant function h(η) = 1, hence ∞ 1 k 1= j0,∆ + jk,∆(x)dλz (x). k! ∆k Xk=1 Z Step 4: Kolmogorov consistency condition. Conclusion. Let N∆ = {η ∈ N | η(X \ ∆)=0} be the set of configurations supported in ∆. Equip N∆ with the trace of the σ-algebra N. Let Q∆ be the uniquely defined probability measure on N∆ with Janossy densities jk,∆, k ∈ N. By Step 3, we have

lim hdPn = hdQ∆ r→∞ ZN ZN∆ for all ∆-local, measurable, exponentially bounded h. If∆1 ⊂ ∆2 are two sets in Xb, then every function h that is ∆1-local is also ∆2-local, and we deduce the consistency property

hdQ∆1 = hdQ∆2 for all bounded measurable ∆1-local h. N N Z ∆1 Z ∆2 Since X is Polish, it follows that there exists a uniquely defined measure P on (N , N) such that P P −1 Q for all ∆ ∈ Xb, the image of under the projection π∆ : N → N∆, η 7→ η∆ is ◦ π∆ = ∆, and the proof of the lemma is easily concluded. Next we check that we can pass to the limit in the structural equation defining G (z). As mentioned above, the passage to the limit is often done with DLR conditions or the Kirkwood-Salsburg equation, but can also be done in the GNZ equation, compare Zessin’s construction for Papangelou kernels that satisfy a Feller property (reminiscent of the quasilocality of specifications) [Zes09]. Let ∗ b b(x) := |f(x, y)|dλz(x), Xb := ∆ ∈ Xb e dλz < ∞ . (B.6) X ∆ Z n Z o Lemma B.3. P P N Let and ( nr )r∈ be as in Lemma B.2. Let F : X × N → R be measurable and ∗ bounded. Assume that (i) F (x, ·) is local, for each x ∈ X, and (ii) there exists ∆ ∈ Xb such that F (·, η) vanishes on X \ ∆, for all η ∈ N . Then

lim Enr F (x, η)dη(x) = E F (x, η)dη(x) , r→∞ X X hZ i hZ i −W (x,η) −W (x,η) lim Enr e F (x, η + δx) dλz(x)= E e F (x, η + δx) dλz(x). r→∞ X X Z h i Z h i Proof. The map η 7→ X F (x, η)dη(x) is clearly local. It is also tame (hence exponentially bounded), since it is bounded by η(∆) sup |F |. Therefore Lemma B.2 yields the ﬁrst equality. R For the second equality, set

−W (x,η) G(η) := e F (x, η + δx)dλz(x). X Z Because of W ≥ 0, the map G is bounded by λz(∆) sup |F |, however in general it is not local (unless the interaction has ﬁnite range), therefore we approximate by local functions. For B ∈ Xb, set ∞ k 1 (k) GB (η) := 1+ f(x, yi)dη (y) F (x, η + δx)dλz (x). X k! k B i=1 ! Z kX=1 Z Y 38 SABINE JANSEN

In view of |f(x, y)| = | exp(−v(x, y)) − 1|≤ 1, we may bound ∞ k 1 (k) η(B) 1+ f(x, yi) dη (y) F (x, η + δx) dλz(x) ≤ e λz(∆) sup |F |, X k! k Z k=1 ZB i=1 ! X Y hence GB is well-deﬁned and exponentially bounded. GB is clearly local. As a consequence,

lim En [GB ]= E[GB], (B ∈ Xb). (B.7) r→∞ r

Next we bound the expected value of G − GB with respect to Pn and P. Using the expansion for n exp(−W (x, η)) from (A.2) and the boundρ ¯k ≤ 1 (see (B.2)), we obtain ∞ k 1 k En |G − GB| ≤ sup |F | f(x, yi) dλz (y) dλz(x). (B.8) k! Xk k Z∆ k=1 Z \B i=1 ! X Y Notice that the right-hand side does not depend on n. The same bound holds true for the expected ∗ value with respect to P. Because of ∆ ∈ Xb deﬁned in (B.6), we have ∞ k 1 k f(x, yi) dλz (y) dλz(x)= exp |f(x, y)| dλz(y) dλz(x) < ∞. k! Xk X Z∆ k=1 Z i=1 ! Z∆ Z X Y Thus by dominated convergence applied to the right-hand side of (B.8), for every increasing sequence (Bj )j∈N with Bj ր X, we have

lim sup En |G − GBj | =0, j→∞ n∈N similarly for expectation with respect to P. Therefore given ε> 0, we can ﬁnd B ∈ Xb such that ε ε sup En |G − GB | ≤ , E |G − GB | ≤ . n∈N 3 3 The proof is concluded with the help of (B.7) and a straightforward ε/3-argument.

Proof of Theorem B.1. Let (Λn)n∈N be a cofinal sequence of non-empty bounded sets, (Pn)n∈N the finite-volume Gibbs measures with empty boundary conditions, and (Pnr )r∈N the convergent subsequence from Lemma B.2. We show that the accumulation point P is in G (z). The finite- volume Gibbs measures satisfy

−W (x,η) En F (x, η)dη(x) = En e F (x, η + δx) dλz(x), (B.9) Λ Λ hZ n i Z n h i for all F : X × N → R+ measurable. Suppose that F is as in Lemma B.3, i.e., F is bounded, F (x, ·) is local, and F (·, η) vanishes outside a bounded set ∆ ∈ Xb satisfying (B.6). Since ∆ ⊂ Λn for all suﬃciently large n, we can replace integration over Λn in (B.9) by integration over X, and then take limits along the subsequence (nr). Lemma B.2 shows that

−W (x,η) E F (x, η)dη(x) = E e F (x, η + δx) dλz(x). (B.10) X X hZ i Z h i The identity is extended to all non-negative measurable F with a π-λ theorem for σ-ﬁnite measures. Consider the measures on (X × N , X ⊗ N) deﬁned by

−W (x,η) C (B)= E 1lB(x, η)dη(x) , µ(B)= E e 1lB(x, η + δx) dλz(x). X X Z Z h i h i ∗ C is the Campbell measure of P. Applying Eq. (B.10) to F (x, η) = 1l∆(x)1lA(x) with ∆ ∈ Xb and A ∈ A a local event, we ﬁnd that ∗ C (∆ × A)= µ(∆ × A) (∆ ∈ Xb , A ∈ A).

Because ofρ ¯1 ≤ 1, we have

C (∆ × N )= E[η(∆)] = ρ¯1dλz(∆) ≤ λz(∆) < ∞ Z∆ for all ∆ ∈ Xb, hence C is σ-ﬁnite. The measure µ is σ-ﬁnite as well, since µ(∆×N )= C (∆×N ) < ∗ ∞ for all ∆ ∈ Xb and X can be written as a countable union of sets in Xb . To conclude, we note CLUSTEREXPANSIONSFORGIBBSPOINTPROCESSES 39

∗ that the collection of sets of the form ∆ × A with ∆ ∈ Xb and A ∈ A is a π-system that generates X ⊗ N. Since the measures C and µ are σ-ﬁnite and coincide on a generating π-system, they must be equal. It follows that (B.10) extends to all indicators of measurable sets and then to all measurable non-negative functions. Thus P satisﬁes (GNZ) and we have proven P ∈ G (z). In particular, G (z) 6= ∅.

Appendix C. Random connection model Here we provide some details for the random connection model discussed in Section 2.6. The- orem C.1 below is a variant of a standard result for translationally invariant random connection model. The theorem follows by a straightforward adaptation of a construction with pruned branching random walk from [MPS97], the details are left to the reader. Let η be a Poisson point process with intensity λz, defined on some probability space (Ω, F, P). q For simplicity let us assume that λz has no atoms so that η is simple. For q ∈ X write η = η + δx. Assume we are given a measurable map ϕ : X × X → [0, 1] with ϕ(x, y) = ϕ(y, x) and X ϕ(x, y)dλz (y) < ∞; for example, ϕ(x, y)= |f(x, y)| with f(x, y) satisfying (2.31). The random connection model is a random graph G whose vertices are the points of η. Roughly, it is constructed R κ as follows: Conditional on η = i=1 δxi , the events {{xi, xj } ∈ E(G)}, i < j, are independent and the probability that {xi, xj } belongs to the graph is given by ϕ(xi, xj ). For q ∈ X, a random q P graph G whose vertices are the points of η+δq is defined, intuitively, in a similar fashion; we write C (q, η) for the connected component of q in Gq. Precise definitions are found in [MR96, LZ17] for the translationally invariant case in Rd and in [LNS18] for the general setup. q Remember the branching process (ηn)n∈N0 introduced above Proposition 2.12. It is a well-known result [MR96, Chapter 6] that extinction of a dominating branching process implies absence of percolation in the random connection model. q Theorem C.1. Suppose that for λz-almost all q, (ηn)n∈N goes extinct with probability 1. Then for λz-almost all q ∈ X, P(|C (q, η)| < ∞)=1. The proof is a straightforward adaptation of the proof in [MR96, MPS97] and therefore omitted. Combining Theorem C.1 with Proposition 2.12, we obtain the following corollary.

Corollary C.2. Let z : X → R+ be a measurable function with X zdλ < ∞ for all B ∈ Xb, and f(x, y) = exp(−v(x, y)) − 1 with v : X × X → [0, ∞) ∪ {∞} measurable. Suppose that the R suﬃcient convergence condition (KPUt) with t = 0 holds true. Then in the random connection model with connectivity probability ϕ(x, y) = |f(x, y)|, driven by a Poisson point process with intensity measure λz, we have P(|C (q, η)| < ∞)=1 for λz-almost all q ∈ X.

We leave as an open problem whether a similar result holds true for the convergence criterion (FPt). Acknowledgments. The generalization to the continuum set-up of the convergence condition by Fernández and Procacci grew out of discussions with Dimitrios Tsagkarogiannis. I am also indebted to Christoph Hofer-Temmel and Matthias Schulte for alerting me to expansion techniques in the theory of point processes. The present work benefitted from the stimulating atmosphere at the Oberwolfach mini-workshop Cluster expansions: from combinatorics to analysis via probability (February 2017). I gratefully acknowledge support by the DFG scientific network Cumulants, concentration, and superconcentration. References

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