2.1. Connectivity graph

Let $d\geq 2$ , $\lambda_{S} > 0$ , and let $X_{S}$ be a homogeneous Poisson point process in the state space $\mathbb{R}^{d}$ with intensity $\lambda_S \in (0,\infty)$ . Consider the Poisson–Voronoi tessellation (PVT) S associated with $X_S$ . In particular, S is stationary and isotropic. By analogy with a telecommunications network, S will be called the street system from now on.

Denote by $E \,{:}\,{\raise-1.5pt{=}}\, (e_{i})_{i \geq 1}$ the edge-set of S and by $V \,{:}\,{\raise-1.5pt{=}}\, (v_{i})_{i \geq 1}$ the vertex-set of S. In other words, the elements of E are the 1-facets of S and the elements of V are the 0-facets of S. Furthering the analogy with a telecommunications network, the elements of E (respectively V) will be called streets (respectively crossroads).

Let $\Lambda(dx) \,{:}\,{\raise-1.5pt{=}}\, \nu_{1}(S \cap dx)$ , where $\nu_{1}$ denotes the one-dimensional Hausdorff measure of $\mathbb{R}^{d}$ . Observe that $\Lambda$ is a stationary random measure on $\mathbb{R}^d$ ( $\Lambda(B)$ is the total edge length of S contained in any Borel set $B \subset \mathbb{R}^d$ ) with finite non-null intensity $\gamma\,{:}\,{\raise-1.5pt{=}}\,\mathbb{E}[\Lambda[0,1]^d] \in (0,\infty)$ . Note that $\gamma$ may also be interpreted as the total edge length per unit volume: for this reason, we call $\gamma$ the street intensity. In the planar case $(d=2)$ , it is known that $\gamma = 2\sqrt{\lambda_S}$ (see [Reference Chiu, Stoyan, Kendall and Mecke18, Section 9.7.2]). More general formulae for $\gamma$ in any dimension $d \geq 2$ can be found in [Reference MØller33, Reference MØller34, Reference Muche35].

For $\lambda>0$ consider a Cox point process $X^{\lambda}$ driven by the random intensity measure $\lambda \Lambda$ . In other words, conditioned on a given realization of the street system S, $X^{\lambda}$ is a Poisson point process with mean measure $\lambda \Lambda$ . In accordance with the telecommunication interpretation of the PVT as a street system, the points of $X^{\lambda}$ model locations of users (equipped with mobile devices). In particular, the number of users on a given street $e \in E$ is a Poisson random variable with mean $\lambda\nu_{1}(e)$ , and the numbers of users on two disjoint subsets of E are independent random variables.

We consider yet another point process on the PVT, namely a doubly stochastic Bernoulli point process Y on the set of crossroads V with parameter p: conditioned on $\Lambda$ (or, equivalently, on the PVT S), points of Y are placed on the crossroads V of S independently with probability p. The points of Y will be called (fixed) relays. We also assume that the processes of users and of relays are conditionally independent given their random support, i.e. $X^{\lambda} \╨ Y \, \vert \, \Lambda$ . We denote by $Z \,{:}\,{\raise-1.5pt{=}}\, X^{\lambda} \cup Y$ the superposition of users and relays.

We consider an undirected connectivity graph $\mathcal{G}$ with the set of vertices given by the points of the point process Z, and an edge $Z_{i} \leftrightsquigarrow Z_{j}$ , $i \neq j$ , if and only if $Z_{i}$ and $Z_{j}$ are located on the same street and of mutual Euclidean distance less than r:

(1) \begin{equation}\forall \, i \neq j, \qquad Z_{i} \leftrightsquigarrow Z_{j} \Leftrightarrow\left\{\begin{array}{l}\exists \, e \in E, \, Z_{i} \in e \ \text{and} \ Z_{j} \in e, \\ \lVert Z_{i} - Z_{j} \rVert \leq r.\end{array}\right.\end{equation}

Figure 1 illustrates a realization of our network model and of its connectivity graph in the planar case ( $d=2$ ).

Figure 1:

Illustration of the connectivity graph $\mathcal{G}$ in a bounded simulation window of the plane $\mathbb{R}^2$ . The dashed blue lines illustrate the PVT supporting the Cox point process of users represented by red circular points. The Bernoulli point process of relays is illustrated by the green triangular points. Finally, the edges of the largest connected component of $\mathcal{G}$ in the simulation window are the thicker solid segments highlighted in orange, while the edges of smaller connected components are the thinner solid segments highlighted in black. These connected components illustrate the connectivity mechanism given by (1).

In this paper, we study the percolation properties of the graph $\mathcal{G}$ . More precisely, we identify regimes (i.e. sets of model parameters $p,\lambda,r$ ) where $\mathcal{G}$ percolates (i.e. has an infinite component) with positive probability.

2.2. Dimensionless model parameters

While our original percolation model parameters are $\gamma$ , p, $\lambda$ , and r, it is customary to introduce the following dimensionless model parameters by proceeding as follows. Such parameters turn out to be much more convenient for numerical simulations, as they are also scale-invariant.

Denote by $\bar{l}$ the mean length of the typical street (that is to say, the mean length of the typical point of the process of 1-facets of S; see [Reference Okabe, Boots, Sugihara and Chiu37]). A general formula for $\bar{l}$ in any dimension $d\geq 2$ is available in [Reference MØller33, Reference Muche35]. In particular, there exists a positive constant $\kappa\,{:}\,{\raise-1.5pt{=}}\,\kappa(d)$ such that $\bar{l}=\kappa(d)\lambda_S^{-1/d}$ . Now, introduce the following dimensionless parameters:

\begin{equation*}U\,{:}\,{\raise-1.5pt{=}}\, \lambda \bar{l} \quad \text{and} \quad H\,{:}\,{\raise-1.5pt{=}}\,\frac{\bar{l}}{r}.\end{equation*}

U corresponds to the mean number of users per typical street, while H is the mean number of hops (of length r) required to traverse the typical street. It is easily shown that for all $d \geq 2$ , $\bar{l}=\infty \Leftrightarrow \gamma = 0$ and $\bar{l}=0 \Leftrightarrow \gamma = \infty$ . In what follows, we will thus denote by $\mathcal{G}_{p,U,H}$ the connectivity graph $\mathcal{G}$ as a function of the model parameters (p, U, H) in the domain $p \in \left[0;\ 1\right]$ , $U\ge 0$ , $H\ge 0$ , with $H=0$ interpreted as $r=\infty$ for some $0<\gamma<\infty$ and $U=0$ interpreted as $\lambda=0$ for some $0<\gamma<\infty$ . Note that since S is stationary and isotropic, percolation of the connectivity graph $\mathcal{G}$ does not actually depend on the parameter $\gamma$ . If needed, we can thus fix a value $\gamma$ once and for all (e.g. $\gamma=1$ ), so that there is a one-to-one map $(p,\lambda,r) \mapsto (p,U,H)$ between the original and the dimensionless parameters.

2.4. Numerical observations in the planar case

In what follows, for the sake of visualization, we briefly recall some numerical findings in the planar case (i.e. setting $d=2$ ) that were obtained in an earlier publication [Reference Le Gall, Błaszczyszyn, Cali and En-Najjary30], to which we refer the reader for further information regarding the simulation methodology. Note that in the planar case, it is known (see e.g. [Reference Okabe, Boots, Sugihara and Chiu37]) that we have

\begin{equation*} \gamma = 2\sqrt{\lambda_S} \qquad and \qquad \bar{l}=\frac{2}{3\sqrt{\lambda_S}};\end{equation*}

as a consequence of the two previous formulae, we have $\kappa(2)=\frac{2}{3}$ , and the length of the typical edge can be written as $\bar{l}=4/(3\gamma)$ .

The estimates for the critical values $p^*$ and $H_c$ were found to be close to $p^*\approx 0.713$ and $H_c\approx 0.743$ . Recall that $H_c$ concerns a model which, as far as we know, has not yet been studied in the literature. As for the estimate for $p^*$ , our value differs only slightly from the most recent estimate available in the literature [Reference Becker and Ziff6], which is $p^* \approx 0.71410 \pm 0.00002$ . While the authors providing this estimate proceeded with Monte Carlo simulations with periodic boundary conditions and investigated the growth of the largest cluster, we chose a crossing-window method to obtain our estimate of $p^*$ .

For $H<H_c$ define

\begin{equation*} p_c(H) \,{:}\,{\raise-1.5pt{=}}\, \inf \lbrace p > 0 \, : \, P(p,0,H)> 0 \rbrace.\end{equation*}

This is hence the lower boundary of the strictly supercritical range of (p, H). Some estimated values of the function $p_c(H)$ are presented in Figure 2.

Finally, Figure 4 presents some estimated values of $U_c(p,H)$ for some selected parameters (p, H) (mainly in the critical range).

Figure 4:

Numerical estimates of $U_c(p,H)$ for some values of (p, H).

3.1. General approach

Theorems 1–3 and Corollary 2 have been stated in terms of the dimensionless parameters (p, U, H) introduced in Subsection 2.2. This allowed us to introduce the different connectivity regimes and their frontiers in a more applied fashion, as illustrated by Figures 2 and 3.

However, when working on the proofs of the aforementioned results, it will be much more convenient to come back to the original parameters $(\gamma, p, \lambda,r)$ . This is mostly due to the fact that H, being inversely proportional to r, is less easy to work with when we are considering particular events related to connectivity in the network graph. Moreover, recalling that percolation of the connectivity graph $\mathcal{G}$ does not depend on the parameter $\gamma$ , we can set $\gamma=1$ without loss of generality in what follows. Switching back to the original network parameters $(\gamma=1,p,\lambda,r)$ , we will therefore now refer to the connectivity graph as $\mathcal{G}=\mathcal{G}_{p,\lambda,r}$ and prove the following equivalent formulations of Theorems 1–3.

Reformulation 1. (Reformulation of Theorem 1.) For small enough $r \in \left(0,\infty\right]$ and small enough $\lambda>0$ (with the thresholds for r and $\lambda$ not depending on one another), $\mathcal{G}_{1,\lambda,r}$ does not percolate; consequently, neither does $\mathcal{G}_{p,\lambda,r}$ for any $p \in \left[0,1\right]$ .

Reformulation 2. (Reformulation of Theorem 2.) For any $r \in \left(0,\infty\right]$ , for large enough $p \in (0,1)$ and $\lambda < \infty$ (the thresholds for p and $\lambda$ depend on r but not on one another), $\mathcal{G}_{p,\lambda,r}$ percolates.

Reformulation 3. (Reformulation of Theorem 3.) For large enough $p<1$ and large enough $r<\infty$ (with the thresholds for p and r not depending on one another), $\mathcal{G}_{p,0,r}$ percolates.

The general idea of the proofs of the above reformulations is to use a coarse-graining argument. This consists in the following steps:

1. 1. Map the percolation of $\mathcal{G}$ to a discretized percolation process on a rescaled integer lattice.

2. 2. As needed, relate the percolation (or absence of percolation) of $\mathcal{G}$ to the percolation (or absence of percolation) of the discretized process.

3. 3. Prove that the discretized process features only short-range dependencies. More precisely, we will refer to the notion of k-dependence, which will be introduced in due course; see Definition 5.

4. 4. Apply a result from Liggett, Schonmann and Stacey ([Reference Liggett, Schonmann and Stacey31, Theorem 0.0]) to conclude that the discretized process is dominated by a Bernoulli percolation process, and thereby draw conclusions on $\mathcal{G}$ .

3.2. Preparation

We begin by introducing some notation and definitions that will be useful for the purposes of our development.

We will use the following convenient notation for the length $\vert s \vert$ of a street segment s (which is a connected, topologically closed subset of some $e \in E$ ). For $A \subset \mathbb{R}^{d}$ and $B \subset \mathbb{R}^{d}$ , we denote the Euclidean distance between A and B (as customary) by

\begin{equation*}\text{dist}(A,B) \,{:}\,{\raise-1.5pt{=}}\, \inf \lbrace \lVert x - y \rVert_{2} , \, x \in A, \, y \in B \rbrace.\end{equation*}

For $x \in \mathbb{R}^{d}$ , $a>0$ we denote by $Q_a(x) \,{:}\,{\raise-1.5pt{=}}\, x + \left[-a/2,a/2\right]^{d}$ the d-dimensional cube of side length a centered at x. We note that this is exactly the definition of the closed ball $\mathscr{B}(x,a/2)$ with center x and radius $a/2$ for the infinite norm of $\mathbb{R}^d$ :

\begin{equation*} Q_a(x) = \lbrace y \in \mathbb{R}^d \,:\, \lVert y-x \rVert_{\infty} \leq a/2 \rbrace = \mathscr{B}(x,a/2).\end{equation*}

For simplicity, whenever $a=n \in \mathbb{N} \setminus \lbrace 0 \rbrace$ , we will write $Q_n$ to mean $Q_n(0)$ .

We denote by $\textbf{M}$ the space of Borel measures on $\mathbb{R}^{d}$ , equipped with the evaluation $\sigma$ -algebra [Reference Last and Penrose29, Section 13.1], which is the smallest $\sigma$ -algebra making the mappings $\textbf{M} \ni \Xi \mapsto \Xi(B)$ measurable for all Borel sets $B \subset \mathbb{R}^{d}$ . For a (possibly random) Borel measure $\mu$ on $\mathbb{R}^{d}$ and $A \subset \mathbb{R}^{d}$ , we denote the restriction of $\mu$ to A by $\mu_A({\cdot}) \,{:}\,{\raise-1.5pt{=}}\, \mu(A \cap \cdot)$ . We also adapt the definition of the support of a measure as follows: let $\mu$ be a (possibly random) Borel measure on $\mathbb{R}^{2}$ . The support of $\mu$ is the following set:

\begin{equation*}\text{supp}(\mu) \,{:}\,{\raise-1.5pt{=}}\, \lbrace x \in \mathbb{R}^{d} \, :\, \forall \epsilon > 0, \, \mu(Q_{\epsilon}(x)) > 0 \rbrace.\end{equation*}

We will also need the concepts of stabilization and asymptotic essential connectedness, both introduced in [Reference Hirsch, Jahnel and Cali26] for investigating spatial dependencies of random measures.

For the sake of simplicity, we slightly modify the definition of asymptotic essential connectedness given in [Reference Hirsch, Jahnel and Cali26], using the following definition instead.

Definition 2. Let $\Xi$ be a random measure on $\mathbb{R}^d$ . Then $\Xi$ is asymptotically essentially connected if there exists a random field $R = \lbrace R_{x} \rbrace_{x \in \mathbb{R}^{d}}$ such that $\Xi$ is stabilizing with R as in Definition 1, and if for all $n \geq 1$ , whenever

\begin{equation*}\displaystyle \sup_{y \in Q_{2n} \cap \mathbb{Q}^{d}} R_{y} < n/2,\end{equation*}

the following assertions hold:

1. (1) $\text{supp}(\Xi_{Q_{n}}) \neq \emptyset$ ;

2. (2) $\text{supp}(\Xi_{Q_{n}})$ is contained in a connected component of $\text{supp}(\Xi_{Q_{2n}})$ .

The following result is stated in [Reference Hirsch, Jahnel and Cali26, Example 3.1] for a slightly modified version of Definition 2. It is easy to check that it adapts to our case as follows.

Proposition 3. Let $\Lambda = \nu_{1}(S \cap dx)$ , where S is the PVT generated by a homogeneous stationary Poisson point process. Then $\Lambda$ is stabilizing and asymptotically essentially connected with the following stabilization field:

\begin{equation*}\forall x \in \mathbb{R}^{d}, \qquad R_x \,{:}\,{\raise-1.5pt{=}}\, \inf \lbrace \lVert x - X_{S,i} \rVert_2 , \, X_{S,i} \in X_{S} \rbrace,\end{equation*}

where $X_S$ is the Poisson point process generating S.

For simplicity, for $x \in \mathbb{R}^{d}$ and $n \in \mathbb{N}\setminus \lbrace 0 \rbrace$ we define

\begin{equation*}R(Q_n(x)) \,{:}\,{\raise-1.5pt{=}}\, \sup_{y \in Q_n(x) \cap \mathbb{Q}^{d}} R_y.\end{equation*}

Finally, we define the openness and closedness of crossroads and street segments (including whole streets) as follows.

We say that s is closed if s is not open.

We are now ready to proceed with the proofs of Theorems 1–3.

3.3. Proof of Proposition 1

To prove that Z is mixing, we will work on the canonical space; it will thus suffice to show that

(2) \begin{equation} \lim_{\lVert x \rVert_2 \rightarrow \infty} \mathbb{P}(A \cap S_xB) = \mathbb{P}(A)\mathbb{P}(B)\end{equation}

for all events $A,B \in \sigma(Z)$ that are measurable with respect to the sigma-algebra $\sigma(Z)$ generated by Z, where $\lbrace S_x \rbrace_{x \in \mathbb{R}^d}$ denotes the natural shift on $\mathbb{R}^d$ .

Note first that by [Reference Daley and Vere-Jones19, Lemma 12.3.II] (and as done in the proof of [Reference Daley and Vere-Jones19, Proposition 12.3.VI]), it suffices to check the mixing condition (2) for local events, i.e. events of the form $A \in \sigma(Z \cap W_A)$ and $B \in \sigma(Z \cap W_B)$ , where $W_A$ and $W_B$ are compact observation windows in $\mathbb{R}^d$ . Thus, let A and B be such events. We will show that for all $\epsilon >0$ , we can find x with $\lVert x \rVert_2$ sufficiently large so that $ \vert \mathbb{P}(A \cap S_xB) - \mathbb{P}(A)\mathbb{P}(B) \vert \leq \epsilon$ .

Take any $\epsilon >0$ . By Condition 2 in the definition of stabilization (Definition 1), we can find sufficiently large $n \geq 1$ such that $\mathbb{P}(R(Q_n) \geq n)\leq \epsilon/3$ . Moreover, such n can be chosen so as to satisfy $W_A \subseteq Q_n$ and $W_B \subseteq Q_n$ . Fix such n.

Since $A \in \sigma(Z \cap W_A)$ , A depends only on the configuration of Z inside $W_A$ . In the same way, B depends only on the configuration of Z inside $W_B$ , and $S_xB$ depends only on the configuration of Z inside $S_xW_B = W_B -x$ . Since $B \subseteq Q_n$ , we have that $S_xW_B \subseteq Q_n -x \,\,{\raise-1.5pt{=}}{:}\, Q_n(\!-x)$ . Take x with $\lVert x \rVert_2 > 6n\sqrt{2}$ . Then we have $Q_n \cap Q_n(\!-x) = \emptyset$ and thus $W_A \cap S_xW_B = \emptyset$ , so that the events A and $S_xB$ depend on the configuration of Z in disjoint sets. Since the conditional distribution of Z given the random support $\Lambda$ is that of a superposition of a Bernoulli process and of a Poisson point process, the events A and $S_xB$ are conditionnally independent given $\Lambda$ . Hence,

(3) \begin{equation} \mathbb{P}(A \cap S_xB) = \mathbb{E} \Big[\mathbb{E}\left(\mathbb{1} \lbrace A \rbrace \mathbb{1} \lbrace S_xB \rbrace \, \vert \, \Lambda \right) \Big] = \mathbb{E} \Big[\mathbb{E} (\mathbb{1} \lbrace A \rbrace \, \vert \, \Lambda) \mathbb{E} \left( \mathbb{1} \lbrace S_xB \rbrace \, \vert \, \Lambda \right) \Big].\end{equation}

Now, since $A \in \sigma(Z \cap W_A)$ with $W_A \subseteq Q_n$ , we can write $\mathbb{E}(\mathbb{1} \lbrace A \rbrace \, \vert \Lambda ) = f(\Lambda_{Q_n})$ as a bounded deterministic function of $\Lambda_{Q_n}$ . In the same way, we can write $\mathbb{E}(\mathbb{1} \lbrace S_xB \rbrace \, \vert \Lambda ) = g(\Lambda_{Q_n(\!-x)})$ as a bounded deterministic function of $\Lambda_{Q_n(\!-x)}$ . By (3), we thus get

(4) \begin{align} & \mathbb{P}(A \cap S_x B)= \mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x})\right] \\ \begin{split}&=\mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace R(Q_n) < n \rbrace \mathbb{1} \lbrace R(Q_n(\!-x))< n \rbrace \right] \\ &\quad +\mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace (R(Q_n) \geq n) \cup ( R(Q_n(\!-x)) \geq n) \rbrace \right].\end{split}\end{align}

Let us first deal with the second term appearing in the right-hand side of (4). Using the fact that both f and g, being conditional expectations of indicator functions, are bounded above by 1, we get

(5) \begin{align} & \mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace (R(Q_n) \geq n) \cup ( R(Q_n(\!-x)) \geq n) \rbrace \right]\nonumber \\ & \qquad\leq \mathbb{P}\left[ (R(Q_n) \geq n) \cup ( R(Q_n(\!-x)) \geq n) \right]\nonumber \\ & \qquad \leq \mathbb{P}\left[R(Q_n) \geq n\right] + \mathbb{P}\left[R(Q_n(\!-x)) \geq n\right],\end{align}

where we have used the union bound in (5). Now, by stationarity of the stabilization field $\lbrace R_y \rbrace_{y \in \mathbb{R}^d}$ , we get that the right-hand side in (5) is equal to $2\mathbb{P}\left[R(Q_n) \geq n\right]$ . In all, we thus get

(6) \begin{equation}\mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x}) \mathbb{1} \lbrace (R(Q_n) \geq n) \cup ( R(Q_n(\!-x)) \geq n) \rbrace \right] \leq 2\epsilon/3.\end{equation}

We now deal with the first term appearing in the right-hand side of (4). Note that since $\lVert x \rVert_2 > 6n\sqrt{2}$ , the set $\varphi \,{:}\,{\raise-1.5pt{=}}\, \lbrace 0,-x \rbrace \subset \mathbb{R}^d$ satisfies $\forall y \in \varphi, \text{dist}(y,\phi\setminus \lbrace y \rbrace) > 3n$ , and so, by Condition 3 in the definition of stabilization, the random variables $f(\Lambda_{Q_n})\mathbb{1}\lbrace R(Q_n) <n \rbrace$ and $g(\Lambda_{Q_n(\!-x)})\mathbb{1}\lbrace R(Q_n(\!-x)) <n \rbrace$ are independent. Thus, the first term appearing in the right-hand side of (4) becomes

(7) \begin{align}\mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace R(Q_n)\right. & \left.\lt n \rbrace \mathbb{1} \lbrace R(Q_n(\!-x)) \lt n \rbrace \right] \nonumber\\ & = \mathbb{E}\left[ f(\Lambda_{Q_n}) \mathbb{1} \lbrace R(Q_n) < n \rbrace\right]\mathbb{E}\left[ g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace R(Q_n(\!-x)) < n \rbrace \right].\nonumber\\\end{align}

Now, using the fact that $f(\Lambda_{Q_n})\,\,{\raise-1.5pt{=}}{:}\,\mathbb{E}\left( \mathbb{1} \lbrace A \rbrace \, \vert \, \Lambda \right)$ and noting that the event $\lbrace R(Q_n) < n \rbrace$ is $\Lambda$ -measurable, we can put everything back into a single expectation and get

\begin{align*} \mathbb{E}\left[ f(\Lambda_{Q_n}) \mathbb{1} \lbrace R(Q_n) < n \rbrace\right] &= \mathbb{E}\left[ \mathbb{E}\left( \mathbb{1} \lbrace A \rbrace \, \vert \, \Lambda \right) \mathbb{1} \lbrace R(Q_n) < n \rbrace\right] \\ &= \mathbb{E}\left[ \mathbb{E}\left( \mathbb{1} \lbrace A \rbrace \mathbb{1} \lbrace R(Q_n) < n \rbrace \, \vert \, \Lambda \right)\right] \\ &= \mathbb{E}\left[ \mathbb{1} \lbrace A \rbrace \mathbb{1} \lbrace R(Q_n) < n \rbrace \right] \\ &= \mathbb{P}(A \cap \lbrace R(Q_n) < n \rbrace ).\end{align*}

In the same way, we get

\begin{equation*}\mathbb{E}\left[ g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace R(Q_n(\!-x)) < n \rbrace\right] = \mathbb{P}(S_xB \cap \lbrace R(Q_n(\!-x)) < n \rbrace ).\end{equation*}

Thus, (7) yields

\begin{align*} \mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace R(Q_n)\right. & \left. < n \rbrace \mathbb{1} \lbrace R(Q_n(\!-x)) < n \rbrace \right] \\ &=\mathbb{P}(A \cap \lbrace R(Q_n) < n \rbrace )\mathbb{P}(S_xB \cap \lbrace R(Q_n(\!-x)) < n \rbrace ) \\ &=\mathbb{P}(A \cap \lbrace R(Q_n) < n \rbrace )\mathbb{P}\circ S_x(B \cap \lbrace R(Q_n) < n \rbrace ) \\ & = \mathbb{P}(A \cap \lbrace R(Q_n) < n \rbrace )\mathbb{P}(B \cap \lbrace R(Q_n) < n \rbrace ),\end{align*}

where we have used the stationarity assumption to get the last line. Finally, using the fact that $\mathbb{P}(R(Q_n)<n)\geq 1-\epsilon/3$ , we get

\begin{equation*}\vert \mathbb{P}(A \cap \lbrace R(Q_n) < n \rbrace )\mathbb{P}(B \cap \lbrace R(Q_n) < n \rbrace ) - \mathbb{P}(A)\mathbb{P}(B) \vert \leq \epsilon/3\end{equation*}

and thus

(8) \begin{equation} \vert \mathbb{E}\left[ f(\Lambda_{Q_n})g(\Lambda_{Q_n(\!-x)}) \mathbb{1} \lbrace R(Q_n) < n \rbrace \mathbb{1} \lbrace R(Q_n(\!-x)) < n \rbrace \right] - \mathbb{P}(A)\mathbb{P}(B) \vert \leq \epsilon/3.\end{equation}

Using (6) and (8) to put everything back together in (4) and using the triangle inequality, we finally get

\begin{equation*} \lvert \mathbb{P}(A \cap S_xB) - \mathbb{P}(A)\mathbb{P}(B) \rvert \leq 2\epsilon/3 + \epsilon/3 = \epsilon\end{equation*}

for sufficiently large x, as required. This concludes the proof of Proposition 1.

3.4. Proof of Theorem 1

As mentioned earlier, proving Theorem 1 is equivalent to proving Reformulation 1. This in turn is equivalent to showing that $\mathcal{G}$ does not percolate when $p=1$ and $\lambda, r$ are sufficiently small but positive. We will use a coarse-graining argument and introduce a discrete site percolation model on the integer lattice $\mathbb{Z}^d$ , constructed in such a way that if it does not percolate, then neither does $\mathcal{G}$ . Proving the absence of percolation of the integer lattice model will then be done by appealing to its local dependence.

To this end, for $n \geq 1$ , we say a site $z \in \mathbb{Z}^{d}$ is n-good if the following conditions are satisfied:

1. (1) $R(Q_n(nz)) < n$ ;

2. (2) for any $e \in E$ , if $s_{z,e} \,{:}\,{\raise-1.5pt{=}}\, e \cap Q_n(nz)\not=\emptyset$ , then $s_{z,e}$ is closed.

We say a site $z \in \mathbb{Z}^{d}$ is n-bad if it is not n-good.

Our first claim is the following.

Proof. Assume $\mathcal{G}$ percolates and denote by $\mathcal{C}$ an unbounded (connected) component of $\mathcal{G}$ . Define $\mathcal{Z}=\mathcal{Z}_n\,{:}\,{\raise-1.5pt{=}}\,\lbrace z\in \mathbb{Z}^d\,:\, \mathcal{C} \cap Q_n(nz) \neq \emptyset \rbrace$ . Since $\mathcal{C}$ is unbounded we have $\#(\mathcal{Z})=\infty$ . Observe that for all $z\in\mathcal{Z}$ , z is n-bad, since Condition 2 of n-goodness is not satisfied (there exists an open street segment intersecting $Q_n(nz)$ ). Also, $\mathcal{Z}$ is almost surely connected in $\mathbb{Z}^d$ (in the following sense: $z,z'\in\mathbb{Z}^d$ , $z\not=z'$ are connected in $\mathbb{Z}^d$ if $\lVert z-z' \rVert_{1}=1$ ). This follows from the fact that the probability that some edge $e\in E$ of the PVT intersects $\mathbb{Z}^d$ is equal to zero. (This is actually also true for the Voronoi tessellation generated by any stationary point process, as a consequence of the fact that such a process does not have points which are equidistant to a given, fixed location; see e.g. [8, Lemma 11.2.3].) Hence, the process of n-bad sites percolates.

By Lemma 1, it suffices to prove that the process of n-bad sites does not percolate (for some n) when $\lambda$ and r are sufficiently small but positive. This will be done using the fact that it is a 3-dependent percolation model on the integer lattice $\mathbb{Z}^d$ .

As previously stated, we have the following.

Proof. As a starting point, note that for every $z \in \mathbb{Z}^{d}$ , $\zeta_{z} = 1 - \mathbb{1} \lbrace z\,{\rm{is}}\,n - {\rm{good}} \rbrace$ . It is therefore equivalent to prove that the process of n-good sites is 3-dependent.

For $z \in \mathbb{Z}^{d}$ , set $\xi_{z} = \mathbb{1} \lbrace z\,{\rm{is}}\,n - {\rm{good}} \rbrace$ . Let $ \lbrace z_{1}, \ldots z_{q} \rbrace \subset \mathbb{Z}^{d}$ be such that for any $i \neq j$ , $\lVert z_i - z_j \rVert_{\infty} > 3$ . We want to show that the random variables $${({\xi _{{z_i}}})_{1 \le i \le q}}$$ are independent. Since we are dealing with indicator functions, this is equivalent to showing that

\begin{equation*}\mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) = \prod_{i=1}^{q} \mathbb{E}(\xi_{z_i}).\end{equation*}

Now, we have

(9) \begin{align} \nonumber \mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) &= \mathbb{E}\left[\mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \Bigg\vert \Lambda \right)\right] \\ \nonumber &= \mathbb{E}\Bigg[\mathbb{E}\Bigg(\prod_{i=1}^{q}\mathbb{1}\lbrace R(Q_n(nz_i)) < n \rbrace \prod_{i=1}^{q} \mathbb{1} \lbrace \forall \, e \in E, \, s_{z_i,e} \, \text{is closed} \rbrace \Bigg\vert \Lambda \Bigg) \Bigg] \\ &= \mathbb{E}\Bigg[\prod_{i=1}^{q} \mathbb{1} \lbrace R(Q_n(nz_i)) <n \rbrace \mathbb{E}\Bigg( \prod_{i=1}^{q} \mathbb{1} \lbrace \forall \, e \in E, \, s_{z_i,e} \, \text{is closed} \rbrace \Bigg\vert \Lambda \Bigg) \Bigg],\end{align}

where we have used $\Lambda$ -measurability of the random variables $\lbrace R_x \rbrace_{x \in \mathbb{R}^d}$ in (9).

For $1 \leq i \leq q$ , set $A_{z_i} \,{:}\,{\raise-1.5pt{=}}\, \lbrace \forall \, e \in E, \, s_{z_i,e} \, \text{is closed} \rbrace$ . According to Definition 4, for a given $1 \leq i \leq q$ , the event $A_{z_i}$ depends only on the configuration of the random measure $\Lambda$ and of the Cox point process $X^{\lambda}$ inside the d-dimensional cube $Q_n(nz_i)$ . Therefore, given $\Lambda$ , the events $\lbrace A_{z_i} \,:\, 1 \leq i \leq q \rbrace$ depend only on $X^{\lambda} \cap Q_n(nz_i)$ , $1 \leq i \leq q$ . Since we have $\lVert z_i - z_j \rVert_{\infty} > 3$ for all $i \neq j$ , the d-dimensional cubes $Q_n(nz_i)$ are disjoint. Moreover, given $\Lambda$ , $X^{\lambda}$ has the distribution of a Poisson point process. Thus, by the Poisson independence property, the events $${({A_{{z_i}}})_{1 \le i \le q}}$$ are conditionally independent given $\Lambda$ . Hence (9) yields

(10) \begin{align} \mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) &= \mathbb{E}\Bigg[\prod_{i=1}^{q} \mathbb{1} \lbrace R(Q_n(nz_i)) <n \rbrace \prod_{i=1}^{q} \mathbb{E}\Bigg( \mathbb{1} \lbrace \forall \, e \in E, \, s_{z_i,e} \, \text{is closed} \rbrace \Bigg\vert \Lambda \Bigg) \Bigg].\end{align}

Recall that the restriction of the random measure $\Lambda$ to the d-dimensional cube $Q_n(x)$ is denoted by $\Lambda_{Q_n(x)} ({\cdot}) \,\,{\raise-1.5pt{=}}{:}\, \Lambda(Q_n(x) \cap \, \cdot)$ , and set

\begin{equation*}f(\Lambda_{Q_n(x)}) \,{:}\,{\raise-1.5pt{=}}\, \mathbb{E}\Bigg( \mathbb{1} \lbrace \forall \, e \in E, \, s_{x,e} \, \text{is closed} \rbrace \Bigg\vert \Lambda \Bigg).\end{equation*}

Then f is a deterministic bounded and measurable function of $\Lambda_{Q_n(x)}$ . Moreover, the set $\varphi \,{:}\,{\raise-1.5pt{=}}\, \lbrace nz_1,\ldots,nz_q \rbrace \subset \mathbb{R}^{d}$ is a finite subset of $\mathbb{R}^d$ satisfying

\begin{equation*}\forall i \neq j, \qquad \lVert nz_i - nz_j \rVert_{\infty} > 3n.\end{equation*}

Since the infinite norm is always bounded above by the Euclidean norm, we have $\lVert nz_i - nz_j \rVert_{2} > 3n $ for all $i \neq j$ , and so $\varphi$ satisfies

\begin{equation*}\forall x \in \varphi, \, \text{dist}(x, \varphi \setminus \lbrace x \rbrace) > 3n.\end{equation*}

Hence, by Condition 3 in the definition of stabilization (Definition 1), the random variables appearing in the right-hand side of (10) are independent. This yields

\begin{align*} \mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) &= \prod_{i=1}^{q} \mathbb{E}\Bigg(\mathbb{1} \lbrace R(Q_n(nz_i)) <n \rbrace \prod_{i=1}^{q} \mathbb{E}\Bigg( \mathbb{1} \lbrace \forall \, e \in E, \, s_{z_i,e} \, \text{is closed} \rbrace \Bigg\vert \Lambda \Bigg) \Bigg) \\ &=\prod_{i=1}^{q} \mathbb{E}\left(\xi_{z_i} \right), \end{align*}

thus concluding the proof of the lemma.

Now we prove that the probability that an arbitrary site, which by stationarity can be chosen to be the origin $\boldsymbol{0} \in \mathbb{Z}^{d}$ , is n-bad can be made arbitrarily small by first taking some large enough finite n and then small enough positive $\lambda,r$ , as stated in the following lemma.

Lemma 3.

\begin{equation*} \lim_{n \uparrow \infty}\lim_{\lambda, r \downarrow 0} \mathbb{P}(\boldsymbol{0}\ is\ n\hbox{-}bad) = 0.\end{equation*}

Proof. Note that we have

\begin{align*}\mathbb{P}(\boldsymbol{0} \, \text{is \textit{n}-bad}) &= \mathbb{P}\Big(\lbrace R(Q_n) \geq n \rbrace \cup \lbrace \exists \, e \in E\,:\, e \cap Q_n\not=\emptyset \text{ and open} \rbrace \Big) \\ & \leq \mathbb{P}\left( R(Q_n) \geq n \right) + \mathbb{P}\left( \exists \, e \in E\,:\, e \cap Q_n\not=\emptyset \text{ and open} \right)\\ &\le\mathbb{P}\left( R(Q_n) \geq n \right) \text{(a)}\\ &\quad + \mathbb{P}\left( \exists \, e \in E\ :\ 0<|e \cap Q_n|\le r \right)\text{(b)}\\ &\quad + \mathbb{P}\left( \exists \, e \in E\ :\ e \cap Q_n \text{ satisfies Cond. 2 in Definition 4}\right).\text{(c)}\end{align*}

Take any $\epsilon>0$ . By the stabilization property of the PVT (Proposition 3) we have $\lim_{n \uparrow \infty} \mathbb{P}(R(Q_n) \geq n) = 0$ , and so we can fix n large enough to make the probability in (a) smaller than $\epsilon/3$ . Then $Q_n$ intersects almost surely zero or a finite number of edges $e\in E$ . Hence the probability in (b) converges to 0 when $r\to 0$ , and consequently we can take r small enough to make the probability in (b) smaller than $\epsilon/3$ . Finally, for given n (and independently of r), we can take $\lambda$ small enough to make the probability in (c) smaller than $\epsilon/3$ . Indeed, this latter probability is dominated by the probability that $X^\lambda(Q_n)\ge 1$ and thus converges to 0 when $\lambda\to 0$ for any finite n. This concludes the proof of Lemma 3.

By Lemmas 2 and 3, using [Reference Liggett, Schonmann and Stacey31, Theorem 0.0], for large enough $n<\infty$ and small enough $r>0,\lambda>0$ , the process of n-bad sites is stochastically dominated from above by an independent site percolation model on the integer lattice where the probability of having an open site is arbitrarily small. Hence this independent site percolation model is subcritical. Consequently, we can make the process of n-bad sites non-percolating. By Lemma 1 the same is true for $\mathcal{G}$ , which concludes the proof of Theorem 1.

3.5. Proof of Theorem 2

We shall prove that for large enough $p<1$ the model $\mathcal{G}$ percolates with positive probability for all $r>0$ and large enough $\lambda<\infty$ (depending on r).

As in the proof of Theorem 1, we will use a coarse-graining argument. To this end, consider the following percolation model on the integer lattice $\mathbb{Z}^d$ . For $n \geq 1$ , we say a site $z \in \mathbb{Z}^{d}$ is n-good if the following conditions are satisfied:

1. (1) $R(Q_{6n}(nz)) < 6n$ .

2. (2) There exists a full street (not just a street segment!) entirely contained in $Q_n(nz)$ . By abuse of notation, we write this as $E \cap Q_n(nz) \neq \emptyset$ .

3. (3) There exists $e \in E \cap Q_n(nz)$ such that e is open, in the sense of Definition 4. In other words, there exists an open street which is fully included in the cube $Q_n(nz)$ .

4. (4) All crossroads in $Q_{6n}(nz)$ are open, in the sense of Definition 3.

5. (5) Every two open edges $e,e' \in E \cap Q_{3n}(nz)$ are connected by a path in $\mathcal{G}\cap Q_{6n}(nz)$ .

We say a site $z \in \mathbb{Z}^{d}$ is n-bad if it is not n-good.

The n-good sites have been defined so as to satisfy the following implication.

Proof. Let $\mathcal{C}$ be an infinite connected component of n-good sites. Consider $z,z'\in \mathcal{C}$ such that $\lVert z-z' \rVert_1=1$ . Without loss of generality, assume $z = (a_1,a_2, \ldots, a_d)$ for some $a_1, a_2, \ldots, a_d \in \mathbb{Z}$ and $z' = (a_1+1,a_2, \ldots, a_d)$ . By Condition 2 in the definition of n-goodness, we can find open edges $e\in E \cap Q_n(nz)$ and $e'\in E \cap Q_n(nz')$ . Since

\begin{gather*}Q_n(nz) = \left[na_1-n/2,na_1+n/2\right] \times \ldots \times \left[na_d-n/2,na_d+n/2\right], \\ Q_n(nz') = \left[na_1+n/2,na_1+3n/2\right] \times \left[na_2-n/2,na_2+n/2\right] \times \ldots \times \left[na_d-n/2,na_d+n/2\right], \\ Q_{3n}(nz) = \left[na_1-3n/2,na_1+3n/2\right] \times \ldots \times \left[na_d-3n/2,na_d+3n/2\right], \\ Q_{6n}(nz) = \left[na_1-3n,na_1+3n\right] \times \ldots \times \left[na_d-3n,na_d+3n\right],\end{gather*}

we have $Q_{n}(nz') \subset Q_{3n}(nz)$ , and so $e' \in E \cap Q_{n}(nz')$ implies $e' \in E \cap Q_{3n}(nz)$ . Since we also have $e \in E \cap Q_{n}(nz) \subset E \cap Q_{3n}(nz)$ and e, e ^′ are both open, by Conditions 4 and 5 in the definition of n-goodness, e and e ^′ are connected by a path $\mathcal{L}$ in $\mathcal{G}\cap Q_{6n}(nz)$ . Therefore, the path $\mathcal{L}$ also connects e and e ^′ in $\mathcal{G}$ , thus giving rise to an infinite connected component in $\mathcal{G}$ . This concludes the proof of Lemma 4.

Proof. In the same way as in the proof of Lemma 2, it suffices to prove that for all finite $\psi = \lbrace z_{1}, \ldots z_{q} \rbrace \subset \mathbb{Z}^{d}$ such that $\forall i \neq j, \lVert z_i - z_j \rVert_{\infty} > 18$ , we have

\begin{equation*}\mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) = \prod_{i=1}^{q} \mathbb{E}(\xi_{z_i}).\end{equation*}

Denote respectively by $A_z,B_z,C_z,D_z,F_z$ the events that Conditions 1, 2, 3, 4, 5 in the definition of n-goodness hold for $z \in \mathbb{Z}^{d}$ . We thus have

\begin{equation*}\forall z \in \mathbb{Z}^{d}, \qquad \xi_{z} = \mathbb{1}\lbrace A_z \rbrace \mathbb{1}\lbrace B_z \rbrace \mathbb{1}\lbrace C_z \rbrace \mathbb{1}\lbrace D_z \rbrace \mathbb{1}\lbrace F_z \rbrace.\end{equation*}

Note first that whenever $z \in \mathbb{Z}^{d}$ , the indicators $\mathbb{1} \lbrace A_z \rbrace$ and $\mathbb{1} \lbrace B_z \rbrace$ are $\Lambda$ -measurable. Thus, we have

\begin{align}\nonumber \mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) &= \mathbb{E}\left[\mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \Bigg\vert \Lambda \right)\right]\\ \nonumber &= \mathbb{E}\left[\prod_{i=1}^{q} \mathbb{1}\lbrace A_{z_i} \rbrace \mathbb{1}\lbrace B_{z_i}\rbrace\mathbb{E}\left( \prod_{i=1}^{q}\mathbb{1}\lbrace C_{z_i} \cap D_{z_i} \cap F_{z_i} \rbrace \Bigg\vert \Lambda \right)\right].\end{align}

Now note that conditioned on $\Lambda$ , for each $1 \leq i \leq q$ , the event $ C_{z_i} \cap D_{z_i} \cap F_{z_i} $ depends only on the configuration of $X^{\lambda}$ and Y inside of the d-dimensional cube $Q_{6n}(nz_i)$ . Since $\psi$ satisfies $\lVert z_i - z_j \rVert_{\infty} > 18$ for all $i \neq j$ , we have $\lVert nz_i - nz_j \rVert_{\infty} > 18n$ for all $i \neq j$ . As a matter of fact, the cubes $\lbrace Q_{6n}(nz_i)) \, : \, 1 \leq i \leq q\rbrace$ are disjoint, i.e.

\begin{equation*}\forall i \neq j, \qquad Q_{6n}(nz_i) \cap Q_{6n}(nz_j) = \emptyset.\end{equation*}

By the complete independence of Poisson and Bernoulli processes (recall that, given $\Lambda$ , $X^\lambda$ has the distribution of a Poisson point process and Y the distribution of a Bernoulli point process), we have

(11) \begin{align}\nonumber \mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) &= \mathbb{E}\left[\prod_{i=1}^{q} \mathbb{1}\lbrace A_{z_i} \rbrace \mathbb{1}\lbrace B_{z_i}\rbrace\mathbb{E}\left( \prod_{i=1}^{q}\mathbb{1}\lbrace C_{z_i} \cap D_{z_i} \cap F_{z_i} \rbrace \Bigg\vert \Lambda \right)\right] \\ \nonumber &= \mathbb{E}\left[\prod_{i=1}^{q} \mathbb{1}\lbrace A_{z_i} \rbrace \mathbb{1}\lbrace B_{z_i}\rbrace \prod_{i=1}^{q}\mathbb{E}\left( \mathbb{1}\lbrace C_{z_i} \cap D_{z_i} \cap F_{z_i} \rbrace \Bigg\vert \Lambda \right)\right]\\ \nonumber &=\mathbb{E}\left[\prod_{i=1}^{q} \mathbb{1}\lbrace A_{z_i} \rbrace \prod_{i=1}^{q}\mathbb{E}\left( \mathbb{1}\lbrace B_{z_i} \cap C_{z_i} \cap D_{z_i} \cap F_{z_i} \rbrace \Bigg\vert \Lambda \right)\right]\\ &=\mathbb{E}\left[\prod_{i=1}^{q} \mathbb{1}\lbrace R(Q_{6n}(nz_i)) < 6n \rbrace f(\Lambda_{Q_{6n}(nz_i)})\right],\end{align}

where $f(\Lambda_{Q_{6n}(x)}) \,{:}\,{\raise-1.5pt{=}}\, \mathbb{E}\left(\mathbb{1}\lbrace B_{x} \cap C_{x} \cap D_{x} \cap F_{x} \rbrace \, \vert \, \Lambda \right)$ , a bounded measurable deterministic function of $\Lambda_{Q_{6n}(x)}$ , and where by $\Lambda$ -measurability of the events $ \lbrace B_{z_i} \, : \, 1 \leq i \leq q \rbrace$ , we put their indicators into the conditional expectation given $\Lambda$ . Now, the set $\varphi \,{:}\,{\raise-1.5pt{=}}\, \lbrace nz_1,\ldots,nz_p \rbrace \subset \mathbb{R}^{d}$ is finite and satisfies

\begin{equation*}\forall i \neq j, \qquad \lVert nz_i - nz_j \rVert_{\infty} > 18n.\end{equation*}

Since the infinite norm is always bounded above by the Euclidean norm, we have $\lVert nz_i - nz_j \rVert_{2} > 18n $ for all $i \neq j$ , and so $\varphi$ satisfies

\begin{equation*}\forall x \in \varphi, \qquad \text{dist}(x, \varphi \setminus \lbrace x \rbrace) > 18n=3 \times 6n.\end{equation*}

We can therefore apply Condition 3 in Definition 1 (with n replaced by 6n) to get that the random variables appearing in the right-hand side of (11) are independent. Hence

\begin{align*} \mathbb{E}\left(\prod_{i=1}^{q} \xi_{z_i} \right) &= \prod_{i=1}^{q} \mathbb{E}\left[\mathbb{1}\lbrace R(Q_{6n}(nz_i) < 6n \rbrace f(\Lambda_{Q_{6n}(nz_i)})\right]\\&= \prod_{i=1}^{q} \mathbb{E}(\xi_{z_i}),\end{align*}

which concludes the proof of Lemma 5.

Lemma 6. For any $r>0$ we have

\begin{equation*} \lim_{n \uparrow \infty}\lim_{ p \uparrow 1,\,\lambda \uparrow \infty} \mathbb{P}(\boldsymbol{0} \ is\ n\hbox{-}good) = 1.\end{equation*}

Proof. We shall prove that

\begin{equation*} \lim_{n \uparrow \infty}\lim_{p \uparrow 1,\;\lambda \uparrow \infty} \mathbb{P}(\boldsymbol{0} \, \text {is} \ n\hbox{-}bad) = 0.\end{equation*}

Take any $\epsilon>0$ . Denote respectively by A, B, C, D, F the events that Conditions 1, 2, 3, 4, 5 in the definition of n-goodness hold for $z=\boldsymbol{0}$ . Denote also by $\tilde A$ the event that $R(Q_{6n})< n/2$ . Note that $\tilde A\subset A$ and thus we have

\begin{align*} \mathbb{P}(\boldsymbol{0} \, \text{is} \n-{\rm bad}) &= \mathbb{P}(A^c \cup B^c \cup C^c \cup D^c \cup F^c)\\&\le \mathbb{P}(\tilde A^c \cup B^c \cup C^c \cup D^c \cup F^c)\\ &\le \mathbb{P}(\tilde A^c) + \mathbb{P}(B^c) + \mathbb{P}(B\cap C^c) + \mathbb{P}(D^c) + \mathbb{P}(\tilde A\cap D \cap F^c).\end{align*}

First, partitioning the cube $Q_{6n}$ into $12^d$ subcubes $(K_i)_{1 \leq i \leq 12^d}$ of side length $n/2$ , we get

\begin{align} \nonumber \mathbb{P}(\tilde A^c) &= \mathbb{P}(R(Q_{6n}) \geq n/2) \\ \nonumber &= \mathbb{P}\left(\bigcup_{i=1}^{12^d} \left\lbrace R(K_i) \geq n/2 \right\rbrace \right) \\ \nonumber &\leq 12^d \; \mathbb{P}(R(Q_{n/2}) \geq n/2) \qquad \text{by stationarity of the} R{\rm s.}\end{align}

Therefore, by Condition 2 of Definition 1, we get $\lim_{n \uparrow \infty} \mathbb{P}(\tilde A^c) = 0$ . Also,

\begin{align*} \mathbb{P}(B^c)=\mathbb{P}(E \cap Q_n = \emptyset),\end{align*}

and thus $\lim_{n \uparrow \infty} \mathbb{P}( B^c) = 0$ . Fix n large enough so that $\mathbb{P}(\tilde A^c)\le \epsilon/5$ and $\mathbb{P}( B^c)\le \epsilon/5$ . For such n, $Q_n$ , $Q_{3n}$ , and $Q_{6n}$ intersect almost surely zero or a finite number of edges and vertices.

Let us now deal with the quantity $\mathbb{P}(B \cap C^c)$ . We have

\begin{align*} \mathbb{P}( B\cap C^c)= \mathbb{P}(E \cap Q_n \neq \emptyset\,\text{ and }\, \forall \, e\in E\cap Q_n\, :\, e \, \text{is closed} ).\end{align*}

This latter probability converges to 0 when $\lambda\to\infty$ (for fixed n and $r>0$ ). Hence, for large enough $\lambda<\infty$ (depending on n, r) we have $\mathbb{P}(B\cap C^c)\le \epsilon/5$ . Similarly,

\begin{align*} \mathbb{P}(D^{c}) &=\mathbb{P}(\exists v\in V\cap Q_{6n}\,:\, v \text{ is closed})\end{align*}

converges to 0 when $p\uparrow1$ (for fixed n and $r>0$ ); hence, for large enough $p<1$ , we have $\mathbb{P}(D^c)\le \epsilon/5$ .

Regarding the event $\tilde A\cap D\cap F^c$ , note that under the event $\tilde A$ , we have

\begin{equation*}R(Q_{6n}) < n/2 < 3n/2.\end{equation*}

Hence, by asymptotic essential connectedness (see Definition 2), we have that $\text{supp}(\Lambda_{Q_{3n}}) \neq \emptyset$ , and moreover, there exists a connected component $\Delta$ of $\text{supp}(\Lambda_{Q_{6n}})$ such that $\text{supp}(\Lambda_{Q_{3n}}) \subset \Delta \subset \text{supp}(\Lambda_{Q_{6n}})$ . Therefore

\begin{equation*}\tilde A\cap D\cap F^c\subset (\exists e\in E\cap Q_{6n}\,:\, e \text{ is closed}).\end{equation*}

Clearly, for fixed n, r and independently of p,

\begin{equation*}\lim_{\lambda\to\infty} \mathbb{P}(\exists e\in E\cap Q_{6n}\,:\, e \text{ is closed})=0.\end{equation*}

Hence, we can find $\lambda<\infty$ large enough (depending on n, r) so that $\mathbb{P}(\tilde A\cap D\cap F^c)\le \epsilon/5$ . Since $\epsilon>0$ was arbitrary, this concludes the proof of Lemma 6.

By Lemmas 5 and 6, using [Reference Liggett, Schonmann and Stacey31, Theorem 0.0], the process of n-good sites is stochastically dominated from below by a supercritical Bernoulli process for large enough $n<\infty$ , $\lambda<\infty$ , $p<1$ . Thus, we can make the process of n-good sites percolating. By Lemma 4, the connectivity graph $\mathcal{G}$ with these values of $\lambda,p$ percolates, which concludes the proof of Theorem 2.

3.6. Proof of Proposition 2

We first prove that $p^* <1$ as a consequence of Theorem 2.3. Then, using an adapted path-count argument, very much as in classical i.i.d. percolation on the square grid (see e.g. [Reference Meester and Roy32, Theorem 1.1]), we show that $p^*>0$ .

$\boxed{p^*<1:}$ By Theorem 2, we can find large enough $p \in (0,1)$ so that $P(p,U,H)>0$ for any $H \in \left[0,\infty \right)$ and large enough $U < \infty$ depending on the chosen H. Choosing $H=0$ , we thus obtain the existence of some $p \in (0,1)$ such that $P(p,U,0)>0$ for some $U < \infty$ . Now, as noted in Section 2.3.1, the case $H=0$ corresponds to PVT site percolation, and the percolation probability $P(p,U,H)\,\,{\raise-1.5pt{=}}{:}\,P_{PVT}(p)$ does not depend on U. Thus, $P_{PVT}(p) >0$ for some $p < 1$ , and so $p^*<1$ .

$\boxed{p^*>0:}$ It is known that the degree of all vertices (i.e. 0-facets) of a d-dimensional PVT generated by a homogeneous Poisson point process in $\mathbb{R}^d$ is almost surely equal to $d+1$ , as a consequence of the following facts:

• Such a PVT is almost surely regular and normal (see [Reference Okabe, Boots, Sugihara and Chiu37, Proposition PV2]).

• The cells of such a PVT are almost surely polytopes, i.e. convex and bounded polyhedra (see [Reference Okabe, Boots, Sugihara and Chiu37, Proposition PV1]).

• Each s-facet of a Voronoi cell in $\mathbb{R}^d$ lies on an s-dimensional hyperplane whose points are equidistant of $d-s+1$ atoms of the Poisson point process having generated the PVT (this is a consequence of [Reference Okabe, Boots, Sugihara and Chiu37, Property IV3] and [Reference Okabe, Boots, Sugihara and Chiu37, Property IV4]).

Moreover, S, being a regular and normal tessellation, is locally finite. This observation combined with the degree bound allows us to use an adapted path-count argument, as follows.

First, denote by ${\tilde{\mathcal G}}_p\,\,{\raise-1.5pt{=}}{:}\,\mathcal{G}_{p,0,0}$ the random graph arising from the PVT site percolation process with parameter p, and note that the percolation of ${\tilde{\mathcal G}}_p$ is equivalent to the percolation of $\mathcal{G}_{p,U,0}$ whenever $p \in \left[0,1\right]$ and $U \geq 0$ (indeed, the fact that $H=0$ makes the percolation independent of the Cox points in our model). Hence

(12) \begin{equation}P_{PVT}(p)=\mathbb{P}\left( \tilde{\mathcal{G}}_p \, \text{ has an infinite connected component}\right).\end{equation}

Denote by $\Phi$ the point process of crossroads (i.e. vertices of the PVT S), and denote by $\mathbb{P}^0$ its Palm probability. Since Y is a doubly stochastic Bernoulli point process supported by the crossroads, the conditional distribution of Y given S is the same under the stationary probability $\mathbb{P}$ and under the Palm probability $\mathbb{P}^0$ . As a matter of fact, for every crossroad $v \in V$ , we have $\mathbb{P}^0(Y(\lbrace v \rbrace) >0 \, \vert \, S) = \mathbb{P}(Y(\lbrace v \rbrace) >0 \, \vert \, S) = p$ . Moreover, conditionally to the realization of the PVT S, the states of distinct crossroads (i.e. open or closed) remain independent.

Fix some $n \geq 1$ . A self-avoiding path $\gamma$ of length n starting from the typical crossroad $\boldsymbol{0}$ is a sequence of crossroads $\boldsymbol{0}= v_1, \ldots, v_n \in V$ with $v_i \neq v_j$ for $i \neq j$ and such that $v_i$ and $v_{i+1}$ are adjacent in S whenever $1 \leq i \leq n-1$ . If the typical crossroad $\boldsymbol{0}$ belongs to an infinite connected component in $\tilde{\mathcal{G}_p}$ (which we denote by $\boldsymbol{0} \leadsto \infty$ ), there must exist such a path with a Bernoulli point present at all crossroads of the path. Denote this event by $A_n$ .

Then we have

\begin{equation*} \mathbb{P}^0(\boldsymbol{0} \leadsto \infty) \leq \mathbb{P}^0(A_n).\end{equation*}

Let $SAP_n$ denote the set of self-avoiding paths of length n starting from the typical crossroad $\boldsymbol{0}$ . By the union bound, we have

\begin{align*} \mathbb{P}^0(A_n) &\leq \mathbb{E}^0 \left[ \sum_{(v_1, \ldots, v_n) = \gamma \in SAP_n} \mathbb{P}^0 \left( \bigcap_{i=1}^n \lbrace Y(\lbrace v_i \rbrace) >0 \rbrace \, \Big\vert \, S\right)\right] \\ &= \mathbb{E}^0 \left[ \sum_{(v_1, \ldots, v_n) = \gamma \in SAP_n} p^n \right] \\ &= \mathbb{E}^0 \left[ \#(SAP_n)p^n \right],\end{align*}

where $\#(SAP_n)$ denotes the cardinality of $SAP_n$ , and where we have used the conditional independence of the states of the vertices as well as the conditional distribution of Y given S to get the first equality. Now, using the fact that for every $v \in V$ , $\deg v = d+1$ almost surely, we get that $\#(SAP_n) \leq (d+1) \times d^{n-1}$ . Hence

\begin{equation*} \mathbb{P}^0(\boldsymbol{0} \leadsto \infty) \leq (d+1) \times d^{n-1} p^n = \frac{d+1}{d} (dp)^{n}.\end{equation*}

When $p<1/d$ , the quantity in the right-hand side converges to 0 as $n \uparrow \infty$ . Hence, for $p<1/d$ , we have $\mathbb{P}^0(\boldsymbol{0} \leadsto \infty) = 0$ .

To conclude that $\tilde{\mathcal{G}}_p$ does not percolate, we proceed as follows. For a crossroad $v \in V$ , denote by $\lbrace v \leadsto \infty \rbrace$ the event that v belongs to an infinite connected component of the PVT site percolation graph $\tilde{\mathcal{G}}_p$ . By Markov’s inequality, we have

\begin{align*} \mathbb{P}\left( \tilde{\mathcal{G}}_p \, \text{ has an infinite connected component}\right) &=\!:\, \mathbb{P}(\exists v \in V \, : \, v \leadsto \infty) \\ &\leq \mathbb{E}\left[ \# \lbrace v \in V \, : \, v \leadsto \infty \rbrace \right],\end{align*}

and so, by (12), we get

(13) \begin{equation} P_{PVT}(p) \leq \mathbb{E}\left[ \# \lbrace v \in V \, : \, v \leadsto \infty \rbrace \right].\end{equation}

Denote by $\lambda_0$ the intensity of the point process $\Phi$ of crossroads of S and fix some $p<1/d$ . By the Campbell–Little–Mecke–Matthes theorem (see [Reference Baccelli, Błaszczyszyn and Karray2, Theorem 6.1.28]), we have

\begin{align*} \mathbb{E}\left[ \# \lbrace v \in V \, : \, v \leadsto \infty \rbrace \right] &= \mathbb{E} \left[ \int_{\mathbb{R}^d} \mathbb{1}\lbrace x \leadsto \infty \rbrace \Phi(dx) \right] \\ &= \lambda_0 \int_{\mathbb{R}^d} \mathbb{E}^0 \left[ \mathbb{1} \lbrace \boldsymbol{0} \leadsto \infty \rbrace \right]dx \\ &= \lambda_0 \int_{\mathbb{R}^d} \mathbb{P}^0(\boldsymbol{0} \leadsto \infty)dx \\ &=0,\end{align*}

where we have used the fact that $p < 1/d \Rightarrow \mathbb{P}^0(\boldsymbol{0} \leadsto \infty) = 0$ to get the last equality. By (13), we have $P_{PVT}(p)=0$ whenever $p<1/d$ , and thus $p^* \geq 1/d >0$ .

3.7. Proof of Theorem 3

Proving Theorem 3 amounts to showing that $\mathcal{G}$ percolates with positive probability when $\lambda=0$ , $p<1$ is sufficiently large, and $r<\infty$ is sufficiently large. We thus assume throughout the rest of this subsection that $\lambda=0$ and that r and p are the varying parameters, and we still refer to $\mathcal{G}$ for the associated connectivity graph.

Say a street $e \in E$ is hard-geometric-open if its length is smaller than the connectivity threshold r: $\vert e \vert \leq r$ . If not, say e is hard-geometric-closed.

Once again, we will use a coarse-graining argument. Since the development is very similar to the one presented in the previous subsection, we only give details on what modifications should be applied to the proof of Theorem 2 to prove Theorem 3.

To this end, we consider now the following percolation model on the integer lattice $\mathbb{Z}^d$ . For $n \geq 1$ , say a site $z \in \mathbb{Z}^{d}$ is n-good if it satisfies the following conditions:

1. (1) $R(Q_{6n}(nz)) < 6n$ .

2. (2) $E \cap Q_n(nz) \neq \emptyset$ ; i.e. the cube $Q_n(nz)$ contains a full street.

3. (3) There exists $e \in E \cap Q_n(nz)$ such that $\vert e \vert \le r$ . In other words, there exists a hard-geometric-open street that is fully included in the cube $Q_n(nz)$ .

4. (4) All crossroads in $Q_{6n}(nz)$ are open, in the sense of Definition 3.

5. (5) Every two hard-geometric-open streets $e,e' \in E \cap Q_{3n}(nz)$ (i.e. such that $\vert e \vert \leq r$ and $\vert e' \vert \leq r$ ) are connected by a path in $\mathcal{G}\cap Q_{6n}(nz)$ .

We say a site $z \in \mathbb{Z}^{d}$ is n-bad if it is not n-good.

Note that this new definition of n-goodness is exactly the same as the one given in the proof of Theorem 2 but with Conditions 3 and 5 replaced by $\hat{3}$ and $\hat{5}$ . In other words, openness is replaced by hard-geometric-openness.

Since we are now dealing with hard-geometric openness and all the other conditions are unchanged, it is straightforward to prove the following by adapting the proof of Lemma 4.

In the same way, we get the result below.

Proof. It suffices to adapt the proof of Lemma 5 as follows.

Denote respectively by $A_z,B_z,\hat{C}_z,D_z,\hat{F}_z$ the events that Conditions 1, 2, $\hat{3}$ , 4, and $\hat{5}$ in the definition of n-goodness hold for $z \in \mathbb{Z}^{d}$ .

Note first that whenever $z \in \mathbb{Z}^{d}$ , the indicators $\mathbb{1} \lbrace A_z \rbrace$ , $\mathbb{1} \lbrace B_z \rbrace$ , $\mathbb{1} \lbrace \hat{C}_z \rbrace$ are all $\Lambda$ -measurable. Doing exactly the same calculations as in the proof of Lemma 5, we arrive at dealing with the quantity

\begin{align}\nonumber \mathbb{E}\left(\prod_{i=1}^{q} \xi'_{z_i} \right) &= \mathbb{E}\left[\prod_{i=1}^{q} \mathbb{1}\lbrace A_{z_i} \rbrace \mathbb{1}\lbrace B_{z_i}\rbrace \mathbb{1} \lbrace \hat{C}_{z_i} \rbrace \mathbb{E}\left( \prod_{i=1}^{q}\mathbb{1}\lbrace D_{z_i} \cap \hat{F}_{z_i} \rbrace \Bigg\vert \Lambda \right)\right].\end{align}

Now note that conditioned on $\Lambda$ , for each $1 \leq i \leq q$ , the event $ D_{z_i} \cap \hat{F}_{z_i} $ depends only on the configuration of Y inside of the cube $Q_{6n}(nz_i)$ . We can thus proceed as in the aforementioned proof by using the complete independence of Y (recall that, given $\Lambda$ , Y has the distribution of a Bernoulli point process). Finally, it is clear that $\mathbb{1} \lbrace \hat{C}_z \rbrace$ is a bounded deterministic function of $\Lambda_{Q_n(nz)}$ and that

\begin{equation*}\mathbb{E}\left( \mathbb{1}\lbrace D_{z} \cap \hat{F}_{z} \rbrace \Bigg\vert \Lambda \right)\end{equation*}

is a bounded deterministic function of $\Lambda_{Q_{6n}(nz)}$ . This allows us to proceed exactly as in the proof of Lemma 5 and conclude.

Finally, for the hard-geometric model, we still have the following.

Lemma 9.

\begin{equation*} \lim_{n \uparrow \infty}\lim_{ p\uparrow 1, r \uparrow \infty} \mathbb{P}(\boldsymbol{0} \ is\ n\hbox{-}good) = 1.\end{equation*}

Proof. Again, we shall prove that

\begin{equation*}\lim_{n \uparrow \infty}\lim_{p \uparrow 1, r \uparrow \infty} \mathbb{P}(\boldsymbol{0} \, \text{is} n-{\rm bad}) = 0.\end{equation*}

Take any $\epsilon>0$ . We adapt the proof of Lemma 6 as follows. Denote respectively by $A,B,\hat{C},D,\hat{F}$ the events that Conditions 1, 2, $\hat{3}$ , 4, and $\hat{5}$ in the definition of n-goodness hold for $z=\boldsymbol{0}$ . Also denote by $\tilde A$ the event that $R(Q_{6n})< n/2$ . As in the aforementioned proof, we have

\begin{align*} \mathbb{P}(\boldsymbol{0} \, \text{is \textit{n}-bad}) &\le \mathbb{P}(\tilde A^c) + \mathbb{P}(B^c) + \mathbb{P}(B \cap \hat{C}^c) + \mathbb{P}(D^c) + \mathbb{P}(\tilde A \cap D \cap \hat{F}^c).\end{align*}

In the above inequality, we deal with the first, second, and fourth quantities as before, and so we can fix n large enough such that $\mathbb{P}(\tilde A^c)\le \epsilon/5$ and $\mathbb{P}(B^c)\le \epsilon/5$ . For such n, $Q_n$ , $Q_{3n}$ , and $Q_{6n}$ intersect almost surely zero or finitely many edges and vertices. We can then fix $p<1$ large enough so that $\mathbb{P}(D^c)\le \epsilon/5$ .

Let us now deal with the quantity $\mathbb{P}(B \cap \hat{C}^c)$ . We have

\begin{align*} \mathbb{P}(B \cap \hat{C}^c) &= \mathbb{P}(E \cap Q_n \neq \emptyset\;\text{and}\; \forall \, e\in E\cap Q_n\; \vert e \vert > r ) \\ &= \mathbb{E}\left(\mathbb{1}\lbrace E \cap Q_n \neq \emptyset \rbrace \prod_{e \in E\cap Q_n} \mathbb{1}\lbrace \vert e \vert > r \rbrace \right).\end{align*}

Note first that on the event $\lbrace E \cap Q_n \neq \emptyset \rbrace$ , the latter product is non-empty. Moreover, since $E\cap Q_n$ contains finitely many edges (recall that n is fixed) and since we have

\begin{equation*}\forall e \in E, \qquad \lim_{r \uparrow \infty} \mathbb{1}\lbrace \vert e \vert > r \rbrace = 0 \quad \text{almost surely,}\end{equation*}

by dominated convergence we have that the latter expectation converges to 0 when $r\to\infty$ (for fixed n). Therefore, $\lim_{r \uparrow \infty} \mathbb{P}(B \cap \hat{C}^c)=0$ (for fixed n).

Regarding the event $\tilde A\cap D \cap \hat{F}^c$ , we proceed as before and use asymptotic essential connectedness to get

\begin{equation*} \tilde A\cap D\cap \hat{F}^c \subset (\exists e\in E\cap Q_{6n}\,:\, e \text{ is hard-geometric-closed}). \end{equation*}

Clearly, for fixed n,

\begin{align*}\lim_{r\to\infty} \mathbb{P}(\exists e\in E\cap Q_{6n}\,:\, e \text{ is hard-geometric-closed})& =\lim_{r\to\infty} \mathbb{P}(\exists e\in E\cap Q_{6n}\,:\, \vert e \vert >r) \\ &=0.\end{align*}

Hence, we can find $r<\infty$ large enough (depending on n) so that $\mathbb{P}(B \cap \hat{C}^c)\le \epsilon/5$ and $\mathbb{P}(\tilde A \cap D \cap \hat{F}^c)\le \epsilon/5$ . Since $\epsilon >0$ was arbitrary, this concludes the proof of Lemma 9.

By Lemmas 8 and 9, using [Reference Liggett, Schonmann and Stacey31, Theorem 0.0], the process of n-good sites is stochastically dominated from below by a supercritical Bernoulli process for large enough $n<\infty$ , $p<1$ , $r<\infty$ . Thus, we can make the process of n-good sites percolating. By Lemma 7, the connectivity graph $\mathcal{G}$ with these values of p and r percolates, which concludes the proof of Theorem 3.