5.1. Proof of Theorem 1

In this section, we carry out the proof under Condition 1 of the theorem. The proofs under Conditions 2 and 3 are rather easy extensions of this proof that use some additional arguments [Reference Hirsch, Jahnel and Cali16, Reference Tóbiás28], which are omitted from our paper but included in the extended online version [Reference Jahnel and Tóbiás19].

Assume for the rest of this section that Condition 1 holds. For fixed $\lambda$ and $\gamma$ , in order to show that ${G_{\gamma}({\textbf{X}}^\lambda)}$ percolates, it suffices to verify that a subgraph of it contains an infinite cluster. Our proof consists of four steps. First, for $\gamma,\lambda>0$ , we define a subgraph that is included in a Cox–Gilbert graph (with constant connection radii). Second, we map this subgraph to a lattice percolation model and show that this discrete model percolates for large $\lambda$ for a suitable choice of auxiliary parameters. In particular, since $\Lambda$ is assumed only to be stabilizing, the connection radius of the Gilbert graph must be large enough so that the graph percolates for large $\lambda$ . In this step, we are able to employ multiple arguments of [Reference Dousse9, Reference Hirsch, Jahnel and Cali16, Reference Tóbiás28]. Our interference-control assertion, Proposition 5.1, is presented here. Third, using the subgraph, we make a choice of $\gamma>0$ such that percolation in the discrete model implies percolation in the SINR graph ${G_{\gamma}({\textbf{X}}^\lambda)}$ , which is done analogously to [Reference Dousse9]. Fourth, we carry out the proof of Proposition 5.1, combining arguments of [Reference Dousse9, Reference Tóbiás28] for SINR graphs with constant powers and arguments used in [Reference Löffler22] for Poisson SINR graphs with random powers.

STEP 1. A subgraph of the SINR graph.

We first present a general construction of a subgraph of ${G_{\gamma}({\textbf{X}}^\lambda)}$ for $\gamma,\lambda>0$ . Let $r_o>d_o$ . Since both $P_o$ and $\mathrm{supp}(\ell)$ are unbounded, we have

\begin{equation*} p(r_o): = \mathbb{P}\big( \ell^{-1}\big(\tau {N_o}/P_o\big) \geq r_o \big) = \mathbb{P}\Big( P_o \geq\tau {N_o}/\ell(r_o) \Big) >0. \end{equation*}

Let us define the independent thinning

\begin{equation*} X^{\lambda,-} := \{ x_i \in X^\lambda \colon P_o \geq \tau {N_o}/\ell(r_o) \} \end{equation*}

of $X^\lambda$ with survival probability $p(r_o)$ . According to [Reference Kingman20, Colouring Theorem], $X^{\lambda,-}$ is a CPP with directing measure $\lambda p(r_o)\Lambda$ . Now, let us define a subgraph $G^-_{\gamma}({\textbf{X}}^\lambda)$ of ${G_{\gamma}({\textbf{X}}^\lambda)}$ as follows. The vertex set is $X^{\lambda,-}$ , and two vertices $x_i,x_j \in X^{\lambda,-}$ , $x_i \neq x_j$ , are connected by an edge if and only if

(5.2) \begin{equation} \mathrm{SINR}^{-}(x_i,x_j,{\textbf{X}}^\lambda): = \frac{\big(\tau {N_o}/\ell(r_o)\big) \ell(|x_i-x_j|)}{{N_o}+\gamma \sum_{k \in \mathbb{N} \setminus \{ i, j\}} P_k \ell(|x_k-x_j|)} > \tau \end{equation}

and the analogously defined $\mathrm{SINR}^-(x_j,x_i,{\textbf{X}}^\lambda)$ also exceeds $\tau$ . Note that for $x_i,x_j \in X^{\lambda,-}$ , in the numerator of $\mathrm{SINR}(x_i,x_j,{\textbf{X}}^\lambda)$ , for the power of $x_i$ we have $P_i \geq \tau {N_o}/\ell(r_o)$ , whereas the denominators of (5.1) and (5.2) are equal, and the same holds with the roles of i and j interchanged. Hence, $G^-_{\gamma}({\textbf{X}}^\lambda)$ is indeed a subgraph of ${G_{\gamma}({\textbf{X}}^\lambda)}$ for any $\gamma \geq 0$ . As for $\gamma=0$ , $G^-_{0}({\textbf{X}}^\lambda)$ equals the Cox–Gilbert graph $g_{r_o}(X^{\lambda,-})$ with connection radius $r_o$ and vertex set $X^{\lambda,-}$ . In words, in order to obtain $G^-_{\gamma}({\textbf{X}}^\lambda)$ from ${G_{\gamma}({\textbf{X}}^\lambda)}$ , one first thins out vertices with small powers, so as to get rid of vertices with small values of the connection radius $r_\textrm{B}^i$ , where

(5.3) \begin{align}r_\textrm{B}^i=\ell^{-1}(\tau {N_o}/ P_i).\end{align}

Then one bounds the powers of the remaining vertices by $\tau {N_o}/\ell(r_o)$ from below.

STEP 2. Mapping the subgraph to a lattice-percolation problem and percolation on the lattice.

Now we are in a position to adapt to the setting of [Reference Tóbiás28, Section 3.2.2] and use strong connectivity of $g_{r_o}(X^{\lambda,-})$ in case $r_o$ is sufficiently large and $\lambda$ is chosen according to $r_o$ . Together with an interference-control argument presented below, this will allow us to verify Condition 1 of Theorem 2.1.

First, let us recall the definition of rescalings of a Gilbert graph, which were also used in [Reference Tóbiás28]. For $c>0$ and a Gilbert graph G with connection radius $r>0$ , deterministic vertex set $V \subset \mathbb{R}^d$ , $d \geq 1$ , and edge set $E=\{ (x,y) \in V\times V \colon x \neq y, |x-y|<r \}$ , the graph cG is defined with vertex set $cV=\{cx\colon x\in V\}$ and edge set $cE=\{ (cx,cy) \colon (x,y) \in E \}$ . It is easy to see that cG is a Gilbert graph with vertex set cV and connection radius cr. For Gilbert graphs with random vertex sets (e.g., if the vertex set is given by a random simple point process), rescalings of the graph are defined realization-wise. From the proof of Theorem 2.9 (Convergence in Bounded Domains) in [Reference Hirsch, Jahnel and Cali16, Section 7.1], we know that in the coupled limit $\widetilde r \uparrow \infty, \, \widetilde \lambda \downarrow 0$ and $\widetilde \lambda \widetilde r^d = \widetilde \varrho>0$ , we have that $\widetilde r^{-1} g_{\widetilde r}(X^{\widetilde \lambda})$ converges weakly to the graph $g_{1}(Y^{\widetilde \varrho})$ , where $Y^{\varrho}$ is a homogeneous PPP with intensity $\varrho$ . Let us note that in [Reference Hirsch, Jahnel and Cali16, Section 7.1] this convergence is formulated equivalently for the Boolean model, and the crucial point is that the convergence is guaranteed only in compact domains.

Let $\varrho_\textrm{c}(1)$ be such that the Poisson–Gilbert graph $g_{1}(Y^{\varrho_\textrm{c}(1)})$ is critical. Then, due to the scale invariance of Poisson–Gilbert graphs [Reference Meester and Roy25, Section 2.2], for $\varrho>\varrho_\textrm{c}(1)$ , we can choose a smaller intensity $\varrho'<\varrho$ such that $g_{1}(Y^{\varrho'})$ is still supercritical. Now, for $r>d_o$ , we define $r_o(r)=r(\varrho/\varrho')^{1/d}$ , $\lambda(r)=\varrho' r^{-d} (p(r_o(r)))^{-1}$ , and $p(r)=\tau {N_o}/\ell(r_o(r))$ . Noting that $g_r(X^{\lambda(r),-})$ is a Cox–Gilbert graph with connection radius r and stabilizing intensity $p(r_o(r)) \lambda(r)=\varrho' r^{-d}$ , we have that $r^{-1}g_r(X^{\lambda(r),-})$ converges to the supercritical graph $g_1(Y^{\varrho'})$ on compact sets, as r tends to infinity.

Furthermore, recalling that R denotes the stabilization radii of $\Lambda$ , we put $R(Q)=\sup_{x \in Q \cap \mathbb{Q}^d} R_x$ for any measurable set $Q \subseteq \mathbb{R}^d$ .

Using these notions, we construct a renormalized percolation process on $\mathbb{Z}^d$ as follows. For $n \geq 1$ and $r>d_o$ , the site $z \in \mathbb{Z}^d$ is (r, n)-good if the following conditions hold:

1. 1. $R(Q_{6rn}(rnz))<rn/2$ .

2. 2. $X^{\lambda(r),-} \cap Q_{rn}(rnz) \neq \varnothing$ .

3. 3. Every pair $x_i,x_j \in X^{\lambda(r),-} \cap Q_{3rn}(rnz)$ is connected by a path in $g_r(X^{\lambda(r),-}) \cap Q_{6rn}(rnz)$ .

The site $z \in \mathbb{Z}^d$ is (r, n)-bad if it is not (r, n)-good. Note that the process of (r, n)-good sites is 7-dependent thanks to the definition of stabilization. The following lemma is verified in [Reference Tóbiás28, Section 3.2.2]. However, since in [Reference Tóbiás28] it is not formulated as a lemma, and two different proofs are presented for $d=2$ and $d \geq 3$ , we provide a self-contained proof here for the reader’s convenience.

Lemma 5.1. ([Reference Tóbiás28].) Assume that the general conditions of Theorem 2.1 plus Condition 1 hold. Then, for all sufficiently large $\lambda>0$ and for all $n \geq 1$ and $r >d_o$ with rn sufficiently large, there exists $q_A=q_A(\lambda,rn) <1$ such that for any $N \in \mathbb{N}$ and pairwise distinct $z_1,\ldots,z_N \in\mathbb{Z}^d$ ,

(5.4) \begin{equation} \mathbb{P}(z_1,\ldots,z_N\ \text{are all }(\textit{r},\textit{n})\text{-bad}) \leq q_A^N. \end{equation}

Furthermore, for any $\varepsilon>0$ , one can choose $\lambda$ and rn sufficiently large so that $q_A <\varepsilon$ .

Proof. For $z \in \mathbb{Z}^d$ , we write $J_{n,r}(z)$ for the event that z satisfies Parts 2 and 3 of the definition of (r, n)-goodness. Then, for any n, r under consideration, the process of (r, n)-good sites is 7-dependent by the definition of stabilization. Furthermore, we write $F_n(z)$ for the event that in the definition of (1, n)-goodness, the PPP $Y^{\varrho'}$ with intensity $\varrho'=\lambda(r)r^d$ satisfies Part 2 with $X^{\lambda(r),-}$ replaced by $Y^{\varrho'}$ and Part 3 with $g_r(X^{\lambda(r),-})$ replaced by $g_1(Y^{\varrho'})$ everywhere. The probability of $F_n(z)$ is independent of the choice of z and tends to 1 as $n \to\infty$ thanks to the arguments of [Reference Hirsch, Jahnel and Cali16, Section 5.2], since the constant directing measure of the PPP $Y^{\varrho'}$ is certainly asymptotically essentially connected. Using a union bound and the well-known scale invariance of Poisson–Gilbert graphs, namely that for $\widetilde\lambda,\widetilde r>0$ , $\widetilde r^{-1} g_{\widetilde r} (X^{\widetilde\lambda})$ equals $g_1(X^{\widetilde \lambda \widetilde r^d})$ in distribution, we conclude that for $z \in \mathbb{Z}^d$ ,

\begin{equation*} \mathbb P(z\text{ is }(n,r)\text{-bad}) \leq \mathbb P \Big( R(Q_{6nr}(nrz)) \geq nr/2 \Big) +\mathbb P(F_n(z)^{\mathrm c})+|\mathbb P(F_{n}(z)^{\mathrm c})-\mathbb P(J_{n,r}(z)^{\mathrm c})|. \end{equation*}

This can be made arbitrarily close to zero by choosing first n large and then r large according to n, due to the weak convergence of $r^{-1} g_r(X^\lambda)$ to $g_1(Y^{\varrho'})$ on $Q_{6n}(nz)$ as $r \to \infty$ , $\lambda(r) \to 0$ , $r^d \lambda(r) = {\varrho'}$ .

Hence, applying [Reference Liggett, Schonmann and Stacey23, Theorem 0.0], for all sufficiently large n and large enough r chosen according to n, the 7-dependent process of (r, n)-good sites is stochastically dominated from below by a supercritical independent site percolation process. Moreover, the probability that a site of the independent site percolation process is closed can be made arbitrarily close to 0 via further increasing nr. This implies the lemma.

We proceed similarly to [Reference Dousse9, Reference Tóbiás28] by defining ‘shifted’ versions of the path-loss function $\ell$ . For $a\ge 0$ , define

(5.5) \begin{equation} \ell_{a}(r) = \ell(0) \unicode{x1D7D9} \big\{ r< a\sqrt d/2 \big\} + \ell\big( r-a\sqrt d/2 \big) \unicode{x1D7D9} \big\{ r \geq a\sqrt d/2 \big\}. \end{equation}

Note that $\ell_0=\ell$ . Now, we define the shot-noise processes

\begin{eqnarray*}& I_{a}(x)=\sum_{i \in \mathbb{N}} P_i \ell_{a}(|x-x_i|), \qquad I(x)=\sum_{i \in \mathbb{N}} P_i \ell(|x-x_i|), \qquad x \in \mathbb{R}^d,\end{eqnarray*}

and note that $I_{0}(x)=I(x)$ . By the triangle inequality, for $a \geq 0$ , $I(x) \leq I_{a}(z)$ holds for any $z \in \mathbb{R}^d$ and $x \in Q_a(z)$ . Now, the interference-control argument consists in verifying the following proposition. For $z \in \mathbb{Z}^d$ , let us write $B_{r, n, M}(z)=\{ I_{6rn}(rnz) \leq M \}$ .

Proposition 5.1. Assume that the general conditions of Theorem 2.1 plus Condition 1 hold. Then, for all $\lambda>0$ , for all $n \geq 1$ and $r>d_o$ with rn sufficiently large, and for all $M>0$ sufficiently large, there exists $q_B=q_B(\lambda,rn,N) <1$ such that for all $N \in \mathbb{N}$ and for all pairwise distinct $z_1,\ldots,z_N \in \mathbb{Z}^d$ we have

(5.6) \begin{equation} \mathbb{P}(B_{r, n, M}(z_1)^\textrm{c} \cap \ldots \cap B_{r, n, M}(z_N)^\textrm{c}) \leq q_B^N. \end{equation}

Furthermore, for any $\varepsilon>0$ and $\lambda>0$ , one can choose rn and M sufficiently large so that $q_B < \varepsilon$ .

The proof of this proposition is postponed until Step 4. Once we have proved Proposition 5.1, we can derive the following corollary using a standard argument (see e.g. the proof of [Reference Dousse9, Proposition 3] or that of [Reference Tóbiás28, Proposition 3.1]). For $z \in \mathbb{Z}^d$ let us define $C_{r, n, M}(z)=\{ z\text{ is }(r, n)\text{-good} \}\cap \{ I_{6rn}(rnz) \leq M \}$ .

Corollary 5.1. Assume that the general conditions of Theorem 2.1 plus Condition 1 hold. Then, for all sufficiently large $\lambda>0$ , for all $r>d_o$ and $n \geq 1$ with rn sufficiently large, and for all $M>0$ sufficiently large, there exists $q_C =q_C(\lambda,rn,M)<1$ such that for all $N \in \mathbb{N}$ and for all pairwise distinct $z_1,\ldots,z_N \in \mathbb{Z}^d$ we have

(5.7) \begin{equation} \mathbb{P}(C_{r, n, M}(z_1)^\textrm{c} \cap \ldots \cap C_{r, n, M}(z_N)^\textrm{c}) \leq q_C^N. \end{equation}

Furthermore, for any $\varepsilon>0$ , one can choose $\lambda,rn,M$ sufficiently large so that $q_C < \varepsilon$ .

STEP 3. Percolation in the subgraph of the SINR graph.

Having Corollary 5.1 and employing a Peierls argument (cf. [Reference Grimmett15, Section 1.4]), we conclude that for $\lambda, rn,M$ sufficiently large, the process of (r, n)-good sites $z \in \mathbb{Z}^d$ such that $I_{6rn}(rnz) \leq M$ percolates. Thanks to exactly the same arguments as in the proof of Theorem 2.6 in [Reference Hirsch, Jahnel and Cali16, Section 5.2], this implies percolation of the Cox–Gilbert graph $G^-_{0}({\textbf{X}}^{\lambda(r)})=g_{r_o(r)}(X^{\lambda(r),-})$ . From this point of the proof it is classical to derive that $G^-_{\gamma}({\textbf{X}}^{\lambda(r)})$ percolates for small $\gamma>0$ ; see [Reference Dousse9, Section 3.3]. For the convenience of the reader, let us give the details here. We define

\begin{equation*} \gamma' = \frac{{N_o}}{p(r)M} \Big( \frac{\ell(r)}{\ell(r_o(r))} - 1 \Big) =\frac{\ell(r_o(r))}{\tau M}\Big( \frac{\ell(r)}{\ell(r_o(r))} - 1 \Big) >0, \end{equation*}

where the strict inequality holds because $r_o(r)>r>d_o$ and $\ell$ has unbounded support. Then we have

\begin{equation*} \frac{p(r) \ell(r)}{{N_o}+\gamma' p(r) M} = \tau. \end{equation*}

Now, let $x_i,x_j \in X^{\lambda(r),-}$ be situated in $Q_{rn}(rnz)$ and $Q_{rn}(rnz')$ , respectively, for some sites $z,z' \in\mathbb{Z}^d$ included in the same infinite cluster of the process of (r, n)-good sites $z \in \mathbb{Z}^d$ satisfying $I_{6rn}(rnz) \leq M$ , with $|x_i-x_j| < r$ . Then, for $\gamma<\gamma'$ , we have

\begin{equation*} \mathrm{SINR}(x_i,x_j,\textbf{X}^\lambda) \geq \mathrm{SINR}^{-}(x_i,x_j,\textbf{X}^\lambda) > \frac{p(r) \ell(r)}{{N_o}+\gamma' p(r) M} = \tau. \end{equation*}

Thus, $x_i$ and $x_j$ are connected by an edge in $G^-_{\gamma}({\textbf{X}}^\lambda)$ . Hence, ${G_{\gamma}({\textbf{X}}^\lambda)}$ also percolates. Thus, we can conclude Theorem 2.1 as soon as we have verified Proposition 5.1.

STEP 4. Proof of Proposition 5.1: the interference-control argument.

Similarly to [Reference Tóbiás28, Section 3.1.1], we split the interference into two parts. For $x \in \mathbb{R}^d$ , $n \geq 1$ , and $r>0$ , we put

\begin{align*}& I_{6rn}^{\mathrm{in}} (x) = \sum_{x_i \in X^\lambda \cap Q_{12rn\sqrt d}(x)} \ell_{6rn}(|x_i-x|),\\[3pt] & I_{6rn}^{\mathrm{out}} (x) = \sum_{x_i \in X^\lambda \setminus Q_{12rn\sqrt d}(x)} \ell_{6rn}(|x_i-x|).\end{align*}

Then, for $M>0$ , if $I_{6rn}(x)>M$ , then $I_{6rn}^{\mathrm{in}} (x)>M/2$ or $I_{6rn}^{\mathrm{out}} (x)>M/2$ . Using a union bound and the fact that M can be chosen arbitrarily large in Proposition 5.1, it suffices to conclude the proposition both with $B_{r, n, M}(z_i)$ replaced by $B_{r, n, M}^{\mathrm{in}}(z_i)$ and with $B_{r, n, M}(z_i)$ replaced by $B_{r, n, M}^{\mathrm{out}}(z_i)$ everywhere in (5.6) for all $i \in \{1,\ldots,N\}$ , where for $z \in \mathbb{Z}^d$ we write $B_{r, n, M}^{\mathrm{in}}(z)= \big\{ I_{6rn}^{\mathrm{in}}(rnz) \le M \big\}$ and $B_{r, n, M}^{\mathrm{out}}(z)= \big\{ I_{6rn}^{\mathrm{out}}(rnz) \le M \big\}$ . Indeed, having these assertions, we can combine them similarly to Corollary 5.1.

We now verify Proposition 5.1 with $B_{r,n,M}(\cdot)$ replaced by $B_{r,n,M}^{\mathrm{in}}(\cdot)$ everywhere. For this assertion, instead of the assumption that $P_o$ and $\Lambda(Q_1)$ have some exponential moments, it suffices to assume they have a first moment (for $\Lambda(Q_1)$ this is automatic since $\mathbb{E}[\Lambda(Q_1)]=1$ by assumption). To be more precise, we prove the following lemma.

Lemma 5.2. Assume that the general conditions of Theorem 2.1 plus Condition 1 hold. Furthermore, let $\Lambda$ be stabilizing and let $\mathbb{E}[P_o]<\infty$ . Then, for all $\lambda>0$ , for all $n \geq 1$ and $r>d_o$ with rn sufficiently large, and for all $M>0$ sufficiently large, there exists $q_B=q_B(\lambda,rn,N) <1$ such that for all $N \in \mathbb{N}$ and for all pairwise distinct $z_1,\ldots,z_N \in \mathbb{Z}^d$ we have

\begin{equation*} \mathbb{P}\big(B_{r,n,M}^{\mathrm{in}}(z_1)^\textrm{c} \cap \ldots \cap B_{r,n,M}^{\mathrm{in}}(z_N)^\textrm{c}\big) \leq q_B^N. \end{equation*}

Furthermore, for any $\varepsilon>0$ and $\lambda>0$ , one can choose rn and M sufficiently large so that $q_B < \varepsilon$ .

Proof. We use the following auxiliary discrete percolation process. A site $z \in \mathbb{Z}^d$ is (r,n)-tame if the following hold:

1. 1. $R\big(Q_{12rn\sqrt d}(rnz)\big)<rn/2$ .

2. 2. $I_{6rn}^{\mathrm{in}}(rnz)\leq M $ .

A site $z \in \mathbb{Z}^d$ is (r, n)-wild if it is not (r, n)-tame. The process of (r, n)-tame sites is $\lceil 12 \sqrt d + 1\rceil$ -dependent according to the definition of stabilization. Thus, it follows from dependent-percolation theory [Reference Liggett, Schonmann and Stacey23, Theorem 0.0] that, in order to verify Lemma 5.2, it suffices to show that for all $\lambda>0$ , $\mathbb P(o\textrm{\ is}\ $ $(r,n)\hbox{-}\textrm{wild})$ can be made arbitrarily close to zero by choosing first rn sufficiently large and then M large enough accordingly. We have

\begin{equation*} \mathbb P(o\text{ is (\textit{r},\textit{n})-wild}) \leq \mathbb P\big(R\big(Q_{12rn\sqrt d}(rnz)\big) \geq rn/2\big) + \mathbb P\big(I_{6rn}^{\mathrm{in}}(rnz)> M\big). \end{equation*}

The first term can be made arbitrarily small by choosing rn large enough, thanks to the definition of stabilization. Moreover, by the definition of $\ell_a$ (see (5.5)),

\begin{equation*} I_{6rn}^{\mathrm{in}}(o) = \sum_{x_i \in X^\lambda \cap Q_{12rn\sqrt d}(o)} P_i \ell_{6rn}(|x_i|) \leq \ell(0) \sum_{x_i \in X^\lambda \cap Q_{12rn\sqrt d}(o)} P_i. \end{equation*}

In particular, using that the point process $\textbf{X}^\lambda$ is independently marked with $P_i$ having marginal distribution $\mu$ , and that $\Lambda$ is stationary with $\mathbb{E}[\Lambda(Q_1)]=1$ , it follows that

\begin{equation*} \mathbb E\big[I^{\mathrm{in}}_{6rn}(o)\big] \leq \ell(0) \lambda \mathbb{E}[P_o] \mathbb{E}\big[\Lambda\big(Q_{12rn\sqrt d}\big)\big]=\Big(12rn\sqrt d\Big)^d \ell(0) \lambda \mathbb{E}[P_o]. \end{equation*}

Thus, for any $n \geq 1$ and $r>0$ , $\mathbb P\big(I_{6rn}^{\mathrm{in}}(o)>M\big)$ can be made arbitrarily small by choosing M large enough, given that $\mathbb{E}[P_o]<\infty$ . Thus, the statement of the lemma follows.

It remains to verify Proposition 5.1 with $B_{r,n,M}(\cdot)$ replaced by $B_{r,n,M}^{\mathrm{out}}(\cdot)$ everywhere. More precisely, thanks to the exponential-moment and b-dependence assumption on $\Lambda$ , the proof can be completed analogously to the proof of [Reference Tóbiás28, Proposition 3.3] starting from [Reference Tóbiás28, Equation (3.15)], as soon as we have verified the following lemma. (In [Reference Tóbiás28] it is also assumed that $\ell(0) \leq 1$ , but since M can be made arbitrarily large in Proposition 5.1, $\ell(0) \leq 1$ can be assumed without loss of generality since for $\ell$ continuous, the function $\widetilde \ell=\ell/\ell(0)$ satisfies $\widetilde \ell(0)=1$ , and for $a \geq 0$ , we have $ \ell_{a}=\ell(0)\widetilde \ell_a$ and hence $I_a(x)=\ell(0)\sum_{i \in \mathbb{N}} P_i \widetilde \ell_a(|x-x_i|)$ .)

Lemma 5.3. Under the general assumptions of Theorem 2.1 plus Condition 1, there exists a constant $c_o=c_o(\mu,\ell)>0$ such that for all sufficiently small $s>0$ , for all $\lambda>0$ , for all $n \geq 1$ and $r>d_o$ with $rn>0$ sufficiently large, and for all large enough $M>0$ , for all $N\in\mathbb{N}$ and pairwise distinct $z_1,\ldots,z_N \in \mathbb{Z}^d$ we have

(5.8) \begin{equation}\begin{split}\mathbb{P}\big(B^{\mathrm{out}}_{r,n,M}(z_1)^\textrm{c}& \cap \ldots \cap B^{\mathrm{out}}_{r,n,M}(z_N)^\textrm{c}\big)\\[3pt] \leq &\, \mathbb{E}\left[ \exp\left( c_o \lambda s \sum_{i=1}^N \int_{\mathbb{R}^d\setminus Q_{12rn\sqrt d}(rnz_i)} \ell_{6rn}(|rnz_i-x|) \Lambda(\textrm{d} x) \right) \right].\end{split}\end{equation}

Indeed, the right-hand side of (5.8) is the same as that of [Reference Tóbiás28, Equation (3.15)], and the assumptions on $\Lambda$ in the two proofs are also the same.

Proof of Lemma 5.2. We start with an estimate originating from [Reference Dousse9, Section 3.2]. By Markov’s inequality, for any $s>0$ ,

(5.9) \begin{align}& \mathbb P\big( B^{\mathrm{out}}_{r,n,M}(z_1)^\textrm{c} \cap \ldots \cap B^{\mathrm{out}}_{r,n,M}(z_N)^\textrm{c}\big) = \mathbb P\big( I_{6rn}^{\mathrm{out}}(rnz_1)>M,\ldots,I_{6rn}^{\mathrm{out}}(rnz_N)>M \big)\nonumber\\[3pt] & \leq \mathbb P \left( \sum_{i=1}^N I_{6rn}^{\mathrm{out}}(rnz_i) > N M \right)\nonumber\\[3pt] & \leq \varepsilon^{-sNM} \mathbb{E} \left[ \exp\left( s \sum_{i=1}^N \sum_{x_k \in X^\lambda\setminus Q_{12rn\sqrt d}(nz_i)} P_k \ell_{6rn}(|rnz_i-x_k|) \right) \right]. \end{align}

The randomness of the power values $P_k$ prevents us from continuing the proof analogously to [Reference Dousse9, Reference Tóbiás28]. On the other hand, similarly to [Reference Löffler22, Section 4.3] in the Poisson case, we can argue as follows. According to the Marking Theorem [Reference Kingman20, Section 5.2], the independently marked CPP $\textbf{X}^\lambda=\{ (x_i,P_i)\}_{i \in \mathbb{N}}$ is a CPP in $\mathbb{R}^d \times [0,\infty)$ with directing measure $\Lambda \otimes \mu$ , where we recall that $\mu=\mathbb{P} \circ P_o^{-1}$ is the distribution of $P_o$ . Hence, applying the Laplace functional of a CPP (cf. [Reference Kingman20, Sections 3.2 and 6]) to the function $f \colon \mathbb{R}^d \times [0,\infty) \to [0,\infty)$ ,

\begin{equation*} f(x,p)=s\sum_{i=1}^N p \ell_{6rn}(|x-rnz_i|) \unicode{x1D7D9} \big\{ x \in \mathbb{R}^d \setminus Q_{12rn\sqrt d}(rnz_i) \big\}, \end{equation*}

we obtain

(5.10) \begin{align}&\mathbb{E} \left[ \exp\left( s \sum_{i=1}^N \sum_{x_k \in X^\lambda\setminus Q_{12rn\sqrt d}(rnz_i)} P_k \ell_{6rn}(|rnz_i-x_k|) \right) \right]\nonumber\\[3pt] &= \mathbb{E} \left[ \exp \left( \lambda \int_{\mathbb{R}^d\setminus Q_{12rn\sqrt d}(rn z_i) } \int_{0}^{\infty} \left( \exp \left( sp \sum_{i=1}^N \ell_{6rn}(|rnz_i-x|)\right) -1 \right) \mu(\textrm{d} p)\Lambda(\textrm{d} x) \right) \right]. \end{align}

Thanks to the exponential-moment assumption on $P_o$ from Condition 1, the moment-generating function

\begin{equation*} \alpha \mapsto \mathbb{E}[\exp(\alpha P_o)]=\int_0^{\infty} \varepsilon^{\alpha p} \mu(\textrm{d} p) \end{equation*}

is infinitely differentiable at $\alpha=0$ , with first derivative $\int_0^{\infty} p \mu(\textrm{d} p)=\mathbb{E}[P_o]<\infty$ . Note that $\sum_{i=1}^N \ell_{6rn}(|rnz_i-x|) $ is uniformly bounded in $x\in \mathbb{R}^d$ , rn, N, and pairwise distinct $z_1,\dots, z_N$ ; see [Reference Tóbiás28, Lemma 3.6]. Consequently, for any $C>1$ , the following holds for all sufficiently small $s>0$ (depending on C):

(5.11) \begin{equation} \int_{0}^{\infty} \left( \exp \left( sp \sum_{i=1}^N\ell_{6rn}(|rnz_i-x|) \right)-1 \right) \mu(\textrm{d} p) \leq Cs \mathbb{E}[P_o] \sum_{i=1}^N \ell_{6rn}(|rnz_i-x|). \end{equation}

For such s, plugging (5.11) back into (5.10), starting from (5.9) we obtain

(5.12) \begin{multline} \mathbb{P}\big(B^{\mathrm{out}}_{r,n,M}(z_1)^\textrm{c} \cap \ldots \cap B^{\mathrm{out}}_{r,n,M}(z_N)^\textrm{c}\big)\\ \leq \mathbb{E}\left[ \exp\left( C \mathbb{E}[P_o] \lambda s \sum_{i=1}^N \int_{\mathbb{R}^d\setminus Q_{12rn\sqrt d}(rnz_i)} \ell_{6rn}(|rnz_i-x|)\Lambda(\textrm{d} x) \right) \right], \end{multline}

which is (5.8) with $c_o=C \mathbb{E}[P_o]$ . With this we conclude the proof of the lemma.

5.2. Proof of Proposition 4.2

The strategy of the proof of Proposition 4.2 is the following. We first show that up to $\mathbb{P}$ -null sets, clusters are either finite or infinite in both directions, i.e., they contain no vertex of degree 1 in the case where they are infinite; see Lemma 5.4 below. Next, we assume for a contradiction that there exists an infinite cluster with positive probability. We then introduce a procedure that removes points from the infinite cluster that is closest to the origin in a certain sense. Thanks to elementary properties of the SINR graph, in the resulting configuration, the infinite cluster still remains infinite, but it contains a vertex of degree 1. Hence, the probability that the process takes values in the set of the resulting configurations is zero. What remains to show afterwards is that the probability that the process takes place in the set of original configurations is also zero, which leads to the desired contradiction. At this point it will be useful to compare the resulting configuration with an independent thinning of the original configuration in a certain ball, and this is where we make use of the fact that the underlying point process is a stationary CPP.

We assume throughout the proof that $\gamma \geq 1/(2\tau)$ , so that degrees in ${G_{\gamma}({\textbf{X}}^\lambda)}$ are bounded by 2, and that $X^\lambda$ is nonequidistant (for all $\lambda>0$ ). We can also assume that $\mathbb{P}(P_o>0)>0$ in what follows, since otherwise the statement is trivially true. We start the proof with the following lemma, which excludes infinite paths that have an endpoint in the case where the degrees are bounded by 2, in a substantially more general setting.

Lemma 5.4. Let $g({\textbf{X}})$ be a random graph based on a stationary marked point process ${\textbf{X}}=\{(x_i,P_i)\}_{i\in \mathbb{N}}$ with values in $\mathbb{R}^d \times Z$ , where the mark space $(Z,\mathcal{Z})$ is an arbitrary measurable space, $X=\{x_i\}_{i\in \mathbb{N}}$ is the vertex set, and the degree of all $x_i\in X$ , $\deg(x_i)$ , is bounded by 2, almost surely. Let X have a finite intensity and consider the point process of degree-one points in infinite clusters,

\begin{equation*} \mathcal{X}_0=\sum_{i\in \mathbb{N}}\delta_{x_i}\unicode{x1D7D9}\{\deg(x_i)=1,\, x_i \text{ is part of an infinite cluster in }g({\textbf{X}})\}.\end{equation*}

Then $\mathbb{P}(\mathcal{X}_0(\mathbb{R}^d)=0)=1$ .

We will apply this lemma to the SINR graph $g({\textbf{X}})={G_{\gamma}({\textbf{X}}^\lambda)}$ with $\lambda$ arbitrary, $\gamma \geq 1/(2\tau)$ , and $Z=[0,\infty)$ . The proof is based on a variant of the mass-transport principle (cf. [Reference Coupier, Dereudre and Le Stum4, Section 4.2] for instance).

Proof of Lemma 5.4. First, using the union bound and stationarity, it is enough to show that $\mathbb{E}[\mathcal{X}_0(Q_1)]=0$ . Let us define the point process of points in infinite clusters that are at distance equal to $k\in\mathbb{N}_o$ from a point in $\mathcal{X}_0$ ,

\begin{equation*}\mathcal{X}_k=\sum_{i\in \mathbb{N}}\delta_{x_i}\unicode{x1D7D9}\{x_i\text{ is part of an infinite cluster and has graph distance }k\text{ from }\mathcal{X}_0\}.\end{equation*}

Thanks to the degree bound, every infinite cluster has at most one point in $\mathcal{X}_0$ , and $\mathbb{E}[\mathcal{X}_k(Q_1)]=\mathbb{E}[\mathcal{X}_0(Q_1)]$ for all $k\in\mathbb{N}_o$ , by stationarity. However, $\sum_{k\ge 0}\mathbb{E}[\mathcal{X}_k(Q_1)]\le \mathbb{E}[X(Q_1)]<\infty$ , and thus $\mathbb{E}[\mathcal{X}_0(Q_1)]=0$ .

Let us denote by $(\mathcal{C}_i)_{0\le i< L}$ the L-many infinite clusters in ${G_{\gamma}({\textbf{X}}^\lambda)}$ , where $L\in \mathbb{N}\cup\{ \infty\}$ . For the proof of Proposition 4.2, it then suffices to show that

(5.13) \begin{equation} \mathbb{P}(L\ge 1)=0. \end{equation}

We view ${\textbf{X}}^\lambda$ as the canonical process ${\textbf{X}}^\lambda(\boldsymbol\omega)=\boldsymbol\omega$ on the set $\textbf{N}$ of marked point configurations $\boldsymbol \omega$ in $\mathbb{R}^d \times [0,\infty)$ such that $\omega=\{ x_i \colon (x_i,p_i) \in \boldsymbol \omega \}$ is an infinite, locally-finite, nonequidistant point configuration on $\mathbb{R}^d$ . The set of such point configurations $\omega$ will be denoted by $\textrm{N}$ . Note that $\textbf{N}$ and N are equipped with the corresponding evaluation $\sigma$ -fields.

Now we introduce an ordering in $\mathbb{R}^d\times [0,\infty)$ , which orders the points of the set according to the received signal power at a given point $y \in \mathbb{R}^d$ (or equivalently, according to the received SINR values $\mathrm{SINR}(\cdot,y,\boldsymbol \omega)$ at y).

When talking about the marked CPP ${\textbf{X}}^\lambda$ , we will always assume that transmitters are associated with their own transmitted signal powers, and hence we will say ‘ $x_i$ transmits a stronger signal to $x_j$ than $x_l$ does’ instead of ‘ $(x_i,P_i)$ transmits a stronger signal to $x_j$ than $(x_l,P_l)$ does’, for any $i,j,l \in \mathbb{N}$ such that $i \neq j$ and $l \neq j$ . It is easy to see that for ${\textbf{X}}^\lambda$ such that $X^\lambda$ is nonequidistant, almost surely the following holds. For all $i \in \mathbb{N}$ , the relation ‘ $x_i$ transmits a stronger signal to $x_j$ than $x_l$ does’ is a total ordering (i.e., irreflexive, antisymmetric, and transitive, with any two elements being comparable) on the set $\{ (i,l) \in \mathbb{N}^2 \colon i \neq j \text{ and } l \neq j \}$ , which we call the ordering of signal-weighted distance from receiver $x_i$ . This fact indeed relies on the tiebreaking mechanism (ii): e.g., if $\ell$ is constant on some interval (which is possible under the assumption of Proposition 4.2 and even under the stronger assumption of Theorem 2.1), then (i) does not define a total ordering on its own.

For $\boldsymbol\omega \in \textbf{N}$ and $x_o \in \omega$ , we can consider the vector $\textbf{V}(x_o,\boldsymbol \omega)=(\textbf{V}_n(x_o,\boldsymbol \omega))_{n \in \mathbb{N}_0}$ of the marked points of $\boldsymbol \omega$ ordered increasingly according to signal-weighted distance from receiver $x_o$ . Then, we define $\mathrm{V}_i(x_o,\boldsymbol \omega)$ as the first component of $\textbf{V}_i(x_o,\boldsymbol \omega)$ , which we call the ith nearest neighbor of $x_o$ in signal-weighted order. In particular, $\mathrm{V}_0(x_o,\boldsymbol \omega)=x_o$ . Note that if the distribution $\mu$ is concentrated in one point $p>0$ , i.e., $\mu=\delta_p$ , then the ith nearest neighbor of $x_o$ in signal-weighted order is just the ith nearest neighbor of $x_o$ with respect to Euclidean distance.

Now, if $x_o$ has degree two in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega))$ , then $x_o$ must be connected by an edge to both $\mathrm{V}_1(x_o,\boldsymbol\omega)$ and $\mathrm{V}_2(x_o,\boldsymbol\omega)$ , since the degree bound applies already for the edges towards $x_o$ . Moreover, both $\mathrm{V}_1(x_o,\boldsymbol\omega)$ and $\mathrm{V}_2(x_o,\boldsymbol\omega)$ must also have $x_o$ as one of their first two nearest neighbors in signal-weighted order; that is,

\begin{equation*} x_o \in\big \{ \mathrm{V}_1(\mathrm{V}_i(x_o,\boldsymbol\omega),\boldsymbol\omega), \mathrm{V}_2(\mathrm{V}_i(x_o,\boldsymbol\omega),\boldsymbol\omega) \big\} \end{equation*}

for all $i\in\{1,2\}$ . These signal-weighted nearest neighbor relations hold almost surely, in particular for every nonequidistant configuration $\boldsymbol\omega$ . The goal of using the configuration space $\textbf{N}$ is to entirely exclude configurations that violate the degree bound or the signal-weighted nearest neighbor relations or that are not nonequidistant.

In the event $\{L\ge 1\}$ , let $\textbf{Z}=(Z,R)$ denote the closest point to the origin that has degree two and is contained in an infinite cluster. Without loss of generality, we will assume that this cluster is always equal to $\mathcal{C}_0$ . Now, Proposition 4.2 immediately follows once we have verified the following proposition.

Proposition 5.2. Consider the event $\{L\ge 1\}$ and define the random variable

\begin{equation*} I = \inf \{ i \geq 3 \colon \mathrm{V}_i(Z,{\textbf{X}}^\lambda) \in \mathcal{C}_0 \}. \end{equation*}

Then, under the assumptions of Proposition 4.2, for any $i \geq 3$ , we have

(5.14) \begin{equation} \mathbb{P}(\{ L\ge 1 \}\cap \{ I=i \})=0. \end{equation}

Proof of Proposition 4.2. Using a union bound and noting that $\{ L\ge 1 \}\subset \{I<\infty\}$ , Proposition 5.2 implies $\mathbb{P}(L\ge 1)=0$ , which is (5.17), and thus the proof of Proposition 4.2 is finished.

Proof of Proposition 5.2. For $\boldsymbol\omega\in \{L\ge 1\}$ , by definition, we have that $Z(\boldsymbol\omega)$ is connected by an edge to both $\mathrm{V}_1(Z(\boldsymbol\omega), \boldsymbol\omega)$ and $\mathrm{V}_2(Z(\boldsymbol\omega), \boldsymbol\omega)$ in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega))$ . Furthermore, thanks to the degree bound of two, in the event $\{L\ge 1\}$ , $\mathrm{V}_1(Z(\boldsymbol\omega), \boldsymbol\omega)$ and $\mathrm{V}_2(Z(\boldsymbol\omega), \boldsymbol\omega)$ have no further joint neighbor in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega))$ , since otherwise $\mathcal{C}_0(\boldsymbol\omega)$ has a loop and cannot be infinite by the degree bound. Thus, for any $i \geq 3$ , there exists $l \in \{1,2\}$ such that $\mathrm{V}_i(Z(\boldsymbol\omega), \boldsymbol\omega)$ and $\mathrm{V}_l(Z(\boldsymbol\omega), \boldsymbol\omega)$ are not connected by an edge in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega))$ . Let us denote the corresponding $\mathrm{V}_l(Z(\boldsymbol\omega), \boldsymbol\omega)$ by $M_i(\boldsymbol\omega)$ , and define $M_i(\boldsymbol\omega)=\mathrm{V}_1(Z(\boldsymbol\omega), \boldsymbol\omega)$ if neither $\mathrm{V}_1(Z(\boldsymbol\omega), \boldsymbol\omega)$ nor $\mathrm{V}_2(Z(\boldsymbol\omega), \boldsymbol\omega)$ is connected to $\mathrm{V}_i(Z(\boldsymbol\omega), \boldsymbol\omega)$ by an edge. The element of $\{ \mathrm{V}_1(Z(\boldsymbol\omega), \boldsymbol\omega),\mathrm{V}_2( Z(\boldsymbol\omega), \boldsymbol\omega) \}$ not equal to $M_i(\boldsymbol\omega)$ is denoted by $N_i(\boldsymbol\omega)$ . We will write Q for the signal power transmitted by $M_i(\boldsymbol\omega)$ .

Let us fix $i \geq 3$ . Let $\boldsymbol \omega \in \{L\ge 1\}$ be such that $I(\boldsymbol \omega)=i$ . Let us define a thinned configuration

\begin{align*}\boldsymbol \omega^i =\boldsymbol \omega\setminus \{(M_i(\boldsymbol\omega),Q), \textbf{V}_3(Z(\boldsymbol\omega),\boldsymbol\omega), \dots, \textbf{V}_{i-1}(Z(\boldsymbol\omega),\boldsymbol \omega)\}.\end{align*}

We claim for $\mathbb{P}$ -almost all $\boldsymbol \omega \in \{L\ge 1\}\cap \{I=i\}$ that also $\boldsymbol \omega^i \in \{L\ge 1\}$ . For this, first note that the removal of finitely many points and their associated edges from an infinite cluster does not change the property that the cluster is infinite. However, the removal of points can still change the edge structure of the remaining points. In order to exclude this, we use the following fundamental property of the SINR graph. Assume that $\boldsymbol \omega,\boldsymbol \omega'$ are elements of $\textbf{N}$ such that $\boldsymbol \omega \subseteq \boldsymbol \omega'$ . Then, for all $x,y \in \omega$ , if $\mathrm{SINR}(x,y,\boldsymbol \omega') >\tau$ , then $\mathrm{SINR}(x,y,\boldsymbol \omega) >\tau$ , which is clear from (5.1). In words, if we remove some vertices from an SINR graph, then edges of the SINR graphs between the remaining points stay preserved.

Our next claim is that for $\boldsymbol\omega\in \{L\ge 1\}\cap \{I=i\}$ , $\boldsymbol\omega^i$ is contained in

\begin{equation*} B = \{\boldsymbol\eta\colon L(\boldsymbol\eta)\ge 1 \text{ and }\mathcal{C}_0(\boldsymbol\eta) \text{ contains a point of degree one}\} \subset \{L\ge 1\}. \end{equation*}

The proof of this claim in the simplest case $i=3$ is illustrated in Figure 3. For general $i\ge 3$ , recall that Z cannot have degree higher than two in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega^i))$ , whereas it has degree at least one and its cluster $\mathcal{C}_0(\boldsymbol\omega^i)$ is infinite in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega^i))$ . Note also that the edge between $Z(\boldsymbol\omega)$ and $N_i(\boldsymbol\omega)$ still exists in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega^i))$ . Furthermore, if $Z(\boldsymbol\omega)$ has degree two in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega^i))$ , then it is connected to the second-nearest neighbor of $Z(\boldsymbol\omega)$ in signal-weighted order in $\boldsymbol\omega^i$ , which is $\mathrm{V}_2(Z(\boldsymbol\omega),\boldsymbol\omega^i)=\mathrm{V}_i(Z(\boldsymbol\omega),\boldsymbol\omega)$ , whereas $\mathrm{V}_1(Z(\boldsymbol\omega),\boldsymbol\omega^i)=N_i(\boldsymbol\omega)$ . Now, since $\boldsymbol\omega \notin B$ , $\boldsymbol\omega \in \{L\ge 1\}$ , and $\mathrm{V}_i(Z(\boldsymbol\omega),\boldsymbol\omega) \in \mathcal{C}_0(\boldsymbol\omega)$ , it follows that $\mathrm{V}_i(Z(\boldsymbol\omega),\boldsymbol\omega)$ has degree equal to two in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega))$ . Furthermore, it is neither connected to $M_i(\boldsymbol\omega)$ by an edge nor to $Z(\boldsymbol\omega)$ in this graph. Hence, both edges adjacent to $\mathrm{V}_i(Z(\boldsymbol\omega),\boldsymbol\omega)$ also exist in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega^i))$ . But since $\mathrm{V}_i(Z(\boldsymbol\omega),\boldsymbol\omega)$ has degree at most two in $G_\gamma({\textbf{X}}^\lambda(\boldsymbol\omega^i))$ , it follows that $Z(\boldsymbol\omega)$ and $\mathrm{V}_i(Z(\boldsymbol\omega),\boldsymbol\omega)$ are not connected by an edge in this graph. Hence, $\boldsymbol\omega^i \in B$ , which implies the claim.

Figure 3.

A visualization of the case $I(\boldsymbol\omega)=3$ for some realization $\boldsymbol\omega\in \{L \geq 1\}$ . $\mathrm{V}_3=\mathrm{V}_3(Z(\boldsymbol \omega),\boldsymbol\omega)$ is contained in the infinite cluster $\mathcal{C}_0=\mathcal{C}_0(\boldsymbol\omega)$ of the SINR graph $G_{\gamma}(\boldsymbol \omega)$ including $Z=Z(\boldsymbol \omega)$ , and it is not a neighbor of $M_3=M_3(\boldsymbol\omega)$ , which in this example equals $\mathrm{V}_1=\mathrm{V}_1(Z(\boldsymbol\omega),\boldsymbol\omega)$ , while $\mathrm{V}_2=\mathrm{V}_2(Z(\boldsymbol \omega),\boldsymbol\omega)=N_3=N_3(\boldsymbol\omega)$ . Hence, if $\mathrm{V}_3$ has degree two in $\mathcal{C}_0$ , then there are various possibilities respecting the degree bound of two to connect $\mathrm{V}_3$ to $\mathcal{C}_0$ so that it is not connected to $M_3$ by an edge. $\mathrm{V}_3$ can be either a direct neighbor of $\mathrm{V}_2$ (dashed line), or a later point of the path from Z to infinity starting with the edge from Z to $\mathrm{V}_2$ (dash-dotted lines), or a non-direct neighbor of $\mathrm{V}_1$ on the path from Z to infinity starting with the edge from Z to $\mathrm{V}_1$ (dotted lines). Now, if $M_3$ is removed from the realization, both edges adjacent to $\mathrm{V}_3$ are preserved. Also, all edges from Z to infinity starting with the edge from Z to $\mathrm{V}_2$ are preserved, so that Z is still contained in an infinite cluster, but the edge from Z to $\mathrm{V}_1$ is removed. In the resulting configuration, the second-nearest neighbor of Z in signal-weighted order is $\mathrm{V}_3$ , and hence this is the only point of the configuration that could be connected to Z by an edge. But $\mathrm{V}_3$ still cannot have degree 3 or more, so it cannot be connected to Z, which implies that in the new configuration Z is in an infinite cluster containing a point of degree one.

Note that by Lemma 5.4, the set B is a $\mathbb{P}$ -null set, i.e.,

(5.15) \begin{equation} \mathbb{P}\big(\big\{ \boldsymbol \omega^i \colon \boldsymbol \omega \in \{L\ge1\}\cap \{ I=i \} \big\}\big)=0. \end{equation}

This implies (5.14) and concludes the proof of Proposition 5.2 as soon as the following lemma is verified.

By Lemma 5.5—where we show that if the collection of thinned configurations is contained in a $\mathbb{P}$ -null set, then the non-thinned configurations also form a $\mathbb{P}$ -null set—we see that (5.15) implies (5.14), which concludes the proof of Proposition 5.2.

Proof of Lemma 5.5. Let us fix $i \geq 3$ and assume that $\mathbb{P}(\{L\ge1\}\cap \{ I=i \})>0$ . Then, by continuity of measures, there exists $K>0$ such that

\begin{equation*} \mathbb{P}\big( \big\{ \boldsymbol \omega \in \{L\ge1\}\cap\{I=i\} \colon \mathrm{V}_j(Z(\boldsymbol\omega), \boldsymbol \omega) \in B_K(o),\ \forall j \in \{ 1,\ldots,i\}\big \}\big)>0, \end{equation*}

where $B_K(o)$ denotes the open Euclidean ball of radius K in $\mathbb{R}^d$ . Hence, there exists $n \geq i$ such that $\mathbb{P}(C_{i,K,n})>0$ , where

\begin{equation*}\begin{aligned} C_{i,K,n} =\big\{ \boldsymbol \omega \in \{L\ge1\}\cap \{I=i\} \colon &\#\big( \omega \cap B_K(o) \big)=n+1 \\[3pt] &\text{and }\mathrm{V}_j(Z(\boldsymbol\omega),\boldsymbol \omega) \in B_K(o), \forall j \in \{ 1,\ldots,i\}\big\}. \end{aligned} \end{equation*}

Conditional on the event $C_{i,K,n}$ , the marked point process $(\textbf{X}^\lambda \setminus \{\textbf{Z} \})\cap B_K(o)$ has precisely n points.

Now, for some fixed $q \in (0,1)$ , we can represent $\textbf{X}^\lambda$ as ${\textbf{X}}^{\lambda,1} \cup {\textbf{X}}^{\lambda,2}$ as follows. For $K>0$ , let $B_K(o)$ denote the open $\ell^2$ -ball of radius K around o. Let ${\textbf{X}}^{\lambda,1}$ be given as the union of $\textbf{X}^\lambda \setminus (B_K(o)\times [0,\infty))$ and the independent thinning of $\textbf{X}^\lambda \cap (B_K(o)\times [0,\infty))$ with survival probability q, and let ${\textbf{X}}^{\lambda,2}$ be the complementary thinning. That is, conditional on $\textbf{X}^\lambda$ , ${\textbf{X}}^{\lambda,1} \cap (B_K(o)\times [0,\infty))$ contains each point of $\textbf{X}^\lambda \cap (B_K(o)\times [0,\infty))$ with probability q independent of the other points of this point process, and it contains no other points. Note further that ${\textbf{X}}^{\lambda,2}$ and ${\textbf{X}}^{\lambda,1} \cap (B_K(o)\times [0,\infty))$ are independent thinnings of $\textbf{X}^\lambda \cap (B_K(o)\times [0,\infty))$ with survival probabilities $1-q$ and q, respectively; moreover, ${\textbf{X}}^{\lambda,1}=\textbf{X}^\lambda \setminus {\textbf{X}}^{\lambda,2}$ , ${\textbf{X}}^{\lambda,2} \setminus (B_K(o)\times [0,\infty))=\varnothing$ , and ${\textbf{X}}^{\lambda,1} \setminus (B_K(o)\times [0,\infty))=\textbf{X}^\lambda \setminus (B_K(o)\times [0,\infty))$ . In order to provide a precise construction of the thinned processes, we choose a sequence $(J_m)_{m \in\mathbb{N}}$ of i.i.d. Bernoulli random variables with parameter q that is independent of $\textbf{X}^\lambda$ , and given the realization $\boldsymbol \omega=\textbf{X}^\lambda(\boldsymbol \omega) = (\textbf{V}_i(Z(\boldsymbol\omega),\boldsymbol \omega))_{i \in \mathbb{N}_0}$ , the realizations of ${\textbf{X}}^{\lambda,1}(\boldsymbol \omega)$ and ${\textbf{X}}^{\lambda,2}(\boldsymbol \omega)$ are defined as follows, depending also on $(J_m)_{m \in\mathbb{N}}$ :

\begin{equation*}\begin{aligned}{\textbf{X}}^{\lambda,1}(\boldsymbol \omega) & ={\textbf{X}}^{\lambda,1}(\boldsymbol \omega,(J_m)_{m \in \mathbb{N}}) = \{ \textbf{V}_m(Z(\boldsymbol\omega),\boldsymbol \omega) \colon J_m = 1, \textbf{V}_m(Z(\boldsymbol\omega),\boldsymbol \omega) \in B_K(o) \}\\[3pt] & \quad \cup \{ \textbf{Z}(\boldsymbol\omega) \} \cup \{\textbf{V}_m(Z(\boldsymbol\omega),\boldsymbol \omega) \colon \textbf{V}_m(Z(\boldsymbol\omega), \boldsymbol \omega) \in \mathbb{R}^d \setminus B_K(o) \} \end{aligned} \end{equation*}

and

\begin{equation*}{\textbf{X}}^{\lambda,2}(\boldsymbol \omega)={\textbf{X}}^{\lambda,2}(\boldsymbol \omega,(J_m)_{m \in \mathbb{N}}) = \{ \textbf{V}_m(Z(\boldsymbol\omega), \boldsymbol \omega) \colon J_m = 0, \textbf{V}_m(Z(\boldsymbol\omega),\boldsymbol \omega) \in B_K(o) \}. \end{equation*}

It is clear that the respective projections $X^{\lambda,1}$ and $X^{\lambda,2}$ of ${\textbf{X}}^{\lambda,1}$ and ${\textbf{X}}^{\lambda,2}$ to the $\mathbb{R}^d$ -coordinate are nonequidistant; furthermore, ${\textbf{X}}^{\lambda,1}$ can be represented as a random variable with values in $\textbf{N}$ , defined on an enlarged probability space $(\Omega',\mathcal{F}',\mathbb P')$ governing both the point process ${\textbf{X}}^\lambda$ and the sequence $(J_m)_{m \in \mathbb{N}}$ . In particular, $\mathbb{P}'({\textbf{X}}^\lambda \in \cdot) = \mathbb{P}({\textbf{X}}^\lambda \in \cdot)$ .

The next property of stationary and nonequidistant CPPs is crucial for completing the proof of Lemma 5.5.

To be more precise, the absolute continuity is meant in this lemma in the following way, with respect to the probability space $(\Omega',\mathcal{F}',\mathbb P')$ on which ${\textbf{X}}^{\lambda,1}$ and ${\textbf{X}}^\lambda$ are jointly defined with $\mathbb P'({\textbf{X}}^\lambda \in \cdot)=\mathbb P({\textbf{X}}^\lambda \in \cdot)$ . Let $G \in \mathcal{F}'$ be any event such that $\mathbb{P}'({\textbf{X}}^{\lambda,1} \in G)>0$ ; then we have $\mathbb{P}'({\textbf{X}}^\lambda \in G)>0$ .

Proof of Lemma 5.6. Let F be an element of the evaluation $\sigma$ -algebra of $\textbf{N}$ such that $\mathbb{P}'({\textbf{X}}^{\lambda,1} \in F)>0$ . We have to show that $\mathbb{P}(\textbf{X}^\lambda \in F)>0$ as well. Under the assumption that $\mathbb{P}'({\textbf{X}}^{\lambda,1} \in F)>0$ , by continuity of measures, we can find $K,l \in \mathbb{N}$ such that

(5.16) \begin{equation} \varepsilon:=\mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in F, \# \big({\textbf{X}}^{\lambda,1} \cap (B_K(o) \times [0,\infty))\big) = l\big) >0. \end{equation}

In other words, we have $0<\varepsilon = \mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in G\big)$ , where $G = \{ \boldsymbol \omega \in F \colon \# (\boldsymbol \omega \cap ( B_K(o) \times$ $[0,\infty)))=l \}$ . Thus,

\begin{equation*}\begin{aligned} \mathbb{P}\big(\textbf{X}^\lambda \in F\big) &\geq \mathbb{P}'\big(\textbf{X}^\lambda \in G, {\textbf{X}}^{\lambda,1} = \textbf{X}\big) \geq \mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in G\big)\mathbb{P}'\big({\textbf{X}}^{\lambda,1} = \textbf{X}^\lambda | {\textbf{X}}^{\lambda,1} \in G\big) \\[3pt] &= \varepsilon \mathbb{P}'\big({\textbf{X}}^{\lambda,1} = \textbf{X}^\lambda | {\textbf{X}}^{\lambda,1} \in G\big),\end{aligned}\end{equation*}

and further,

(5.17) \begin{equation}\begin{aligned} \mathbb{P}'\big({\textbf{X}}^{\lambda,1} = \textbf{X}^\lambda | {\textbf{X}}^{\lambda,1} \in G\big) & \geq \mathbb{P}'\big({\textbf{X}}^{\lambda,1} = \textbf{X}, {\textbf{X}}^{\lambda,1} \in G\big) = \mathbb{P}'\big({\textbf{X}}^{\lambda,2} = \varnothing, {\textbf{X}}^{\lambda,1} \in G\big).\end{aligned} \end{equation}

According to (5.16), we have

\begin{equation*} 0<\varepsilon = \mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in G\big) = \sum_{n=0}^{\infty}a_n,\end{equation*}

where $a_n=\mathbb{E}'\big[ \mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in G | \Lambda\big) \unicode{x1D7D9} \{ \Lambda(B_K(o)) \in [n,n+1) \} \big]$ , and thus there exists $m\in\mathbb{N}_0$ with $a_m>0$ . Now, conditional on $\Lambda$ , ${\textbf{X}}^{\lambda,1}$ is an i.i.d. marked PPP, and hence a PPP on $\mathbb{R}^d \times [0,\infty)$ , which also implies that the complementary thinnings ${\textbf{X}}^{\lambda,1}$ and ${\textbf{X}}^{\lambda,2}$ are independent given $\Lambda$ ; see the Colouring Theorem and the Marking Theorem in [Reference Kingman20]. Hence, we obtain

\begin{equation*}\begin{aligned}\mathbb{P}'\big({\textbf{X}}^{\lambda,2} = \varnothing, &{\textbf{X}}^{\lambda,1} \in G\big) = \mathbb{E}'\big[\mathbb{P}'\big({\textbf{X}}^{\lambda,2} = \varnothing|\Lambda\big) \mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in G|\Lambda\big) \big] \\[3pt] & = \sum_{n=0}^{\infty} \mathbb{E}'\big[ \varepsilon^{-(1-q)\Lambda(B_K(o))}\mathbb{P}'\big({\textbf{X}}^{\lambda,1} \in G |\Lambda\big) \unicode{x1D7D9}\{ \Lambda(B_K(o)) \in [n,n+1) \} \big]\\[3pt] & \geq \sum_{n=0}^{\infty} \varepsilon^{-(1-q)(n+1)} a_n\ge \varepsilon^{-(1-q)(m+1)} a_m>0,\end{aligned}\end{equation*}

which verifies the lemma that the distribution of ${\textbf{X}}^{\lambda,1}$ is absolutely continuous with respect to that of ${\textbf{X}}^\lambda$ .

Given Lemma 5.6, we now finish the proof of Lemma 5.5. Thanks to the assumption that $\mathbb{P}(C_{i,K,n})>0$ and using the definition of ${\textbf{X}}^{\lambda,1}$ ,

(5.18) \begin{equation}\begin{aligned} \mathbb P' \big( {\textbf{X}}^{\lambda,1} \in& \{ \boldsymbol \omega^i \colon \boldsymbol \omega \in \{L\ge 1\} \cap \{ I=i \} \}\big) \geq \mathbb P' \big( {\textbf{X}}^{\lambda,1} \in \{ \boldsymbol \omega^i \colon \boldsymbol \omega \in C_{i,K,n} \} \big) \\[3pt] & \geq \mathbb P' \big( {\textbf{X}}^{\lambda,1} \in \{ \boldsymbol \omega^i \colon\boldsymbol \omega \in C_{i,K,n}\}, {\textbf{X}}^\lambda \in C_{i,K,n} \big) \\[3pt] & = \mathbb{P}(C_{i,K,n}) \mathbb P'\big({\textbf{X}}^{\lambda,1} \in \{\boldsymbol \omega^i \colon \boldsymbol \omega \in C_{i,K,n} \}| {\textbf{X}}^\lambda \in C_{i,K,n} \big) \\[3pt] & = \mathbb{P}(C_{i,K,n}) q^{n-i+2} (1-q)^{i-2}>0.\end{aligned} \end{equation}

Finally, by Lemma 5.6, under $\mathbb P'$ the distribution of ${\textbf{X}}^{\lambda,1}$ is absolutely continuous with respect to that of ${\textbf{X}}^\lambda$ . Hence, it follows from (5.18) that

\begin{equation*} \mathbb P \big( {\textbf{X}}^\lambda \in \{ \boldsymbol \omega^i \colon \boldsymbol \omega \in \{L\ge 1\} \cap \{ I=i \} \} \big)>0, \end{equation*}

which implies the lemma.

5.3. Proof of Theorem 2.3

This proof is similar to that of Theorem 2.1, Condition 1, but simpler. The new proof ingredient that we use here is the strong connectivity of any supercritical Poisson Boolean model [Reference Tóbiás28, Theorems 2 and 5] in the case of $d\geq 2$ , which allows us to improve the result that $\lambda^*_\textrm{c}<\infty$ to $\lambda^*_\textrm{c}=\lambda_\textrm{c}(r_\textrm{B})$ . First we introduce an adequate discrete percolation model and then we control the interferences.

Throughout the proof, $X^\lambda=\{ x_i \}_{i \in \mathbb{N}}$ denotes a homogeneous PPP with intensity $\lambda$ in $\mathbb{R}^d$ , and we write $X^\lambda$ instead of ${\textbf{X}}^\lambda$ since marks are non-random. Let us introduce the notion and elementary properties of Boolean models with (constant) radius $r>0$ . The Poisson Boolean model $B(X^\lambda,r)$ (with constant connection radii r) is defined as

\begin{equation*} B(X^\lambda,r)= \bigcup_{i \in \mathbb{N}} B_{r}(x_i)=X^\lambda \oplus B_r(o). \end{equation*}

Connecting any two different points $x_i,x_j \in X^\lambda$ by an edge whenever

(5.19) \begin{equation} |x_i-x_j|<2r, \end{equation}

we obtain the Poisson–Gilbert graph $g_{2r}(X^\lambda)$ with connection radius 2r. Percolation in this Gilbert graph is equivalent to the existence of an unbounded connected component in $B(X^\lambda,r)$ , which we also refer to as percolation. Thus, one can speak about subcritical, critical, and supercritical Poisson Boolean models.

Recall the definition of the radius $r_\textrm{B}$ from (2.4), and let us fix $\lambda>\lambda_\textrm{c}(r_\textrm{B})$ for the remainder of this section. Thanks to scale invariance of Poisson Boolean models [Reference Meester and Roy25, Section 2.2] and the well-behavedness of $\ell$ , we can fix $r\in (d_o,r_\textrm{B})$ such that the Poisson Boolean model $B(X^\lambda,r/2)$ associated to $g_{r}(X^\lambda)$ is still supercritical. The next lemma is an immediate consequence of the results in [Reference Penrose and Pisztora26, Section 1].

Using Lemma 5.7, we construct a renormalized percolation process on $\mathbb{Z}^d$ . For $z \in \mathbb{Z}^d$ , let $\Xi_n(z)$ denote the union of all connected components of $B(X^\lambda,r/2) \cap Q_n(z)$ that are of diameter at least $n/3$ . For $n \geq 1$ , we say that the site $z \in \mathbb{Z}^d$ is n-good if

1. i. $\Xi_n(nz) \neq \varnothing$ , and

2. ii. for any $z' \in \mathbb{Z}^d$ with $|z-z'|_\infty \leq 1$ , it holds that all pairs of connected components C of $\Xi_n(nz)$ and C $^{\prime}$ of $\Xi_n(nz')$ are contained in the same connected component of $B(X^\lambda,r/2) \cap Q_{6n}(nz)$ .

The site $z \in \mathbb{Z}^d$ is n-bad if z is not n-good. We have the following lemma.

Lemma 5.8. Under the assumptions of Theorem 2.3, for all $n \geq 1$ sufficiently large, there exists $q_{A}=q_A(\lambda,n) \in (0,1)$ such that for any $N \in \mathbb{N}$ and pairwise distinct $z_1,\ldots,z_N \in \mathbb{Z}^d$ we have

\begin{equation*} \mathbb{P}(z_1,\ldots,z_N\text{ are all \textit{n}-bad}) \leq q_A^N. \end{equation*}

Furthermore, for any $\varepsilon>0$ , for all large enough n one can choose $q_A$ such that $q_A<\varepsilon$ .

Proof. For $z \in \mathbb{Z}^d$ , $\unicode{x1D7D9} \{ z\text{ is \textit{n}-good} \}$ is measurable with respect to $X^{\lambda} \cap(Q_{6n}(nz) \oplus B_{r/2}(o))$ , which is contained in $X^\lambda \cap Q_{7n}(nz)$ for all n large enough; hence for all sufficiently large n the process of n-good sites is 7-dependent thanks to the independence property of the PPP $X^\lambda$ . Hence, by a standard argument (using dependent percolation theory [Reference Liggett, Schonmann and Stacey23], as in the proof of Lemma 5.2), it suffices to verify that

(5.20) \begin{equation} \limsup_{n\uparrow\infty}\mathbb{P}(o\text{ is \textit{n}-bad})=0. \end{equation}

The limit (5.20) can be verified along the lines of the proof of [Reference Hirsch, Jahnel and Cali16, Theorem 2.6] using an adequate interpretation of the Poisson Boolean model. More precisely, in view of Definition 3.2, the assertion of Lemma 5.7 is equivalent to the statement in [Reference Hirsch, Jahnel and Cali16, Section 2.1] that the (for all sufficiently large $b>0$ ) b-dependent directing random measure $\Lambda$ given as $\Lambda(\textrm{d} x)= \lambda_1 \unicode{x1D7D9} \{ x \in B(X^\lambda,r/2) \} \textrm{d} x$ is asymptotically essentially connected, where $\lambda_1>0$ is such that $\mathbb{E}[\Lambda(Q_1)]=1$ .

The other essential proof ingredient is the interference control. We recall the ‘shifted’ path-loss functions $\ell_a$ (5.5) and the shot-noise processes $I_a(x),I(x)$ from Section 5.1, and also that by the triangle inequality, for $a \geq 0$ , $I(x) \leq I_{a}(z)$ holds for any $z \in \mathbb{R}^d$ and $x \in Q_a(z)$ .

For $n \geq 1$ and $M>0$ , we say that $z \in \mathbb{Z}^d$ is (n,M)-tame if $I_{7n}(nz) \leq M$ and (n,M)-wild otherwise. Then we have the following assertion, which holds for all $\lambda$ such that $B(X^\lambda,r/2)$ is supercritical.

Lemma 5.9. ([Reference Tóbiás28].) Under the assumptions of Theorem 2.3, for fixed $n \geq 1$ , for all sufficiently large $M>0$ , there exists $q_B=q_B(\lambda,n,M)\in (0,1)$ such that for any $N \in \mathbb{N}$ and pairwise distinct $z_1,\ldots,z_N \in \mathbb{Z}^d$ we have

\begin{equation*} \mathbb{P}(z_1,\ldots,z_N\text{ are all (\textit{n},\ \textit{M})-wild}) \leq q_B^N. \end{equation*}

Furthermore, for $\varepsilon>0$ , for any $n \geq 1$ , for all sufficiently large M one can choose $q_B$ such that $q_B < \varepsilon$ .

Proof. Clearly, the Lebesgue measure $\Lambda$ is asymptotically essentially connected and b-dependent for any $b>0$ , and $\Lambda(Q_1)$ has all exponential moments. Hence the lemma can be proven very similarly to [Reference Tóbiás28, Proposition 3.3] under the condition (2b) in [Reference Tóbiás28, Theorem 2.4]. The only difference is that in [Reference Tóbiás28], $I_{6n}(nz)$ was considered instead of $I_{7n}(nz)$ , but this makes no qualitative difference for the proof. Furthermore, the additional condition in [Reference Tóbiás28, Theorem 2.4] that $\ell(0) \leq 1$ can be assumed to hold without loss of generality, for the same reason as in the proof of Proposition 5.1 (see the beginning of Step 4 in the proof of Theorem 2.1).

Equipped with these results, we can now prove our main theorem.

Proof of Theorem 2.3. For $n \geq 1$ and $M>0$ , we say that the site $z \in \mathbb{Z}^d$ is (n, M)-nice if it is both n-good and (n, M)-tame. We claim that for all sufficiently large n and accordingly chosen large enough M, the process of (n, M)-nice sites percolates. Indeed, this follows from combining the estimates of Lemmas 5.7 and 5.9, similarly to Corollary 5.1, and carrying out a Peierls argument.

We claim that this assertion implies percolation in $G_{\gamma}(X^\lambda)$ for small $\gamma>0$ . Indeed, let n, M be so large that the process of (n, M)-nice sites percolates, and such that $Q_{6n}(o) \oplus B_{r/2}(o) \subseteq Q_{7n}(o)$ . Using a standard argument (cf. [Reference Dousse9] or Step 3 in the proof of Theorem 2.1, Condition 1), one can choose $\gamma>0$ sufficiently small so that for any (n, M)-tame site z, all connections in $g_r(X^\lambda) \cap Q_{7n}(nz)$ also exist in $G_{\gamma}(X^\lambda) \cap Q_{7n}(nz)$ .

Now, analogously to [Reference Hirsch, Jahnel and Cali16, Section 5.2], we can argue as follows. Let $\mathcal{C}$ be an infinite connected component of the process of sites that are (n, M)-nice. Let $z,z' \in \mathcal{C} $ and let $\{ z_0=z,z_1,\ldots,z_{k-1},z_k=z' \}$ be a path in $\mathcal{C}$ connecting z and z $^{\prime}$ . Then, thanks to n-goodness, for any $j=0,\ldots,k$ and for any $x_j \in X^\lambda$ such that $B_{r/2}(x_j) \cap Q_n(nz_j) \subseteq \Xi_n(nz_j)$ we have that $x_j$ and $x_{j+1}$ are in the same connected component of $B(X^\lambda,r/2)\cap Q_{6n}(nz_j)$ . In other words, $x_j$ and $x_{j+1}$ are connected in the Poisson–Gilbert graph $g_r(X^\lambda)$ via a path in $Q_{7n}(nz_j)$ , where the additional unit of n comes from the fact that the centers of balls in the Boolean model might lie in a neighboring box. Hence, using (n, M)-tameness, we conclude that all edges of this path in $g_r(X^\lambda)$ also exist in $G_{\gamma}(X^\lambda)$ . Thus, $G_{\gamma}(X^\lambda)$ also percolates. Since $\lambda>\lambda_\textrm{c}(r_\textrm{B})$ was arbitrary, the theorem follows.