2.1. General notation

We let $[n] \,:\!=\{1,\ldots, n\}$ and $[n]_0 \,:\!= [n] \cup \{0\}$ . For a set V, we write $\binom{V}{2} \,:\!=\{E \subseteq V\,:\, |E|=2\}$ . For $I = \{i_1, i_2, \ldots, i_\kappa\} \subset \mathbb N$ , let $\vec x_I=\big(x_{i_1}, \ldots, x_{i_\kappa}\big)$ . For compact intervals $[a,b]\subset \mathbb R$ , we write $\vec x_{[a,b]} = \vec x_I$ with $I=[a,b]\cap \mathbb N$ . If $a=1$ , we write $\vec x_{[b]} = \vec x_{[1,b]}$ . By some abuse of notation, we are going to interpret $\vec x_{[a,b]}$ both as an ordered vector and as a set.

If not specified otherwise, $\Lambda$ denotes a bounded, measurable subset of $\mathbb R^d$ .

2.2. Graph theory

We recall that a (simple) graph $G=(V,E) = (V(G), E(G))$ is a tuple with vertex set (or set of points, sites, nodes) V and edge set (or set of bonds) $E \subseteq \binom{V}{2}$ . In this paper, we will always consider graphs with $V \subset \mathbb R^d$ , and for $x,y\in\mathbb R^d$ , an edge $\{x,y\}$ will sometimes be abbreviated xy.

If $xy\in E$ , we write $x \sim y$ (and say that x and y are adjacent). We extend this notation and write $x \sim W$ for $x \in V$ and $W \subseteq V$ if there is $y \in W$ such that $x \sim y$ ; also, we write $A \sim B$ if there is $x\in A$ such that $x \sim B$ . For $W \subseteq V$ , we define the W-neighborhood $N_W(x) = \{y \in W\,:\, x \sim y\}$ and the W-degree of a vertex $x\in V$ as $\deg_{W}\!(x) = |N_W(x)|$ , and we write $N(x) = N_V(x)$ as well as $\deg\!(x) = \deg_V\!(x)$ . For two sets $A,B \subseteq V$ , we write $E(A,B) = \{xy \in E(G)\,:\, x\in A, y \in B\}$ .

Given a graph $G=(V,E)$ and $W \subseteq V$ , we denote by $G[W] \,:\!=\, (W, \{e \in E\,:\, e \subseteq W\})$ the subgraph of G induced by W. Given two simple graphs G, H, we let $G \oplus H \,:\!=\, (V(G) \cup V(H), E(G) \cup E(H))$ .

Connectivity. Given a graph G and two of its vertices $x,y\in V(G)$ , we say that x and y are connected if there is a path between x and y—that is, a sequence of vertices $x=v_0, v_1, \ldots, v_k=y$ for some $k\in\mathbb N_0$ such that $v_{i-1}v_i \in E(G)$ for $i \in [k]$ . We write $x \longleftrightarrow y$ in G or simply $x \longleftrightarrow y$ . We call $\mathscr {C}(x)=\mathscr {C}(x;\,G) = \{y \in V(G)\,:\, x \longleftrightarrow y\}$ the cluster (or connected component) of x in G. If there is only one cluster in G, we say that G is connected.

For $x \longleftrightarrow y$ in G, we let $\textsf {Piv}(x,y;\,G)$ denote the set of pivotal vertices for the connection between x and y. That is, $v\notin \{x,y\}$ is in $\textsf {Piv}(x,y;\,G)$ if every path from x to y in G passes through v. We say that x is doubly connected to y in G (and write $x \Longleftrightarrow y\textrm { in } G$ ) if $\textsf {Piv}(x,y;\,G)=\varnothing$ . We remark that in the physics literature, pivotal points are usually known as nodal points.

In the pathological case $x=y$ , we use the convention $x \longleftrightarrow x$ in G and set $\textsf {Piv}(x,x;\,G)=\varnothing$ for any graph G with $x\in V(G)$ (equivalently, $x \Longleftrightarrow x\textrm { in } G$ ).

We observe that the pivotal points $\{u_1,\ldots, u_k\}$ can be ordered in a way such that every path from x to y passes through the pivotal points in the order $(u_1, \ldots, u_k)$ . We define $\textsf{PD}(x,y,G) = \textsf{PD}(G)$ to be the pivot decomposition of G, that is, a partition of the vertex set V into a sequence, $(x,V_0, u_1, V_1, \ldots, u_k, V_k, y)$ , where $(u_1, \ldots, u_k)$ are the ordered pivotal points and $V_i$ is the (possibly empty) set of vertices that can be reached only by passing through $u_i$ and that is still connected to x after the removal of $u_{i+1}$ . See Figure 1.

Figure 1.

A schematic sketch of the pivot decomposition $(u_0,V_0, \ldots, V_7, u_8)$ of G, setting $x=u_0$ and $y=u_{k+1}$ .

Classes of graphs. Given a (locally finite) set $X \subset \mathbb R^d$ , we let $\mathcal G(X)$ be the set of graphs with vertex set X. We let $\mathcal C(X)$ be the set of connected graphs on X. Moreover, for $x,y \in X$ , we let $\mathcal D_{x,y}(X)\subseteq \mathcal C(X)$ be the set of non-pivotal graphs, i.e., the set of connected graphs such that $\textsf {Piv}(x,y;\,G) = \varnothing$ .

Given m bags $X_1, \ldots, X_m \subset \mathbb R^d$ with $|X_i \cap X_j| \leq 1$ for all $1 \leq i<j \leq m$ , we let $\mathcal G(X_1,\ldots, X_m)$ denote the set of m-partite graphs on $X_1, \ldots, X_m$ , i.e., the set of graphs G with $V(G) = \cup_{i=1}^m X_i$ and $E(G[X_i]) =\varnothing$ for $i\in[m]$ . Note that we allow bags to have (at most) one vertex in common, which is a slight abuse of the notation in graph theory, where m-partite graphs have disjoint bags.

The notion of ( $\pm$ )-graphs. We introduce a ( $\pm$ )-graph as a triple

\begin{equation*}G^\pm = (V(G),E^+(G), E^-(G)) = (V,E^+,E^-),\end{equation*}

where V is the vertex set and $E^+,E^- \subseteq \binom{V}{2}$ are disjoint. In other words $G^\pm$ is a graph where every edge is of exactly one of two types (plus or minus). We set $E \,:\!=\, E^+ \cup E^-$ and associate to $G^\pm$ the two simple graphs $G^{|\pm|} \,:\!=\, (V,E)$ and $G^+\,:\!=\,(V^+,E^+)$ , where $V^+ \,:\!=\, \{x \in V\,:\, \exists e \in E^+\,:\, x \in e\}$ are the vertices incident to at least one $({+})$ -edge.

We extend all the notions for simple graphs to ( $\pm$ )-graphs. In particular, given $X \subset \mathbb R^d$ , we let $\mathcal G^\pm(X)$ be the set of ( $\pm$ )-graphs on X. Moreover, $\mathcal C^\pm(X)$ are the ( $\pm$ )-connected graphs on X, that is, the graphs such that $G^{|\pm|}$ is connected. Similarly, $\mathcal C^+(X)\subset \mathcal C^\pm(X)$ are the $({+})$ -connected graphs, that is, those where $G^+$ is connected and $V(G)=V^+$ . For $x,y\in X$ , we denote by $\mathcal D^\pm_{x,y}(X)$ the set of those ( $\pm$ )-connected graphs on X where $\textsf {Piv}(x,y;\,G^{|\pm|})= \varnothing$ , and by $\mathcal D^+_{x,y}(X)\subset \mathcal D^\pm_{x,y}(X)$ the set of those ( $\pm$ )-connected graphs on X where $\textsf {Piv}\big(x,y;\,G^+\big)=\varnothing$ . We also define the $({{\pm}})$ -pivot decomposition $\textsf{PD}^\pm\big(x,y,G^\pm\big) = \textsf{PD}^\pm\big(G^\pm\big) = \textsf{PD}(G^{|\pm|})$ and the $({+})$ -pivot decomposition $\textsf{PD}^+\big(x,y,G^\pm\big) = \textsf{PD}^+\big(G^\pm\big) = \textsf{PD}\big(G^{+}\big)$ . Lastly, we write $x \overset{+}{\longleftrightarrow} y$ if there is a path from x to y in $E^+$ .

Given a ( $\pm$ )-graph G and a simple graph H, we define

\begin{equation*}G \oplus H \,:\!=\, (V(G) \cup V(H), E^+(G), E^-(G) \cup E(H)).\end{equation*}

Weights. Given a simple graph G, a ( $\pm$ )-graph H on $X \subset \mathbb R^d$ , and the connection function $\varphi$ , we define the weights

\begin{equation*} \textbf{w} (G) \,:\!=\, ({-}1)^{|E(G)|} \prod_{\{x,y\} \in E(G)} \varphi(x-y), \qquad \textbf{w}^\pm (H) \,:\!=\, ({-}1)^{|E^- (H)|} \prod_{\{x,y\} \in E(H)} \varphi(x-y). \end{equation*}

2.3. The random connection model

The RCM $\xi$ can be formally constructed as a point process, that is, a random variable taking values in the space of locally finite counting measures $(\textbf N, \mathcal N)$ on some underlying metric space $\mathbb X$ . There are various ways to choose $\mathbb X$ . One option is to let $\mathbb X = \mathbb R^d \times \mathbb M$ for an appropriate mark space $\mathbb M$ (see [Reference Meester and Roy18]); another way can be found in [Reference Heydenreich, van der Hofstad, Last and Matzke10, Reference Last and Ziesche15]. In any case, one can reconstruct from $\xi$ the point process $\eta$ on $\mathbb R^d$ which makes up the vertex set of $\xi$ . We treat $\eta$ both as a counting measure and as a set, giving meaning to statements of the form $x \in \eta$ .

If $e=\{x,y\}$ is an edge, then we write $\varphi(e)= \varphi(x-y)$ . For a bounded set $\Lambda\subset \mathbb R^d$ , we write $\eta_\Lambda=\eta\cap\Lambda$ and let $\xi_\Lambda$ denote the RCM restricted to $\Lambda$ , that is, $\xi[\eta_\Lambda]$ . The two-point function restricted to $\Lambda$ is defined as $\tau_\lambda^\Lambda(x,y) = \mathbb P_\lambda\big(x \longleftrightarrow y\textrm { in } \xi_\Lambda^{x,y}\big)$ for $x,y \in \Lambda$ and zero otherwise.

For $V \subset W$ , there is a natural way to couple the models $\xi^V$ and $\xi^W$ , which is by deleting from $\xi^W$ all points in $W \setminus V$ along with their incident edges. We implicitly assume throughout this paper that this coupling for different sets of added points is used.

The Mecke equation. Since it is used repeatedly throughout this paper, we state the Mecke equation, a standard tool in point process theory, in its version for the RCM (see [Reference Last and Ziesche15]). For $m \in \mathbb N$ and a measurable function $f\,:\, \textbf N \times \mathbb R^{dm} \to \mathbb R_{\geq 0}$ , the Mecke equation states that

(2.1) \begin{equation} \mathbb E_\lambda \Bigg[ \sum_{\vec x_{[m]} \in \eta^{(m)}} f\big(\xi, \vec{x}_{[m]}\big)\Bigg] = \lambda^m \int \mathbb E_\lambda\big[ f\big(\xi^{x_1, \ldots, x_m}, \vec x_{[m]}\big)\big] \, \textrm{d} \vec{x}_{[m]}, \end{equation}

where $\eta^{(m)}=\big\{\vec x_{[m]} \in \eta^m\,:\, x_i \neq x_j \text{ for } i \neq j\big\}$ are the pairwise distinct tuples.

Rescaling. It is a standard trick in continuum percolation to rescale space in order to normalize a quantity of interest, which is $\int \varphi(x) \, \textrm{d} x$ in our case. We refer to [Reference Meester and Roy18, Section 2.2]. As a consequence, we may without loss of generality assume that $\int \varphi(x) \, \textrm{d} x=1$ .

The BK inequality. We say that $A\in\mathcal{N}$ lives on $\Lambda$ if $\unicode{x1D7D9}_{A}(\mu) = \unicode{x1D7D9}_{A}(\mu_\Lambda)$ for every $\mu \in \textbf N$ . We call an event $A \in \mathcal N$ increasing if $\mu \in A$ implies $\nu\in A$ for each $\nu\in\textbf{N}$ with $\mu\subseteq\nu$ . Let $\mathcal{R}$ denote the ring of all finite unions of half-open rectangles with rational coordinates. For two increasing events $A,B\in\mathcal{N}$ we define

(2.2) \begin{equation} A \circ B \,:\!=\, \{ \mu\in\textbf{N}\,:\, \exists K,L \in \mathcal{R} \text{ s.t. }\, K \cap L = \varnothing \text{ and } \mu_K \in A,\ \mu_L\in B \}. \end{equation}

Informally, this is the event that A and B take place in spatially disjoint regions. It is proved in [Reference Heydenreich, van der Hofstad, Last and Matzke10, Theorem 2.1] that for two increasing events A and B living on $\Lambda$ , we have

\begin{equation*}\mathbb P_\lambda(A\circ B) \leq \mathbb P_\lambda(A) \mathbb P_\lambda(B).\end{equation*}

The RCM on a fixed vertex set. Given some (finite) set $X \subset \mathbb R^d$ and a function $\varphi\,:\,\mathbb R^d\to [0,1]$ , we will often have to deal with the following random graph: its vertex set is X, and two vertices $x,y\in X$ are adjacent with probability $\varphi(x-y)$ , independently of other pairs of vertices. This is simply the RCM conditioned to have the vertex set X. To highlight the difference from $\xi$ , which depends on the PPP $\eta$ , we denote this random graph by $\Gamma_\varphi(X)$ . If $Y \subset X$ , then we write $\Gamma_\varphi(Y)$ for $\Gamma_\varphi(X)[Y]$ . Since there is no dependence on $\lambda$ , we write $\mathbb P$ for the probability measure of the RCM with fixed vertex set.

4.1. Motivation and rough outline

The expansion of the direct-connectedness function in powers of the activity given by [Reference Coniglio, DeAngelis and Forlani6], without proofs and convergence bounds, is

(4.1) \begin{equation} g_\lambda^\Lambda(x_1, x_2) = \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \sum_{\substack{G \in \mathcal D_{x_1,x_2}^\pm\big(\vec x_{[n+2]}\big)\,:\, \\ x_1 \overset{+}{\longleftrightarrow} x_2}} \textbf{w}^\pm(G) \, \textrm{d} \vec x_{[3, n+2]}. \end{equation}

It is obtained from the expansion of the pair-connectedness function in Proposition 3.1 by discarding graphs that have pivotal points (i.e., graphs G where $\textsf{Piv}^\pm(G)$ is nonempty). Before we pass to the thermodynamic limit, we perform a re-summation and find another representation of $g_\lambda^\Lambda$ which has the conjectured advantage of increasing the domain of convergence.

Let $G=(V, E^+, E^-) \in \mathcal C^\pm\big(\vec x_{[n+2]}\big)$ be a $({{\pm}})$ -graph appearing in the expansion (3.6). Thus $V= \{ x_i\,:\, 1\leq i \leq n+2\}$ , the graph $G^+$ is connected, $x_1$ and $x_2$ belong to $V^+=V\big(G^+\big)$ , every vertex $y\in V(G) \setminus V\big(G^+\big)$ is linked by at least one $({-})$ -edge to $V^+$ , and there are no edges between two vertices in $ V \setminus V^+$ . We impose the additional constraint that $G^{|\pm|} = \big(\vec x_{[n+2]}, E^+\cup E^-\big)$ has no pivotal points for paths from $x_1$ to $x_2$ .

Since $x_1$ and $x_2$ are connected by a path of $({+})$ -edges, G admits a $({+})$ -pivot decomposition $\vec W= (u_0, V_0, \ldots, u_k, V_k, u_{k+1})$ (with $u_0=x_1$ and $u_{k+1}=x_2$ ), where $k \in \mathbb N_0$ is the number of pivotal points in $\textsf{Piv}^+(x_1, x_2;\,G)$ . Then, G decomposes into a core graph $G_{\text{core}} = \big(V\big(G^+\big), E^+, E_{\text{core}}^-\big)$ , with $E_{\text{core}}^-$ the set of $({-})$ -edges of G with both endpoints in $V_i \cup \{u_i, u_{i+1}\}$ for some $i \in [k]_0$ , and a shell graph $H = \big(V, \varnothing, E^-\setminus E^-_{\text{core}}\big)$ . By our choice of $E^-_{\text{core}}$ , we have $\textsf{PD}^\pm\big(G_{\text{core}}\big) = \textsf{PD}^+\big(G_{\text{core}}\big)= \vec W$ . Clearly

\begin{equation*} \textbf{w}^\pm (G) = \textbf{w}^\pm \big(G_{\text{core}}\big) \textbf{w}^\pm (H). \end{equation*}

In the right-hand side of (4.1), we restrict to graphs that also appear in (3.6) and rewrite the resulting sum as a double sum over core graphs and shell graphs. This gives rise to the series

\begin{equation*}\sum_{r=0}^\infty \frac{\lambda^r}{r!} \int_{\Lambda^r} \sum_{\vec W}\sum_{G_{\text{core}}} \textbf{w}^\pm \big(G_{\text{core}}\big)\Bigg( \sum_{m=0}^\infty \frac{\lambda^m}{m!} \int_{\Lambda^m} \sum_H \textbf{w}^\pm (H) \, \textrm{d} \vec y_{[m]} \Bigg) \, \textrm{d} \vec x_{[3,r+2]}. \end{equation*}

The outer sum is over potential pivot decompositions $\vec W$ of core vertices $\vec x_{[r+2]}$ , the second sum over $({{\pm}})$ -graphs $G_{\text{core}}= \big(\vec x_{[r+2]}, E^+,E_{\text{core}}^-\big)$ that are $({+})$ -connected and for which $\vec W$ is both the $({{\pm}})$ -pivot decomposition and the $({+})$ -pivot decomposition (in other words, the simple graph $\big(\vec x_{[r+2]}, E^+\big)$ is connected and $\textsf{PD}^\pm(x_1,x_2,G) =\textsf{PD}^+(x_1,x_2,G)=\vec W$ ). The inner sum is over $({{\pm}})$ -graphs $H = (V(H),\varnothing, E^-(H))$ with vertex set $\vec x_{[r+2]}\cup \vec y_{[m]}$ and $({-})$ -edges $\{y_i,x_j\}$ such that every vertex $y_i$ is linked to at least one vertex $x_j$ , under the additional constraint that $(\vec x_{[r+2]} \cup \vec y_{[m]}, E^+, E_{\text{core}}^-\cup E^-(H))$ has no $({{\pm}})$ -pivotal points for paths from $x_1$ to $x_2$ . Let us denote the series associated to such graphs H by $h_\lambda^\Lambda\big(G_{\text{core}}\big)$ :

(4.2) \begin{equation} h_\lambda^\Lambda\big(G_{\text{core}}\big) = \sum_{m=0}^\infty \frac{\lambda^m}{m!} \int_{\Lambda^m} \sum_H \textbf{w}^\pm (H) \, \textrm{d} \vec y_{[m]}. \end{equation}

The right-hand side of (4.2) depends on $G_{\text{core}}$ only through the pivot decomposition $\vec W$ . We obtain the representation

(4.3) \begin{equation}g_\lambda^\Lambda(x_1,x_2) = \sum_{r=1}^\infty \frac{\lambda^r}{r!}\int_{\Lambda^r} \sum_{\vec W} \sum_{G_{\text{core}}} \textbf{w}^\pm \big(G_{\text{core}}\big) h_\lambda^\Lambda\big(G_{\text{core}}\big) \, \textrm{d} \vec x_{[3,r+3]}. \end{equation}

This expression, written in a slightly different form (see Definition 4.2), forms the starting point of this section. The main results of this section are the following:

1. 1. Let $G_{\text{core}}$ be a $({{\pm}})$ -graph as above. Then the corresponding power series $h_\lambda^\Lambda\big(G_{\text{core}}\big)$ is absolutely convergent for all intensities $\lambda \geq 0$ (Proposition 4.1). In addition, $h_\lambda^\Lambda\big(G_{\text{core}}\big)$ can be expressed in terms of probabilities involving the random connection model on the fixed vertex set $V\big(G_{\text{core}}\big)$ and of Poisson processes in $\Lambda$ . This alternative expression is used to show that the (pointwise) limit

   \begin{equation*} h_\lambda\big(G_{\text{core}}\big) = \lim_{\Lambda\nearrow \mathbb R^d} h_\lambda^\Lambda\big(G_{\text{core}}\big) \end{equation*}
   exists for all $\lambda>0$ (Lemma 4.5).

2. 2. Then we show in Theorem 4.1 that

   \begin{equation*} \sum_{r=0}^\infty \frac{\lambda^r}{r!}\int_{(\mathbb R^d)^r} \sum_{\vec W} \sum_{G_{\text{core}}} \textbf{w}^\pm \big(G_{\text{core}}\big) \Bigl|h_\lambda\big(G_{\text{core}}\big)\Bigr| \, \textrm{d} \vec x_{[3,r+3]}< \infty. \end{equation*}
   This allows us to define
   \begin{equation*} g_\lambda(x_1,x_2)\,:\!=\, \sum_{r=0}^\infty \frac{\lambda^r}{r!}\int_{(\mathbb R^d)^r} \sum_{G_{\text{core}}} \textbf{w}^\pm \big(G_{\text{core}}\big) h_\lambda\big(G_{\text{core}}\big) \, \textrm{d} \vec x_{[3,r+3]} \end{equation*}
   and to pass to the limit in (4.3), showing that
   \begin{equation*} \lim_{\Lambda\nearrow \mathbb R^d} g_\lambda^\Lambda(x_1,x_2) = g_\lambda(x_1,x_2)\end{equation*}
   as part of Theorem 4.1.

4.2. Definition

Here we introduce the precise definitions of core graphs and shell graphs as well as of the functions $h_\lambda^\Lambda$ and $g_\lambda^\Lambda$ . We follow the ideas outlined in the previous section but make two small changes. First, shell graphs H are defined not as $({{\pm}})$ -graphs with minus edges only but right away as standard graphs. Second, a close look reveals that the shell function $h_\lambda^\Lambda\big(G_{\text{core}}\big)$ defined in (4.2) depends on the core graph only via $\vec W$ ; accordingly we view $h_\lambda^\Lambda$ as a function of a sequence of sets. In addition we drop the index from the core graph; thus the graph G in Definition 4.1 below corresponds to $G_{\text{core}}$ in the previous section (see Figure 2).

Figure 2.

In the first line, we see an example of two ( $\pm$ )-graphs; the ( $+$ )-edges are depicted by dotted lines and the (minus;)-edges by dashed lines. Notice that both graphs are ( $+$ )-connected. However, the solid black vertex—which is ( $+$ )-pivotal for the $x_1$ – $x_2$ connection in both graphs—is ( $\pm$ )-pivotal for the $x_1$ – $x_2$ connection in the graph on the left but not in the graph on the right. Hence, the graph on the left is a core graph according to Definition 4.2 but the graph on the right is not. In the second line, the simple graph on the left is a shell graph for the core graph above, since the ( $\pm$ )-graph given by their sum (depicted on the right) is ( $\pm$ )-doubly connected; in particular there are no ( $\pm$ )-pivotal points for the $x_1$ – $x_2$ connection.

We define $h_\lambda^\Lambda$ and $g_\lambda^\Lambda$ by expansions similar to (4.2) and (4.3) and postpone the proof of convergence to Proposition 4.3 and Theorem 4.4. By some abuse of language, we refer to the series (4.6) as the direct-connectedness function, and we use the same letter $g_\lambda$ as in (1.2). This is justified a posteriori by the proof of Theorem 1.1, where we show that the series is indeed the expansion for the direct-connectedness function $g_\lambda$ defined as the unique solution of the OZE (1.2).

The 0-shell function $h^{(0)}$ is understood to be given in terms of shell graphs without satellite vertices, i.e.,

\begin{equation*} h^{(0)}\big(\vec W\big) = \sum_{\substack{H \in \mathcal G^{\varnothing, \vec W}_{\text{shell}}}} {\textbf{w}}(H). \end{equation*}

Note that because of translation-invariance, $g_\lambda^\Lambda(x_1,x_2) = g_\lambda^\Lambda(\textbf{0}, x_2-x_1) = g_\lambda^{\Lambda}(x_2-x_1) $ .

4.3. Analysis of the shell functions: laces

If we take a look at the graphs that are summed over in the shell function, we note that the associated minimal structures have a form which is very reminiscent of graphs that are known as laces and famously appear in the analysis of, for example, self-avoiding walks [Reference Brydges and Spencer3, Reference Slade22]. They are also the namesake of the lace-expansion technique.

Proposition 4.1 is the central result of this section. It allows us to bound the shell function by the probability that the points in a PPP $\eta$ are not connected to the core vertices W. Moreover, we introduce laces and partition the shell graphs with respect to them. For every lace, we obtain a precise expression for its contribution to the shell function.

To prove Proposition 4.1, we will need quite a few definitions (see Definitions 4.3, 4.4, and 4.5) and some intermediate results thereon.

Proposition 4.1. (Bounds on the shell functions) Let $\lambda\geq 0$ and let $\Lambda\subset \mathbb R^d$ be bounded. Let $u_0, \ldots, u_{k+1} \in \Lambda$ for $k\in\mathbb N_0$ , let $V_0, \ldots, V_k \subset \Lambda$ be finite sets, and set $\vec W=(u_0, V_0, \ldots, V_k, u_{k+1})$ . Then

(4.7)

Moreover,

(4.8) \begin{equation} \sum_{m \geq 0}\frac{\lambda^m}{m!} \int_{\Lambda^m} \big| h^{(m)} \big(\vec W, \vec y_{[m]}\big) \big| \, \textrm{d}\vec y_{[m]} \leq \frac{1}{\sqrt{5}} {\text{e}}^{3 \lambda|W|} \big( 3+\sqrt{5} \big)^{|W|}. \end{equation}

Proposition 4.1 consists of two parts, and it is (4.8) that guarantees the well-definedness of the shell function $h_\lambda^\Lambda$ of Definition 4.2.

Proposition 4.1 is easy to prove for $k=0$ , and we mostly focus on $k\geq 1$ . Throughout the remainder of this section, we fix a pivot decomposition $\vec W = (u_0, V_0, \ldots, V_k, u_{k+1})$ and recall that $\overline{V}_i = V_i \cup \{u_i,u_{i+1}\}$ .

We now work towards a deeper understanding of the shell graphs H summed over in (4.4).

Thus, the graph $\hat H$ has no nearest-neighbor bonds, and $\alpha\beta$ with $\vert\alpha-\beta\vert\geq 2$ is a bond in $E(\hat H)$ if and only if $\{u_\alpha\}\cup V_\alpha$ and $V_{\beta-1}\cup\{u_{\beta}\}$ are connected by a direct or indirect stitch. See Figure 3 for an illustration. We may now apply the standard vocabulary of lace expansion (for self-avoiding walks) to the graph $\hat H$ [Reference Slade22, Section 3.3].

Figure 3.

In the first line, we see a schematic shell graph $H_1$ . Its skeleton $\hat H_1$ is already a lace, namely L. The skeleton of the graph $H_2$ in the second line is not a lace, but $H_2 \in \langle\!\langle L \rangle\!\rangle$ . The structure of L is indicated in $H_2$ and in $\hat H_2$ by the thicker edges.

In the context of lace expansions, usually the word ‘connected’ is used instead of ‘irreducible’, but ‘connected’ is clearly misleading in our setup; Brydges and Spencer originally called those graphs ‘primitive’ [Reference Brydges and Spencer3]. We observe that the skeleton graphs $\hat H$ arising from our shell graphs H are precisely the irreducible graphs (and so $G\oplus H$ being 2-connected corresponds to the skeleton $\hat H$ being irreducible).

We map irreducible graphs to laces by following a standard procedure [Reference Slade22, Section 3.3], performed backwards. That is, we define bonds $\alpha^{\prime}_j\beta^{\prime}_j$ with $\beta^{\prime}_1>\beta^{\prime}_2>\cdots$ inductively as follows: we set

\begin{equation*} \beta^{\prime}_1 \,:\!=\,k+1,\qquad \alpha^{\prime}_1 \,:\!=\, \min\!\big\{\alpha\,:\,\ \alpha\beta^{\prime}_1\in E(\hat H)\big\}, \end{equation*}

and

\begin{equation*} \alpha^{\prime}_{j+1} = \min\!\big\{\alpha\,:\,\ \exists \beta > \alpha^{\prime}_j\text{ with } \alpha \beta\in E(\hat H)\big\}, \qquad \beta^{\prime}_{j+1} = \max\!\big\{\beta\,:\,\ \alpha^{\prime}_{j+1} \beta \in E(\hat H)\big\}.\end{equation*}

The procedure terminates when $\alpha^{\prime}_j = 0$ . At the end, we let $\alpha_j\beta_j$ be a relabeling of the bonds $\alpha^{\prime}_j \beta^{\prime}_j$ from left to right.

It is well known that the algorithm maps irreducible graphs to laces; moreover, the set of irreducible graphs that are mapped to a given lace L can be characterized as follows.

In other words, H is in the span of L if the above algorithm maps $\hat H$ to L. See Figure 3.

Given $\vec W$ and a lace L, we define

(4.9) \begin{equation} h_\lambda^\Lambda(\vec W;\, L) \,:\!=\, \sum_{m \geq 0} \frac{\lambda^m}{m!} \int_{\Lambda^m} \sum_{H \in \langle\!\langle L \rangle\!\rangle\,:\, \mathcal S(H) = \vec y_{[m]}} \textbf{w}(H) \, \textrm{d} \vec y_{[m]}. \end{equation}

The series $h_\lambda^\Lambda\big(\vec W;\,L\big)$ converges absolutely for every fixed $\lambda$ . This is shown as part of the proof of (4.8) in Proposition 4.1. Now,

\begin{equation*}h_\lambda^\Lambda\big(\vec W\big) = \sum_{L \in \mathcal L_k} h_\lambda^\Lambda(\vec W;\, L). \end{equation*}

The following characterization of compatible bonds will be useful. We recall that the bonds of a lace with m bonds can be labeled as $\alpha_j\beta_j$ with

\begin{equation*} 0 = \alpha_1 < \alpha_2 < \beta_1\leq \alpha_3 < \beta_2\leq \cdots \leq \alpha_m < \beta_{m-1}< \beta_m = k+1;\end{equation*}

see [Reference Slade22, Equations (3.15) and (3.16)].

To prove the second result of Proposition 4.1, we need the following counting lemma, which may be of independent interest.

Lemma 4.2. (On the number of laces.) Let $f_i$ be the ith Fibonacci number with $f_1 =0$ , $f_2=1$ . Then

\begin{equation*} |\mathcal L_k| = 1 + \sum_{i=1}^{k} \binom{k}{i} f_i \qquad and,\ as\ k \to\infty, \qquad |\mathcal L_k| \sim \frac{1}{\sqrt{5}} \bigg( \frac{3+\sqrt{5}}{2} \bigg)^k.\end{equation*}

Proof. We first choose i vertices in $\{1, \ldots, k\}$ and then count the laces that use exactly those vertices. To this end, let $A_i$ be the set of laces L with $V(L)=\{0, \ldots, i+1\}$ so that every vertex is the endpoint of at least one stitch. We claim that $|A_i|=f_i$ for $i \geq 1$ . Clearly, $|A_0|=1$ , $|A_1|=0$ , $|A_2|=1$ . See Figure 4 for an illustration.

Figure 4.

Schematic illustration of $A_i$ from the proof of Lemma 4.2 for $i=0,2,3,4,5,6$ .

Let $i \geq 3$ . We now establish the Fibonacci recursion. First, note that the bond incident to 0 (the ‘first’ bond) must always have 2 as the second endpoint. Now, depending on whether or not the third bond is incident to 2, the remaining lace lives on $\{1, 2, \ldots, i+1\}$ or on $\{1, 3, 4, \ldots, i+1\}$ , and so $|A_i| = |A_{i-1}| + |A_{i-2}|$ .

The asymptotic behavior follows from the fact that $f_n \sim \Phi^n /\sqrt{5}$ , where $\Phi = \tfrac 12 \big(1+\sqrt{5}\big)$ is the golden ratio.

We can now work towards finding an explicit expression for $h_\lambda^\Lambda\big(\vec W;\,L\big)$ for a fixed lace. The next lemma is in the spirit of Observation 3.1 and will help us find probabilistic factors in the shell function.

Proof. The first part of the statement is rather straightforward. If $Y=\{y\}$ , then $\mathcal G(A\cup C, \{y\})$ is the set of star graphs (with center y). Observe first that

\begin{equation*} \sum_{H \in\mathcal G(A \cup C, \{y\})\,:\, y \sim A} \textbf{w}(H) = \Bigg( \sum_{H^{\prime} \in\mathcal G(A, \{y\})\,:\, y \sim A} \textbf{w}(H^{\prime}) \Bigg) \Bigg( \sum_{H^{\prime\prime} \in\mathcal G(C, \{y\})} \textbf{w}\big(H^{\prime\prime}\big) \Bigg).\end{equation*}

The first sum is over all star graphs in $\mathcal G(A,\{y\})$ except the empty one, the second is over all star graphs in $\mathcal G(C,\{y\})$ , and so

\begin{equation*} \sum_{H \in\mathcal G(A \cup C, \{y\})\,:\, y \sim A} \textbf{w}(H) = - \bigg(1- \prod_{x \in A}(1- \varphi(y,x))\bigg) \prod_{x \in C}(1- \varphi(y,x)) = -\mathbb P(A \sim y \nsim C).\end{equation*}

It is an easy induction to prove that for general Y, the sum factors into a product over sums over star graphs. For the second statement, assume again that $Y = \{y\}$ and observe that

\begin{align*} \sum_{\substack{H \in\mathcal G(A \cup B \cup C, \{y\})\,:\, \\ A \sim y \sim B}} \textbf{w}(H) &= \bigg( \sum_{H \in\mathcal G(A \cup C, \{y\})\,:\, y \sim A} \textbf{w}(H) \bigg) \bigg( \sum_{H \in\mathcal G(B, \{y\})\,:\, y \sim B} \textbf{w}(H) \bigg) \\ &= \mathbb P(A \sim y \sim B, y \nsim C), \end{align*}

where the last identity is due to independence. The statement easily extends to general Y (again, the sum factors).

For the third statement, note that we sum over every graph except the empty one.

Since the explicit expression for $h_\lambda^\Lambda\big(\vec W;\,L\big)$ is a lengthy product of probabilities, we first introduce some notation to represent the factors of this product compactly. Let A, B be two subsets of $[k+1]_0$ . We define the set of all possible direct stitches in H leading to bonds $\alpha \beta\in E(\hat H)$ with $\alpha\in A$ , $\beta \in B$ as

\begin{equation*} \Upsilon(A,B) \,:\!=\, \big\{ xy \subset W\,:\,\ \exists \alpha \in A, \beta\in B \text{ with } \alpha<\beta-1 \text{ and } x\in \{u_\alpha\}{\cup}V_\alpha,\ y \in V_{\beta-1}{\cup} \{u_\beta\} \big\},\end{equation*}

and we write $\Upsilon(A) = \Upsilon(A,A)$ . We define

\begin{equation*} q_{\alpha,\beta} \,:\!=\, \prod_{xy \in \Upsilon([\alpha,\beta)) \cup \Upsilon((\alpha,\beta])} (1-\varphi(x-y))\end{equation*}

and, for $\alpha_1<\alpha_2<\alpha_3$ ,

\begin{equation*} q_{\alpha_1, \alpha_2, \alpha_3} \,:\!=\, \prod_{xy \in \Upsilon([\alpha_1+1,\alpha_2),[\alpha_2, \alpha_3))} (1-\varphi(x-y)).\end{equation*}

Note that these products encode the sum over all $\textbf{w}$ -weighted graphs on the set of edges multiplied over.

To lighten notation, for $0 \leq\alpha\leq \beta\leq k+1$ , set

\begin{gather*} [u_\alpha]\!] \,:\!=\,\{u_\alpha\}\cup V_\alpha,\qquad [\![u_\beta] \,:\!=\, V_{\beta-1} \cup \{u_\beta\},\\ [u_\alpha,u_\beta]\,:\!=\, \{u_\alpha\} \cup V_\alpha \cup \cdots \cup V_{\beta-1}\cup \{u_\beta\}.\end{gather*}

We extend this notation further: for $a,b \in \{u_0, \ldots, u_{k+1}\}$ , let $(a,b) \,:\!=\, [a,b] \setminus \{a, b\}$ , let $[a,b) \,:\!=\, [a,b] \setminus \{b\}$ , and let $(a,b]\,:\!=\, [a,b]\setminus\{a\}$ . We set $(\!(a,b)\!) \,:\!=\, [a,b] \setminus ([a]\!] \cup [\![b]) $ and define sets $(\!(a,b]$ etc. accordingly.

Moreover, define

\begin{equation*} Q_{\alpha,\beta} = \mathbb P_\lambda\big(\nexists y \in \eta_\Lambda \text{ s.t. } [u_\alpha]\!] \sim y \sim [\![u_\beta], y \nsim [u_{\alpha+1}, u_{\beta-1}] \big) \end{equation*}

for $\beta \geq \alpha+2$ . We extend this notation by writing

\begin{equation*} Q_{A,B} = \prod_{\alpha \in A} \prod_{\beta \in B} Q_{\alpha,\beta}\end{equation*}

for sets of pivotal points A, B; we abbreviate $Q_{a,[b,c]} = Q_{\{a\},[b,c]}$ .

We are now ready to state Lemma 4.4, for which we recall the definition of $h_\lambda^\Lambda\big(\vec W;\,L\big)$ in (4.9).

Lemma 4.4. (The shell function of a lace.) Let $\lambda \geq 0$ and let $\Lambda \subset \mathbb R^d$ be bounded. Let $W\subset \mathbb R^d$ be a core vertex set with pivot decomposition $\vec W=(u_0, V_0, \ldots, u_{k+1})$ . Let L be a lace with vertex set $[k+1]_0$ and m bonds $\alpha_i\beta_i$ , $i \in [m]$ . Then, setting $\alpha_{m+1}=k$ , we have

(4.10)

Moreover,

(4.11) \begin{equation} \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \Big| \sum_{H \in \langle\!\langle L \rangle\!\rangle:\mathcal S(H) = \vec y_{[n]}} {\textbf{w}}(H) \Big| \, \textrm{d} \vec y_{[n]} \leq 2^m {\text{e}}^{3\lambda|W|}. \end{equation}

Proof. We abbreviate $\eta=\eta_\Lambda$ , $h = h_\lambda^\Lambda$ , and prove the statement by induction on m.

Base case. Let $m=1$ , which means that $\alpha_1=0$ and $\beta_1=k+1$ . Set $A=[u_0]\!], B=[u_1, u_k]$ , and $C=[\![u_{k+1}]$ . See Figure 5 for an illustration of A, B, C.

Figure 5.

Illustration of the induction proof of Lemma 4.4. The lace L is sketched using dashed lines. The left picture shows the base case $m=1$ , where $u_0=s_1$ and $u_{k+1}=t_1$ . To the right, the first three stitches of L are (partially) sketched. The sets C, D are defined as $C = [s_2,t_1)\!)$ and $D=[\![t_1]$ .

Note first that the edge set $\Upsilon([k+1]_0) \setminus E(A,C)$ , that is, the possible direct stitches between points of W except the direct ones between A and C, do not determine membership of H in $\langle\!\langle L \rangle\!\rangle$ . Any such edge xy may or may not be present, resulting in a factor $(1-\varphi(x-y))$ that can be extracted. In total, this produces the factor $q_{0,k+1}$ , and we can restrict to considering graphs $H\in\langle\!\langle L \rangle\!\rangle$ that do not possess any such edge. The remaining graphs H only have edges that are incident to $A\cup C\cup \mathcal S(H)$ .

We split this set of remaining graphs H into those that have a direct stitch between A and C and those that do not. Among the former, the sum over graphs factors into graphs $H^{\prime} \in \mathcal G(A,C)$ (the direct stitches) and graphs $H^{\prime\prime} \in \mathcal G(W, \mathcal S(H))$ . With Lemma 4.3,

(4.12)

For now the power series are treated as formal power series; convergence is proven later. To treat the sum in (4.12), we define

\begin{equation*} \mathcal S_1 \,:\!=\, \{y\,:\, A \sim y \sim C\}, \quad \mathcal S_2\,:\!=\, \{y\,:\, C \nsim y \sim (A \cup B)\}, \quad \text{and } \mathcal S_3\,:\!=\, \{ y\,:\, C \sim y \nsim A \}. \end{equation*}

With these definitions, we can partition $\vec y = \mathcal S(H)= \mathcal S_1 \cup \mathcal S_2 \cup \mathcal S_3$ . Moreover, we know that $\mathcal S_1 \neq \varnothing$ . Re-summing and then applying Lemma 4.3, the sum over n in (4.12) becomes

(4.13) \begin{align}& \sum_{n_1,n_2,n_3 \geq 0} \frac{\lambda^{n_1+n_2+n_3}}{n_1!n_2!n_3!} \int_{\Lambda^{n_1+n_2+n_3}} \sum_{\substack{H \in \langle\!\langle L \rangle\!\rangle: \\ \mathcal S_i (H)=\vec y_ {i,[n_i]} \forall i \in [3]}} \textbf{w}(H) \, \textrm{d} \big(\vec y_{1,[n_1]}, \vec y_{2,[n_2]}, \vec y_{3,[n_3]} \big) \notag\\[-3pt] =& \Bigg( \sum_{n \geq 1} \frac{\lambda^n}{n!} \int_{\Lambda^n} \sum_{\substack{H \in \mathcal G(A \cup B \cup C,\vec y_{[n]}): \\ \forall i \in [n]: A \sim y_i \sim C }} \textbf{w}(H) \, \textrm{d} \vec y_{[n]} \Bigg) \Bigg( \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \sum_{\substack{H \in \mathcal G(A \cup B,\vec y_{[n]}): \\ \forall i \in [n]: y_i \sim (A \cup B) }} \textbf{w}(H) \, \textrm{d} \vec y_{[n]} \Bigg) \notag\\[-3pt] & \qquad \times \Bigg( \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \sum_{\substack{H \in \mathcal G(B \cup C,\vec y_{[n]}): \\ \forall i \in [n]: y_i \sim C }} \textbf{w}(H) \, \textrm{d} \vec y_{[n]} \Bigg) \notag\\[-3pt] =& \Bigg(\sum_{n \geq 1} \frac{\lambda^n}{n!} \bigg(\int_\Lambda \mathbb P(A \sim y \sim C, y \nsim B) \textrm{d} y \bigg)^n \Bigg) \Bigg(\sum_{n \geq 0} \frac{\lambda^n}{n!} \bigg({-}\int_\Lambda \mathbb P( y \sim (A \cup B)) \, \textrm{d} y \bigg)^n \Bigg) \notag\\[-3pt] & \qquad \times \Bigg(\sum_{n \geq 0} \frac{\lambda^n}{n!} \bigg({-}\int_\Lambda \mathbb P( C \sim y \nsim B) \, \textrm{d} y \bigg)^n \Bigg). \end{align}

Recognizing the exponential series in the expression above, we can rewrite the probabilities with respect to $\mathbb P$ (appearing in the exponents) as probabilities with respect to $\mathbb P_\lambda$ associated to $\xi^y$ , e.g., $\mathbb P( y \sim (A \cup B))= \mathbb P_\lambda( y \sim (A \cup B) \text{ in }\xi^y)$ . Then we can apply the univariate Mecke formula (see (2.1) for $m=1$ ) to rewrite (4.13) as

\begin{align*} & \Big( \text{e}^{\mathbb E_\lambda[|\{y \in \eta: A \sim y \sim C, y\nsim B \}|] } - 1\Big) \text{e}^{-\mathbb E_\lambda[|\{y \in \eta: y \sim (A \cup B) \}|]} \text{e}^{-\mathbb E_\lambda[|\{y \in \eta: C \sim y \nsim B \}|]} \\ =& \Big(1- \text{e}^{-\mathbb E_\lambda[|\{y \in \eta: A \sim y \sim C, y\nsim B \}|] } \Big) \text{e}^{-\mathbb E_\lambda[|\{y \in \eta: y \sim (A \cup B) \}|]} \text{e}^{-\mathbb E_\lambda[|\{y \in \eta: C \sim y \nsim (A \cup B) \}|]} \\ =& (1-Q_{0,k+1}) \text{e}^{-\mathbb E_\lambda[|\{y \in \eta: y \sim (A \cup B \cup C) \}|]}. \end{align*}

Since , we can plug this back into (4.12) and obtain

on the level of formal power series. Now we prove convergence and check that the previous computational steps are justified not only on the level of formal power series. We first revisit Equation (4.13). On the left-hand side, let us put absolute values inside the integral (but outside the sum over shell graphs H). The resulting expression is bounded by the middle part of (4.13), again with absolute values inside the integral. Each integrand is bounded in absolute value by a probability; hence it is smaller than or equal to 1. The resulting series are exponential series and, in particular, absolutely convergent. As a consequence, Equation (4.13) is justified and the last sum in (4.12) is bounded as

(4.14) \begin{align} \sum_{n \geq 0} \frac{\lambda^n}{n!} &\int_{\Lambda^n} \left| \sum_{\substack{H \in \mathcal G(W,\vec y_{[n]}): \\ \deg(y_i) \geq 1 \forall i \in[n], \\ \exists i: A \sim y_i \sim C}} \textbf{w}(H) \right| \, \textrm{d} \vec y_{[n]} \notag \\ & \leq \text{e}^{\mathbb E_\lambda[|\{y \in \eta: A \sim y \sim C, y\nsim B \}|] } \text{e}^{\mathbb E_\lambda[|\{y \in \eta: y \sim (A \cup B) \}|]} \text{e}^{\mathbb E_\lambda[|\{y \in \eta: C \sim y \nsim B \}|]} \notag\\ &\leq \text{e}^{\mathbb E_\lambda[|\{y \in \eta: y \sim A\}|] } \text{e}^{\mathbb E_\lambda[|\{y \in \eta: y \sim (A \cup B) \}|]} \text{e}^{\mathbb E_\lambda[|\{y \in \eta: y \sim C\}|]} \notag\\ & \leq \text{e}^{2 \lambda|W|}, \end{align}

where for the last inequality we use the fact that the expected number of direct neighbors of any fixed element of W with respect to $\eta$ is given by $\lambda\int\varphi(x)\, \textrm{d} x$ , as well as the rescaling introduced in Section 2.3 ensuring that $\int\varphi(x)\, \textrm{d} x=1$ ; compare this bound to the one used in (3.4). For the other contribution to $h\big(\vec W;\,L\big)$ , we notice that

(4.15) \begin{align} & \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n}\left|\Bigg(\sum_{H^{\prime} \in \mathcal G(A,C): E(H) \neq \varnothing} \textbf{w}(H^{\prime}) \Bigg) \left(\sum_{\substack{H^{\prime\prime} \in \mathcal G(W,\vec y_{[n]}): \\ y_i \sim W \forall i \in[n]}} \textbf{w}\big(H^{\prime\prime}\big) \right) \right| \, \textrm{d} \vec y_{[n]} \notag\\ &\qquad \leq \mathbb P(A\sim C)\, \text{e}^{ \mathbb E_\lambda[|\{ y \in \eta: y \sim W \}|]} \leq \text{e}^{\lambda|W|}, \end{align}

by the same argument as in (4.14).

Combining (4.14) and (4.15) with (4.12) and $0\leq q_{0,k+1} \leq 1$ , we deduce

\begin{equation*} \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \left| \sum_{H \in \langle\!\langle L \rangle\!\rangle: \mathcal S(H) = \vec y_{[n]}} \textbf{w}(H)\right| \, \textrm{d} \vec y_{[n]} \leq \text{e}^{\lambda|W|} + \text{e}^{ 2\lambda|W|} \leq 2\text{e}^{ 2\lambda|W|} < \infty. \end{equation*}

Inductive step. For the inductive step, let $m>1$ . We write the lace L in terms of its vertices $(s_i,t_i)$ in W (that is $s_i=u_{\alpha_i}$ and $t_i=u_{\beta_i}$ ) and let L ^′ be the lace on $W^{\prime} \,:\!=\, W \setminus [s_1, s_2)$ obtained from L by deleting the first stitch. We note that if $H \in \langle\!\langle L \rangle\!\rangle$ , then $H[[s_2,u_{k+1}]]\in \langle\!\langle L^{\prime} \rangle\!\rangle$ . Observe that

(4.16) \begin{equation} h\big(\vec W;\,L\big) = h\big(\vec {W^{\prime}};\,L^{\prime}\big) \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \sum_{\substack{H \in \mathcal G(\overline V_0, \ldots, \overline V_{\alpha_3-1}, \vec y_{[n]})\,:\, \\ H \oplus L^{\prime} \in \langle\!\langle L \rangle\!\rangle}} \textbf{w}(H) \, \textrm{d} \vec y_{[n]}. \end{equation}

Again we first prove (4.10) and carry out computations on the level of formal power series; we prove convergence (and thus (4.11)) at the end. We can apply the induction hypothesis to $h(\vec {W^{\prime}};\,L)$ ; it remains to deal with the second factor. We partition the vertices in $[s_1, s_3]$ as $A = [s_1]\!]$ , $B=(\!(s_1,s_2)$ , $C=[s_2,t_1)\!)$ , $D = [\![t_1]$ , and $E= (t_1,s_3]$ (see Figure 5). If $m=2$ , we let $E=(t_1,u_k]$ .

The graphs summed over in (4.16) must satisfy the following restraints: there must be at least one direct or indirect stitch between A and D, and there cannot be any (direct or indirect) edge between A and E. In particular, the remaining direct stitches may or may not be there, and thus can be extracted as the factor $q_{\alpha_1, \alpha_2, \alpha_3}$ .

We partition $\mathcal S(H) = \cup_{i=1}^4 \mathcal S_i$ , where

\begin{align*}\mathcal S_1 &= \{y\,:\, A \sim y \sim D, N(y) \subseteq [s_1,t_1] \}, \qquad \mathcal S_2 = \{y\,:\, A \sim y \sim C, N(y) \subseteq [s_1, t_1)\!) \}, \\ \mathcal S_3 &= \{y\,:\, \varnothing \neq N(y) \subseteq [s_1, s_2)\}, \qquad \mathcal S_4 = \{y\,:\, B \sim y \sim (C \cup D \cup E), N(y) \subseteq (\!(s_1, s_3]\}.\end{align*}

Again, we intend to split the sum over graphs into those that have at least one direct stitch between A and D, and those that do not. We can thus rewrite the second factor in (4.16) as

(4.17) \begin{align} & q_{\alpha_1, \alpha_2, \alpha_3} \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \sum_{\substack{H \in \mathcal G(\overline V_0, \ldots, \overline V_{\alpha_3-1}, \vec y_{[n]}): \\ H \oplus L^{\prime} \in \langle\!\langle L \rangle\!\rangle, \notag\\ \forall e\in E(H): e \cap (A\cup D \cup \vec y_{[n]}) \neq \varnothing}} \textbf{w}(H) \, \textrm{d} \vec y_{[n]}\\ & \quad = q_{\alpha_1, \alpha_2, \alpha_3} \prod_{i=2}^{4}\Bigg( \sum_{n_i \geq 0} \frac{\lambda^{n_i}}{n_i!} \int_{\Lambda^{n_i}} \sum_{\substack{H \in \mathcal G(W, \vec y_{i,[n_i]}): \\ \mathcal S(H) = \mathcal S_i}} \textbf{w}(H) \, \textrm{d} \vec y_{[i,[n_i]} \Bigg) \notag\\ & \qquad \times \Bigg[-\mathbb P(A\sim D) \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n}\sum_{\substack{H \in \mathcal G([s_1,s_3],\vec y_{[n]}): \\ y_i \sim A\cup B \forall i \in[n]}} \textbf{w}(H) \, \textrm{d} \vec y_{[n]} \notag\\ &\qquad\quad \qquad +\sum_{n \geq 1} \frac{\lambda^n}{n!} \int_{\Lambda^n}\sum_{\substack{H \in \mathcal G([s_1,s_3],\vec y_{[n]}): \\ y_i \sim A\cup B \forall i \in[n]}} \textbf{w}(H) \, \textrm{d} \vec y_{[n]} \Bigg] \notag\\ &\qquad = q_{\alpha_1, \alpha_2, \alpha_3} \Big({-}\mathbb P(A\sim D) \text{e}^{\mathbb E_\lambda[|\{y \in \eta: y \in \mathcal S_1\}|]} + \text{e}^{\mathbb E_\lambda[|\{y \in \eta: y \in \mathcal S_1\}|]}-1 \Big) \notag\\ & \quad \qquad \times \exp\!\Big\{\mathbb E_\lambda[|\{y \in \eta: y \in \mathcal S_2\}|] - \mathbb E_\lambda[|\{y \in \eta: y \in \mathcal S_3\}|] + \mathbb E_\lambda[|\{y \in \eta: y \in \mathcal S_4\}|]\Big\}, \end{align}

where the last identity was obtained using Lemma 4.3. Note that the factor $h\big(\vec{W^{\prime}};\,L^{\prime}\big)$ contains the factor . Together with this factor, (4.17) equals

(4.18)

It remains to rewrite the argument in the expectation of the exponent in (4.18). Note that

\begin{align*} &- |\{y \in \eta\,:\, A \sim y \sim [s_2,u_{k+1}], y\nsim B \}| - |\{y \in \eta\,:\, B \sim y \sim [s_2,u_{k+1}] \}| \\ &+ |\{y \in \eta\,:\, A \sim y \sim (C \cup D), y \nsim B\}| + |\{y \in \eta\,:\, B \sim y \sim (C \cup D \cup E) \}| \\ =& - |\{y \in \eta\,:\, A \sim y \sim (t_1,u_{k+1}], y\nsim (B \cup C \cup D) \}| \\ &- |\{y \in \eta\,:\, B \sim y \sim (s_3,u_{k+1}], y \nsim (C \cup D \cup E) \}|. \end{align*}

This gives two exponential terms. The first is

\begin{align*} &\exp\!\Big\{-\mathbb E_\lambda\big[\big|\big\{y \in \eta\,:\, [u_{\alpha_1}]\!] \sim y \sim (u_{\beta_1},u_{k+1}], y \nsim \big(\!\big(u_{\alpha_1},u_{\beta_1}\big]\big\}\big|\big]\Big\}\\ = \, & \prod_{j=\beta_1+1}^{k+1} \exp\!\Big\{{-\mathbb E_\lambda\big[\big|\big\{y \in \eta\,:\, [u_{\alpha_1}]\!] \sim y \sim [\![u_j], y \nsim \big(\!\big(u_{\alpha_1},u_j\big)\!\big)\big\}\big|\big]}\Big\} \\ = \, & Q_{\alpha_1,(\beta_1,k+1]}.\end{align*}

Similarly, the second exponential term equals $Q_{(\alpha_1,\alpha_2), (\alpha_3,k+1]}$ .

Again, we prove convergence and justify the previous computational steps. Revisiting the left-hand side of (4.17), we insert absolute values inside the integral (and outside the sum over graphs H). As in the base case, this is bounded by the middle part of (4.17) with absolute values in the integrals, and each integrand is a probability. With the Mecke equation, we obtain

\begin{align*} &\sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^n} \bigg| \sum_{\substack{H \in \mathcal G(\overline V_0, \ldots, \overline V_{\alpha_3-1}, \vec y_{[n]}):\, \\ H \oplus L^{\prime} \in \langle\!\langle L \rangle\!\rangle, \notag\\ \forall e\in E(H): e \cap (A\cup D \cup \vec y_{[n]}) \neq \varnothing}} \textbf{w}(H) \bigg| \, \textrm{d} \vec y_{[n]} \notag \\ & \leq 2 \exp\!\Big\{\mathbb E_\lambda[|\{y \in \eta\,:\, y \in \mathcal S_1\}|] + \mathbb E_\lambda[|\{y \in \eta\,:\, y \in \mathcal S_2\}|] \\ & \qquad \qquad + \mathbb E_\lambda[|\{y \in \eta\,:\, y \in \mathcal S_3\}|] + \mathbb E_\lambda[|\{y \in \eta\,:\, y \in \mathcal S_4\}|]\Big\} \notag \\ & \leq 2 \text{e}^{3\lambda |A \cup B|},\end{align*}

arguing as in (4.14) for the last inequality.

Note that by the induction hypothesis, the term $h\big(\vec{W^{\prime}};\,L^{\prime}\big)$ with absolute values in the respective integrals is bounded by $2^{m-1} \text{e}^{3 \lambda|W^{\prime}|}$ . Since $A\cup B$ and W ^′ are disjoint, this proves (4.11).

Proof of Proposition 4.1. Again, we abbreviate $\eta=\eta_\Lambda$ and $h=h_\lambda^\Lambda$ . First, consider $k=0$ , i.e., pivot decompositions with no pivotal points. Then there are no direct stitches, and we have

Moreover,

\begin{equation*} \sum_{m \geq 0} \frac{\lambda^m}{m!} \int_{\Lambda^m} | h^{(m)} \big(\vec W, \vec y_{[m]}\big) | \, \textrm{d}\vec y_{[m]} = \text{e}^{\mathbb E_\lambda[|\{ y \in \eta:y \sim W\}|]} \leq \text{e}^{\lambda|W|}, \end{equation*}

using the same bound as in (4.15). Since this proves the proposition for $k=0$ , we turn to $k\geq 1$ and we first prove (4.7).

We rewrite $h\big(\vec W\big)$ by explicitly writing out the sum over laces L in terms of the endpoints of their stitches in W (note that any lace can have at most k stitches). We first exhibit this for $k=2$ , where $\vec W=(u_0,V_0,u_1,V_1,u_2,V_2,u_3)$ and there are two different laces. With the abbreviation $\tilde Q_{i,j} = Q_{i,j} + \mathbb P([u_i]\!] \sim [\![u_j])$ ,

(4.19)

Clearly, this is unnecessarily complicated for $k=2$ , as the sum over $\alpha_2$ contains only one term and $q_{0,2}=1$ . However, this turns out to be convenient for general k. We use the convention that $Q_{[a,b],\varnothing} = Q_{\varnothing, [a,b]} =1$ . Carefully rearranging the sum over all laces yields

Note that if $\beta=k+1$ for some i, then the double sum following the corresponding indicator breaks down to 0. Also, only the innermost bracketed term contains two factors of $q_{a,b,c}$ .

We now show that, starting with the innermost square brackets, the bracketed terms are bounded by 1 in absolute value.

To lighten notation, we write the innermost sum as $\sum_{\alpha=b_1}^{b_2-1} R(\alpha)$ . We split the factor

\begin{equation*}1-\tilde Q_{\alpha_k,k+1} = (1-Q_{\alpha_k,k+1}) - \mathbb P([u_{\alpha_k}\!] \sim [\![ u_{k+1}]).\end{equation*}

This yields two sums

\begin{equation*}\sum_{\alpha=b_1}^{b_2-1} R(\alpha) = \sum_{\alpha=b_1}^{b_2-1} R^{\prime}(\alpha) - \sum_{\alpha=b_1}^{b_2-1} R^{\prime\prime}(\alpha),\end{equation*}

where R ^′ and R ^′′ are both nonnegative. Now, with the estimate $Q_{(\alpha_{k-1},\alpha),k+1} \leq Q_{[\beta_{k-2},\alpha),k+1} = Q_{[b_1,\alpha),k+1}$ , we can bound

(4.20) \begin{align} \sum_{\alpha=b_1}^{b_2-1} R^{\prime}(\alpha) &\leq \sum_{\alpha=b_1}^{b_2-1} (1- Q_{\alpha,k+1}) Q_{[b_1,\alpha),k+1} \notag \\ &= (1-Q_{b_1,k+1}) + Q_{b_1,k+1} \sum_{\alpha=b_1+1}^{b_2-1} (1- Q_{\alpha,k+1}) Q_{[b_1+1,\alpha),k+1}, \end{align}

which is readily proven to be at most 1 by induction. Moreover,

(4.21) \begin{equation} \sum_{\alpha=b_1}^{b_2-1} R^{\prime\prime}(\alpha) \leq \sum_{\alpha=b_1}^{b_2-1} q_{\alpha,k+1} \mathbb P([u_\alpha]\!] \sim [\![ u_{k+1}]). \end{equation}

The above summands can be rewritten as the probability of the event that the direct stitch $(\alpha,k+1)$ is present, while all direct stitches $(j,k+1)$ for $j\in (\alpha,k+1]$ are not. Hence, these are disjoint events for different values of $\alpha$ , and so the sum is at most 1.

In total, we rewrote $\sum_{\alpha=b_1}^{b_2-1} R(\alpha)$ as the difference of two nonnegative values, both at most 1, proving our claim.

To deal with the summands for $2 \leq i <k$ , we write the double sum as

\begin{equation*}\sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} R(\alpha, \beta)\end{equation*}

and split the term $1-\tilde Q_{\alpha_i,\beta_i} = (1- Q_{\alpha_i,\beta_i}) - \mathbb P([u_{\alpha_i}\!] \sim [\![ u_{\beta_i}])$ so that

(4.22) \begin{equation} \sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} R(\alpha, \beta) = \sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} R^{\prime}(\alpha, \beta) - \sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} R^{\prime\prime}(\alpha, \beta) \end{equation}

for nonnegative summands R ^′, R ^′′. We prove a bound on the sum over $R^{\prime}(\alpha,\beta)$ by induction on $k-b_2$ . If $b_2=k$ , then the bound is the same as for the bound (4.20). For $b_2<k$ , we first bound $Q_{(\alpha_i,\alpha],(\beta,k+1]} \leq Q_{[b_1,\alpha],(\beta,k+1]}$ and then extract the summand for $\beta=k+1$ , yielding

(4.23) \begin{align}\sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} R^{\prime}(\alpha, \beta) & \leq \sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} (1-Q_{\alpha,\beta}) Q_{[b_1,\alpha],(\beta,k+1]} Q_{[b_1,\alpha),\beta} \notag\\ &\leq \sum_{\alpha=b_1}^{b_2-1} (1-Q_{\alpha,k+1}) Q_{[b_1,\alpha),k+1} \notag\\ & \qquad + Q_{[b_1,b_2-1],k+1} \sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k} (1-Q_{\alpha,\beta}) Q_{[b_1,\alpha],(\beta,k]} Q_{[b_1,\alpha),\beta}. \end{align}

By the induction hypothesis, the double sum in (4.23) is at most 1. Therefore,

\begin{align*}\sum_{\alpha=b_1}^{b_2-1} \sum_{\beta = b_2+1}^{k+1} R^{\prime}(\alpha, \beta) & \leq \sum_{\alpha=b_1}^{b_2-2} (1-Q_{\alpha,k+1}) Q_{[b_1,\alpha),k+1} \\ & \qquad + (1-Q_{b_2-1,k+1}) Q_{[b_1,b_2-1),k+1} + Q_{b_2-1,k+1} Q_{[b_1,b_2-1),k+1} \\ &= \sum_{\alpha=b_1}^{b_2-2} (1-Q_{\alpha,k+1}) Q_{[b_1,\alpha),k+1} + Q_{[b_1,b_2-2],k+1} \\ &= 1, \end{align*}

where the last identity is now an easy induction.

Turning to the second summand in (4.22), by the same argument used to treat (4.21), the summands $R^{\prime\prime}(\alpha,\beta)$ are probabilities of events which are disjoint for different values of $(\alpha,\beta)$ , and so they sum to at most 1.

The observation that the bracket term for $i=1$ is handled analogously finishes the proof of (4.7).

We proceed to prove (4.8) for $k>1$ . By combining Lemma 4.2 with Lemma 4.4, we obtain

\begin{align*} \sum_{m\geq 0} \frac{\lambda^m}{m!} \int_{\Lambda^m} \big| h^{(m)}\big(\vec W,\vec y_{[m]}\big) \big| \, \textrm{d}\vec y_{[m]} & \leq \sum_{L \in \mathcal L_k}\sum_{m \geq 0} \frac{\lambda^m}{m!} \int_{\Lambda^m} \Big| \sum_{H \in \langle\!\langle L \rangle\!\rangle: \mathcal S(H) = \vec y_{[m]}} {\textbf{w}}(H) \Big| \, \textrm{d} \vec y_{[m]} \\ & \leq \frac{1}{\sqrt{5}} \bigg( \frac{3+\sqrt{5}}{2} \bigg)^{k} 2^k \text{e}^{3\lambda|W|}. \end{align*}

Using the bound $k \leq |W|$ finishes the proof.

Lemma 4.5. (Thermodynamic limit of the shell function.) For every $\lambda \geq 0$ , the pointwise limit

\begin{equation*} \lim_{\Lambda \nearrow \mathbb R^d} h_\lambda^\Lambda\big(\vec W\big) = h_\lambda\big(\vec W\big) \end{equation*}

along $\mathbb R^d$ -exhausting sequences exists.

Proof. Let $(\Lambda_n)_{n\in\mathbb N}$ be an $\mathbb R^d$ -exhausting sequence. For fixed $\vec W=(u_0, V_0, \ldots, u_{k+1})$ , note that

\begin{equation*} h_\lambda^{\Lambda_n}\big(\vec W\big) = \sum_{L\in\mathcal L_k} h_\lambda^{\Lambda_n}\big(\vec W;\,L\big). \end{equation*}

For each lace L, the limit

\begin{equation*} h_\lambda\big(\vec W;\,L\big) = \lim_{n\to \infty} h_\lambda^{\Lambda_n}\big(\vec W;\,L\big) \end{equation*}

exists and does not depend on the precise choice of $\mathbb R^d$ -exhausting sequence. This is clear from the representation for $h_\lambda^\Lambda\big(\vec W;\,L\big)$ proven in Lemma 4.4. In particular, $h_\lambda^\Lambda\big(\vec W;\,L\big)$ is given as the finite product of $\Lambda$ -independent factors and factors that describe the probability of certain point processes containing no points (namely, and the factors $Q_{i,j}$ ). As probabilities that are decreasing in the volume, the latter admit a $\Lambda \nearrow \mathbb R^d$ limit. It follows that the limit of the shell function exists as well and is given by

\begin{equation*} h_\lambda\big(\vec W\big) = \sum_{L\in\mathcal L_k} h_\lambda\big(\vec W;\,L\big). \end{equation*}

4.4. The direct-connectedness function in infinite volume

In this section, we consider the limit $\lim_{\Lambda \nearrow \mathbb R^d} g_\lambda^{\Lambda}$ with $g_\lambda^\Lambda$ as in (4.6) and give sufficient conditions under which it exists, thereby proving the two convergence statements from Theorem 1.1.

The candidate limit is given by the analogue of (4.6) with $\Lambda$ replaced by $\mathbb R^d$ ; the existence of $h_\lambda^{\mathbb R^d}\equiv h_\lambda$ has been checked in Lemma 4.5. Thus,

(4.24) \begin{equation} g_\lambda(x_1,x_2) = \sum_{r=0}^\infty \frac{\lambda^r}{r!} \int_{(\mathbb R^d)^r} \sum_{\vec W} \left( \sum_{G\in\mathcal G^{\vec W}_{\text{core}}} \textbf{w}^\pm(G)\right) h_\lambda\big(\vec W\big) \, \textrm{d} \vec x_{[3,r+2]}, \end{equation}

where the inner sum is over core graphs G on $\vec x_{[r+2]}$ with pivot decomposition $\vec W$ , i.e., over $({+})$ -connected graphs G on $\vec x_{[r+2]}$ with $\textsf{PD}^+(x_1,x_2,G)= \textsf{PD}^\pm(x_1,x_2,G)=\vec W$ . Remember the quantities $0<\tilde \lambda_*\leq \lambda_*$ introduced before Theorem 1.1. We will see in (4.25) that the sum over core graphs for a given pivot decomposition is a probability, hence in particular nonnegative.

Theorem 4.1. (The thermodynamic limit of $g_\lambda^\Lambda$ : pointwise convergence.) If $\lambda< \lambda_\ast$ , then

\begin{equation*} \sum_{r=0}^\infty \frac{\lambda^r}{r!} \int_{(\mathbb R^d)^r} \sum_{\vec W} \left( \sum_{G\in\mathcal G^{\vec W}_{\textrm{core}}} {\textbf{w}}^\pm(G)\right) \bigl| h_\lambda\big(\vec W\big)\bigr| \, \textrm{d} \vec x_{[3,r+2]}<\infty \end{equation*}

for all $x_1,x_2\in \mathbb R^d$ . Moreover, for every $\mathbb R^d$ -exhausting sequence $(\Lambda_n)_{n\in \mathbb N}$ , we have the pointwise convergence

\begin{equation*} \lim_{n\to \infty} g_\lambda^{\Lambda_n} (x_1,x_2) = g_\lambda(x_1,x_2) \end{equation*}

with $g_\lambda$ given in (4.24) (equivalently, Equation (4.6) with $\Lambda$ replaced by $\mathbb R^d$ ).

Theorem 4.2. (Integrability and convergence in the $L^1$ -norm.) If $\lambda< \tilde \lambda_\ast$ , then for all $x_1\in \mathbb R^d$ ,

\begin{equation*} \int_{\mathbb R^d} |g_\lambda(x_1,x_2)|\, \textrm{d} x_2\leq \sum_{r=0}^\infty\frac{\lambda^r}{r!} \int_{\mathbb R^d}\Bigg( \int_{(\mathbb R^d)^r} \sum_{\vec W} \Bigg( \sum_{G\in\mathcal G^{\vec W}_{\textrm{core}}} {\textbf{w}}^\pm(G)\Bigg) \bigl|h_\lambda\big(\vec W\big)\bigr| \, \textrm{d} \vec x_{[3,r+2]} \Bigg) \, \textrm{d} x_2 <\infty. \end{equation*}

Proof of Theorem 4.1. We consider a summand in (4.24) for fixed $\vec W$ and set $x_1=u_0$ as well as $x_2=u_{k+1}$ . Let $\vec W=(u_0,V_0, \ldots, V_k, u_{k+1})$ . Remember $\overline{V}_i = \{u_{i}\}\cup V_i\cup \{u_{i+1}\}$ . A first important observation is the fact that the weight of a core graph with pivot decomposition $\vec W$ factors into the product over the k $({{\pm}})$ -subgraphs induced by the vertex sets $\overline{V}_i$ . The sum over core graphs thus factors as

(4.25) \begin{align} \sum_{\substack{G \in \mathcal C^+(W)\,:\, \\ \textsf{PD}^+(G)= \textsf{PD}^\pm(G)=\vec W}} \textbf{w}^\pm(G) &= \prod_{i=0}^{k} \Bigg( \sum_{H \in \mathcal D^+_{u_i,u_{i+1}}(\overline V_i)} \textbf{w}^\pm(H) \Bigg) \notag\\ &= \prod_{i=0}^{k} \mathbb P \Big(\Gamma_\varphi(\overline V_i) \in \mathcal D_{u_i, u_{i+1}} \Big) \notag\\ &= \mathbb P \Bigg(\bigcap_{i=0}^k \{\Gamma_\varphi(\overline V_i) \in \mathcal D_{u_i, u_{i+1}} \} \Bigg). \end{align}

Hence, the core can be written as a probability. Combining this with Proposition 4.1, we get

Above, we used independence as well as the fact that for $V \subseteq W$ , the two random graphs $\Gamma_\varphi(V)$ and $\xi^W[V]$ are identical in distribution. The inequality holds true for bounded $\Lambda$ as well as $\Lambda = \mathbb R^d$ .

We now go back to (4.6) and rearrange the sum by first summing over the number of pivotal points k, giving

(4.26) \begin{align} & \sum_{r=0}^\infty \frac{\lambda^r}{r!} \int_{\Lambda^r} \sum_{\vec W} \Bigg( \sum_{G\in\mathcal G^{\vec W}_{\text{core}}} {\textbf{w}}^\pm(G)\Bigg) \bigl| h_\lambda^\Lambda\big(\vec W\big)\bigr| \, \textrm{d} \vec x_{[3,r+2]} \notag \\[4pt] & = \sum_{k \geq 0} \lambda^k \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^{k+n}} \sum_{\vec W} \Bigg( \sum_{G\in\mathcal G^{\vec W}_{\text{core}}} \textbf{w}^\pm (G) \Bigg) \big|h_\lambda^\Lambda\big(\vec W\big)\big| \, \textrm{d} \vec v_{[n]} \, \textrm{d}\vec u_{[k]}. \end{align}

In the second term, the sum is over pivot decompositions $\vec W=(u_0, V_0, \ldots, V_k, u_{k+1})$ where $u_0=x_1$ , $u_{k+1} =x_2$ , and $\cup_{i=0}^k V_i = \{v_1, \ldots, v_n\}$ .

When rewriting the integrand of (4.26) as a probability, the event that $u_i$ and $u_{i+1}$ are 2-connected for $i\in [k]_0$ in disjoint vertex sets $V_i$ becomes the event that these connection events occur disjointly within W; see Section 2 and recall the definition (2.2). The inner series can thus be bounded as

(4.27) \begin{align} & \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^{n}} \sum_{\vec W} \Bigg( \sum_{G\in\mathcal G^{\vec W}_{\text{core}}} \textbf{w}^\pm (G) \Bigg) |h_\lambda^\Lambda\big(\vec W\big)| \, \textrm{d} \vec v_{[n]}\notag \\ \leq & \sum_{n \geq 0} \frac{\lambda^{n}}{n!} \int_{\Lambda^{n}} \mathbb P_\lambda \bigg( \Big\{\mathscr {C}\Big(u_0, \xi_\Lambda^{\vec u_{[k]}, \vec v_{[n]}}\Big) = \vec u_{[k]} \cup \vec v_{[n]} \Big\} \notag \\ & \qquad\qquad\cap\Big( \big\{u_0 \Longleftrightarrow u_1\textrm { in } \xi^{u_0,u_1,\vec v_{[n]}}\big\} \circ \cdots \circ \big\{u_k \Longleftrightarrow u_{k+1}\textrm { in } \xi^{u_k,u_{k+1},\vec v_{[n]}} \big\} \Big) \Bigg) \, \textrm{d} \vec v_{[n]} \notag \\ = \, & \mathbb P_\lambda\big( \big\{u_0 \Longleftrightarrow u_1\textrm { in } \xi^{u_0,u_1}\big\} \circ \cdots \circ \big\{u_k \Longleftrightarrow u_{k+1}\textrm { in } \xi^{u_k,u_{k+1}} \big\}\big), \end{align}

where the identity is due to the Mecke equation and the fact that by summing over $\vec v$ , we were partitioning over what the joint cluster of $\vec u_{[0,k+1]}$ is. We can now use the BK inequality ([Reference Heydenreich, van der Hofstad, Last and Matzke10, Theorem 2.1]) to bound (4.27) by

(4.28) \begin{equation} \prod_{i=0}^{k} \mathbb P_\lambda \Big(u_i \Longleftrightarrow u_{i+1}\textrm { in } \xi_\Lambda^{u_i,u_{i+1}} \Big) \leq \prod_{i=0}^{k} \sigma_\lambda(u_{i+1}-u_i). \end{equation}

Inserting this back into (4.26),

(4.29) \begin{align} &\sum_{k \geq 0} \lambda^k \sum_{n \geq 0} \frac{\lambda^n}{n!} \int_{\Lambda^{k+n}} \sum_{\vec W} \Bigg( \sum_{G\in\mathcal G^{\vec W}_{\text{core}}} \textbf{w}^\pm (G) \Bigg) \big|h_\lambda^\Lambda\big(\vec W\big)\big| \, \textrm{d} \vec v_{[n]} \, \textrm{d}\vec u_{[k]} \notag\\ \leq & \sum_{k \geq 0} \lambda^k \sigma_\lambda^{\ast (k+1)}(x_2-x_1). \end{align}

The last expression is finite for $\lambda<\lambda_\ast$ , by the definition of $\lambda_\ast$ . The pointwise convergence of $g_\lambda^{\Lambda_n}$ to $g_\lambda$ follows by dominated convergence.

Proof of Theorem 4.2. If we integrate over $x_2$ in (4.29), this yields the upper bound

\begin{equation*} \lambda^{-1} \sum_{k \geq 1} \bigg( \lambda \int\sigma_\lambda(x) \, \textrm{d} x \bigg)^k, \end{equation*}

which is finite for $\lambda<\tilde\lambda_\ast$ , by definition of $\tilde\lambda_\ast$ . The theorem follows by Fubini–Tonelli and the triangle inequality.