2. Main results

2.1. Setting

Let $(\mathbb X,\textrm{dist})$ be a complete separable metric space, $\mathcal X$ the Borel $\sigma$ -algebra, and $\mathcal X_\textsf{b}$ the collection of bounded Borel sets. Let $\lambda$ be a reference measure on $\mathbb X$ that is locally finite, i.e. $\lambda(B)< \infty$ for all $B\in \mathcal X_\textsf{b}$ . A locally finite counting measure is a measure $\eta$ on $(\mathbb X,\mathcal X)$ with $\eta(B) \in {\mathbb{N}}_0$ for all $B\in \mathcal X_\textsf{b}$ . Let $\mathcal N$ and $\mathcal N_\textsf{f}$ be the spaces of locally finite and finite counting measures, respectively. Each non-zero $\eta\in \mathcal N$ can be written as $ \eta =\sum_{j=1}^\kappa \delta_{x_j}$ with $\kappa \in {\mathbb{N}}\cup \{\infty\}$ and $x_1,x_2,\ldots \in \mathbb X$ . We define equip $\mathcal N$ with the $\sigma$ -algebra $\mathfrak N$ generated by the integer-valued random variables $\eta \mapsto \eta(B)$ , $B\in \mathcal X_\textsf{b}$ . The nth factorial measure of $\eta = \sum_{j=1}^ \kappa \delta_{x_j} \in \mathcal N$ is denoted by $\eta^{(n)}$ . It is the unique measure on $\mathbb X^n$ such that, for all measurable non-negative $g \colon \mathbb X^n\to {\mathbb{R}}_+$ , we have

\begin{equation*}\int_{\mathbb X^n} g\, {\textrm{d}} \eta^{(n)} = \sum_{i_1,\ldots,i_n\leq \kappa}\kern-22pt{\phantom{A^A}}^{\neq} g(x_{i_1},\ldots,x_{i_n})\end{equation*}

with summation over pairwise distinct indices $i_1,\ldots,i_n$ .

Fix a non-negative pair potential v, i.e. a measurable function $v \colon \mathbb X\times \mathbb X\to {\mathbb{R}}_+\cup \{\infty\}$ that is symmetric, $v(x,y) = v(y,x)$ . Set $H_0=0$ and

\begin{equation*}H_n(x_1,\ldots,x_n) = \sum_{1\leq i \lt j \leq n} v(x_i,x_j)\quad (x_1,\ldots,x_n\in \mathbb X)\end{equation*}

so that $H_1(x_1) = 0$ . Further define

\begin{equation*}W(x;\ \eta) =\int_\mathbb X v(x,y)\,{\textrm{d}} \eta(\kern1pt y)\quad (x\in \mathbb X,\ \eta\in \mathcal N).\end{equation*}

Cartesian products such as $\mathbb X^n$ or $\mathbb X\times \mathfrak N$ are equipped with product $\sigma$ -algebras, e.g. $\mathcal X^{\otimes n}$ , $\mathcal X\otimes \mathcal N$ . We adopt the conventions $\log 0 = -\infty$ , $0\log 0 =0$ , $0^0 =1$ , $\exp({-}\infty) =0$ , and ${\exp(\infty) = \infty}$ .

Definition 2.1. Let $z \colon \mathbb X\to {\mathbb{R}}_+$ be measurable map with $\int_B z\,{\textrm{d}} \lambda \lt \infty$ for all $B\in \mathcal X_\textsf{b}$ . A probability measure $\textsf{P}$ on $(\mathcal N,\mathfrak N)$ is a Gibbs measure with pair interaction v(x, y) and activity z if

\begin{equation}\textsf{E}\bigg[ \int_{\mathbb X} F(x,\eta) \,{\textrm{d}} \eta(x)\bigg] = \int_\mathbb X \textsf{E}[ F(x,\eta+ \delta_x)\,{\textrm{e}}^{-W(x;\ \eta)}] z(x) \,{\textrm{d}} \lambda(x)(\textrm{GNZ})\end{equation}

for all measurable $F \colon \mathbb X\times \mathcal N \to[0,\infty)$ . The set of Gibbs measures is denoted $\mathscr{G}(z)$ .

The GNZ equation named after Georgii, Nguyen, and Zessin is equivalent to the DLR conditions (named after Dobrushin, Lanford, and Ruelle) more familiar to mathematical physicists; see [Reference Georgii22] and [Reference Nguyen and Zessin50]. It is a probabilistic cousin of the Kirkwood–Salsburg equations [Reference Ruelle60, 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

\begin{equation*}(Lg)(\eta) = \int_{\mathbb X} [ g(\eta - \delta_x) - g(\eta)] \,{\textrm{d}} \eta(x) + \int_\mathbb X z(x) \,{\textrm{e}}^{- W(x;\ \eta)} [ g(\eta+ \delta_x) - g(\eta)] \,{\textrm{d}} \lambda(x),\end{equation*}

sometimes called continuum Glauber dynamics; see [Reference Preston54] and [Reference Kondratiev and Lytvynov34] for the construction of such processes and [Reference Fernández, Ferrari and Garcia18], [Reference Bertini, Cancrini and Cesi4], [Reference Kondratiev, Kuna and Ohlerich35], [Reference Schuhmacher and Stucki62], and [Reference Fernández, Groisman and Saglietti19] for some applications. When the interaction vanishes ( $v\equiv 0$ ), the GNZ equation reduces to the Mecke formula for the Poisson point process with intensity measure $\lambda_z$ and, in the context of Poisson–Malliavin calculus, the operator L and the generated semi-group are called the Ornstein–Uhlenbeck operator and the Mehler semi-group [Reference Last40].

The nth factorial moment density $\rho_n(x_1,\ldots,x_n)$ , called n-point correlation function in statistical mechanics, is the Radon–Nikodým derivative $\rho_n = {{\textrm{d}} \alpha_n}/{{\textrm{d}} \lambda^n}$ of the nth factorial moment measure $\alpha_n$ of $\textsf{P}$ [Reference Daley and Vere-Jones13, Reference Last and Penrose41]. Thus

\begin{equation*}\textsf{E} \bigg[ \int_{\mathbb X^n} g\, {\textrm{d}} \eta^{(n)}\bigg] = \int_{\mathbb X^n} g \rho_n \,{\textrm{d}} \lambda^n\end{equation*}

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 2.1. Let $\textsf{P}\in \mathscr{G}(z)$ . Then the n-point functions $\rho_n$ exist and satisfy

(2.1) \begin{equation}\rho_n(x_1,\ldots,x_n) =z(x_1)\cdots z(x_n)\, {\textrm{e}}^{-H_n(x_1,\ldots,x_n)} \int_\mathcal N \,{\textrm{e}}^{- \sum_{i=1}^n W(x_i;\ \eta)} \,{\textrm{d}} \textsf{P}(\eta)\end{equation}

for all $n\in {\mathbb{N}}$ and $\lambda^n$ -almost all $(x_1,\ldots,x_n)\in \mathbb X^n$ .

The lemma allows us to adopt (2.1) 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

(2.2) \begin{equation} \rho_n(x_1,\ldots,x_n) \leq z(x_1)\cdots z(x_n).\end{equation}

The factorial cumulant densities $\rho_n^{\textsf{T}}(x_1,\ldots,x_n)$ , called truncated correlation functions in statistical mechanics, are uniquely defined by the requirement that they are symmetric and satisfy, for all $n\in {\mathbb{N}}$ ,

\begin{equation*}\rho_n(x_1,\ldots,x_n) = \sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n} \prod_{k=1}^r \rho_{\#V_k}^{\textsf{T}} ( (x_i)_{i\in V_k})\end{equation*}

where $\mathcal P_n$ is the collection of set partitions of $\{1,\ldots,n\}$ . For example, $\rho_1^{\textsf{T}}(x_1) = \rho_1(x_1)$ and $\rho_2^{\textsf{T}}(x_1,x_2) = \rho_2(x_1,x_2) - \rho_1(x_1) \rho_1(x_2)$ .

Example 2.1. (Mixture of hard spheres and Poisson exclusion process with random radii.) Let $\mathbb X = {\mathbb{R}}^d \times {\mathbb{R}}_+$ , equipped with the Euclidean distance, and $\lambda$ the Lebesgue measure. Consider the interaction $v((x,r),(y,R)) \,:\!= \infty\,\textrm{1\kern-3.5pt1}_{\{|x-y|\leq R+r\}}$ so that

\begin{equation*}{\textrm{e}}^{- v((x,r),(y,R))} = \textrm{1\kern-3.5pt1}_{\{ B_d(x,r) \cap B_d(y,R) = \varnothing\}}\end{equation*}

with $B_d(x,r)\subset {\mathbb{R}}^d$ the closed ball of radius r centered at x. Let $\textsf{P}\in \mathscr{G}(z)$ . Any $\eta\in \mathcal N$ can be written as $\eta = \sum_{j=1}^\kappa \delta_{(x_j,r_j)}$ , and we have $B_d(x_i,r_i)\cap B_d(x_j,r_j) =\varnothing$ for all $i\neq j$ , $\textsf{P}$ -almost surely. The measure describes a mixture of spheres of different radii, distinct spheres are not allowed to overlap. $\textsf{P}$ is the distribution of a marked point process on ${\mathbb{R}}^d$ , with marks in ${\mathbb{R}}_+$ . A priori, the points are Poisson-distributed with intensity $z_0(x) = \int_0^\infty z(x,r) \,{\textrm{d}} r$ (assuming that each $z_0(x)$ is finite), and sphere at point x has a random radius with probability density function $r\mapsto p(x,r)$ given by $p(x,r) = z(x,r) /z_0(x)$ . We can think of $\textsf{P}$ , 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 [Reference Hofer-Temmel and Houdebert30], [Reference Torquato64] 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 [Reference Fernández and Procacci17] for hard-core interactions and discrete spaces $\mathbb X$ to non-negative interactions and more general spaces $\mathbb X$ .

Define Mayer’s f-function

\begin{equation*}f(x,y) \,:\!= {\textrm{e}}^{ - v(x,y)} -1\quad (x,y\in \mathbb X).\end{equation*}

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 interested 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 [Reference Fernández, Ferrari and Garcia18].

Theorem 2.1. Assume $f(x,y) \leq 0$ for $\lambda^2$ -almost all $(x,y)\in \mathbb X^2$ and $\int_B z\,{\textrm{d}} \lambda \lt \infty$ for all $B\in \mathcal X_\textsf{b}$ . Suppose that there exists a measurable function $a \colon \mathbb X\to {\mathbb{R}}_+$ and some $t>0$ such that

\begin{equation*}\sum_{k=1}^\infty \dfrac{{\textrm{e}}^{tk} }{k!}\int_{\mathbb X^k} \prod_{j=1}^k |\kern2pt f(x_0,y_j)| \prod_{1\leq i \lt j \leq k} (1+ f(\kern1pt y_i,y_j)) \prod_{j=1}^k z(\kern1pt y_j) \,{\textrm{e}}^{a(\kern1pt y_j)} \,{\textrm{d}} \lambda^k(\kern1pt{\textit{\textbf{y}}}) \leq {\textrm{e}}^{a(x_0)} - 1(\textsf{FP}_t)\end{equation*}

for $\lambda$ -almost all $x_0\in \mathbb X$ . Then $\#\mathscr{G}(z) = 1$ .

The theorem is proved in Section 3; results for $t=0$ are provided in Section 2.3. In view of $1+f(y_i,y_j)\leq 1$ , a sufficient condition for ( $\textsf{FP}_t$ ) to hold true is that

\begin{equation*}{\textrm{e}}^t\int_\mathbb X |\kern2pt f(x_0,y)|\,{\textrm{e}}^{a(\kern1pt y)} z(\kern1pt y) \,{\textrm{d}} \lambda(\kern1pt y) \leq a(x_0)(\textsf{KPU}_t)\end{equation*}

for $\lambda$ -almost all $x_0\in \mathbb X$ . Condition ( $\textsf{KPU}_t$ ) with $t=0$ is discussed in depth in [Reference Ueltschi65]; it is a continuum version of the Kotecký–Preiss condition for polymer systems [Reference Kotecký and Preiss37]. Condition ( $\textsf{KPU}_t$ ) with $t\gt{0}$ is a special case of Assumption 3 in [Reference Poghosyan52], who considered both non-negative and attractive potentials. Condition ( $\textsf{FP}_t$ ) with $t=0$ corresponds to the criterion for discrete polymer models by Fernández and Procacci [Reference Fernández and Procacci17] and for hard spheres by Fernández, Procacci, and Scoppola [Reference Fernández, Procacci and Scoppola20].

Example 2.2. (Hard spheres / Poisson exclusion process.) Choose $\mathbb X = {\mathbb{R}}^d$ with the Euclidean norm $|{\cdot}|$ and the Lebesgue measure $\lambda = \textrm{Leb}$ . Let $v(x,y) =\infty \textrm{11}_{\{|x-y|\leq r\}}$ with $r\gt{0}$ . Suppose that

(2.3) \begin{equation} z\,\textrm{Leb}(B(0,r))\lt{1} /\textrm{e}\simeq 0.36787 {.} \end{equation}

Because of $1/\textrm e = \max_{a\gt{0}} a \exp({-}a)$ , we can find $a,t\gt{0}$ such that

\begin{equation*}{\textrm{e}}^t \int_{\mathbb R^d} |\kern2pt f(x,y)| \,{\textrm{e}}^a z\,{\textrm{d}} y = {\textrm{e}}^t \textrm{Leb}\, (B(0,r)) \,{\textrm{e}}^a \leq a\end{equation*}

and conditions ( $\textsf{KPU}_t$ ) and ( $\textsf{FP}_t$ ) holds true for some constant function $a(\kern1pt y) = a$ . The criterion (2.3) is well known [Reference Ruelle60, Chapter 4]. As shown in [Reference Fernández, Procacci and Scoppola20], for a two-dimensional gas of hard spheres, condition ( $\textsf{FP}_t$ ) allows one to improve (2.3) to $z\,\textrm{Leb}(B(0,r))<0.5107$ .

The condition ( $\textsf{FP}_t$ ) 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 $\mathcal G_n$ be the collection of graphs $G=(V,E)$ with vertex set $V=\{1,\ldots,n\}=[n]$ and edge set $E\subset \{\{i,j\}\mid i,j\in V,\, i \neq j\}$ . We write $E(G) =E$ for the edge set of G. We define graph weights

\begin{equation*}w(G;\ x_1,\ldots,x_n)\,:\!= \prod_{\{i,j\} \in E(G)} f(x_i,x_j){.}\end{equation*}

Let $\mathcal C_n\subset \mathcal G_n$ be the collection of connected graphs with vertex set [n] and

\begin{equation*}\varphi_n^{\textsf{T}}(x_1,\ldots,x_n)\,:\!= \sum_{G\in \mathcal C_n} w(G;\ x_1,\ldots,x_n)\end{equation*}

the nth Ursell function. Let $\lambda_z$ be the measure on $\mathbb X$ that is absolutely continuous with respect to $\lambda$ with Radon–Nikodým derivative $z({\cdot})$ . For $h \colon \mathbb X\to {\mathbb{R}} \cup\{\infty\}$ , we introduce the integrability condition

(2.4) \begin{equation} \int_\mathbb X |{\textrm{e}}^{-h(x)}-1| \,{\textrm{e}}^{a(x)} \,{\textrm{d}} \lambda_z(x) \lt \infty.\end{equation}

Theorem 2.2. Under the conditions of Theorem 2.1, the log-Laplace functional of the unique Gibbs measure $\textsf{P}\in \mathscr{G}(z)$ satisfies

(2.5) \begin{equation} \log \textsf{E} [ {\textrm{e}}^{-\int_\mathbb X h{\textrm{d}} \eta} ]= \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\mathbb X^n} ( {\textrm{e}}^{-\sum_{j=1}^n h(x_j)} - 1) \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}})\end{equation}

for all measurable $h \colon \mathbb X\to [{-}t,\infty) \cup \{\infty\}$ or $h \colon \mathbb X\to \{ s \in \mathbb C\mid \, {\textrm{Re}\,} s \geq - t \}$ that satisfy (2.4). Moreover,

(2.6) \begin{equation} \sum_{n=2}^\infty \dfrac{{\textrm{e}}^{t(n-1)}}{(n-1)!}\int_{\mathbb X^{n-1}} |\varphi_n^{\textsf{T}}(x_1,\ldots,x_n)| \,{\textrm{d}} \lambda_z(x_2)\cdots {\textrm{d}} \lambda_z(x_n) \leq {\textrm{e}}^{a(x_1)}-1\end{equation}

for $\lambda$ -almost all $x_1\in \mathbb X$ .

The convergence of the right-hand side of (2.5) follows from the inequality (2.6) and the estimate

(2.7) \begin{equation} |{\textrm{e}}^{-\sum_{j=1}^n h(x_j)} - 1| = \bigg| \sum_{j=1}^n ({\textrm{e}}^{- h(x_j)} - 1) \,{\textrm{e}}^{ - \sum_{i=1}^{j-1} h(x_i)}\bigg| \leq {\textrm{e}}^{(n-1) t} \sum_{j=1}^n |{\textrm{e}}^{- h(x_j)}-1|.\end{equation}

Remark 2.1. (Cumulants.) Complex parameters allow us to extract bounds on cumulants via contour integrals. For example, let $\Lambda\in \mathcal X$ . Applying Theorem 2.2 to $- h = s N_\Lambda$ with $|s|\leq t$ , we obtain

\begin{equation*}\log \textsf{E}[{\textrm{e}}^{sN_\Lambda}] = \sum_{\ell=1}^\infty \dfrac{\kappa_\ell}{\ell!} s^\ell\end{equation*}

with

\begin{equation*}|\kappa_\ell| = \ell ! \bigg| \dfrac{1}{2\pi \textrm{i}}\oint \log \textsf{E} [{\textrm{e}}^{sN_\Lambda}] \dfrac{{\textrm{d}} s}{s^{\ell+1}}\bigg| \leq \dfrac{\ell!}{t^\ell} \int_{\Lambda} \,{\textrm{e}}^a \,{\textrm{d}} \lambda_z.\end{equation*}

See [Reference Bryc7] for applications to central limit theorems.

Remark 2.2. (Signed Lévy measure.) Equation (2.5) can be rewritten as

(2.8) \begin{equation} \log \textsf{E} [ {\textrm{e}}^{-\int_\mathbb X h{\textrm{d}} \eta} ]= \int_{\mathcal N_\textsf{f}} ({\textrm{e}}^{-\int_\mathbb X h\,{\textrm{d}} \eta} - 1) \,{\textrm{d}} \Theta(\eta)\end{equation}

where the signed measure $\Theta$ on $\mathcal N_\textsf{f}$ is defined by

\begin{equation*}\int_{\mathcal N_\textsf{f}} g\,{\textrm{d}} \Theta = \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} g(\delta_{x_1}+\cdots + \delta_{x_n}) \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation*}

Log-Laplace transforms such as (2.8) – but with non-negative measures $\Theta$ – appear naturally in the context of cluster point processes and infinitely divisible point processes [Reference Daley and Vere-Jones13, Chapter 10.2]. Following Nehring, Poghosyan, and Zessin [Reference Nehring, Poghosyan and Zessin49] we may call $\Theta$ the pseudo- or signed Lévy measure of $\textsf{P}$ (compare (2.11) below).

Theorem 2.3. Under the conditions of Theorem 2.1, the truncated correlation functions (factorial cumulant densities) of the unique Gibbs measure $\textsf{P} \in \mathscr{G}(z)$ satisfy

(2.9) \begin{align} \rho_n^{\textsf{T}}(x_1,\ldots,x_n) = \prod_{j=1}^n z(x_j) \bigg(\varphi_{n}^{\textsf{T}}(x_1,\ldots,x_n)+ \sum_{k=1}^\infty \dfrac{1}{k!}\int_{\mathbb X^k} \varphi_{n+k}^{\textsf{T}}(x_1,\ldots,x_n,y_1,\ldots,y_k) \,{\textrm{d}} \lambda_z^k({\textit{\textbf{y}}})\bigg)\end{align}

for all $n\in {\mathbb{N}}$ and $\lambda^n$ -almost all $(x_1,\ldots,x_n) \in \mathbb X^n$ , with

(2.10) \begin{equation} \sum_{k=0}^\infty \dfrac{1}{k!}\int_{\mathbb X^{n-1+k}} |\varphi_n^{\textsf{T}}(x_1,\ldots,x_{n+k})| \,{\textrm{d}} \lambda_z(x_2)\cdots {\textrm{d}} \lambda_z(x_{n+k}) \leq \dfrac{(n-1)!}{({\textrm{e}}^t- 1)^{n-1}}\, {\textrm{e}}^{a(x_1)}.\end{equation}

The theorem is proved in Section 5. The bound (2.10) proves the convergence of the expansion (2.9) in a suitable $L^1$ -norm, in agreement with the spaces used by Kuna and Tsagkarogiannis [Reference Kuna and Tsagkarogiannis39] and Last and Ziesche [Reference Last and Ziesche42] for the two-point function $\rho_2^{\textsf{T}}$ . Pointwise estimates better suited to estimating decay of correlations and mixing properties can be found in [Reference Ueltschi65]; see also [Reference Hanke27, Corollary 5.2] and the references therein for the two-point function.

Remark 2.3. (Positivity of t.) For the bound (2.10) the strict positivity of t is essential. Something similar happens for the factorial cumulants of random variables. Let N be an ${\mathbb{N}}_0$ -valued random variable that is infinitely divisible with finite Lévy measure $\nu =\sum_{n\in {\mathbb{N}}} \nu_n \delta_n$ . Then

(2.11) \begin{equation} \log {\mathbb{E}}[{\textrm{e}}^{-\lambda N} ] = \sum_{n=1}^ \infty ({\textrm{e}}^{-\lambda n} - 1) \nu_n \quad (\lambda\geq 0).\end{equation}

The factorial cumulants $(\alpha_n)_{n\in {\mathbb{N}}}$ , if they exist, are defined as the coefficient in the asymptotic expansion

\begin{equation*}\log {\mathbb{E}}[{\textrm{e}}^{-\lambda N} ] = \sum_{n=1}^ \infty \dfrac{\alpha_n}{n!} ( {\textrm{e}}^{-\lambda} - 1)^n \quad (\lambda \searrow 0).\end{equation*}

If the series $\sum_n \nu_n r^n$ has a radius of convergence ${\textrm{e}}^t>1$ , then $u\mapsto \log {\mathbb{E}}[(1+u)^N]$ is analytic in $|u|<{\textrm{e}}^t-1$ , which leads to bounds on its Taylor coefficients $\alpha_n$ . If on the other hand $\sum_n \nu_n r^n$ diverges for $r>1$ , then nothing can be said about the $\alpha_n$ without additional information on the $\nu_n$ .

The correlation functions are associated with graphs subject to slightly weaker connectivity constraints. For $k \leq n$ , let $\mathcal D_{k,n}\subset \mathcal G_n$ be the set of multi-rooted graphs G with vertices $1,\ldots,n$ , roots $1,\ldots,k$ , such that every non-root vertex $j\in \{k+1,\ldots,n\}$ is connected to some root vertex $i\in\{1,\ldots,k\}$ by a path in G. Note that $\mathcal D_{n,n} = \mathcal G_n$ and $\mathcal D_{1,n} = \mathcal C_n$ . Set

(2.12) \begin{equation} \psi_{k,n}(x_1,\ldots,x_n) = \sum_{G\in \mathcal D_{k,n}}w(G;\ x_1,\ldots,x_n).\end{equation}

Theorem 2.4. Under the conditions of Theorem 2.1, the correlation functions (factorial moment densities) of the unique Gibbs measure $\textsf{P} \in \mathscr{G}(z)$ are given by

(2.13) \begin{equation} \rho_n(x_1,\ldots,x_n) = \prod_{j=1}^n z(x_j) \bigg(\!\psi_{n,n}(x_1,\ldots,x_n)+\! \sum_{k=1}^\infty \dfrac{1}{k!}\!\int_{\mathbb X^k}\! \psi_{n,n+k}(x_1,\ldots,x_n,y_1,\ldots,y_k) \,{\textrm{d}} \lambda_z^k({\textit{\textbf{y}}})\!\bigg)\end{equation}

with

(2.14) \begin{align}& |\psi_{n,n}(x_1,\ldots,x_n)|+ \sum_{k=1}^\infty \dfrac{{\textrm{e}}^{tk}}{k!}\int_{\mathbb X^k} |\psi_{n,n+k}(x_1,\ldots,x_n,y_1,\ldots,y_k)| \,{\textrm{d}} \lambda_z^k({\textit{\textbf{y}}}) \notag \\& \qquad \leq \prod_{1\leq i{<}j\leq n}(1+f(x_i,x_j)) \,{\textrm{e}}^{\sum_{i=1}^n a(x_i)}. \end{align}

The theorem is proved at the end of Section 4.

Let us briefly comment on the order in which Theorems 2.1, 2.2, 2.3, and 2.4 are proved. Theorem 2.1 about the uniqueness of Gibbs measures is proved first with the help of Kirkwood–Salsburg equations, in Section 3. Theorem 2.4 is proved 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 2.2 in Section 5, and finally Theorem 2.3 is deduced from Theorem 2.2, also in Section 5.

2.3. About the case $t=0$

The strict positivity $t\gt{0}$ in condition ( $\textsf{FP}_t$ ) ensures that a certain linear operator ${\textit{\textbf{K}}}_z$ has operator norm $\|{\textit{\textbf{K}}}_z\|\leq {\textrm{e}}^{-t}<1$ , 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 ( $\textsf{FP}_t$ ) with $t=0$ stays useful. Roughly, the condition ensures that the cluster expansions for the correlation functions of Gibbs measures in finite volume $\Lambda$ (with empty boundary conditions) are absolutely convergent, uniformly in the volume $\Lambda$ . The uniformity in the volume guarantees the existence of the infinite-volume limit. This reasoning is fairly standard in mathematical physics.

Let $\Lambda\in \mathcal X_\textsf{b}$ be a bounded set (thus $\lambda_z(\Lambda) \lt \infty$ , ‘finite volume’). Assume $\lambda_z(\Lambda)>0$ . Let

\begin{equation*}\mathcal N_\Lambda = \{\eta\in\mathcal N\mid \text{for all}\ A\in \mathcal X \colon \eta(A) = \eta(A\cap \Lambda) \}\end{equation*}

be the space of point configurations in $\Lambda$ . The space $\mathcal N_\Lambda$ is equipped with the trace of the $\sigma$ -algebra $\mathfrak N$ . It is not too difficult to see that there is a uniquely defined measure $\textsf{P}_\Lambda$ such that

\begin{equation*}\textsf{E}_\Lambda\bigg[\int_\Lambda F(x,\eta) \,{\textrm{d}} \eta(x)\bigg] = \int_\Lambda \textsf{E}_\Lambda [ F(x,\eta+ \delta_x) \,{\textrm{e}}^{- W(x,\eta)} ] z(x) \,{\textrm{d}} \lambda(x)\end{equation*}

for all measurable $F \colon \Lambda\times \mathcal N\to [0,\infty)$ . Moreover, the measure $\textsf{P}_\Lambda$ is absolutely continuous with respect to the distribution of the Poisson point process with intensity measure $ z\textrm{11}_\Lambda\,{\textrm{d}} \lambda$ , with a Radon–Nikodým derivative proportional to $\exp\kern-1pt(\! - H(\eta))$ , where $H(\delta_{x_1}+\cdots+ \delta_{x_n}) = H_n(x_1,\ldots,x_n)$ . The measure $\textsf{P}_\Lambda$ is usually called a finite-volume Gibbs measure with empty boundary conditions (because $H(\eta)$ does not include interactions with points outside $\Lambda$ ).

We formulate analogues of Theorems 2.2 and 2.4 before commenting on Theorem 2.1. As explained earlier, Theorem 2.3 really needs the condition $t>0$ and it has no direct counterpart for $t=0$ (see however [Reference Ueltschi65]).

Theorem 2.5. Assume $v(x,y)\geq 0$ for $\lambda^2$ -almost all $(x,y)\in \mathbb X^2$ and $\int_B z \,{\textrm{d}} \lambda<\infty$ for all $B\in \mathcal X_\textsf{b}$ . Suppose that there exists a measurable function $a \colon \mathbb X\to {\mathbb{R}}_+$ such that condition ( $\textsf{FP}_t$ ) holds true with $t=0$ . Then

(2.15) \begin{equation} \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^{n}} |\varphi_n^{\textsf{T}}(x_0,\ldots,x_n)| \,{\textrm{d}} \lambda_z(x_1)\cdots {\textrm{d}} \lambda_z(x_n) \leq {\textrm{e}}^{a(x_0)}-1\end{equation}

for $\lambda$ -almost all $x_0\in \mathbb X$ . Moreover, for every non-empty $\Lambda \in \mathcal X_\textsf{b}$ , and all measurable $h \colon \Lambda\to [0,\infty]$ (or h complex-valued with non-negative real part) with $\int_\Lambda |{\textrm{e}}^{-h} - 1| \,{\textrm{e}}^a \,{\textrm{d}} \lambda_z <\infty$ , we have

(2.16) \begin{equation} \log \textsf{E}_\Lambda [ {\textrm{e}}^{-\int_\Lambda h\,{\textrm{d}} \eta}]= \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\Lambda^n} ( {\textrm{e}}^{-\sum_{j=1}^n h(x_j)} - 1) \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation}

Theorem 2.6. Under the conditions of Theorem 2.5, the bound (2.14) holds true for $t=0$ and integration over $\mathbb X^n$ , and the n-point functions $\rho_{n,\Lambda}$ of $\textsf{P}_\Lambda$ are given by (2.13) with integrals over $\Lambda^n$ instead of $\mathbb X^n$ .

Theorems 2.5 and 2.6 are proved in Section 6. Note that the convergence estimate (2.15) really has integrals over $\mathbb X^{n}$ (not $\Lambda^{n}$ ), and the expansions for the log-Laplace transform and correlation functions depend on $\Lambda$ only through the domains of integration. As a consequence, we can pass to the infinite-volume limit. Let $(\Lambda_n)_{n\in{\mathbb{N}}} $ be a sequence of non-empty sets in $\mathcal X_\textsf{b}$ with $\Lambda_n \nearrow \Lambda$ . Then, for all measurable $h \colon \Lambda\to [0,\infty]$ (or h complex-valued with non-negative real part) with $\int_\mathbb X |{\textrm{e}}^{-h} - 1| \,{\textrm{e}}^a \,{\textrm{d}} \lambda_z <\infty$ , we have

(2.17) \begin{equation} \lim_{n\to \infty} \log \textsf{E}_{\Lambda_n} [ {\textrm{e}}^{-\int_{\Lambda_n} h\,{\textrm{d}} \eta}]= \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\mathbb X^n} ( {\textrm{e}}^{-\sum_{j=1}^n h(x_j)} - 1) \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation}

A similar convergence for correlation functions $\rho_{n,\Lambda_n}$ holds true as well; the precise statement is left to the reader. In view of (2.17), we should expect that the sequence $(\textsf{P}_{\Lambda_n})_{n\in {\mathbb{N}}}$ converges in some suitable sense to a uniquely defined probability measure $\textsf{P}$ on $(\mathcal{N},\mathfrak{N})$ . The measure $\textsf{P}$ should be in $\mathscr{G}(z)$ and its log-Laplace functional (restricted to the functions h under consideration) should be given by the right-hand side of (2.17). 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 2.1.

Mathematical physicists often work with the grand-canonical partition function

\begin{equation*}\Xi_\Lambda(z) = 1+ \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\Lambda^n} \,{\textrm{e}}^{- H_n(x_1,\ldots,x_n)} \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}){,}\end{equation*}

instead of log-Laplace transforms.

Theorem 2.7. Under the conditions of Theorem 2.5, if $\Lambda\in \mathcal X_\textsf{b}$ with $\int_\Lambda \,{\textrm{e}}^a \,{\textrm{d}} \lambda_z <\infty$ , then

\begin{equation*}\log \Xi_\Lambda(z) = \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\mathbb X^n}\varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}){,} \end{equation*}

with absolutely convergent integrals and series.

The theorem is proved in Section 6. For $\int_\Lambda \,{\textrm{e}}^a \,{\textrm{d}} \lambda_z<\infty$ and measurable, non-negative h, we have

\begin{equation*}\textsf{E}_\Lambda [{\textrm{e}}^{-\int_\Lambda h \,{\textrm{d}} \eta}] = \log \Xi_\Lambda( z\,{\textrm{e}}^{-h}) - \log \Xi_\Lambda(z), \end{equation*}

and the formula (2.16) is easily recovered.

2.4. Attractive pairwise interactions

The results proved in this article cover non-negative pairwise interactions only, for the reader’s convenience we summarize some known results on attractive or multi-body interactions. Consider first pairwise possibly attractive interactions. Let $v \colon \mathbb X\times \mathbb X\to {\mathbb{R}}\cup\{\infty\}$ with $v(x,y) = v(y,x)$ on $\mathbb X\times \mathbb X$ . The pair potential v is called stable [Reference Ruelle60, Chapter 3.2] if, for some constant $B\geq 0$ and all $n\in {\mathbb{N}}$ and $x_1,\ldots, x_n\in \mathbb X$ , we have

\begin{equation*}\sum_{1\leq i \lt j \leq n} v(x_i,x_j) \geq - B n.\end{equation*}

Sometimes it is more convenient to allow for x-dependent B, and replace the previous condition by

(2.18) \begin{equation} \sum_{1\leq i \lt j \leq n} v(x_i,x_j) \geq - \sum_{i=1}^n B(x_i)\end{equation}

for some measurable function $B \colon \mathbb X\to {\mathbb{R}}_+$ and all $n\in {\mathbb{N}}$ and all $x_1,\ldots, x_n\in \mathbb X$ ; see [Reference Poghosyan and Ueltschi53]. A sufficient condition for stability in the sense of (2.18) is local stability, a notion somewhat more popular in stochastic geometry. Suppose that

(2.19) \begin{equation} \int_\mathbb X v(x,y) \,{\textrm{d}} \eta(\kern1pt y) \geq - C(x)\end{equation}

for some function $C \colon \mathbb X\to {\mathbb{R}}_+$ and all finite configurations $\eta = \sum_{i=1}^k \delta_{y_i}$ such that $\eta + \delta_x$ has finite total energy. Then, for all $n\in {\mathbb{N}}$ and $x_1,\ldots, x_n\in \mathbb X$ , we have

\begin{equation*}\sum_{1\leq i \lt j \leq n} v(x_i,x_j) = \dfrac12 \sum_{i=1}^n \sum_{\substack{1\leq j \leq n:\\ j \neq i}} v(x_i,x_j) \geq - \dfrac12 \sum_{i=1}^n C(x_i).\end{equation*}

(Note that the inequality is trivial when the left-hand side is infinite.) Thus local stability (2.19) implies stability in the sense of (2.18).

The following result is due to Procacci and Yuhjtman [Reference Procacci and Yuhjtman57]. Assume that v is stable with stability function B(x), and that

(2.20) \begin{equation} \int_\mathbb X ( 1- {\textrm{e}}^{- |v(x,y)|} ) \,{\textrm{e}}^{B(\kern1pt y) + a(\kern1pt y)} \,{\textrm{d}} \lambda_z(\kern1pt y) \leq a(x)\end{equation}

for some measurable function $a \colon \mathbb X\to {\mathbb{R}}_+$ and $\lambda$ -almost all $x\in \mathbb X$ . Then

\begin{equation*}\sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} | \varphi_n^{\textsf{T}}(q,x_1,\ldots, x_n)|\, {\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \leq {\textrm{e}}^{B(q)} ({\textrm{e}}^{a(q)} - 1)\end{equation*}

for $\lambda$ -almost all $q\in \mathbb X^n$ , and a suitable analogue of Theorem 2.7 (hence Theorems 2.5 and 2.6) holds true. The proof in [Reference Procacci and Yuhjtman57] is written for $\mathbb X= {\mathbb{R}}^d$ , $\lambda$ the Lebesgue measure, and translationally invariant pair potentials, building on a novel tree-graph inequality. The result is easily extended to the general set-up by using the formulation of the tree-graph inequality from [Reference Ueltschi, Fernández, Jansen and Tsagkarogiannis66]. For further results on attractive pairwise interactions, see [Reference Poghosyan and Ueltschi53] and the references therein.

The condition (2.20) extends ( $\textsf{KPU}_t$ ) for $t=0$ ; we leave as an open problem the extension of ( $\textsf{FP}_t$ ) with $t=0$ to stable pair potentials. We also leave as an open question the extension of (2.20) to $t\gt{0}$ , that is, whether multiplying the left-hand side of (2.20) with ${\textrm{e}}^t$ for some x-independent $t\gt{0}$ yields a sufficient condition for the uniqueness of the infinite-volume Gibbs measure as in Theorem 2.1, and mention instead that related uniqueness criteria based on Kirkwood–Salsburg equations are available [Reference Ruelle60, 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 [Reference Fernández and Procacci17], [Reference Poghosyan and Ueltschi53], 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 2.1 and 2.2 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 [Reference Faris16, Reference Fernández and Procacci17] but usually the focus is on sufficient convergence conditions and the equivalence is rarely explicitly stated (see, however, [Reference Jansen32]).

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 [Reference Ueltschi65] implies extinction of a related multi-type, discrete-time branching process with type space $\mathbb X$ . 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.

2.5.1. Convergence of generating functions for trees

Let $\mathcal T_n$ be the set of trees with vertex set $[n]=\{1,\ldots,n\}$ , $\mathcal T_n^\bullet = \mathcal T_n \times [n]$ the rooted trees with vertex set [n], and $\mathcal T_n^\circ$ the set of trees with vertex set $\{0,1,\ldots,n\}$ . 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’ [Reference Bergeron, Labelle and Leroux3], and white circles to ‘ghosts’ that do not come with powers of z in the generating functions. Define a family $(T_q^\circ)_{q\in \mathbb X}$ of generating functionals

(2.21) \begin{equation} T_q^\circ (z) \,:\!= 1+ \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n}\sum_{T\in\mathcal T_n^\circ} |w(T;\ x_0,\ldots,x_n)| \,{\textrm{d}} \lambda_z(x_1)\cdots {\textrm{d}} \lambda_z(x_n),\quad x_0\,:\!=q \end{equation}

and

(2.22) \begin{equation} T_q^\bullet (z) \,:\!= z(q) + \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n}\sum_{(T,r) \in\mathcal T_n^\bullet} |w(T;\ x_1,\ldots,x_n)| \prod_{i=1}^n z(x_i) \,{\textrm{d}} \lambda(x_1)\cdots {\textrm{d}}\delta_q(x_r)\cdots {\textrm{d}}\lambda(x_n). \end{equation}

Note that

(2.23) \begin{equation} T_q^\bullet (z) = z(q) T_q^\circ (z) = z(q)\exp\bigg( \int_\mathbb X |\kern2pt f(q,y)| T_y^\bullet(z)\,{\textrm{d}} \lambda(\kern1pt y) \bigg){;} \end{equation}

see Faris [Reference Faris16, Section 3.1]. In (2.23), both sides are either infinite or finite. If they are finite, the fixed-point problem may have more than one solution, and $T_q^\bullet (z)$ is the smallest solution [Reference Faris16].

Proposition 2.1. Let $z \colon \mathbb X\to {\mathbb{R}}_+$ be a measurable map. The following two conditions are equivalent.

1. (i) $T_q^\circ(z) \lt \infty$ for all $q \in \mathbb X$ .

2. (ii) There exists a measurable function $a \colon \mathbb X\to {\mathbb{R}}_+$ such that $\int_{\mathbb X} |\kern2pt f(q,y)| \,{\textrm{e}}^{a(\kern1pt y)} \,{\textrm{d}}\lambda_z(\kern1pt y) \leq a(q)$ for all $q\in \mathbb X$ .

Proof. The implication (ii) $\Rightarrow$ (i) is proved by induction as in [Reference Poghosyan and Ueltschi53]; see also [Reference Faris16, Theorem 3.1]. The implication (i) $\Rightarrow$ (ii) follows from the fixed-point equation (2.23), which proves that $a(q)\,:\!= \log T_q^\circ(z)$ is an admissible choice.

Let $(T,r)\in \mathcal T_n^\bullet$ . The root r induces a notion of generations and descendants. For $k\in T$ , let $C_k^{(r)} \subset[n]$ be the collection of children of k. Set

\begin{equation*}\tilde w(T,r;\ x_1,\ldots,x_n) = \prod_{k \in T}\Bigg( \prod_{i\in C_k^{(r)}} |\kern2pt f(x_k,x_i)| \prod_{\substack{i,j\in C_k^{(r)}:\\ i \lt j}} (1+f(x_i,x_j))\Bigg).\end{equation*}

Define $\tilde T_q^\bullet$ as in (2.22) but with weights $\tilde w(T,r;\ x_1,\ldots,x_n)$ instead of $|w(T;\ x_1,\ldots,x_n)|$ , and $\tilde T_q^\circ$ as in (2.21) but with weights $\tilde w(T,0;\ q,x_1,\ldots,x_n)$ . Then

(2.24) \begin{equation} \tilde T_q^\bullet(z) = z (q) \tilde T_q^\circ (z) = z(q) \bigg( 1+\sum_{k=1}^\infty \dfrac{1}{k!} \int_{\mathbb X^k} \prod_{1\leq i \lt j \leq k } (1+ f(x_i,x_j)) \prod_{i=1}^k |\kern2pt f(q,x_i)|\tilde T_{x_i}^\bullet (z) \,{\textrm{d}} \lambda^k({\textit{\textbf{x}}}) \bigg).\end{equation}

Proposition 2.2. Let $z \colon \mathbb X\to {\mathbb{R}}_+$ measurable. The following two conditions are equivalent.

1. (i) $\tilde T_q^\circ(z) \lt \infty$ for all $q \in \mathbb X$ .

2. (ii) There exists a measurable function $a \colon \mathbb X\to {\mathbb{R}}_+$ such that condition ( $\textsf{FP}_t$ ) holds true for all $n\in {\mathbb{N}}$ and all $x_0\in \mathbb X$ with $t=0$ .

Proof. The implication (i) $\Rightarrow$ (ii) is again an immediate consequence of the fixed-point equation (2.24) (set $a(q) = \log \tilde T_{q}^\circ(z)$ ). The implication (ii) $\Rightarrow$ (i) can be proved by an induction similar to [Reference Ueltschi65], or by adapting the arguments from [Reference Fernández and Procacci17]. For the reader’s convenience, we provide an alternative proof, based on Kirkwood–Salsburg equations for forests, in Section 7.

The key bound (2.6) is recovered from Proposition 2.2 with the help of the tree-graph inequality

\begin{equation*}|\varphi_n^{\textsf{T}}(x_0,\ldots,x_n) | \leq \sum_{T\in \mathcal T_n^\circ} \tilde w(T,0;\ x_1,\ldots,x_n){,}\end{equation*}

which can be proved as in [Reference Fernández and Procacci17].

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 $T_q^\circ$ only. Assume that

(2.25) \begin{equation} b(q) = \int_{\mathbb X} |\kern2pt f(q,y)| \,{\textrm{d}} \lambda_z(\kern1pt y) \lt \infty\end{equation}

for all $x\in \mathbb X$ . Let $\xi^q$ be a Poisson point process with intensity measure $|\kern2pt f(q;\ y)| \,{\textrm{d}}\lambda_z(\kern1pt y)$ , defined on some probability space $(\Omega,\mathcal F,\mathbb P)$ . It has Laplace functional

\begin{equation*}\mathbb E [{\textrm{e}}^{-\int_\mathbb X h \,{\textrm{d}} \xi^q}] = \exp\bigg( \int_{\mathbb X} ({\textrm{e}}^{- h(x)}- 1) |\kern2pt f(q,x)| \,{\textrm{d}} \lambda_z(x)\bigg){.}\end{equation*}

The condition (2.25) ensures that the expected number of points b(q) of $\xi^q$ is finite, and hence, in particular, $\xi^q$ has only finitely many points, $\mathbb P$ -almost surely.

We can define a discrete-time, multi-type branching process $(\eta^q_n)_{n\in {\mathbb{N}}}$ with type space $\mathbb X$ and ancestor q as a Markov chain with state space $\mathcal N$ as follows: $\eta_0^q\,:\!=\delta_q$ , $\eta_1^q\,:\!= \xi_q$ , and $\eta^q_{n+1}$ is obtained from $\eta^q_n$ by asking, roughly, that each point $x\in \eta^q_n$ has an offspring equal in distribution to $\xi^x$ , with the usual independence assumptions. Precisely, $(\eta^q_n)_{n\in {\mathbb{N}}_0}$ is a Markov chain with

\begin{equation*}\mathbb E [ {\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta^q_{n+1}}\mid \eta^q_n]= \exp\bigg( \int_\mathbb X \bigg(\int_{\mathbb X} |\kern2pt f(x,y)|({\textrm{e}}^{- h(\kern1pt y)}- 1) \,{\textrm{d}} \lambda_z(\kern1pt y)\bigg) \,{\textrm{d}} \eta_{n}^q(x)\bigg) \quad \mathbb P\text{-a.s.}\end{equation*}

for all $n\in {\mathbb{N}}_0$ . For $B\in \mathcal X_\textsf{b}$ , the quantity $\eta_n^q(B)$ 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

\begin{equation*}p(q) \,:\!= \mathbb P( \exists n \in {\mathbb{N}}\colon \eta_n^ q = 0 ){,}\end{equation*}

as a function of q, is the smallest non-negative solution of the fixed-point problem

(2.26) \begin{equation} p(q) = \exp\bigg( \int_{\mathbb X}(p(x)-1) |\kern2pt f(q,x)| \,{\textrm{d}} \lambda_z(x)\bigg) \quad (q\in \mathbb X).\end{equation}

That is, if $\tilde p({\cdot})$ is another non-negative solution, then $p(q) \leq \tilde p(q)$ for all $q\in \mathbb X$ .

For finite-type spaces, this statement is proved in [31, Theorem 4.2.2], the proof for infinite $\mathbb X$ is similar.

The similarity of the fixed-point equations (2.26) and (2.23) 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 2.3. Assume $\int_\mathbb X |\kern2pt f(q,x)| \,{\textrm{d}} \lambda_z(x) \lt \infty$ for all $q\in \mathbb X$ and let p(q) be the extinction probability of the z-dependent branching process with ancestor $\delta_q$ . If $T_q^\circ(z)< \infty$ for all $q\in \mathbb X$ , then $p(q) =1$ for all $q\in \mathbb X$ .

The proof adapts a classical result [Reference Harris28, Chapter III, Theorem 12.1]: if the principal eigenvalue of the so-called expectation operator is smaller than or equal to 1 and some additional regularity conditions hold true, then the branching process goes extinct.

Proof of Proposition 2.3. Consider the kernel on $\mathbb X$ given by $M(q, {\textrm{d}} x)\,:\!= |\kern2pt f(q,x)|\lambda_z(\,{\textrm{d}} x)$ and use the same letter for the associated integral operator

\begin{equation*}(M g)(q)\,:\!= \int_{\mathbb X} |\kern2pt f(q,x)| g(x) \,{\textrm{d}} \lambda_z(x) \quad (q\in \mathbb X,\, g\in \mathscr L^ \infty(\mathbb X, \mathcal X)).\end{equation*}

Here 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 (2.25) guarantees that the expected value $M(q,\mathbb X)$ of the total number of children of a type-q individual is finite for all $q\in \mathbb X$ . However $\sup_{q\in \mathbb X} M(q,\mathbb X) =\sup_{q\in \mathbb X}b(q)$ can be infinite, which is why Theorem 12.1 in [Reference Harris28, 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 weighted supremum norms. By the convergence of $T_q^\circ (z)$ and Proposition 2.1, there is a function $a \colon \mathbb X\to {\mathbb{R}}_+$ such that $\int_{\mathbb X} |\kern2pt f(q,x)|\exp\kern-1pt(\! a(x))\,{\textrm{d}} \lambda_z(x) \leq a(q)$ for all q. We define

\begin{equation*}\|g\|_a\,:\!= \sup_{q\in \mathbb X} |g(q)| \,{\textrm{e}}^{-a(q)}\end{equation*}

and note that M is a contraction with respect to this norm. Indeed, if g is a function with $\|g\|_a<\infty$ , then

\begin{align*}|(M g)(q) |& \leq \int_{\mathbb X} |\kern2pt f(q,x)| \,{\textrm{e}}^{a(x)} \| g\|_{a} \,{\textrm{d}} \lambda_z(x)\leq a(q) \|g\|_a\end{align*}

and

\begin{equation*}\|Mg\|_a \leq \|g\|_a \sup_{q\in \mathbb X} a(q)\, {\textrm{e}}^{- a(q)} \leq \|g\|_a \sup_{\alpha \geq 0}\alpha\, {\textrm{e}}^ {-\alpha} =\dfrac{1}{\textrm e} \|g\|_a.\end{equation*}

The fixed-point equation for the extinction probability and the inequality $\exp\kern-1pt(s) \geq 1+s$ yield

\begin{equation*}p(q) \geq 1 + \int_{\mathbb X} |\kern2pt f(q,x)| (p(x) - 1) \,{\textrm{d}} \lambda_z(x) = 1+ (M(p-\textbf 1))(q)\end{equation*}

with $\textbf 1$ the constant function with value 1. Consequently $\textbf 1 - p \leq M (\textbf 1- p)$ pointwise on $\mathbb X$ and

\begin{equation*}\| \textbf 1 - p\|_a \leq \| M (\textbf 1 - p)\|_a \leq \dfrac{1}{\textrm e} \| \textbf 1 - p\|_a.\end{equation*}

It follows that $\|\textbf 1 - p\|_a =0$ and $p(q) =1$ for all $q\in \mathbb X$ .

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 (2.25), the extinction probability p(q) is equal to the probability that the total offspring $N_\infty^q\,:\!= \sum_{n=0}^ \infty \eta_n^q(\mathbb X)$ 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 over trees on n vertices, rooted at the ancestor. Leaves correspond to individuals without offspring. Hence

\begin{align*}p(q) & = \sum_{n\in {\mathbb{N}}_0} {\mathbb{P}}( N_\infty^q = n) \\ = {\textrm{e}}^{- z(q) b(q)}\bigg( 1+ \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} \sum_{T\in \mathcal T_n^\circ} |w(T;\ q,x_1,\ldots,x_n)| \prod_{i=1}^n \,{\textrm{e}}^{- z(x_i) b(x_i)} \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \bigg) \notag \\ = {\textrm{e}}^{- z(q) b(q)} T_q^\circ( z{\textrm{e}}^{-b z}).\end{align*}

More generally, let $\eta^q\,:\!=\sum_{n=0}^\infty \eta^q_n$ be the total progeny of $\delta_q$ , including the ancestor itself. Then, for all $h \colon \mathbb X\to {\mathbb{R}}$ , we have

\begin{equation*} {\textrm{e}}^{h-bz} T_q^\circ( z {\textrm{e}}^{h-bz}) = \mathbb E\big[ {\textrm{e}}^{\int_\mathbb X h \,{\textrm{d}} \eta^q} \textrm{1\kern-3.5pt1}_{\{N_\mathbb X(\eta^q) \lt \infty\} } \big].\end{equation*}

and

\begin{equation*} T_q^\circ( z) = \mathbb E\big[ {\textrm{e}}^{\int_\mathbb X b z \,{\textrm{d}} \eta^q} \textrm{1\kern-3.5pt1}_{\{N_\mathbb X(\eta^q) \lt \infty\} } \big].\end{equation*}

This equality together with Proposition 2.3 shows that $T_q^\circ (z)$ is finite for all $q\in \mathbb X$ 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 2.3. (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 ${\mathbb{R}}^d$ , and then $b = |B(0,2R)|$ ; compare Example 2.2. Then the generation counts $N_n\,:\!= N_\mathbb X(\eta^q_n)$ , $n\in {\mathbb{N}}_0$ , form a Galton–Watson process with offspring distribution $\textrm{Poi}(z b)$ . The distribution of the total progeny (including ancestor) $N= \sum_{n=0}^\infty N_n$ satisfies

\begin{equation*}\mathbb P(N=n) = \dfrac{(b n)^{n-1}}{n!} z^{n-1} \,{\textrm{e}}^{-n bz} \quad (n\in {\mathbb{N}}){;}\end{equation*}

compare the Borel distribution [Reference Borel6]. The tree generating function becomes

\begin{equation*}T^\circ (z) = \sum_{n=1}^\infty\dfrac{(bn)^{n-1}}{n!} z^{n-1} = \sum_{n\in {\mathbb{N}}} \,{\textrm{e}}^{nb z} {\mathbb{P}}(N=n).\end{equation*}

We have $T^\circ(z)< \infty$ if and only if $b z\leq 1/\textrm{e}$ , a condition stronger than subcriticality $b z \leq 1$ .

We leave open whether the analogue of Proposition 2.3 holds true for the Fernández-Procacci trees from Proposition 2.2, but sketch how some first steps might be implemented. Define

\begin{equation*}B_n(q;\ x_1,\ldots,x_n)\,:\!= \prod_{i=1}^n |\kern2pt f(q;\ x_i)| \prod_{1\leq i {<}j \leq n} (1+ f(x_i,x_j))\end{equation*}

and assume that

\begin{equation*}G_q(z)\,:\!= 1 +\sum_{n=1}^ \infty \dfrac{1}{n!} \int_{\Lambda^n} B_n(q;\ x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \lt \infty\end{equation*}

for all $q\in \mathbb X$ . We may then consider a branching process whose branching mechanism is such that

\begin{equation*}\mathbb E [ {\textrm{e}}^ {- \int_{\mathbb X} h \,{\textrm{d}} \eta_{n+1}} \mid \eta_n = q] = \dfrac{G_q(z{\textrm{e}}^ {-h})}{G_q(z)}.\end{equation*}

The expectation operator is

\begin{equation*} M g(q) = \dfrac{1}{G(z)} \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\Lambda^{n}} B_n(q;\ x_1,\ldots,x_n) \bigg(\sum_{i=1}^n g(x_i)\bigg) \,{\textrm{d}}\lambda_z^n({\textit{\textbf{x}}}).\end{equation*}

The fixed-point problem for extinction probabilities becomes $p(q) = G_q(z p) /G_q(z)$ . The inequality $\prod_{i=1}^n p(x_i)\geq 1 + \prod_{i=1}^n ( p(x_i)-1)$ yields $p (q) \geq 1+ ( M (p - \textbf 1))(q)$ , thus again we have the pointwise inequality $\textbf 1 - p \leq M(\textbf 1 - p)$ . However, it is not clear how to turn the convergence condition $G_q(z{\textrm{e}}^a) \leq{\textrm{e}}^{a(q)}$ into a contraction estimate for the operator M.

2.6. Random connection model and disagreement percolation

Disagreement percolation provides an expansion-free route to proofs of uniqueness and exponential mixing for Gibbs measures; see [Reference Georgii, Häggström and Maes25, Chapter 7] for Gibbs measures on lattices and [Reference Hofer-Temmel29] and [Reference Hofer-Temmel and Houdebert30] as well as [Reference Dereudre14, Section 2.7] for models in ${\mathbb{R}}^d$ , and [Reference Beneš, Hofer-Temmel, Last and Večeřa2] for Gibbs particle processes. It is an interesting question which approach – percolation or convergence of cluster expansions – trumps the other as far as proving uniqueness of the infinite-volume Gibbs measure goes. In full generality, the question remains open. The purpose of this section is, nevertheless, to provide some very partial answers that might help attack the question in a more systematic way in the future.

First we recall some known facts on the relation between percolation and uniqueness of Gibbs measures. If the interaction v(x, y) has finite range R, that is, $v(x,y) =0$ when $\textrm{dist}(x,y)\geq R$ and there is absence of percolation in the Boolean percolation model with deterministic connectivity radius R, then the Gibbs measure is unique. This applies, in particular, to a gas of hard spheres with fixed radius R. A variant for Boolean percolation models with random connectivity radius [Reference Hofer-Temmel and Houdebert30] allows for an extension to mixtures of hard spheres. Uniqueness of Gibbs measures and absence of percolation are linked for finite-range interactions that take negative values as well, as long as some additional conditions are satisfied [Reference Dereudre14]. Thus absence of Boolean percolation implies uniqueness of Gibbs measures.

For a gas of hard spheres, condition ( $\textsf{KPU}_t$ ) with $t=0$ implies extinction of a branching process by Proposition 2.3, which in turn implies absence of percolation [Reference Meester and Roy45, Chapters 3 and 6]. As a consequence, for a gas of hard spheres, condition ( $\textsf{KPU}_t$ ) is more restrictive than asking for absence of Boolean percolation. We do not know, however, if convergence of cluster expansions or condition ( $\textsf{FP}_t$ ) also implies absence of Boolean percolation.

For a non-negative pair potential that has finite range and is everywhere finite, in constrast, if $\beta$ is sufficiently small, conditions ( $\textsf{FP}_t$ ) and ( $\textsf{KPU}_t$ ) can be less restrictive than asking for absence of Boolean percolation (we thank C. Hofer-Temmel and the anonymous referee for this remark). Indeed, asking for absence of Boolean percolation imposes a condition $z\leq z^*$ (or $z{<}z^*$ ) for some finite, $\beta$ -independent $z^*$ . Indeed, consider for simplicity $\mathbb X= {\mathbb{R}}^d$ and a pair potential that is translationally invariant; then condition ( $\textsf{KPU}_t$ ) reads $z< ({\textrm{e}}\, C(\beta))^{-1}$ with

\begin{equation*}C(\beta) = \int_{{\mathbb{R}}^d} (1 - \exp\kern-1pt(\! - \beta v(0,x)))\,{\textrm{d}} x. \end{equation*}

Clearly $\lim_{\beta \to 0} C(\beta) =0$ , and hence for small $\beta \gt{0}$ the condition $z \lt ({\textrm{e}}\, C(\beta))^{-1}$ is less restrictive than the condition $z \lt z^*$ .

We conclude this section with an open question. The only information about the interaction kept by the associated Boolean percolation model is the range of the interaction. The natural question arises whether there is a more refined model whose connectivity might relate to properties of the Gibbs measures. For non-negative interactions, a natural candidate is the random connection model with connectivity probability given by

\begin{equation*}\varphi(x,y)\,:\!= |\kern2pt f(x,y)|\quad (x,y\in \mathbb X).\end{equation*}

See [Reference Meester and Roy45] and [Reference Last and Ziesche42] for definitions in the translationally invariant case in ${\mathbb{R}}^d$ and [Reference Last, Nestmann and Schulte43] for the general set-up. Note that, by the non-negativity of the pair potential, we have $|\kern2pt f(x,y)| = |{\textrm{e}}^{-v(x,y)} - 1|\leq 1$ and may indeed interpret $\varphi(x,y)$ as a probability. The extinction of the branching process from Proposition 2.3, together with arguments from [Reference Meester and Roy45] and [Reference Meester, Penrose and Sarkar46], implies absence of percolation in the random connection model; see Appendix C for a more precise statement. As a consequence, condition ( $\textsf{KPU}_t$ ) for non-negative interactions implies absence of percolation in the random connection model, much in the same way as condition ( $\textsf{KPU}_t$ ) for hard spheres implies absence of percolation in a Boolean model. To the best of our knowledge, however, there is no theorem in the literature that shows that absence of percolation in the random connection model implies uniqueness of the Gibbs measure; we do not know whether there is a result in the spirit of disagreement percolation that would use the random connection model rather than Boolean percolation.

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 [Reference Peccati and Taqqu51, Chapter 7], [Reference Last and Penrose41, Chapter 12.2], and [Reference Last, Penrose, Schulte and Thäle44]. To that end we provide two expressions for the cumulants of random variables of the form $\int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}$ , where $\eta$ is a Poisson point process with intensity measure $\lambda_z$ and $u \colon \mathbb X\times \mathbb X\to {\mathbb{R}}$ is a function that satisfies some integrability conditions. We may view such random variables as double integrals with respect to a Poisson random measure $\eta$ (not compensated). The first formula involves edge-labeled graphs with multiple edges (Proposition 2.4), and the second formula involves precisely the pairs of partitions (Proposition 2.5) that appear in the literature on multiple stochastic integrals [Reference Peccati and Taqqu51].

To the best of our knowledge these formulas are new, but 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 nth coefficient of the cluster expansion is a sum over graphs on n vertices while the mth moment of a double stochastic integral is a sum over multigraphs with m edges.

We start from the following observation. Let $u \colon \mathbb X\times \mathbb X\to {\mathbb{R}}$ and $\beta \in {\mathbb{R}}$ . Suppose that $v= - \beta u$ satisfies the assumptions of Theorem 2.7. Then the latter theorem provides an expansion of the cumulant generating function of $\tfrac12 \beta \int_{\mathbb X^2} u\,{\textrm{d}} \eta^{(2)}$ , with $\eta$ a Poisson point process of intensity $\lambda_z$ . The expansion is not in powers of $\beta$ but rather, if $z(x)\equiv z$ is independent of x, in powers of z. It is a sum over connected graphs, each graph comes with a product of edge weights $\exp\kern-1pt(\! \beta u(x_i,x_j))-1$ . In order to obtain the cumulants, we need to understand the expansion in powers of $\beta$ rather than z. Now, every edge weight is expanded as

\begin{equation*}{\textrm{e}}^{\beta u(x_i,x_j)}-1 = \sum_{m_{ij}=1}^ \infty \dfrac{\beta^ {m_{ij}}}{m_{ij}!}\, u(x_i,x_j)^ {m_{ij}}.\end{equation*}

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 2.2. Let A and V be two non-empty sets and $\mathcal E_2(V) = \{e\subset V\mid \#e =2\}$ .

• A labeled multigraph $\gamma$ with vertex labels V and edge labels A is a map from A to $\mathcal E_2(V)$ . The set of such multigraphs is denoted $\mathcal M(V,A)$ .

• The multiplicity of an edge $\{i,j\} \in \mathcal E_2(V)$ in $\gamma\in \mathcal M(V,A)$ is $m_{ij}(\gamma) = \# \{a \in A\mid \gamma(a) = \{i,j\} \}$ .

• A multigraph $\gamma$ is spanning if every vertex $i\in V$ belongs to some edge in $\gamma$ , i.e. $m_{ij}(\gamma)\geq 1$ for some $j\in V$ . The set of spanning multigraphs is denoted $\mathcal M_s(V,A)$ .

• A multigraph $\gamma$ is connected if, for all $i,j\in V$ , there exist $k\in {\mathbb{N}}$ and a sequence $(a_1,\ldots,a_k)\in A^k$ such that $i\in \gamma(a_1)$ , $j\in \gamma(a_k)$ , and $\gamma(a_r) \cap \gamma(a_{r+1})\neq \varnothing$ for all $r\in \{1,\ldots,k-1\}$ . The set of connected multigraphs is denoted $\mathcal M_c(V,A)$ .

Proposition 2.4. Let $\eta$ be a Poisson point process with intensity measure $\lambda_z$ and $u \colon \mathbb X\times \mathbb X\to {\mathbb{R}}$ a symmetric function. Suppose that

(2.27) \begin{equation} \int_{\mathbb X^n} \prod_{1 \leq i \lt j \leq n} | u(x_i,x_j)|^{m_{ij}(\gamma)} \,{\textrm{d}} \lambda^n({\textit{\textbf{x}}}) \lt \infty \end{equation}

for all $n\in \{2,\ldots,2m\}$ and $\gamma\in \mathcal M_s([n],[m])$ . Then $\mathbb E[(\int_{\mathbb X} |u|\,{\textrm{d}} \eta^{(2)})^m]< \infty$ and the mth moment and the mth cumulant of $\int_{\mathbb X^2 } u \,{\textrm{d}} \eta^{(2)}$ are given by sums over spanning and connected multigraphs as

(2.27) \begin{align*}\mathbb E\bigg[ \bigg( \dfrac12 \int_{\mathbb X^2} u\,{\textrm{d}} \eta^{(2)} \bigg)^m\bigg] & = \sum_{n=2}^{2m} \dfrac{1}{n!}\int_{\mathbb X^n} \sum_{\gamma \in \mathcal M_s([n],[m])}\prod_{1 \leq i \lt j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)} \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) {,} \\ \kappa_m \bigg( \dfrac12 \int_{\mathbb X^2} u\,{\textrm{d}} \eta^{(2)} \bigg) & = \sum_{n=2}^{2m} \dfrac{1}{n!}\int_{\mathbb X^n} \sum_{\gamma \in \mathcal M_c([n],[m])}\prod_{1 \leq i \lt j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)}\,{\textrm{d}} \lambda^n_z({\textit{\textbf{x}}}),\end{align*}

with absolutely convergent integrals.

The proposition is proved in Section 8. The proof does not use the previous convergence theorems, which is why it works under different convergence conditions (compare (2.27) and ( $\textsf{FP}_t$ ) with $v=-\beta u$ ), much in the same way as moments of a random variable may be finite even though exponential moments are infinite. Nevertheless it is instructive to derive the formula for the cumulants from Theorem 2.7 on the expansion of the pressure in finite volume, which we now do. Suppose that $v= -\beta u$ satisfies the conditions of Theorem 2.7, and that $\lambda_z(\mathbb X)< \infty$ so that we may set $\Lambda = \mathbb X$ . Then we have

\begin{equation*}\log \mathbb E\big[ {\textrm{e}}^{\frac12 \beta \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}}\big] = \sum_{n=2}^\infty \dfrac{1}{n!} \int_{\mathbb X} \bigg( \sum_{G\in \mathcal C_n}\prod_{\{i,j\}\in E(G)}( {\textrm{e}}^{\beta u(x_i,x_j)}-1) \bigg) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation*}

Expanding the exponential, we find that for each n, the integrand is a sum over pairs $(G, {\textit{\textbf{m}}})$ that consist of a connected graph G and a vector of integers ${\textit{\textbf{m}}} = (m_{ij})_{i{<}j}$ with $m_{ij}\geq 1$ if and only if $\{i,j\} \in E(G)$ . The pair $(G,{\textit{\textbf{m}}})$ is best thought of as a graph with multiple edges (edges are non-labeled). The vector of multiplicities determines the graph uniquely. Let $C_n$ be the set of multiplicity assignments for which the associated graph is connected. We get

\begin{align*}& \sum_{G\in \mathcal C_n}\prod_{\{i,j\}\in E(G)}( {\textrm{e}}^{\beta u(x_i,x_j)}-1) \\ & \qquad = \sum_{G\in \mathcal C_n}\prod_{\{i,j\}\in E(G)} \bigg( \sum_{m_{ij}=1}^\infty \dfrac{1}{m_{ij}!} ( \beta u(x_i,x_j))^{m_{ij}} \bigg) \\ & \qquad =\sum_{m=1}^\infty \dfrac{\beta^m}{m!} \sum_{\substack{ (m_{ij})_{1 \leq i {<}j \leq n}\\ m_{ij}\in {\mathbb{N}}_0 \\ \sum_{i{<}j} m_{ij} =m} }\ \dfrac{m!}{\prod_{1\leq i {<}j \leq n} m_{ij}!} \prod_{1 \leq i {<} j \leq n} u(x_i,x_j)^ {m_{ij}} \textrm{1\kern-3.5pt1}_{C_n}({\textit{\textbf{m}}}).\end{align*}

The multinomial coefficient is equal to the number of multigraphs with edge labels $\{1,\ldots,m\}$ and prescribed multiplicities $m_{ij}$ . Thus we find

(2.28) \begin{equation} \log \mathbb E\big[ {\textrm{e}}^{\frac12 \beta \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}}\big] = \sum_{m=1}^ \infty \dfrac{\beta^ m}{m!}\sum_{n=2}^\infty\dfrac{1}{n!}\int_{\mathbb X^ n} \sum_{\gamma\in \mathcal M_c([n],[m])}\prod_{1 \leq i \lt j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)} \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation}

Since the cumulants are defined by the relation

\begin{equation*}\log \mathbb E\big[ {\textrm{e}}^{\frac12 \beta \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}}\big] = \sum_{m=1}^ \infty \dfrac{\beta^m}{m!}\, \kappa_m\bigg( \dfrac12 \int_{\mathbb X^2} u\,{\textrm{d}} \eta^{(2)} \bigg) {,}\end{equation*}

we may read them off from (2.28) and obtain the expression from Proposition 2.4.

Next we explain how to go from multigraphs to partition pairs so as to obtain expressions closer to [Reference Peccati and Taqqu51]. Let $\gamma \in \mathcal M_s(V,A)$ be a spanning multigraph on V with edge labels A. We define an associated pair $(\pi,\sigma)$ of partitions as follows.

• First we define a new set S of ‘dedoubled’ vertices: every vertex $v\in V$ gives rise to as many points in S as there are edges to which it belongs. The new vertices s are labeled by the parent vertex $v\in V$ and the edge label a. Precisely, we set $S\,:\!= \{(v,a) \in V\times A \mid v\in \gamma(a)\}$ .

• The partition $\pi$ has the blocks $B_a = \{(v,a)\mid v\in \gamma(a)\}$ , $a \in A$ : it groups dedoubled vertices s connected by an edge. Every block $B_a$ has cardinality exactly 2.

• The partition $\sigma$ lumps together new vertices s that come from a common underlying vertex $v\in V$ . Put differently, the blocks of $\sigma$ are the sets $T_v = \{(a,v)\mid v\in \gamma(a)\}$ , $v\in V$ . Note that the $T_v$ are non-empty because the graph is spanning.

Note that for all $a\in A$ , $v\in V$ , the set $B_a\cap T_v$ is a singleton if the vertex v belongs to the edge with label a, and empty if it does not; in particular, $B_a \cap T_v$ is either empty or a singleton. If the multigraph $\gamma$ is connected, then so is $(\pi,\sigma)$ , in the sense given below.

Definition 2.3. ([Reference Peccati and Taqqu51].) Let S be a finite non-empty set and $(\pi,\sigma)\in \mathcal P(S)\times \mathcal P(S)$ a pair of set partitions of S.

• The pair is non-flat if, for all blocks $B\in \pi$ and $T\in \sigma$ , the intersection $B\cap T$ is either empty or a singleton.

• The pair is connected if, for every strict subset $M\subsetneq S$ , there is a block B of $\pi$ or $\sigma$ such that B intersects both M and $S\setminus M$ .

Equivalently, a pair $(\pi,\sigma)$ is non-flat and connected if $\pi \wedge \sigma$ is the partition into singletons and $\pi \vee \sigma$ is the partition consisting of a single block S, where $\wedge$ and $\vee$ refer to the join and meet in the lattice of set partitions.

Remark 2.4. (Gaussian diagrams.) The pair of partitions $(\pi,\sigma)$ is not only non-flat and connected, but also has the property that every block of $\pi$ 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 [Reference Peccati and Taqqu51] call pairs where instead $\sigma$ 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 graph clearly different from ours.

Let S be a finite set and $\hat \sigma =(T_1,\ldots,T_n)$ an ordered set partition of S with n blocks; let $\sigma = \{T_1,\ldots,T_n\} \in \mathcal P(S)$ be the underlying set partition. For $s\in S$ , let $i(s)\in \{1,\ldots,n\}$ be the label of the block to which s belongs, i.e. $s\in T_{i(s)}$ . The ordered partition $\hat \sigma$ induces an embedding of $\mathbb X^n$ into $\mathbb X^S$ defined by $(x_1,\ldots,x_n)\mapsto (x_{i(s)})_{s\in S}$ . With any given map $g \colon \mathbb X^n\to {\mathbb{R}}$ we associate a new map $g_{\hat \sigma} \colon \mathbb X^S\to {\mathbb{R}}$ by $g_{\hat \sigma} (x_1,\ldots,x_n)\,:\!= g((x_{i(s)})_{s\in S})$ . For example, if $ S= \{1,2,3\}$ and $\hat \sigma = ( \{1,3\}, \{2\})$ , then $g_{\hat \sigma}( x_1,x_2) = g(x_1,x_2,x_1)$ . Changing the order of the blocks in $\hat \sigma$ permutes the variables in $g_{\hat \sigma}$ but leaves the integral $\int_{\mathbb X^n} g_{\hat \sigma} \,{\textrm{d}} \lambda_z^ n$ unchanged; with a slight abuse of notation we write $\int_{\mathbb X^n} g_\sigma \,{\textrm{d}} \lambda_z^n$ instead.

In the following proposition the m-fold tensor $u\otimes\cdots \otimes u$ is the function $(x_1,\ldots,x_{2m}) \mapsto u(x_1,x_2) u(x_3,x_4)\cdots u(x_{2m-1},x_{2m})$ , and $\|\sigma\|$ is the number of blocks of a partition $\sigma$ .

Proposition 2.5. Let $S_m=\{1,\ldots,2m\}$ , and $\pi_m =\{ \{1,2\},\{3,4\},\ldots,\{2m-1,2m\}\}$ . Under the assumptions of Proposition 2.4, the mth cumulant of $\int_{\mathbb X^ 2} u \,{\textrm{d}} \eta^{(2)}$ is given by

\begin{equation*}\kappa_m\bigg( \int_{\mathbb X^2} u \,{\textrm{d}} \eta^ {(2)}\bigg) = \sum_{\substack{\sigma\in \mathcal P(S_m):\\ (\pi_m,\sigma)\, {non\hbox{-}flat, connected}}} \int_{\mathbb X^{\|\sigma\|}} (u\otimes \cdots \otimes u)_\sigma(x_1,\ldots,x_{\|\sigma\|})\,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}})\end{equation*}

with absolutely convergent integrals.

The proposition is proved in Section 8. It is deduced from Proposition 2.4 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 the reason why Proposition 2.4 looks at $\tfrac12 \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}$ while Proposition 2.5 deals with $\int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}$ .

Proposition 2.5 should be contrasted with a similar expression for the cumulants of the random variable

\begin{equation*}I_2(u) = \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)} - \int_{\mathbb X^ 2} u\, {\textrm{d}} ( \eta\otimes \lambda_z ) - \int_{\mathbb X^2} u\, {\textrm{d}} (\lambda_z \otimes \eta) + \int_{\mathbb X^2} u \,{\textrm{d}} (\lambda_z\otimes \lambda_z).\end{equation*}

The cumulants of $I_2(u)$ are given by sums over non-flat, connected diagrams $(\pi_m,\sigma)$ such that every block of $\sigma$ has cardinality at least 2; see [Reference Peccati and Taqqu51] and [Reference Last, Penrose, Schulte and Thäle44]. This corresponds to connected multigraphs for which every vertex $i\in V$ has degree $\sum_{j\in V} m_{ij}(\gamma) \geq 2$ .

We leave open whether the known formulas for the cumulants of $I_2(u)$ have a simple explanation 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 2.1 on the uniqueness of Gibbs measures. Our proof follows a standard strategy [Reference Ruelle60, 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.

Lemma 3.1. Assume $v\geq 0$ , $\lambda^2$ -almost everywhere, and $\int_{\mathbb X} |\kern2pt f(x,y)| \,{\textrm{d}} \lambda_z(x)<\infty$ for $\lambda_z$ -almost all $x\in \mathbb X$ . Let $\textsf{P}\in \mathscr{G}(z)$ . The correlation functions $\rho_n$ of $\textsf{P}$ satisfy the Kirkwood–Salsburg equations

\begin{align*} & \rho_{n+1}(x_0,\ldots,x_n) = z(x_0) \prod_{i=1}^n (1+ f(x_0,x_i)) \notag \\ \quad \times \bigg( \rho_n(x_1,\ldots,x_n) + \sum_{k=1}^\infty \dfrac{1}{k!} \int_{\mathbb X^k} \prod_{j=1}^k f(x_0,y_i)\rho_{n+k}(x_1,\ldots,x_n,y_1,\ldots,y_k) \,{\textrm{d}} \lambda^k({\textit{\textbf{y}}})\bigg)(\textrm{KS})\end{align*}

for all $n\in {\mathbb{N}}_0$ and $\lambda^{n+1}$ -almost all $(x_0,\ldots,x_{n})\in \mathbb X^{n+1}$ , with the convention $\rho_0 =1$ and

\begin{equation*}\prod_{i=1}^0 (1+ f(x_0,x_i)) =1.\end{equation*}

The absolute convergence of the right-hand side of (KS) is ensured by the condition ( $\textsf{FP}_t$ ) and the bound (2.2). The lemma is a consequence of the well-known equivalence of the GNZ and the Dobrushin–Lanford–Ruelle (DLR) conditions [Reference Nguyen and Zessin50] on the one hand, and the DLR conditions and the Kirkwood–Salsburg equation on the other hand [Reference Ruelle61]; see also [Reference Kuna38] and [Reference Kondratiev and Kuna33]. For the reader’s convenience we present a short self-contained proof of the implication (GNZ) $\Rightarrow$ (KS) in Appendix A.

The Kirkwood–Salsburg equations can be rephrased as a fixed-point equation in some suitable Banach space. Let $E_0$ be the space of sequences $\boldsymbol{\rho} = (\rho_n)_{n\in{\mathbb{N}}}$ of real-valued measurable functions $\rho_n \colon \mathbb X^n\to {\mathbb{R}}$ such that

\begin{equation*}|\rho_n(x_1,\ldots,x_n)| \leq C_{\boldsymbol{\rho}} \,{\textrm{e}}^{tn}\prod_{1\leq i {<}j\leq n} (1+ f(x_i,x_j)) \prod_{i=1}^n z(x_i) \,{\textrm{e}}^{a(x_i)}\end{equation*}

for some $C_{\boldsymbol{\rho}} \geq 0$ , all $n\in {\mathbb{N}}$ , and $\lambda^n$ -almost all $(x_1,\ldots,x_n)\in \mathbb X^n$ . Let $\|\boldsymbol{\rho}\|_0$ be the smallest constant $C_{\boldsymbol{\rho}}$ . The quotient E of $E_0$ with the null space $\{ \boldsymbol{\rho} \colon \|\boldsymbol{\rho}\|_0=0\}$ , together with the norm $\| [\boldsymbol{\rho}] \|\,:\!= \|\boldsymbol{\rho}\|_0$ , is a Banach space; by a slight abuse of notation we write $\boldsymbol{\rho}$ instead of $[\boldsymbol{\rho}]$ . For $\boldsymbol{\rho} \in E$ , set

\begin{equation*}(K_z \boldsymbol{\rho})_1(x_0) = z(x_0) \sum_{k=1}^\infty \dfrac{1}{k!} \int_{\mathbb X^k} \prod_{j=1}^k f(x_0,y_j)\rho_{k}(y_1,\ldots,y_k) \,{\textrm{d}} \lambda^k({\textit{\textbf{y}}})\end{equation*}

and if $n\in {\mathbb{N}}$

\begin{align*}&(K_z \boldsymbol{\rho})_{n+1}(x_0,\ldots,x_n)= z(x_0) \prod_{i=1}^n (1+ f(x_0,x_i))\\ \quad\, \times \bigg( \rho_n(x_1,\ldots,x_n) + \sum_{k=1}^\infty \dfrac{1}{k!} \int_{\mathbb X^k} \prod_{j=1}^k f(x_0,y_j)\rho_{n+k}(x_1,\ldots,x_n,y_1,\ldots,y_k) \,{\textrm{d}} \lambda^k({\textit{\textbf{y}}})\bigg).\end{align*}

Let ${\textit{\textbf{e}}}_z =(e_{z,n})_{n\in{\mathbb{N}}} \in E$ be the sequence given by $e_{z,1}(x_1)= z(x_1)$ on $ \mathbb X$ and $e_{z,n} =0$ for $n\geq 2$ . The Kirkwood–Salsburg equation is rewritten as

\begin{equation*}\boldsymbol{\rho} = {\textit{\textbf{e}}}_z + K_z\boldsymbol{\rho}.\end{equation*}

Proof. Suppose that $\boldsymbol{\rho} \in E$ . Then, for $n\in {\mathbb{N}}$ , we have

\begin{equation*}\prod_{i=1}^n (1+ f(x_0,x_i)) |\rho_n(x_1,\ldots,x_n)|\leq \|\boldsymbol{\rho}\| {\textrm{e}}^{tn} \prod_{0\leq i< j \leq n}(1+ f(x_i,x_j)) \prod_{i=1}^n z(x_i) \,{\textrm{e}}^{ a(x_i)}{,}\end{equation*}

and for $n\in {\mathbb{N}}$ , $k\in {\mathbb{N}}_0$ ,

\begin{align*}&\prod_{i=1}^n (1+ f(x_0,x_i))\prod_{i=1}^k |\kern2pt f(x_0,y_i)|\, |\rho_{n+k}(x_1,\ldots,x_n,y_1,\ldots,y_k)| \\ \qquad \leq \|\boldsymbol{\rho}\| \,{\textrm{e}}^{t(n+k)} \prod_{0\leq i {<} j \leq n} (1+ f(x_i,x_j)) \prod_{i=1}^k |\kern2pt f(x_0,y_i)| \prod_{1\leq i {<} j \leq k}(1+ f(y_i,y_j)) \\ \qquad \quad\, \times \prod_{i=1}^n z(x_i)\,{\textrm{e}}^{a(x_i)} \prod_{j=1}^kz(y_j) \,{\textrm{e}}^{a(y_j)}.\end{align*}

Here the mixed terms $1+f(x_i,y_j)$ that arise from bounding $\rho_{n+k}(x_1,\ldots,x_n,y_1,\ldots,y_k)$ have been bounded by 1. In the Kirkwood–Salsburg equation we use the triangle inequality and plug in the previous estimates; this yields

\begin{align*}& |(K_z\boldsymbol{\rho})_{n+1}(x_0,\ldots,x_n)| \leq \|\boldsymbol{\rho}\|\, z(x_0)\, \prod_{0\leq i {<} j \leq n} (1+ f(x_i,x_j)) \,{\textrm{e}}^{tn}\prod_{i=1}^n z(x_i) \,{\textrm{e}}^{a(x_i)} \\ \quad\, \times \bigg( 1 + \sum_{k=1}^\infty \dfrac{{\textrm{e}}^{tk}}{k!} \int_{\mathbb X^k} \prod_{i=1}^k |\kern2pt f(x_0,y_i)| \prod_{1\leq i {<} j \leq k}(1+ f(y_i,y_j)) \,{\textrm{e}}^{\sum_{j=1}^k a(y_j)} \,{\textrm{d}} \lambda^k({\textit{\textbf{y}}})\bigg) \\ \qquad \quad\, \leq {\textrm{e}}^{-t } \|\boldsymbol{\rho}\|\, z(x_0)\, \prod_{0\leq i \lt j \leq n} (1+ f(x_i,x_j))\prod_{i=1}^n z(x_i) \,{\textrm{e}}^{\sum_{i=0}^n a(x_i)}\,{\textrm{e}}^{t (n+1)} .\end{align*}

For $n=0$ ,

\begin{equation*}|(K_z \boldsymbol{\rho})_1(x_0)|\leq {\textrm{e}}^{-t} \|\boldsymbol{\rho}\| ({\textrm{e}}^{a(x_0)}-1)\leq {\textrm{e}}^{-t} \|\boldsymbol{\rho}\| \,{\textrm{e}}^{t+a(x_0)}.\end{equation*}

It follows that $K_z\boldsymbol{\rho} \in E$ and $\|K_z\boldsymbol{\rho}\|\leq{\textrm{e}}^{-t} \|\boldsymbol{\rho}\|$ .

Proof of Theorem 2.1. The existence of a Gibbs measure follows from Theorem B.1 in Appendix B. For the uniqueness, we start from Lemma 2.1 and note that the sequence of n-point functions of a Gibbs measure $\textsf{P}\in \mathscr{G}(z)$ is in E. In view of $\|K_z\|\leq {\textrm{e}}^{-t}<1$ , the operator $(\textrm{id} - K_z)$ is invertible with bounded inverse given by a Neumann series. Therefore the vector of correlation functions $\boldsymbol{\rho} = (\rho_n)_{n\in {\mathbb{N}}}$ , is uniquely determined by the Kirkwood–Salsburg equations and is given by

(3.1) \begin{equation} \boldsymbol{\rho} = (\textrm{id} - K_z)^{-1} {\textit{\textbf{e}}}_z = {\textit{\textbf{e}}}_z+ \sum_{\ell=1}^\infty K_z^\ell {\textit{\textbf{e}}}_z.\end{equation}

By Lemma 2.1, we have $\rho_n(x_1,\ldots,x_n)\leq \prod_{j=1}^n z(x_j)$ and

\begin{equation*} \textsf{E}[ \eta(B)( \eta(B) -1) \cdots ( \eta(B)-n+1)]= \int_{B^n} \rho_n \,{\textrm{d}}\lambda^n \leq \bigg( \int_B z\,{\textrm{d}}\lambda\bigg)^n = \lambda_z(B)^n.\end{equation*}

for all $B\in \mathcal X_\textsf{b}$ and $n \in {\mathbb{N}}_0$ . As a consequence the correlation functions determine the measure $\textsf{P}$ uniquely [Reference Last and Penrose41, Proposition 4.12] and we find $\#\mathscr{G}(z) = 1$ .

4. Weighted graphs. Expansion of correlation functions

In the previous section we have proved that condition ( $\textsf{FP}_t$ ) guarantees the uniqueness of the Gibbs measure $\textsf{P}$ . We proceed with the proof of Theorem 2.4 on the expansion of correlation functions. As noted earlier, the vector of correlation functions $\boldsymbol{\rho} = (\rho_k)_{k\in {\mathbb{N}}}$ is the unique solution of a fixed-point equation $\boldsymbol{\rho} = {\textit{\textbf{e}}}_z + K_z {\textit{\textbf{e}}}_z$ and has the series representation (3.1). It remains to compute the powers $K_z^n {\textit{\textbf{e}}}_z$ , that is, 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 $\textit{\textbf{e}}_z$ . Indeed,

\begin{equation*}\sum_{n =0}^{N+1} K_z^n {\textit{\textbf{e}}}_z = {\textit{\textbf{e}}}_z + K_z \bigg( \sum_{n =0}^{N} K_z^n {\textit{\textbf{e}}}_z\bigg).\end{equation*}

The combinatorial counterpart to the partial sums of the Neumann series (3.1) are sums over graphs truncated at some maximum number N of vertices. Remember the functions $\psi_{k,n+k}$ from (2.12) and the multirooted graphs $\mathcal D_{k,n}$ with k roots and $n\geq k$ vertices. Define

(4.1) \begin{align} &( {\textit{\textbf{S}}}_N(z) )_k(x_1,\ldots,x_k) \,:\!= z(x_1)\cdots z(x_k) \notag \\ \quad\, \times \sum_{n=k}^N \dfrac{1}{(n-k)!} \int_{\mathbb X^{n-k}} \psi_{k,n}(x_1,\ldots,x_k,y_{k+1},\ldots,y_n) \,{\textrm{d}} \lambda_z^{n-k}({\textit{\textbf{y}}})\end{align}

if $1\leq k \leq N$ and $({\textit{\textbf{S}}}_N(z))_k(x_1,\ldots,x_k)\,:\!=0$ if $k\geq N+1$ . The summand for $k=n$ is understood as $\psi_{k,k}(x_1,\ldots,x_k)$ . In order to prove that ${\textit{\textbf{S}}}_N(z)$ is equal to the partial sum $\sum_{n=1}^N K_z^n {\textit{\textbf{e}}}_z$ , we show that ${\textit{\textbf{S}}}_1 (z) = {\textit{\textbf{e}}}_z$ and that ${\textit{\textbf{S}}}_{N+1}(z)$ is obtained from ${\textit{\textbf{S}}}_N(z)$ by a Picard iteration; see Proposition 4.1 below. The proof builds on recursive properties of multirooted graph with respect to removal of a root and as such generalizes the recursive proof with singly rooted graphs from [Reference Ueltschi65]; see also [Reference Friedli and Velenik21, Chapter 5.4].

In addition, we prove that if the condition ( $\textsf{FP}_t$ ) holds true with $t=0$ , then the right-hand side of (4.1) converges pointwise as $N\to \infty$ , and we provide bounds. This ensures convergence of expansions even if $K_z$ has operator norm equal to 1. In that case the limit corresponds to a fixed point of $\boldsymbol{\rho} = {\textit{\textbf{e}}}_z + K_z \boldsymbol{\rho}$ 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 [Reference Bissacot, Fernández and Procacci5].

Define $\tilde {{\textit{\textbf{S}}}}_N$ in a similar way to ${\textit{\textbf{S}}}_N$ but with additional absolute values, that is,

\begin{align*}& ( \tilde{{\textit{\textbf{S}}}}_N(z) )_k(x_1,\ldots,x_k) \,:\!= z(x_1)\cdots z(x_k) \\ \quad\, \times \sum_{n=k}^N \dfrac{1}{(n-k)!} \int_{\mathbb X^{n-k}} |\psi_{k,n}(x_1,\ldots,x_k,y_{k+1},\ldots,y_n)| \,{\textrm{d}} \lambda_z^{n-k}({\textit{\textbf{y}}})\end{align*}

if $1\leq k \leq N$ and $(\tilde{{\textit{\textbf{S}}}}_N(z))_k(x_1,\ldots,x_k)\,:\!=0$ if $k\geq N+1$ . Clearly ${\textit{\textbf{S}}}_1 = \tilde{{\textit{\textbf{S}}}}_1={\textit{\textbf{e}}}_z$ .

Proposition 4.1. Assume $v\geq 0$ , $\lambda_z(B)< \infty$ for all $B\in \mathcal X_\textsf{b}$ , and suppose that condition ( $\textsf{FP}_t$ ) holds true with $t=0$ . Then we have

(4.2) \begin{equation}{\textit{\textbf{S}}}_1(z) = {\textit{\textbf{e}}}_z,\quad {\textit{\textbf{S}}}_{N}(z) = {\textit{\textbf{e}}}_z + K_z {\textit{\textbf{S}}}_{N-1}(z) \quad (N\geq 2)\end{equation}

with

(4.3) \begin{equation}|({\textit{\textbf{S}}}_N(z))_k(x_1,\ldots,x_k)| \leq (\tilde{{\textit{\textbf{S}}}}_N(z))_k(x_1,\ldots,x_k)\leq \prod_{1\leq i {<} j \leq k} (1+ f(x_i,x_j)) \prod_{j=1}^k z(x_j) \,{\textrm{e}}^{a(x_j)}\end{equation}

for all $k,N\in {\mathbb{N}}$ and $\lambda^k$ -almost all $(x_1,\ldots,x_k)\in \mathbb X^k$ .

The equality (4.2) for ${\textit{\textbf{S}}}_N$ is complemented by an inequality for $\tilde{{\textit{\textbf{S}}}}_N$ : define the operator $\tilde K_z$ just as $K_z$ but with $f(x_0,y_j)$ replaced with $|\kern2pt f(x_0,y_j)|$ . Then $\tilde{{\textit{\textbf{S}}}}_1(z) = {\textit{\textbf{e}}}_z$ and

(4.4) \begin{equation} \tilde{{\textit{\textbf{S}}}}_1(z) = {\textit{\textbf{e}}}_z,\quad \tilde{{\textit{\textbf{S}}}}_N(z) \leq {\textit{\textbf{e}}}_z + \tilde K_z \tilde{{\textit{\textbf{S}}}}_{N-1}(z)\quad (N\geq 2)\end{equation}

where ‘ ${\textit{\textbf{g}}}\leq {\textit{\textbf{h}}}$ ’ refers to pointwise inequality $g_k\leq h_k$ of the components. The inequality (4.4) allows for an inductive proof of (4.3).

The proof of Proposition 4.1 builds on two lemmas which hold true for attractive interactions as well. It is convenient to extend the definitions $\mathcal D_{k,n}$ and $\psi_{k,n}$ . Given two finite non-empty sets I, J with $I\subset J$ , let $\mathcal D(I,J)$ as the set of connected graphs G with vertex set J such that every vertex $j\in J$ connects in G to at least one vertex $i\in I$ . Set

\begin{equation*}\psi(I,J;\ (x_j)_{j\in J})\,:\!= \sum_{G\in \mathcal D(I,J)} w(G;\ (x_j)_{j\in J}) \quad ((x_j)_{j\in J}\in \mathbb X^J ).\end{equation*}

Thus $\mathcal D_{k,n} = \mathcal D([k],[n])$ and $\psi_{k,n}({\cdot}) = \psi([k],[n];\ {\cdot})$ . For J a finite possibly empty set, define

\begin{equation*}\psi(\varnothing,J;\ (x_j)_{j\in J}) \,:\!=\begin{cases}1 & J=\varnothing,\\0 & J \neq \varnothing.\end{cases}\end{equation*}

Note that for all permutations $\sigma$ of J,

(4.5) \begin{equation} \psi(I,J;\ (x_{\sigma(j)})_{j\in J}) = \psi(\sigma(I),J;\ (x_j)_{j\in J}).\end{equation}

In particular, $\psi (I, J;\ {\cdot})$ is invariant with respect to permutations that act on $J\setminus I$ only.

The following set of equations is similar to [Reference Poghosyan and Ueltschi53, Lemma 6.2]. In fact Lemma 4.1 and [Reference Poghosyan and Ueltschi53, Lemma 6.2] show that our functions $\psi(I,J;\ {\cdot})$ are equal to the functions $g(I,J\setminus I)$ defined with Ruelle’s algebraic approach in [Reference Poghosyan and Ueltschi53]. We may think of $\mathcal D(I,J)$ 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.

Lemma 4.1. Let I, J be two non-empty finite sets with $I\subset J\subset {\mathbb{N}}_0$ . Fix $(x_j)_{j\in J}\in \mathbb X^J$ . Let $\iota(I) = \min I$ , $I' = I\setminus \{\iota (I)\}$ , and $J'= J\setminus \{\iota (I)\}$ . We have

\begin{equation*}\psi(I,J;\ (x_j)_{j\in J}) =\bigg(\prod_{i\in I'} (1+f(x_{\iota(I)}, x_i))\bigg) \sum_{L\subset J\setminus I} \bigg(\prod_{\ell\in L} f(x_ {\iota(I),\ell})\bigg) \psi (I'\cup L, J';\ (x_j)_{j\in J'}).\end{equation*}

In the sum $L=\varnothing$ is allowed. Products over empty sets are equal to 1. Lemma 4.1 is also true for attractive pairwise interactions.

Proof. For $G\in \mathcal D(I,J)$ , let $L\subset J\setminus I$ the set of vertices $j\in J\setminus I$ that are adjacent to $\iota (I)$ , that is,

\begin{equation*}L = \{j\in J\setminus I\mid \{\iota (I), j\} \in E(G)\}.\end{equation*}

Let G ^′ be the graph obtained from G by removing the vertex $\iota (I)$ and all incident edges. Assume first that $\#I \geq 1$ so that $I'\neq \varnothing$ . Then every vertex of G ^′ must be connected in G ^′ to at least one vertex in $I'\cup L$ , thus $G'\in \mathcal D(I'\cup L, J')$ . The weight of G is given by

\begin{equation*}w(G;\ (x_j)_{j\in J}) = \Bigg(\prod_{\substack{i\in I':\\ \{\iota(I),i\}\in E(G)}} f(x_{\iota(I)},x_i)\Bigg) \bigg( \prod_{\ell \in L} f(x_{\iota (I)},x_\ell)) \bigg) w(G';\ (x_j)_{j\in J}) .\end{equation*}

Conversely, given a possibly empty subset $L\subset J\setminus I$ and a graph $G'\in \mathcal D(I'\cup L, J')$ , we construct a graph $G\in \mathcal D(I,J)$ by adding the vertex $\iota (I)$ , adding all edges $\{\iota(I),\ell\}$ , $\ell \in L$ , and adding none, some, or all of the edges $\{\iota(I),i\}$ , $i\in I'$ . The proof for the case $\#I\geq 1$ 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 $\# J\geq 2$ , then for a given graph G the element $\iota(I)$ has at least one adjacent vertex, i.e. $L\neq \varnothing$ , and the proof is completed as before, taking into account that $\psi(\varnothing, J;\ {\cdot}) =0$ if $J'\neq \varnothing$ . If $\#I = \#J=1$ , then $\psi(I,J;\ (x_j)_{j\in J}) = \psi(\{1\},\{1\};\ x_1) =1$ and the formula from the lemma holds true because $\psi(\varnothing,\varnothing;\ {\cdot}) =1$ .

The partial sums satisfy the following recursive inequality.

Lemma 4.2. For all $N\geq 2$ and $k\in {\mathbb{N}}$ , and all $(x_1,\ldots, x_k)\in \mathbb X^k$ ,

\begin{align*}& (\tilde{{\textit{\textbf{S}}}}_{N}(z))_k(x_1,\ldots,x_k) \leq z(x_1) \prod_{j=2}^k (1+ f(x_1,x_j)) \\ \quad\, \times \sum_{m=0}^{N-k} \dfrac{1}{m!} \int_{\mathbb X^m} \prod_{\ell=k+1}^{k+m} |\kern2pt f(x_1,x_\ell)|(\tilde{{\textit{\textbf{S}}}}_{N-1}(z))_{k+m-1}(x_2,\ldots,x_{k+m}) \,{\textrm{d}}\lambda_z(x_{k+1})\cdots {\textrm{d}} \lambda_z(x_{k+m}) {,}\end{align*}

where the term for $k=1$ , $m=0$ is to be read as $z(x_1)$ .

Thus (4.4) holds true.

Proof. Let $N\geq 2$ . For $k\geq N+1$ we have $(\tilde{{\textit{\textbf{S}}}}_{N}(z))_k(x_1,\ldots,x_k) =0$ and the inequality is trivial. Consider $k\in \{2,\ldots, N\}$ . For $n\in \{k,\ldots,N\}$ and $(x_1,\ldots,x_n)\in \mathbb X^n$ . By Lemma 4.1 applied to $I = \{1,\ldots, k\}$ and $J= \{1,\ldots, n\}$ ,

(4.6) \begin{align} \psi_{k,n}(x_1,\ldots, x_n) = \bigg(\prod_{i=2}^k (1+f(x_{1}, x_i))\bigg) \sum_{L\subset \{k+1,\ldots,n\}} \bigg(\prod_{\ell\in L} f(x_ {1},x_\ell)\bigg) \psi (I'\cup L,J';\ (x_j)_{j\in J'})\end{align}

with $I'= \{2,\ldots,k\}$ and $J' = \{2,\ldots,n\}$ . Let $\sigma\in \mathfrak S_n$ be a permutation that acts on $\{k+1,\ldots,n\}$ only, i.e. $\sigma (i) = i$ for all $i\in \{1,\ldots,n\}$ . By (4.5), we have for every $L\subset\{k+1,\ldots,n\}$ ,

\begin{equation*}\bigg(\prod_{\ell\in L} f(x_ {1},x_{\sigma(\ell)}) \bigg)\psi (I'\cup L,J';\ (x_{\sigma(j)})_{j\in J'}) = \bigg(\prod_{\ell\in \sigma(L)} f(x_ {1},x_\ell) \bigg) \psi(I'\cup \sigma(L),J';\ (x_j)_{j\in J'}).\end{equation*}

We apply the triangle inequality in (4.6), integrate over $x_{k+1},\ldots, x_n$ , note that the contribution of each set L to the integral depends on the cardinality of L alone, and obtain

\begin{align*}&\int_{\mathbb X^{n-k}} |\psi_{k,n}(x_1,\ldots,x_n)| \,{\textrm{d}} \lambda_z(x_{k+1})\cdots {\textrm{d}} \lambda_z(x_n) \leq \bigg(\prod_{i=2}^k (1+f(x_{1}, x_i))\bigg) \\ \qquad \times \sum_{m=0}^{n-k} \binom{n-k}{m} \int_{\mathbb X^{n-k}} \bigg(\prod_{\ell=k+1}^{k+m-1}| f(x_1,x_\ell)|\bigg) |\psi_{k+m-1,n-1}(x_2,\ldots,x_{n})\bigr| \,{\textrm{d}} \lambda(x_{k+1})\cdots {\textrm{d}} \lambda(x_n).\end{align*}

We divide by $(n-k)!$ , sum over $n=k,\ldots, N$ , and find that

(4.7) \begin{align} &\sum_{n=k}^N \dfrac{1}{(n-k)!} \int_{\mathbb X^{n-k}} |\psi_{k,n}(x_1,\ldots,x_n)| \,{\textrm{d}} \lambda_z(x_{k+1})\cdots {\textrm{d}} \lambda_z(x_n) \notag \\ \qquad \leq \bigg( \prod_{i=1}^k (1+ f(x_1,x_i)) \bigg) \sum_{m=0}^{N-k} \bigg( \prod_{\ell=k+1}^{k+m} | f(x_1,x_\ell)|\bigg) \{\cdots \} \,{\textrm{d}} \lambda_z(x_{k+1})\cdots {\textrm{d}} \lambda_z(x_{k+m}))\end{align}

with

\begin{align*}\{\cdots \} &= \sum_{n=k+m}^N \dfrac{1}{(n-k-m)!} \int_{\mathbb X^{n-k}} |\psi_{k+m-1,n-1}(x_2,\ldots,x_{n})| \,{\textrm{d}} \lambda_z(x_{k+m+1})\cdots {\textrm{d}} \lambda_z(x_n).\end{align*}

Changing the summation index from n to $n'= n-1$ , we see that the expression is exactly

\begin{equation*}(\tilde{{\textit{\textbf{S}}}}_{N-1}(z))_{k+\ell-1}(x_2,\ldots,x_k,y_{k+1},\ldots,y_{k+m}),\end{equation*}

except for the missing product $z(x_2)\cdots z(x_k)$ . Hence, multiplying (4.7) with $z(x_1)\cdots z(x_k)$ on both sides, we obtain the required inequality when $k\geq 2$ . The proof for $k=1$ is similar, with a careful consideration of the term $m=0$ which gives rise to $z(x_1)$ .

Proof of Proposition 4.1. First we prove, by induction over N, that the inequality (4.3) holds true for all $N,k\in {\mathbb{N}}$ . For $N=k=1$ , we have $(\tilde{{\textit{\textbf{S}}}}_1)_1(x_1) = z(x_1) \leq z(x_1)\exp\kern-1pt(\! a(x_1))$ and the required inequality holds true. For $N=1$ and $k\geq 2$ , we have $(\tilde{{\textit{\textbf{S}}}}_1)_k(x_1,\ldots, x_k)=0$ and the inequality holds true as well.

For the induction step, let $N\geq 2$ suppose that the bound (4.3) holds true for all $k\in {\mathbb{N}}$ at $N-1$ instead of N. We insert the bound into the right-hand side of the recursive inequality from Lemma 4.2, then bound

\begin{equation*}\prod_{\substack{1\leq i \leq k\\ k+1\leq j \leq m}} (1+ f(x_i,x_j))\leq 1,\end{equation*}

and finally use condition ( $\textsf{FP}_t$ ) with $t=0$ (compare the proof of Lemma 3.2). This yields the bound (4.3) for N and completes the induction step.

Equation (4.3) shows, in particular, that the integrals defining ${\textit{\textbf{S}}}_N$ are absolutely convergent. We may now revisit the induction step, but without absolute values and triangle inequalities. The inequalities then become equalities for ${\textit{\textbf{S}}}_N$ : we find

\begin{equation*}({\textit{\textbf{S}}}_{N}(z))_k = ( K_z{\textit{\textbf{S}}}_{N-1})_k,\quad({\textit{\textbf{S}}}_{N}(z))_1(x_1) = z(x_1) + ( K_z {\textit{\textbf{S}}}_{N-1}(z))_1(x_1)\end{equation*}

and the proof of the proposition is complete.

Theorem 2.4 is a consequence of the Neumann series (3.1) and Proposition 4.1.

Proof of Theorem 2.4. Passing to the limit $N\to \infty$ in the inequality (4.3) from Proposition 4.1, we obtain the estimate (2.14) from Theorem 2.4. The equality (4.2) shows

\begin{equation*}{\textit{\textbf{S}}}_N(z) = {\textit{\textbf{e}}}_z+ K_z {\textit{\textbf{e}}}_z+\cdots + K_z^{N-1} {\textit{\textbf{e}}}_z.\end{equation*}

Thus ${\textit{\textbf{S}}}_N(z)$ is a partial sum of the Neumann series (3.1). The bound (4.3) ensures that each $({\textit{\textbf{S}}}_N)_k(x_1,\ldots,x_k)$ converges pointwise as $N\to \infty$ . But we already know from the proof of Theorem 2.4 that the Neumann series converges in the Banach space E to the vector of correlation functions; thus $\rho_k(x_1,\ldots,x_k) = \lim_{N\to \infty} ({\textit{\textbf{S}}}_N)_k(x_1,\ldots,x_k)$ and we obtain the representation of the correlation functions. The bound (2.14) with $t=0$ follows from Proposition 4.1. For $t>0$ , we note that $z{\textrm{e}}^t$ satisfies ( $\textsf{FP}_t$ ) with $t=0$ so we can apply the inductive bound of Proposition 4.1 to $z{\textrm{e}}^t$ , and the proof is easily concluded.

5. Log-Laplace functional and truncated correlation functions

Here we prove Theorems 2.2 and 2.3. Theorem 2.3 is deduced from Theorem 2.2 by exploiting that the log-Laplace functional at h is simply the generating functional of the truncated correlation functions (factorial cumulant densities) at $u = {\textrm{e}}^{-h} -1$ ; see (5.16) below. Explicit bounds are proved with the complex contour integrals (here $t\gt{0}$ is crucial).

For the proof of Theorem 2.2, we first specialize Theorem 2.4, proved in the previous section, to the one-particle density ( $k=1$ ). As noted earlier, the classes of graphs $\mathcal D_{1,n}$ and $\mathcal C_n$ are equal (if every vertex $j\in \{2,\ldots,n\}$ connects to the vertex 1, then the graph is connected, and vice versa). It follows that $\psi_{1,n} = \varphi_n^{\textsf{T}}$ and

(5.1) \begin{equation} \rho_1(x_1) =z(x_1) \sum_{n=1}^\infty \dfrac{1}{(n-1)!}\int_{\mathbb X^{n-1}} \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z(x_2)\cdots {\textrm{d}}\lambda_z(x_n)\end{equation}

with absolutely convergent integrals and series; moreover, the bound (2.6) stated in Theorem 2.2 is just the special case of the inequality (2.14) in Theorem 2.4.

For the proof of the identity (2.5), the idea is to first prove a differentiated version of it. Formally,

\begin{equation*}\dfrac{{\textrm{d}}}{{\textrm{d}} t} \log \textsf{E}\big[{\textrm{e}}^{- \int_\mathbb X(h+ t g) \,{\textrm{d}} \eta}\big] \big|_{t=0} = \dfrac{\textsf{E}[ (\int_\mathbb X g \,{\textrm{d}} \eta) \exp\kern-1pt(\! - \int_{\mathbb X} h \,{\textrm{d}} \eta)]}{\textsf{E}[ \exp\kern-1pt(\! - \int_{\mathbb X} h \,{\textrm{d}} \eta)] } = \int_{\mathbb X} g(x_1) \rho_1^h(x_1) \,{\textrm{d}}\lambda(x_1),\end{equation*}

with $\rho_1^h(x)$ the one-particle density of a tilted measure $\textsf{P}_h$ , which we write succinctly with notation from variational derivatives as

(5.2) \begin{equation} \dfrac{\delta}{\delta h(x_1)} \log \textsf{E}\big[{\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta}\big] = \rho_1^h(x_1).\end{equation}

The variational derivative of the right-hand of (2.5) is

(5.3) \begin{equation} z(x_1) \,{\textrm{e}}^{-h (x_1)} \sum_{n=1}^\infty \dfrac{1}{(n-1)!}\int_{\mathbb X^{n-1}} \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{e}}^{- \sum_{i=2}^n h(x_i)} \,{\textrm{d}} \lambda_z(x_2)\cdots {\textrm{d}}\lambda_z(x_n).\end{equation}

This is simply the right-hand side of (5.1) with z replaced by $z {\textrm{e}}^{-h}$ . If the tilted measure $\textsf{P}_h$ is a Gibbs measure at tilted activity $z{\textrm{e}}^{-h}$ , we can conclude that the expressions (5.2) and (5.3) are equal, that is, the differentiated form of (2.5) 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 the tilted measure $\textsf{P}_h$ is indeed a Gibbs measure with tilted activity $z{\textrm{e}}^{-h}$ .

Lemma 5.1. Let $\textsf{P}\in \mathscr{G}(z)$ and $h \colon \mathbb X\to {\mathbb{R}}_+$ with $\int_\mathbb X h \,{\textrm{d}} \lambda_z\lt \infty$ . Consider the measure $\textsf{P}_h$ that is absolutely continuous with respect to $\textsf{P}$ , with Radon–Nikodým derivative

\begin{equation*}\dfrac{{\textrm{d}} \textsf{P}_h}{{\textrm{d}} \textsf{P}}(\eta) = \dfrac{\exp\kern-1pt(\! - \int_\mathbb X h \,{\textrm{d}} \eta)}{\int_\mathcal N \exp({-} \int_\mathbb X h\,{\textrm{d}} \gamma)\,{\textrm{d}} \textsf{P}(\gamma)}.\end{equation*}

Then $\textsf{P}_h \in \mathscr{G}(z{\textrm{e}}^{-h})$ .

Proof. From $h\geq 0$ , Jensen’s inequality, and the bound $\rho_1(x) \leq z(x)$ we get

\begin{equation*}0< {\textrm{e}}^{- \int_\mathbb Xh \,{\textrm{d}} \lambda_z} \leq \textsf{E}[ {\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta}] \leq 1 {,}\end{equation*}

so the normalization constant is strictly positive and finite. Let $F \colon \mathbb X\times \mathcal N\to {\mathbb{R}}_+$ be a measurable map. Note that $\int_\mathbb X h \,{\textrm{d}} (\eta + \delta_x) = \int_\mathbb X h \,{\textrm{d}} \eta+ h(x)$ . Then by (GNZ), we have

\begin{equation*}\textsf{E}\bigg[ \int_{\mathbb X} F(x,\eta) \,{\textrm{e}}^{-\int_\mathbb X h\,{\textrm{d}} \eta}\,{\textrm{d}} \eta(x) \bigg]= \int_{\mathbb X}\textsf{E}[ F(x,\eta+\delta_x) \,{\textrm{e}}^{-\int_\mathbb X h \,{\textrm{d}}\eta - W(x;\ \eta)}] z(x) \,{\textrm{e}}^{-h(x)} \,{\textrm{d}} \lambda(x).\end{equation*}

We divide on both sides by $\int_\mathcal N \,{\textrm{e}}^{- \int_\mathbb X h\,{\textrm{d}} \gamma}\,{\textrm{d}} \textsf{P}(\gamma) $ and find that $\textsf{P}_h$ satisfies (GNZ) with z(x) replaced by $z(x)\exp({-}h(x))$ , hence $\textsf{P}_h \in \mathscr{G}(z{\textrm{e}}^{-h})$ .

Remember the function $a \colon \mathbb X\to \mathbb R_+$ from the convergence criterion ( $\textsf{FP}_t$ ).

Lemma 5.2. Let $h \colon \mathbb X\to {\mathbb{R}}_+$ be a bounded measurable function. Suppose that h is bounded and supported in some set of the form $\Lambda\cap \{{\textit{\textbf{x}}} \in \mathbb X\mid a(x) \leq M\}=:\Lambda_M$ with $\lambda_z(\Lambda)\lt\infty$ and $M\in (0,\infty)$ . Then (2.5) holds true.

Proof. Under the assumptions of the lemma, we have

(5.4) \begin{equation} \int_\mathbb X h (x)\,{\textrm{e}}^{a(x)}\,{\textrm{d}} \lambda_z(x)< \infty.\end{equation}

The modified activity $z{\textrm{e}}^{-h}$ satisfies the bound ( $\textsf{FP}_t$ ) as well, with the same a and t as z, so we have a representation for the one-point function $\rho_1^h$ of $\textsf{P}_h$ analogous to (5.1). Let us introduce functions $F,G \colon {\mathbb{R}}_+ \to {\mathbb{R}}$ by

\begin{equation*}F(s) = \log \textsf{E}[{\textrm{e}}^{-s \int_\mathbb X h\,{\textrm{d}} \eta}],\quad G(s) = \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n }({\textrm{e}}^{-s \sum_{i=1}^n h(x_i)} -1 )\varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation*}

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

(5.5) \begin{equation} F'(s) = - \int_{\mathcal N} \bigg(\int_\mathbb X h\,{\textrm{d}} \eta\bigg) \,{\textrm{d}} \textsf{P}_{sh}(\eta) = \int_\mathbb X h(x) \rho_1^{sh}(x) \,{\textrm{d}} \lambda(x)\end{equation}

and, exploiting the symmetry of the Ursell function $\varphi_N^{\textsf{T}}$ ,

(5.6) \begin{align} G'(s) &=- \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n }\bigg(\sum_{i=1}^n h(x_i)\bigg)\,{\textrm{e}}^{-s \sum_{i=1}^n h(x_i)}\varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \notag\\ = - \int_\mathbb X h(x_1) z(x_1) \,{\textrm{e}}^{- s h(x_1)}\notag \\ \qquad \times \bigg(\sum_{n=1}^\infty \dfrac{1}{(n-1)!} \int_{\mathbb X^{n-1}} \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_{z{\textrm{e}}^{-sh}}(x_2)\cdots {\textrm{d}} \lambda_{z{\textrm{e}}^{-sh}}(x_n)\bigg) \,{\textrm{d}} \lambda (x_1).\end{align}

Using the analogue of (5.1) for $\rho_1^{sh}$ , we see that $F'(s) = G'(s)$ for all $s\geq 0$ and it follows that $F(s) =G(s)$ for all $s\geq 0$ . In particular, $F(1) = G(1)$ and (2.5) holds true for bounded h. It remains to justify (5.5) and (5.6). For $s>{\varepsilon}> 0$ we have

(5.7) \begin{equation} \dfrac{1}{{\varepsilon}}|{\textrm{e}}^{-(s\pm {\varepsilon})\alpha} - {\textrm{e}}^{-s\alpha}|\leq \alpha \,{\textrm{e}}^{- (s-{\varepsilon})\alpha}\leq \alpha,\end{equation}

and for $s=0$ we have ${\varepsilon}^{-1}|{\textrm{e}}^{- {\varepsilon} \alpha} - 1|\leq \alpha$ . For the difference quotients of

\begin{equation*}s\mapsto \textsf{E}\bigg[\exp\bigg(- s \int_\mathbb X h \,{\textrm{d}} \eta\bigg)\bigg],\end{equation*}

we apply the inequality to $\alpha = \int_\mathbb X h\,{\textrm{d}} \eta$ , note that

\begin{equation*}\textsf{E}\bigg[\int_\mathbb X h \,{\textrm{d}} \eta\bigg] = \int_\mathbb X h \rho_1\,{\textrm{d}} \lambda \leq \int_\mathbb X h \,{\textrm{d}}\lambda_z \lt \infty {,}\end{equation*}

and conclude with dominated convergence that differentiation and expectation can be exchanged. For the difference quotients of G(s), we apply the bound (5.7) to $\alpha = \sum_{i=1}^n h(x_i)$ and note that, in view of (2.6) and the inequality (2.7), we have

\begin{equation*}\sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} \sum_{j=1}^n h(x_j)|\varphi_n^{\textsf{T}}(x_1,\ldots,x_n)| \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \leq \int_{\mathbb X} h(x) \,{\textrm{e}}^{a(x)}\,{\textrm{d}} \lambda_z(x) \lt \infty.\end{equation*}

and we conclude with dominated convergence that (5.6) holds true. It follows that (2.5) holds true if h satisfies (5.4).

Thus we have proved that

(5.8) \begin{equation} \textsf{E} [{\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta} ] =\exp\bigg( \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} ({\textrm{e}}^{ - \sum_{j=1}^n h (x_j)} - 1) \varphi_n^{\textsf{T}}({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \bigg)\end{equation}

when h is non-negative, bounded, and supported in some set $\Lambda_M = \Lambda\cap \{a\leq M\}$ . 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 $h\geq 0$ , we express the left- and right-hand sides of (5.8) as power series in ${\textrm{e}}^{-h}$ . Roughly, the idea is that if two power series of some variable s coincide and converge on $s\in [0,1]$ , 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 (5.8). Let $\Lambda_M\subset \mathbb X$ be such that $\int_{\Lambda_M} \,{\textrm{e}}^a \,{\textrm{d}}\lambda_z <\infty$ . Set

\begin{equation*}\vartheta_m(x_1,\ldots,x_m) = \sum_{k=0}^\infty \dfrac{1}{k!} \int_{(\mathbb X\setminus \Lambda_M)^k} \varphi_{m+k}^{\textsf{T}}(x_1,\ldots,x_n,y_1,\ldots,y_k) \,{\textrm{d}} \lambda_z^k({\textit{\textbf{y}}}).\end{equation*}

To lighten notation, we suppress the $\Lambda_M$ -dependence from $\vartheta_m$ . By (2.6), we have

(5.9) \begin{align} \sum_{m=1}^\infty \dfrac{{\textrm{e}}^{t m}}{m!} \int_{\Lambda_M^m} |\vartheta_m(x_1,\ldots,x_m)| \,{\textrm{d}} \lambda^m({\textit{\textbf{x}}}) &\leq \sum_{m=1}^\infty \sum_{k=0}^\infty \dfrac{{\textrm{e}}^{t(m+k)}}{m!k!} \int_{\Lambda_M^m\times (\Lambda_M^c)^k} |\varphi_{m+k}^{\textsf{T}}({\textit{\textbf{x}}})| \,{\textrm{d}} \lambda_z^{m+k} ({\textit{\textbf{x}}}) \notag \\ = \sum_{n=1}^\infty \dfrac{{\textrm{e}}^{tn}}{n!}\int_{\mathbb X^n}| \varphi_n^{\textsf{T}}(x_1,\ldots,x_n)| \textrm{1\kern-3.5pt1}_{\{\exists j\colon x_j \in \Lambda_M\}} \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \notag \\ \leq \int_{\Lambda_M} \,{\textrm{e}}^a \,{\textrm{d}} \lambda_z \notag \\ \leq {\textrm{e}}^M \lambda_z(\Lambda) \notag \\ \lt \infty,\end{align}

so the integrals and series in the definition of $\vartheta_m$ are absolutely convergent for $\lambda^m$ -almost all $(x_1,\ldots,x_m)\in \Lambda_M^m$ ; moreover, the constant

\begin{equation*}C_z(\Lambda_M) \,:\!= \sum_{m=1}^\infty \dfrac{1}{m!} \int_{\Lambda_M^m} \vartheta_m (x_1,\ldots,x_m)\,{\textrm{d}} \lambda_z^m({\textit{\textbf{x}}})\end{equation*}

is finite.

Lemma 5.3. Let $h \colon \mathbb X\to [{-}t,\infty)\cup \{\infty\}$ or $h \colon \mathbb X\to \{s\in {\mathbb{C}} \colon {\textrm{Re}\,}\, s\geq - t\}$ . Assume that

\begin{equation*} \int_{\mathbb X} |{\textrm{e}}^{-h} - 1|{\textrm{e}}^a \,{\textrm{d}} \lambda_z <\infty \end{equation*}

and that h is supported in some set $\Lambda_M =\Lambda\cap \{a \leq M\}$ with $\lambda_z(\Lambda)< \infty$ . Then

(5.10) \begin{align} & \exp\bigg( \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} ({\textrm{e}}^{ - \sum_{j=1}^n h (x_j)} - 1) \varphi_n^{\textsf{T}}({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \bigg)\\ \qquad ={\textrm{e}}^{ - C_z(\Lambda_M)} \bigg(1 + \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\Lambda_M^n} \sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n} \prod_{k=1}^r \,{\textrm{e}}^{ - \sum_{i\in V_k} h(x_i)} \vartheta_{\#V_k}({\textit{\textbf{x}}}_{V_k}) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \bigg)=:\mathcal R(h)\notag \end{align}

with absolutely convergent sums and integrals.

Proof. We have already observed that the bound (2.6) holds true, therefore the convergence of the integrals and the sums on the left-hand side of (5.10) follows from the inequality (2.7) and the integrability condition $\int_{\mathbb X} |{\textrm{e}}^{-h} - 1|{\textrm{e}}^a \,{\textrm{d}} \lambda_z <\infty$ . On the left-hand side of (5.10) the only non-zero contributions to integrals come from ${\textit{\textbf{x}}}$ with $x_j \in \Lambda_M$ for some j. Straightforward computations together with the symmetry of the Ursell functions $\varphi_n^{\textsf{T}}$ show that the sum inside the exponential can be rewritten as

(5.11) \begin{align} & \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} ({\textrm{e}}^{ - \sum_{j=1}^n h (x_j)} - 1) \varphi_n^{\textsf{T}}({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) \notag \\ & \qquad = \sum_{m=1}^\infty \dfrac{1}{m!}\int_{\Lambda_M^m} ({\textrm{e}}^{- \sum_{i=1}^m h(x_i)} - 1) \vartheta_m(x_1,\ldots,x_m) \,{\textrm{d}} \lambda_z^m({\textit{\textbf{x}}})\end{align}

The proof of the equation requires some exchange of order of summation, which is justified with (2.7) and estimates similar to (5.9). The right-hand side of (5.11) is actually convergent also without the factor $-1$ . Indeed, from $|\exp\kern-1pt(\! - h)|= \exp({-} {\textrm{Re}\,} \, h) \leq \exp\kern-1pt(\! - t)$ and (5.9), we get

\begin{equation*}\sum_{m=1}^\infty \dfrac{1}{m!}\int_{\Lambda_M^m} | {\textrm{e}}^{- \sum_{i=1}^m h(x_i)} \vartheta_m(x_1,\ldots,x_m)|\, {\textrm{d}} \lambda_z^m({\textit{\textbf{x}}}) \leq \int_{\Lambda_M}\,{\textrm{e}}^a \,{\textrm{d}} \lambda_z \lt \infty.\end{equation*}

Exponentiating (5.11), we find that the left-hand side of (5.10) is given by

\begin{equation*} \exp \bigg( - C_z(\Lambda_M) + \sum_{m=1}^\infty \dfrac{1}{m!}\int_{\Lambda_M^m} | {\textrm{e}}^{- \sum_{i=1}^m h(x_i)} \vartheta_m(x_1,\ldots,x_m) |\, {\textrm{d}} \lambda_z^m({\textit{\textbf{x}}}) \leq \int_{\Lambda_M}\,{\textrm{e}}^a \,{\textrm{d}} \lambda_z\bigg)\end{equation*}

and the proof is concluded with a standard identity on exponential generating functions of set partitions (see (6.3) below).

Next we turn to the left side of (5.8). Let h and $\Lambda_M$ be as in Lemma 5.3 and $(j_{n,\Lambda_M})_{n\in {\mathbb{N}}_0}$ such that

(5.12) \begin{equation} \textsf{E}[{\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta} ] = j_{0,\Lambda_M} + \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\Lambda_M^n} \,{\textrm{e}}^{- \sum_{j=1}^n h(x_j)} j_{n,\Lambda_M}({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^n ({\textit{\textbf{x}}}) =:\mathcal L(h)\end{equation}

whenever h is supported in $\Lambda_M$ and the right-hand side converges (with absolute values in the integrand), for example, if $h\geq 0$ . Thus $j_{0,\Lambda_M}$ is an avoidance probability and $j_{n,\Lambda_M}$ , $n\geq 1$ , are the Janossy densities [Reference Daley and Vere-Jones13, Chapter 5.3], also called density distributions [Reference Ruelle61], of $\textsf{P}$ in $\Lambda_M$ with respect to the reference measure $\lambda_z$ . In our set-up the Janossy densities exist and satisfy

(5.13) \begin{align} j_{n,\Lambda_M}({\textit{\textbf{x}}}) \prod_{i=1}^n z(x_i) = \rho_n(x_1,\ldots,x_n) + \sum_{k=1}^\infty \dfrac{(-1)^k}{k!} \int_{\Lambda_M^k} \rho_n(x_1,\ldots,x_n,y_1,\ldots,y_k) \,{\textrm{d}} \lambda ^k({\textit{\textbf{y}}})\end{align}

with the convention $\rho_0=1$ . The integrals and series are absolutely convergent because of (2.2). Equation (5.13) is the well-known inversion formula for Janossy densities and correlation functions; see [Reference Daley and Vere-Jones13, Chapter 5 (5.4.11)] and the first equation after equation (5.33) in [Reference Ruelle61]. The odd-looking product $z(x_1)\cdots z(x_n)$ in (5.12) appears because the $\rho_n$ are defined with the reference measure $\lambda$ rather than $\lambda_z$ .

Lemma 5.4. Let $\Lambda_M$ , h, and $(\vartheta_m)_{m\in {\mathbb{N}}_0}$ be as in Lemma 5.3. The Janossy densities $(j_{n,\Lambda_M})_{n\in {\mathbb{N}}_0}$ are then given by

(5.14) \begin{equation} j_{0,\Lambda_M} = {\textrm{e}}^{- C_z(\Lambda_M)},\quad j_{n,\Lambda_M}({\textit{\textbf{x}}}) = {\textrm{e}}^{- C_z(\Lambda_M)}\sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n}\vartheta_{\#V_k}({\textit{\textbf{x}}}_{V_k})\end{equation}

for all $n\in {\mathbb{N}}$ and $\lambda_z^n$ -almost all $(x_1,\ldots,x_n)\in \Lambda_M^n$ . Moreover (5.8) extends to all functions h supported in $\Lambda_M$ that take values in $[{-}t,\infty) \cup \{\infty\}$ or in $\{s\in \mathbb C\mid {\textrm{Re}\,}\, s \geq - t\}$ .

Proof. By Lemma 5.2 and (5.8), the functionals $\mathcal L(h)$ and $\mathcal R(h)$ from (5.12) and Lemma 5.3 have to coincide for all non-negative, bounded h supported in $\Lambda_M$ and subject to (2.4), which is enough to ensure that (5.14) holds true. It follows that the identity $\mathcal L(h) = \mathcal R(h)$ extends to the functions h that satisfy the assumptions of Lemma 5.3.

Now we can complete the proof of Theorem 2.2.

Proof of Theorem 2.2. We have already observed that the bound (2.6) follows from Theorem 2.4 and the identity $\mathcal D_{1,n} = \mathcal C_n$ . Let h be a measurable function that satisfies the integrability condition (2.4) and takes values in either $[{-}t,\infty)\cup \{\infty\}$ or $\{s\in \mathbb C \colon {\textrm{Re}\,}\, s\geq t\}$ . For $k\in {\mathbb{N}}$ , set $h_k\,:\!=h \textrm{1\kern-3.5pt1}_{B(0,k)}\textrm{1\kern-3.5pt1}_{\{a\leq k \}}$ . By Lemma 5.4, we have

(5.15) \begin{equation} \log \textsf{E}[ {\textrm{e}}^{-\int_\mathbb X h_k \,{\textrm{d}} \eta}] = \sum_{n=1}^\infty \dfrac{1}{n!}\int_{\mathbb X^n} ({\textrm{e}}^{- \sum_{j=1}^n h_k(x_j)} - 1) \varphi_n^{\textsf{T}}(x_1,\ldots,x_n)\,{\textrm{d}} \lambda^n({\textit{\textbf{x}}}).\end{equation}

In order to pass to the limit on the left side, we write

\begin{equation*}\textsf{E}[ {\textrm{e}}^{-\int_\mathbb X h_k \,{\textrm{d}} \eta}] = 1 + \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\mathbb X^n} \prod_{i=1}^n ({\textrm{e}}^{-h_k(x_i)} - 1) \rho_n({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda^n({\textit{\textbf{x}}}){.}\end{equation*}

A similar identity holds true with $h_k$ replaced by h. Moreover, $h_k\to h$ pointwise with

\begin{equation*}|{\textrm{e}}^{-h_k}-1| =|{\textrm{e}}^{-h} - 1|\textrm{1\kern-3.5pt1}_{B(0,k)}\textrm{1\kern-3.5pt1}_{\{a\leq k \}}\leq |{\textrm{e}}^{-h} - 1|\end{equation*}

and

\begin{equation*} 1 + \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\mathbb X^n} \prod_{i=1}^n |{\textrm{e}}^{-h(x_i)} - 1| \rho_n({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda^n({\textit{\textbf{x}}}) \leq \exp\bigg( \int_\mathbb X |{\textrm{e}}^{- h} - 1|\,{\textrm{d}} \lambda_z\bigg)< \infty.\end{equation*}

Dominated convergence thus yields $\textsf{E}[\!\exp\kern-1pt(\! - \int_\mathbb X h_k \,{\textrm{d}} \eta)]\to \textsf{E}[\!\exp\kern-1pt(\! - \int_\mathbb X h \,{\textrm{d}} \eta)]$ , that is, we can exchange limits and integration on the left-hand side of (5.15). On the right-hand side of (5.15), we can can exchange limits and integration too because of the inequality

\begin{equation*}|{\textrm{e}}^{- \sum_{i=1}^n h_k(x_i)} - 1|\leq {\textrm{e}}^{(n-1)t} \sum_{i=1}^n |{\textrm{e}}^{- h_k(x_i)} - 1|\leq {\textrm{e}}^{(n-1)t} \sum_{i=1}^n |{\textrm{e}}^{- h(x_i)} - 1|\end{equation*}

(remember (2.7)!), the bound (2.6) and dominated convergence. Altogether, passing to the limit $k\to \infty$ in (5.15), we find that (2.5) holds true for h.

Proof of Theorem 2.3. Let $h \colon \mathbb X\to {\mathbb{R}}_+$ . It follows from Corollary 9.5.VIII in [Reference Daley and Vere-Jones13] that

(5.16) \begin{equation} \log \textsf{E}[{\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta}] = \sum_{\ell=1}^\infty \dfrac{1}{\ell!} \int_{\mathbb X^\ell} \prod_{j=1}^\ell ({\textrm{e}}^{- h(x_j)}-1) \rho_\ell^{\textsf{T}}(x_1,\ldots,x_\ell) \,{\textrm{d}} \lambda^\ell({\textit{\textbf{x}}})\end{equation}

whenever the right-hand side is absolutely convergent and h is non-negative with bounded support. On the right-hand side of (2.5) in Theorem 2.2, we express the log-Laplace functional in terms of $\exp\kern-1pt(\! - h)-1$ . We have

\begin{equation*}{\textrm{e}}^{- \sum_{i=1}^n h(x_i)}-1 = \sum_{\substack{I\subset [n]:\\ I\neq \varnothing}}\ \prod_{i\in I} ({\textrm{e}}^{- h(x_i)}-1)\end{equation*}

and, if $|h|\leq t$ ,

(5.17) \begin{equation} \sum_{\substack{I\subset [n]:\\ I\neq \varnothing}}\ \prod_{i\in I} | {\textrm{e}}^{- h(x_i)}-1|= \prod_{i=1}^n ( 1+ |{\textrm{e}}^{- h(x_i)} - 1|) -1 \leq {\textrm{e}}^{(n-1)t}\sum_{j=1}^n |{\textrm{e}}^{-h (x_j)}-1|\end{equation}

(compare (2.7)). Exploiting the symmetry of the Ursell functions, we deduce from (2.5) in Theorem 2.2 that, whenever $|h|\leq t$ and h satisfies condition (2.4), we have

(5.18) \begin{align}\log \textsf{E}[{\textrm{e}}^{- \int_\mathbb X h \,{\textrm{d}} \eta}]&= \sum_{n=1}^\infty \dfrac{1}{n!}\sum_{\ell=1}^n \binom{n}{\ell}\int_{\mathbb X^n} \prod_{i=1}^\ell({\textrm{e}}^{- h (x_i)} - 1) \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}}\lambda_z^n({\textit{\textbf{x}}}) \notag \\ = \sum_{\ell=1}^\infty\dfrac{1}{\ell!} \int_{\mathbb X^\ell} \prod_{j=1}^\ell ({\textrm{e}}^{- h(x_j)}-1)\prod_{j=1}^\ell z(x_j) \notag \\ \quad \, \times \bigg( \sum_{k=0}^\infty \dfrac{1}{k!}\int_{\mathbb X^k} \varphi_{\ell+k}^{\textsf{T}}(x_1,\ldots,x_\ell,y_1,\ldots,y_k) \,{\textrm{d}}\lambda_z^k({\textit{\textbf{y}}}) \bigg) \,{\textrm{d}} \lambda^\ell({\textit{\textbf{x}}})\end{align}

The exchange of the order of summation is justified with (5.17) and absolute convergence. Changing variables to $u(x) = 1 - {\textrm{e}}^{-h(x)}$ and comparing (5.16) and (5.18), we find

\begin{align*}&\int_{\mathbb X^\ell} \prod_{j=1}^\ell u(x_j) \rho_\ell^{\textsf{T}}(x_1,\ldots,x_\ell) \,{\textrm{d}} \lambda^\ell({\textit{\textbf{x}}}) \\ \qquad = \int_{\mathbb X^\ell} \prod_{j=1}^\ell u(x_j) \bigg(\sum_{k=0}^\infty \dfrac{1}{k!}\int_{\mathbb X^k} \varphi_{\ell+k}^{\textsf{T}}(x_1,\ldots,x_\ell,y_1,\ldots,y_k) \,{\textrm{d}}\lambda_z^k({\textit{\textbf{y}}})\bigg) \,{\textrm{d}} \lambda^\ell({\textit{\textbf{x}}})\end{align*}

for all $u \colon \mathbb X\to [0,1]$ that have bounded support and satisfy $\int_\mathbb X u \,{\textrm{e}}^{a}\,{\textrm{d}}\lambda_z< \infty$ . Equation (2.9) follows. For $s\in \mathbb C$ with $|s|\leq {\textrm{e}}^t - 1$ , we have

\begin{align*}&\sum_{\ell=0}^\infty \dfrac{|s|^\ell}{\ell!} \sum_{k=0}^\infty \dfrac{1}{k!} \int_{\mathbb X^{\ell+k}} |\varphi_{\ell+k+1}^{\textsf{T}}(x_0,x_1,\ldots,x_{\ell+k}) |\,{\textrm{d}} \lambda_z(x_1)\cdots {\textrm{d}} \lambda_z(x_{\ell+k} )\\ &\qquad \leq \sum_{m=0}^\infty \dfrac{1}{m!} (1 + |s|)^m \int_{\mathbb X^m} |\varphi_{m+1}^{\textsf{T}}(x_0,x_1,\ldots,x_m) \,{\textrm{d}} \lambda_z(x_1)\cdots {\textrm{d}}\lambda_z(x_m) \\ &\qquad \leq {\textrm{e}}^{a(x_0)}.\end{align*}

Taking contour integrals along the circle centered at the origin with radius ${\textrm{e}}^t - 1$ , we deduce

\begin{align*}\dfrac{1}{\ell!} \sum_{k=0}^\infty \dfrac{1}{k!} \int_{\mathbb X^{\ell+k}} |\varphi_{\ell+k+1}^{\textsf{T}}(x_0,x_1,\ldots,x_{n+k}) |\,{\textrm{d}} \lambda_z(x_1)\cdots {\textrm{d}} \lambda_z(x_{\ell+k} )&\leq \bigg|\dfrac{1}{2\pi \textrm{i}}\oint \dfrac{{\textrm{d}} s}{s^{\ell+1}}\bigg|{\textrm{e}}^{a(x_0)} \\ \leq \dfrac{1}{({\textrm{e}}^t - 1)^\ell}\, {\textrm{e}}^{a(x_0)}\end{align*}

and the bound (2.10) follows.

6. Gibbs measures in finite volume: estimates for $t=0$

Here we prove Theorems 2.7 and 2.6 on the partition function and correlation functions in finite volume. The proof of Theorem 2.7 is fairly standard, once the bound (2.6) is available. The proof of Theorem 2.6 explains the appearance of the class $\mathcal D_{k,n}$ of multirooted graphs directly, without any reference to Kirkwood–Salsburg equations. The theorems are proved in reverse order: first Theorem 2.7, then Theorem 2.6, and finally Theorem 2.5.

Proof of Theorem 2.7. First we note that the bound (2.6) with $t=0$ follows from the identity $\varphi_{n}^{\textsf{T}} = \psi_{1,n}$ and Proposition 4.1 applied to $k=1$ . The bound (2.15) is an immediate consequence of (2.6). The proof of Theorem 2.7 then follows standard arguments [Reference Ruelle60], which we reproduce for the reader’s convenience. Let $\mathcal C(V)$ be the collection of connected graphs with vertex set V (thus $\mathcal C_n = \mathcal C([n])$ ). We have

\begin{equation*}{\textrm{e}}^{- H_n(x_1,\ldots,x_n)} = \prod_{1\leq i {<} j \leq n}(1+ f(x_i,x_j))= \sum_{E\subset \{\{i,j\}\mid 1 \leq i {<} j \leq n\}} \prod_{\{i,j\}\in E} f(x_i,x_j).\end{equation*}

The sum over subsets E can be reinterpreted as a sum over graphs with vertex set $\{1,\ldots,n\}$ and edge sets E so that

(6.1) \begin{equation} {\textrm{e}}^{- H_n(x_1,\ldots,x_n)} = \sum_{G\in \mathcal G_n} w(G;\ x_1,\ldots,x_n).\end{equation}

As the weight of a graph is the product of the weights of its connected components, we deduce

\begin{align*}{\textrm{e}}^{- H_n(x_1,\ldots,x_n)} & = \sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n} \prod_{k=1}^r \sum_{G_k\in \mathcal C(V_k)} w(G_k;\ {\textit{\textbf{x}}}_{V_k}) \\ = \sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n} \prod_{k=1}^r \varphi_{\#V_k}^{\textsf{T}} ({\textit{\textbf{x}}}_{V_k}).\end{align*}

We integrate both sides over $\Lambda^n$ and note that integrals factorize as well; moreover, on the right-hand side they depend on the cardinality of $V_i$ alone. Thus

\begin{equation*}\Xi_\Lambda(z) = 1+\sum_{n=1}^\infty \dfrac{1}{n!}\sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n} A_{\#V_1}\cdots A_{\#V_k}\end{equation*}

with

(6.2) \begin{equation} A_n\,:\!= \int_{\Lambda^n} \bigg( \sum_{G\in \mathcal C_n} w(G;\ x_1,\ldots,x_n) \bigg) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}) = \int_{\Lambda^n} \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation}

It is a general combinatorial fact that

(6.3) \begin{equation} 1+ \sum_{n=1}^\infty \dfrac{1}{n!}\sum_{r=1}^n \sum_{\{V_1,\ldots,V_r\}\in \mathcal P_n} A_{\#V_1}\cdots A_{\#V_k} = \exp\bigg( \sum_{n=1}^\infty \dfrac{1}{n!} A_n\bigg)\end{equation}

whenever

\begin{equation*}\sum_{n=1}^\infty \dfrac{1}{n!}|A_n| \lt \infty.\end{equation*}

The latter condition is satisfied for the concrete choice (6.2) because of (2.6) with $t=0$ , and we obtain

\begin{equation*}\Xi_\Lambda(z) = \exp\bigg( \sum_{n=1}^\infty \dfrac{1}{n!} \int_{\Lambda^n} \varphi_n^{\textsf{T}}(x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}})\bigg).\qed\end{equation*}

Proof of Theorem 2.6. The bound (2.14) with $t=0$ follows from Proposition 4.1. In order to go from Theorem 2.7 to correlation functions we note that the generating functional of the latter is

(6.4) \begin{equation} \textsf{E}_\Lambda\Bigg[\prod_{x\in \Lambda} (1+u(x))^{\eta(\{x\})}\Bigg] = 1 + \sum_{k=1}^\infty \dfrac{1}{k!}\int_{\Lambda^k} \prod_{j=1}^k u(x_j) \rho_{k,\Lambda}({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda^k({\textit{\textbf{x}}}). \end{equation}

The identity holds true for all measurable $u \colon \Lambda\to [{-}1,0]$ ; the right-hand side of (6.4) is absolutely convergent because of the bound (2.2). By the definition of partition functions,

(6.5) \begin{equation} \textsf{E}_\Lambda\Bigg[\prod_{x\in \Lambda} (1+u(x))^{\eta(\{x\})}\Bigg] =\dfrac{ \Xi_\Lambda( z(1+u))}{\Xi_\Lambda(z)}.\end{equation}

In order to evaluate $\Xi_\Lambda(z(1+u))$ , we remember (6.1) and expand

\begin{align*}\prod_{j=1}^n (1+u(x_j)) \sum_{G\in \mathcal G_n} w(G;\ x_1,\ldots,x_n)= \sum_{J\subset [n]} \prod_{j\in J} u(x_j) \sum_{G\in \mathcal G_n} w(G;\ x_1,\ldots,x_n).\end{align*}

Given a non-empty subset $J\subset [n]$ , we establish a one-to-one correspondence between graphs G and pairs $(G_1,G_2)$ that consist of a graph $G_1$ on some subset $V\supset J$ subject to connectivity constraints, and a graph $G_2$ on $[n]\subset V$ , as follows.

For $G\in \mathcal G_n$ and a non-empty subset $J\subset [n]$ , let $V\subset[n]$ be those vertices that connect to some vertex $j\in J$ by a path in G (thus $J\subset V\subset [n]$ ), and $W= [n]\setminus V$ . 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 $\{\{i,j\}\mid i,j\in V,\, \{i,j\}\in E(G)\}$ , similarly for $G|_W$ . Conversely, let $\mathcal D_J(V)$ be the graphs on V for which every vertex in V connects to some vertex $j\in J$ . Given two graphs $G_1\in \mathcal D_J(V)$ , $G_2\in \mathcal G(W)$ such that every vertex in V connects to some elements $j\in J$ by a path in $G_1$ , we obtain a graph $G\in \mathcal G_n$ with edge set $E(G) = E(G_1)\cup E(G_2)$ . In this way we have a one-to-one correspondence, and we find

\begin{align*}& \prod_{j=1}^n (1+u(x_j)) \sum_{G\in \mathcal G_n} w(G;\ x_1,\ldots,x_n)= \sum_{G\in \mathcal G_n} w(G;\ x_1,\ldots,x_n) \\ \qquad + \sum_{\substack{J\subset [n]:\\ J\neq \varnothing}} \sum_{\substack{\{V,W\}\in \mathcal P_n:\\ V\supset J}} \bigg( \prod_{j\in J} u(x_j) \sum_{G_1\in \mathcal D_J(V)} w(G_1;\ {\textit{\textbf{x}}}_V)\bigg) \bigg(\sum_{G_2\in \mathcal G(W)} w(G_2;\ {\textit{\textbf{x}}}_W) \bigg). \end{align*}

Exploiting the symmetry of the graph weights, we deduce

(6.6) \begin{align} & \Xi_\Lambda( z(1+u)) = \Xi_\Lambda(z)\notag \\ & \qquad \times \bigg( 1+ \sum_{k=1}^\infty \dfrac{1}{k!} \int_{\Lambda^k} \prod_{j=1}^k u(x_j)\bigg(\sum_{m=0}^\infty \dfrac{1}{m!}\int_{\Lambda^m} \psi_{k,k+m}(x_1,\ldots,x_k,y_1,\ldots,y_m)\,{\textrm{d}} \lambda_z^m({\textit{\textbf{y}}})\bigg)\bigg) \end{align}

(think $k=\#J$ , $m=\# (V\setminus J)$ ). The expression of the correlation functions follows from (6.4) (6.5), and (6.6).

Proof of Theorem 2.5. The validity of (2.15) has been checked at the beginning of the proof of Theorem 2.7. The representation for the log-Laplace functional builds on Theorem 2.6 and is similar to the proof of Theorem 2.2 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 (2.24) 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) $\Rightarrow$ (i) in Proposition 2.2 that clarifies the relation between Kirkwood–Salsburg equations and inductive convergence proofs.

Let $B_k(q;\ x_1,\ldots,x_k)$ , $k \in {\mathbb{N}}_0$ , be the family of non-negative weight functions given by

\begin{equation*}B_0(q) \,:\!= 1, \quad B_k(q;\ x_1,\ldots,x_k) = \prod_{i=1}^k |\kern2pt f(q;\ x_i)| \prod_{1\leq i {<} j \leq k} (1+ f(x_i,x_j)),\end{equation*}

Set $F_0\,:\!=1$ , and for $k\in {\mathbb{N}}$ and $\textit{\textbf{q}}\in \mathbb X^k$ , set

\begin{equation*} F_k(q_1,\ldots,q_k;\ z) = \prod_{i=1}^k \tilde T_{q_i}^\bullet (z).\end{equation*}

$F_k$ is the generating function for k-forests, that is, k-tuples $F = ((T_1,r_1),\ldots,(T_k,r_k))$ of rooted trees such that their respective vertex sets $V_1,\ldots,V_k$ form a set partition of V. The weight of a forest is the product of the weights of the trees, the root colors are prescribed as $x_{r_i}=q_i$ . Clearly

\begin{equation*}F_{n+1}(q_0,\ldots,q_{n};\ z) = \tilde T_{q_0}^\bullet (z) F_{n}(q_1,\ldots,q_n;\ z).\end{equation*}

Using the non-linear fixed-point equation (2.24) for trees, we obtain a linear set of equations for forests: we have

\begin{align*}F_{n+1}(q_0,\ldots,q_{n};\ z)& = z(q_0) \bigg( B_0(q_0) F_n(q_1,\ldots,q_n;\ z) \\ \quad\, + \sum_{\ell=1}^\infty \dfrac{1}{\ell!} \int_{\mathbb X^\ell} B_\ell(q_0;\ x_1,{\ldots},x_\ell) F_{n+\ell}(q_1,{\ldots},q_n,x_1,{\ldots},x_\ell;\ z) \,{\textrm{d}} \lambda^\ell({\textit{\textbf{x}}})\! \bigg),\end{align*}

a variant of (KS) for forests. Let $F_k^{(N)}(q_1,\ldots,q_k;\ z)$ be the partial sums of the generating functions where we sum only over forests with vertex sets [n], $n\leq N$ . Proceeding as in the proof of Proposition 4.1, we find $F_1^{(1)}(q_1) = z(q_1)$ , $F_k^{(1)} =0$ for $k \geq 2$ , and

\begin{align*}F_{n+1}^{(N+1)}(q_0,\ldots,q_{n};\ z)&= z(q_0) \bigg( B_0(q_0) F_n^{(N)} (q_1,\ldots,q_n;\ z) \\ &\quad\! + \sum_{\ell=1}^\infty \dfrac{1}{\ell!} \!\int_{\mathbb X^\ell}\! B_\ell(q_0;\ x_1,{\ldots},x_\ell) F_{n+\ell}^{(N)} (q_1,{\ldots},q_n,x_1,{\ldots},x_\ell;\ z) \,{\textrm{d}} \lambda^\ell({\textit{\textbf{x}}}) \!\bigg).\end{align*}

An induction over N then shows that if

\begin{equation*}B_0(q)+ \sum_{k=1}^\infty \dfrac{1}{k!}\int_{\mathbb X^k}B_k(q;\ x_1,{\ldots},x_k) \,{\textrm{e}}^{\sum_{i=1}^k a(x_i)} \,{\textrm{d}} \lambda_z^k({\textit{\textbf{x}}}) \leq {\textrm{e}}^{a(x_i)},\end{equation*}

then $F_n^{(N)} (q_1,\ldots,q_n;\ z) \leq \prod_{i=1}^n z(q_i) \,{\textrm{e}}^{a(q_i)}$ for all $N,n, \textit{\textbf{q}}$ . Passing to the limit $N\to \infty$ , we find that the forest generating functions are convergent. In particular, the generating function for trees (i.e. 1-forests) is convergent as well. This proves the implication (ii) $\Rightarrow$ (i) in Proposition 2.2.

8. Cumulants of second-order stochastic integrals

Here we prove Propositions 2.4 and 2.5.

Proof of Proposition 2.4. Pick $\eta = \sum_{i=1}^\kappa \delta_{x_i} \in \mathcal N$ and set $[\kappa]={\mathbb{N}}$ if $\kappa = \infty$ . Assume that u is non-negative or that $\int_{\mathbb X} |u|\,{\textrm{d}}\eta^{(2)} \lt \infty$ . A reasoning similar to the proof of (2.28) shows

\begin{equation*}\bigg( \sum_{i {<}j} u(x_i,x_j)\bigg)^m = \sum_{\gamma\in \mathcal M([\kappa], [m])} \prod_{\{i,j\}\in \mathcal E_2([\kappa])} u(x_i,x_j)^{m_{ij}(\gamma)}.\end{equation*}

Note that the number of edges $\{i,j\}$ with $m_{ij}(\gamma)\geq 1$ is at most m and hence, in particular, finite even when $\kappa = \infty$ , so the product is finite.

For $\gamma\in \mathcal M([\kappa], [m])$ , let $V\subset [\kappa]$ be the set of vertices that belong to some edge of $\gamma$ . The cardinality $n= \#V$ satisfies $2 \leq n \leq 2m$ and the restriction of $\gamma$ to V is a spanning multigraph on V. We have

\begin{equation*}\bigg( \sum_{i {<}j} u(x_i,x_j)\bigg)^m = \sum_{V\subset [\kappa]} \sum_{\gamma\in \mathcal M_s(V, [m])} \prod_{\{i,j\}\in \mathcal E_2([\kappa])} u(x_i,x_j)^{m_{ij}(\gamma)} = \sum_{V\subset [\kappa]} F_{nm}({\textit{\textbf{x}}}_V)\end{equation*}

with

\begin{equation*} F_{nm}(x_1,\ldots,x_n) = \sum_{\gamma\in \mathcal M_s([n], [m])} \prod_{1 \leq i {<} j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)}{,}\end{equation*}

where $F_{nm}$ vanishes unless $2 \leq n \leq 2m$ . Each $F_{nm}$ is integrable with respect to $\lambda_z^n$ by the assumption on u. Therefore

\begin{equation*}\bigg( \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)} \bigg)^m = \sum_{n=2}^{2m}\dfrac{1}{n!} \int_{\mathbb X^n} F_{nm}\,{\textrm{d}} \eta^{(n)}\end{equation*}

with integrals that are almost surely absolutely convergent, and

\begin{equation*}\mathbb E\bigg[ \bigg( \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)} \bigg)^m \bigg] = \sum_{n=2}^{2m} \dfrac{1}{n!} \int_{\mathbb X^n} F_{nm}({\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^n({\textit{\textbf{x}}}).\end{equation*}

This proves the representation for the moments and shows that the moments of order $s \leq m$ of $\int_{\mathbb X^2} |u| \,{\textrm{d}} \eta^{(2)}$ are finite.

For the cumulants, we use the multiplicativity of the graph weights

\begin{equation*}\bar w(\gamma;\ {\textit{\textbf{x}}}) = \prod_{\{i,j\}} u(x_i,x_j)^{m_{ij}(\gamma)}{.}\end{equation*}

Each $\gamma\in \mathcal M_s([n],[m])$ decomposes into connected multigraphs $\gamma_i\in \mathcal M_c(V_i,A_i)$ , $i=1,\ldots,r$ , where $\{V_1,\ldots,V_r\}$ and $\{A_1,\ldots,A_r\}$ form set partitions of [n] and [m], and the weight of $\gamma$ is the product of the weights of the connected components. Therefore

\begin{equation*}\sum_{\gamma\in \mathcal M_s([n],[m])} \bar w(\gamma;\ x_1,\ldots,x_n) = \sum_{r=1}^m \sum_{\substack{\{V_1,\ldots,V_r\}\in \mathcal P_n\\ \{A_1,\ldots,A_r\}\in \mathcal P_m}} \prod_{\ell=1}^r \bigg( \sum_{\gamma_\ell\in \mathcal M_c(V_\ell,A_\ell)} \bar w(\gamma_\ell;\ {\textit{\textbf{x}}}_{V_\ell})\bigg)\end{equation*}

and, formally,

(8.1) \begin{align}& \mathbb E\bigg[ \bigg( \int_{\mathbb X} u \,{\textrm{d}} \eta^{(2)} \bigg)^m \bigg] \notag \\ \quad =\sum_{n=2}^{2m} \dfrac{1}{n!} \sum_{r=1}^n \sum_{\{A_1,\ldots,A_r\}\in \mathcal P_m} \sum_{\substack{n_1,\ldots,n_r\in {\mathbb{N}}_0:\\n_1+\cdots + n_r = n}} \binom{n}{n_1,\ldots,n_r} \sum_{\gamma \in \mathcal M_c([n_\ell],A_\ell)} \int_{\mathbb X^{n_\ell}} \bar w(\gamma;\ {\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^{n_\ell}({\textit{\textbf{x}}}) \notag \\ \quad = \sum_{r=1}^n \sum_{\{A_1,\ldots,A_r\}\in \mathcal P_m} \prod_{\ell=1}^r \bigg( \sum_{n_\ell=2}^{2 \#A_\ell} \dfrac{1}{n_\ell!}\int_{\mathbb X^{n_\ell}} \sum_{\gamma\in \mathcal M_c([n_\ell],A_\ell)}\bar w(\gamma;\ {\textit{\textbf{x}}}) \,{\textrm{d}} \lambda_z^{n_\ell}({\textit{\textbf{x}}})\bigg) \notag \\ \quad = \sum_{r=1}^n \sum_{\{A_1,\ldots,A_r\}\in \mathcal P_m} \kappa_{\#A_1}\cdots \kappa_{\#A_r} \end{align}

with $\kappa_s$ defined by the equation in Proposition 2.4. The identity is formal for now because we need to check the convergence of the integrals in $\kappa_\ell$ . But the convergence can be checked by induction over $s\in \{1,\ldots,m\}$ , using that the moments $\mathbb E[(\int_{\mathbb X^2} |u|\,{\textrm{d}} \eta^{(2)})^s]$ are finite and that (8.1) holds true if we replace m by any $s\leq m$ . The set of equations (8.1) with $s\leq m$ instead of m is exactly the set of equations satisfied by the cumulants (see, for example, [Reference Peccati and Taqqu51]) and the solution of the set of equations is unique, so we identify $\kappa_1,\ldots,\kappa_s$ as the first s cumulants.

Proof of Proposition 2.5. From Proposition 2.4 and the general observation

\begin{equation*} \kappa_m(X) = 2^m \kappa_m\bigg(\dfrac12 X\bigg) {,} \end{equation*}

we get

(8.2) \begin{equation} \kappa_m\bigg( \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}\bigg) =\sum_{n=2}^{2m} \dfrac{2^m}{n!}\int_{\mathbb X^n} \sum_{\gamma \in \mathcal M_c([n],[m])}\prod_{1 \leq i \lt j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)}\,{\textrm{d}} \lambda^n_z.\end{equation}

For $\gamma\in \mathcal M_c([n],[m])$ , let

\begin{equation*}S^\gamma \,:\!= \{ (v,a)\in [n]\times [m]\mid v\in \gamma(a)\}.\end{equation*}

Let $\pi^\gamma$ be the partition of $S^ \gamma$ with blocks $B_a^\gamma = \{ (v,a) \mid v\in \gamma(a)\}$ , $a=1,\ldots,m$ , and $\sigma^\gamma$ the partition with blocks $T^\gamma_v = \{(v,a)\mid a \in A\colon \gamma(a)\ni v\}$ , $v=1,\ldots,n$ . We have already observed that $(\pi^\gamma,\sigma^\gamma)$ is non-flat and connected. The partition $\pi^\gamma$ has m blocks of cardinality 2 each, and $S^\gamma$ has cardinality 2m, so it is natural to map $S^\gamma$ to $\{1,\ldots, 2m\}=S_m$ and work with pairs of partitions on $S_m$ instead.

In order to do this in a systematic way, we note that there are exactly $2^m$ ways of choosing a total order $\prec$ on $S^\gamma$ that respects the ordering of the blocks $B^\gamma_1,\ldots,B^\gamma_m$ , that is, such that for all $1\leq i {<}j \leq m$ and $s_i\in B^\gamma_i$ , $s_j\in B^\gamma_j$ , we have $s_i \prec s_j$ . 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 $2^m$ .

Given an order $\prec$ , we define $\varphi_{\prec} \colon S^\gamma \to \{1,2,\ldots,2m\}= S_m$ as the unique order-preserving bijection. Clearly $\varphi_\prec(B_1) = \{1,2\}$ , $\varphi_{\prec} (B_2) = \{3,4\}$ , etc. Put differently, $\varphi_{\prec}$ maps the partition $\pi^\gamma$ of $S^\gamma$ to the partition $\{\{1,2\}, \{3,4\}, \ldots, \{2m-1,2m\}\} = \pi_m$ of $S_m$ . Let $\hat \sigma \,:\!= (\varphi_{\prec}(T^\gamma_1),\ldots,\varphi_{\prec}(T^\gamma_n))$ be the ordered partition of $S_m$ obtained from $\sigma$ and the map $\varphi_{\prec}$ . Let $\sigma$ be the underlying non-ordered set partition of $S_m$ . It is easily checked that $(\pi_m,\sigma)$ is non-flat and connected.

Summarizing, consider the following two types of objects at fixed n and m:

• pairs $(\gamma, \prec)$ consisting of a multigraph $\gamma\in \mathcal M_c([n],[m])$ and a total order $\gamma$ on $S^\gamma$ that preserves the order of the blocks $B^\gamma _1,\ldots,B^\gamma_m$ ;

• ordered set partitions $\hat \sigma = (T_1,\ldots, T_n)$ of $S_m$ with n blocks such that $(\pi_m,\sigma)$ is non-flat and connected.

We have defined a systematic assignment $(\gamma,\prec)\mapsto \hat \sigma$ , a careful examination shows that this is in fact a one-to-one correspondence.

Let us come back to (8.2). Write $\sum^n_{(\gamma, \prec)}$ and $\sum^n_{\hat \sigma}$ for sums over the types of objects introduced above. As there are $2^m$ choices of order $\prec$ , we may rewrite (8.2) as

\begin{equation*}\kappa_m\bigg( \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}\bigg) =\sum_{n=2}^{2m} \dfrac{1}{n!}\sum^n_{(\gamma, \prec)} \int_{\mathbb X^n} \prod_{1 \leq i \lt j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)}\,{\textrm{d}} \lambda^n_z({\textit{\textbf{x}}}).\end{equation*}

The integral can be rewritten with the help of the ordered partition $\hat \sigma$ associated with $(\gamma,\prec)$ as

\begin{equation*} \int_{\mathbb X^n} \prod_{1 \leq i \lt j \leq n} u(x_i,x_j)^{m_{ij}(\gamma)}\,{\textrm{d}} \lambda^n_z({\textit{\textbf{x}}}) = \int_{\mathbb X^n} (u\otimes\cdots\otimes u)_{\hat \sigma}\, (x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^ n({\textit{\textbf{x}}}),\end{equation*}

hence

\begin{equation*}\kappa_m\bigg( \int_{\mathbb X^2} u \,{\textrm{d}} \eta^{(2)}\bigg) =\sum_{n=2}^{2m} \dfrac{1}{n!}\sum^n_{\hat \sigma}\int_{\mathbb X^n} (u\otimes\cdots\otimes u)_{\hat \sigma}\, (x_1,\ldots,x_n) \,{\textrm{d}} \lambda_z^ n({\textit{\textbf{x}}}).\end{equation*}

To conclude, we note that for a given set partition $\sigma$ of n blocks, each of the $n!$ ordered set partitions $\hat \sigma$ gives rise to an integral with exactly the same value, so we may replace the sum over ordered partition $\hat \sigma$ by a sum over non-ordered partitions $\sigma$ and drop the prefactor $1/n!$ . The proposition follows.