4.1. Proof Proposition 1

Proof Proposition 1. First, let us assume that $\mathbb{P}($ there exists $X_i\in X^\lambda$ such that $\deg (X_i)=\infty)\gt 0$ . Then, by continuity of measures, there exists $R\gt 0$ such that $\mathbb{P}($ there exists $X_i\in X^\lambda\cap Q_{R}$ such that $\deg (X_i)=\infty)\gt 0$ and hence, by translation invariance, also $\mathbb{P}($ there exists $X_i\in X^\lambda\cap Q_1$ such that $\deg\! (X_i)=\infty)\gt 0$ . But the last inequality implies that $\mathbb{E}\big[\sum_{X_i \in X^\lambda \cap Q_1}\deg\!(X_i)\big]= \infty$ . However, in order to derive a contradiction, we can use the Mecke–Slivnyak theorem twice to see that

\begin{align*} \mathbb{E}\Bigg[\sum_{X_i \in X^\lambda \cap Q_1}\deg\!(X_i)\Bigg] &\le \mathbb{E}\Bigg[\sum_{X_i \in X^\lambda \cap Q_1} \sum_{X_j \in X^\lambda \setminus X_i}\textbf{1}\big\{|X_i-X_j| \le \ell_S(X_i) + \ell_S(X_j)\big\}\Bigg] \\ &=\lambda^2\mathbb{E}\bigg[\int_{S\cap Q_1} {\textrm d} x\int_{S }{\textrm d} y\, \textbf{1}\big\{|x-y| \le \ell_S(x) + \ell_S(y)\big\}\bigg].\end{align*}

Next, we bound the maximal reach $\ell_S$ by the street length in a stabilized region. For this, let $K\,:\!= \sup\{|x-y|\colon x\in S, y\in{\textrm{supp}}(\kappa^S(x,{\textrm d} y))\}\lt\infty$ denote the maximal reach of the kernel $\kappa$ , which exists due to the assumption of a uniformly bounded support of $\kappa$ . Using this, we can bound the length of any shortest path starting in $x\in S$ to any $y \in S\cap B_K(x)$ by the total street length of the box in which there is a connection between x and y, due to asymptotic essential connectedness. More precisely, note that if $R(Q_{2n}(x)) \lt n/2$ then $\ell_S(x) \le |S\cap Q_{2n}(x)|+r/2$ , for any $n\gt 2K$ , and we can define

\begin{equation*} N(x) \,:\!= \min_{n\in \mathbb{N}, n\gt K} R(Q_{2n+1}(x))\lt n/2.\end{equation*}

In particular, for all $y \in Q_1(z)$ we have $\ell_S(y) \le |S\cap Q_{2N(z)+1}(z)|+r/2$ , and hence

\begin{align*}&\mathbb{E}\bigg[\int_{S\cap Q_1} {\textrm d} x\int_{S }{\textrm d} y\, \textbf{1}\big\{|x-y| \le \ell_S(x) + \ell_S(y)\big\}\bigg]\\&\le\mathbb{E}\Bigg[\sum_{n \gt K} \textbf{1}\{N(o) = n\} |S\cap Q_1| \int_S {\textrm d} y \,\textbf{1} \big\{|y| \le \ell_S(y) + 1/2+r/2 + |S\cap Q_{2n+1}|\big\} \Bigg]\\&\le\mathbb{E}\Bigg[\sum_{n \gt K}\sum_{m \gt K}\sum_{z\in \mathbb Z^2} \textbf{1}\{N(o) = n\}\textbf{1}\{N(z) = m\} |S\cap Q_1| \\&\qquad\qquad\qquad\qquad\qquad\qquad \int_{S\cap Q_1(z)} {\textrm d} y\, \textbf{1} \big\{|y| \le \ell(y) + 1/2+r/2 + |S\cap Q_{2n+1}|\big\} \Bigg]\\&\le\sum_{n \gt K}\sum_{m \gt K}\sum_{z\in \mathbb Z^2}\mathbb{E}\big[ \textbf{1}\{ N(o) = n\}\textbf{1}\{N(z) = m\}\\&\quad\quad\quad\quad\quad\quad|S\cap Q_1| |S\cap Q_1(z)| \textbf{1} \big\{|z| \le |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\big\} \big],\end{align*}

where we used $|y|\le|z|+ 1/2$ for $y \in Q_1(z)$ in order to eliminate the dependence of the integrals on x and y respectively and thus derive a uniform bound for the integrals. It remains to show that this bound is integrable. By Hölder’s inequality we can bound each summand by

\begin{align*} &\mathbb{P}(N(o) = n)^{{1}/{5}}\mathbb{P}(N(o) = m)^{{1}/{5}} \mathbb{E}\big[|S\cap Q_1|^5\big]^{{2}/{5}} \\ & \quad \mathbb{P}\big(|z| \le |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|\big)^{{1}/{5}}\\ & \qquad\quad\le C_1\exp\!(\!-c_1(n+m))\mathbb{P}(|z| \le |S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|)^{{1}/{5}}\end{align*}

for some constants $C_1,c_1\gt 0$ due to the existence of exponential moments for $|S\cap Q_1|$ as well as the assumption of exponential stabilization. Now, the last factor can be further bounded by Markov’s inequality as

\begin{align*} \mathbb{P}\big(|z| \le |S\cap Q_{2m+1}(z)| & + 1+r + |S\cap Q_{2n+1}|\big)^{1/5}\\ &\le |z|^{-3}\mathbb{E}\big[(|S\cap Q_{2m+1}(z)| + 1+r + |S\cap Q_{2n+1}|)^{15}\big]^{1/5} \\ &\le |z|^{-3}\big(\mathbb{E}\big[(3|S\cap Q_{2m+1}|)^{15}] + (3+3r)^{15} + \mathbb{E}\big[(3|S\cap Q_{2n+1}|)^{15}\big]\big)^{1/5} \\ &\le |z|^{-3}\big((3(2m+1)^2+3(2n+1)^2)^{15}\mathbb{E}\big[(|S\cap Q_{1}|)^{15}] + (3+3r)^{15}\big)^{1/5} \\ &\le C_2 (mn)^{c_2}|z|^{-3},\end{align*}

for some constants $C_2,c_2\gt 0$ . Finally, we can use $|z|\gt|z|_{\infty}$ , where $|\cdot|_\infty$ denotes the $\ell_\infty$ -norm, combined with the fact that for $k\in \mathbb{N}_0$ we have $\sum_{z\in Z}\textbf{1}\{|z|_{\infty}=k\} \le 4k+1$ , to see that

\begin{equation*}\sum_{n \gt K}\sum_{m \gt K}\sum_{k\in \mathbb{N}_0}(4k+1)C_2(mn)^{c_2}C_1\exp\!(\!-c_1(n+m))k^{-3}\lt\infty,\end{equation*}

and hence the proof is finished.

4.2. Proof of Theorem 1

Proof of Theorem 1. For simplicity we only prove the case where $\mu=\delta_v$ . The general case can be verified using a thinning argument where we ignore all devices that have a speed greater than $v_{\rm min}+\varepsilon$ for an arbitrarily small $\varepsilon$ , only leading to a potentially larger value of $\lambda_{\textrm c}$ .

First note that the assumption that $\kappa$ is translation covariant and c-well-behaved for some $c\gt 2{\varrho_{\rm I,min}} v$ ensures that it is possible for the kernel to generate a shortest path from one street to an adjacent street while spending a sufficient amount of time on both streets to allow an infection to be passed on to another device. Further, the requirement that $r\gt{\varrho_{\rm I,min}} v$ ensures that the communication radius is large enough such that an infection transmission can happen between devices with initial and target locations in the vicinity of the crossing. Also recall that we require that the streets of length larger than $2v{\varrho_{\rm I,min}}$ percolate with positive probability.

Since we assume ${\lambda_{\rm W}}$ to be arbitrary, in spirit, we think of white knights being everywhere. We consider open streets as well as open crossings. Let us start with the streets. Let $0\lt\varepsilon\lt\min\!(r/4 ,v({\varrho_{\rm I,min}} -{\varrho_{\rm W,min}})/2)$ and define $n(s)= \lfloor |s|/(2r)\rfloor$ , which will represent the number of devices on the street s on which the infection will be propagated. We call a street $s=(c_1,c_2)$ open if there are devices $X_0^s, \dots, X_{n(s)}^s$ such that:

1. (S1) $|s|\gt v{\varrho_{\rm I,min}}+\varepsilon$ ;

2. (S2) for all t, $X_{i,t}^s \in B_\varepsilon\bigg(c_1+\dfrac{ri}{2}\dfrac{c_2-c_1}{|c_2-c_1|}\bigg)$ for all $i \in \{1,\dots, n(s)\}$ ;

3. (S3) ${\rho_{\rm I}}\big(X_i^s,X_{i+1}^s\big)\lt {\varrho_{\rm W,min}}$ for all $i \in \{1,\dots, n(s)-1\}$ ; and

4. (S4) ${\rho_{\rm I}}\big(X_i^s,X_{i-1}^s\big)\lt {\varrho_{\rm W,min}}$ for all $i \in \{2,\dots, n(s)\}$ .

In words, a street is open if, by Condition (S1), the street is long enough that an infection transmission is possible. By Condition (S2), there is a chain of devices reaching from the crossing $c_1$ to the crossing $c_2$ such that each pair of neighboring devices has distance less than r. In particular, if device $X_0^s$ is infected, Condition (S3) ensures that the infection cannot be stopped by white knights as they are too slow and hence the infection propagates from $X_0^s$ to $X^s_{n(s)}$ . Similarly, Condition (S4) ensures that an infection also propagates from $X^s_{n(s)}$ to $X^s_0$ . Let us call $X^s_{c_1}=X^s_1$ and $X^s_{c_2}=X^s_{n(s)}$ the boundary devices of an open street $s=(c_1,c_2)$ .

Next, for a street $s=(c_1,c_2)$ , denote by $[a,b]_{s,c_1}$ the segment

\begin{equation*}\bigg\{x \in s \colon x = c_1 + d\frac{c_2-c_1}{|c_2-c_1|}\text{ for some } d\in [a,b]\bigg\}\end{equation*}

on s such that $\{0\}_{s,c_1}$ is identified with the crossing $c_1$ . Consider again $0\lt\varepsilon\lt\min(r/4 ,v({\varrho_{\rm I,min}} -{\varrho_{\rm W,min}})/2)$ to be small enough that $v_{\textrm{min}} {\varrho_{\rm I,min}} +\varepsilon\lt c/2 $ . We call a crossing c open at time t if, for any ordered pair of open streets $s_1,s_2$ that share the crossing c, there exists a bridging device $X^{s_1,s_2}$ such that:

1. (C1) $X_{0}^{s_1,s_2}\in A_{s_1}=[v{\varrho_{\rm I,min}}/2+\varepsilon/2,v{\varrho_{\rm I,min}}/2+\varepsilon]_{s_1,c}$ ;

2. (C2) $X^{s_1,s_2}$ has its target location in $B_{s_2}=[v{\varrho_{\rm I,min}}/2,v{\varrho_{\rm I,min}}/2+\varepsilon]_{s_2,c}$ ;

3. (C3) $X_{t}^{s_1,s_2}\in J_{s_1}=[0,\varepsilon/2]_{s_1,c}$ and is on its way back to its starting location;

4. (C4) ${\rho_{\rm I}}\big(X^{s_1,s_2},X_{c}^{s_1}\big)\lt {\varrho_{\rm I,min}}+\varepsilon/(2v)$ ; and

5. (C5) ${\rho_{\rm I}}\big(X^{s_1,s_2},X_{c}^{s_2}\big)\lt {\varrho_{\rm I,min}}+\varepsilon/v$ .

These conditions roughly guarantee that, for sufficiently long streets, the bridging device $X^{s_1,s_2}$ propagates the infection from $s_1$ to $s_2$ . The time the bridging device stays on each of the streets is insufficiently long to allow patching by white knights. This is ensured by Conditions (C1) and (C2), as the distance the devices travel on each street before entering the crossing is at most $v{\varrho_{\rm I,min}}+2\varepsilon\lt v{\varrho_{\rm W,min}}$ , since $\varepsilon$ is sufficiently small. Together then, since $s_2$ is open, due to the configuration of the devices on the open street and the fact that $r\gt v{\varrho_{\rm I,min}} $ , $X^{s_1,s_2}$ will pass the infection on to the boundary device $X^{s_2}_c$ of the chain of transmitting devices on $s_2$ . This is ensured by Condition (C5). Finally, Conditions (C3) and (C4) ensure that $X^{s_1,s_2}$ becomes infected if $X^{s_1}_c$ becomes infected at time t, as it is on its way back to its starting location and will spend at least a time of ${\varrho_{\rm W,min}}+\varepsilon/(2v)$ in the vicinity of $X^{s_1}_c$ .

The following statement guarantees that there exists a uniform bound for the probability that a given crossing is open for all sufficiently large times. In particular, this bound converges to 1 as $\lambda$ tends to infinity. Recall that we denoted by H the range of dependence of $\kappa$ , and define $L= {H+v{\varrho_{\rm I,min}}/2+\varepsilon}$ .

We present the proof of the lemma later in this section.

Consider $\tau_{s_1,c}$ , the time that a boundary device associated with the crossing c is infected; then, for $\tau_{s_1,c}\gt T_0$ and conditioned on S, we can couple the event that c is open at time $\tau_{s_1,c}$ to a Bernoulli random variable $\mathcal{B}_c$ with parameter $1-\exp\!(\!-\lambda C_c)$ , and note that the $\mathcal{B}_c$ are independent conditioned on S and only depend on S in the region $S\cap B_L(c)$ . For $\tau_{s_1,c}= \infty$ , i.e. if the boundary device never becomes infected, we assume that the crossing is open nonetheless. We say that a crossing is open if $\mathcal{B}_c=1$ . In order to make use of this observation, we need to ensure that the infection survives up until time $T_0$ . However, note that survival for long but finite times has a positive probability and can be constructed locally by an appropriate isolation procedure for the typical device.

Now we consider the model at time $t\ge T$ and show that the graph of open streets and open crossings percolates with positive probability via stabilization arguments. For this, let us fix an a such that ${\varrho_{\rm I,min}} \lt a/2 \lt a_{\rm c}/2$ and recall that we have the random field of stabilizing radii $\{R_x\}_{x\in\mathbb{R}^2}$ for the street system at our disposal, in addition to the random field $\{R^a_x\}_{x\in\mathbb{R}^2}$ that acts as a replacement for the asymptotically essentially connectedness of the thinned graph $S^a$ . Then, we say that $z \in \mathbb Z^2$ is n-open if

1. (a) $R(Q_{3n}(nz))\lt n$ ;

2. (b) $R^a(Q_{n}(nz))\lt n/2$ ;

3. (c) every street fully contained in $Q_{3n}(nz)\cap S^a$ is open; and

4. (d) every crossing in $Q_{3n}(nz)$ is open.

Condition (b) limits the size of cells of points in $Q_{n}(nz)$ to a diameter of at most $n/2$ with respect to $S^a$ . Further, under the condition, there exists a unique giant component in $Q_{n}(nz)\cap S^a$ , built by the boundaries of the cells, that is connected in $Q_{3n}(nz)$ . If, now, adjacent sites $z_1$ and $z_2$ both satisfy Condition (b), they form a joint connected component, since there exist cells that are simultaneously in $Q_{n}(nz_1)$ and in $Q_{n}(nz_2)$ . Due to Conditions (c) and (d), the infection is able to propagate through open streets and open crossings unpatched.

Now, we can use the domination by product measures theorem [Reference Liggett, Schonmann and Stacey34, Theorem 0.0] to establish Bernoulli site percolation of n-open sites on $\mathbb Z^2$ since Condition (a) guarantees 7-dependence of $S^a$ , and under Condition (a) Conditions (b) and (c) can also be decided within the 7-dependent region. By choosing $n\gt L$ , the crossing-dependent constant $C_{\rm c}$ from Lemma 1 is also 7-dependent like the street system. Hence, using translation invariance, it suffices to show that

\begin{equation*}\liminf_{n\uparrow \infty}\liminf_{\lambda \uparrow \infty} \mathbb{P}( o \text{ is }\,\textit{n}\, \text{-open} )=1.\end{equation*}

To show this, we denote by A(n), B(n), $C(n,\lambda)$ , and $D(n,\lambda)$ the events that Conditions (a), (b), (c), and (d) are not satisfied for $z=o$ . Then, $\mathbb{P}( o \text{ is} \textit{n}\text{ -open})\ge 1-\mathbb{P}(A(n)\cup B(n))-\mathbb{P}(C(n,\lambda))-\mathbb{P}(D(n,\lambda))$ . By assumption, $\limsup_{n\uparrow \infty}\mathbb{P}(A(n)\cup B(n)) = 0$ independently of $\lambda$ . For fixed n, using Markov inequality and dominated convergence, we now also have $\limsup_{\lambda \uparrow \infty}(\mathbb{P}(C(n,\lambda)) +\mathbb{P}(D(n,\lambda))) = 0$ , as $\lambda$ increases, where we used that the expected number of streets and crossings is finite.

Finally, due to the translation invariance of the model, the existence of an infinite cluster of open sites implies that an infinite cluster of good streets and good crossings exists, and, thus, there is a positive probability that the typical device transmits the infection to infinity.

Proof of Lemma 1. First note that the conditions are only dependent on the bridging devices, which have their initial positions and their destinations close to the crossing and move over the crossing, and are therefore independent for distinct crossings. This guarantees the independence of the events.

In order to bound the probability of openness at late times, we compute the intensity of devices that satisfy Conditions (C1), (C2), and (C3) at time t and then show that, as t tends to infinity, this intensity converges to a positive crossing-dependent value. In turn, the probability that no such devices exist can be seen as a Poisson void probability.

Let us start by deriving expressions for the intensity of devices at time t in the desired interval, i.e. devices that start on $s_1$ in the interval $A_{s_1}$ and have a target location in $B_{s_2}$ , on $s_2$ . Recall that the streets $s_1$ and $s_2$ have a joint crossing c. Note that, by the displacement theorem for Poisson point processes, the number of devices that satisfy the conditions is a Poisson-distributed random variable and its parameter is given by $\lambda p_S(s_1,s_2)$ , where

\begin{equation*}p_S(s_1,s_2)=\int_{A_{s_1}}{\textrm d} x \int_{B_{s_2}}\kappa^S(x,{\textrm d} z)\,\textbf{1}\big\{0\le r_t(|x,z|_S)-|z,c|_S \le \varepsilon/2\big\},\end{equation*}

where $|z,x|_S$ denotes the shortest distance between x and z on S, and $r_t(|z,x|_S)$ is uniquely defined as $vt=|z,x|_S n_t+r_t(|z,x|_S)$ with $n_t$ the largest odd number such that $0\le r_t(|z,x|_S)\le 2|z,x|_S$ . In words, $n_t(|z,x|_S)$ denotes the number of times that the device has traveled from x to z and back, and since we want the device to be on the way back, we require $n_t(|z,x|_S)$ to be odd. Then, $r_t(|z,x|_S)$ is the distance from z on the way back. Subtracting the distance from z to the joint crossing c, we obtain the traveled distance on $s_1$ , which has to lie in the interval $(0,\varepsilon)_{s_1,c}$ .

We will show that there is a time $T_0$ such that, for all $t \ge T_0$ , we can bound $p_S(s_1,s_2)$ from below by a constant that only depends on $S\cap B_{L}(c)$ .

Let us define for $x\in A_{s_1}$ by

\begin{equation*} 0\lt\eta_x=\eta_x\big(s_1,s_2,S\cap B_{L}(x),\kappa^S(x,{\textrm d} y)\big)= \inf_{y \in B_{s_2}}\kappa^S (x,y)\end{equation*}

the minimal density for the kernel, starting in x and having a destination in $B_{s_2}$ . This value $\eta_x$ is positive as the kernel is c-continuous, $B_{s_2} \subset B_c(x)$ , and we use our assumption that $2v {\varrho_{\rm I,min}} +2\varepsilon\lt c$ .

In order to simplify the notation for $A_{s_1}$ and $B_{s_2}$ let us abbreviate $b=v{\varrho_{\rm W,min}}/2$ and associate bijectively the interval $A=[b+\varepsilon/2,b+\varepsilon]$ to $A_{s_1}$ such that $x\mapsto {x}_{s_1,c}$ . We associate $B=[b,b+\varepsilon]$ in a similar way to $B_{s_2}$ . Then,

\begin{equation*}p_S(s_1,s_2)\ge \int_{A}{\textrm d} x\, \eta_x\int_{B}{\textrm d} z\,\textbf{1}\big\{0\le r_t(x+z)-z \le \varepsilon/2\big\},\end{equation*}

where we also dropped the singular part in $\kappa^S(x,{\textrm d} y)$ . Now, let us define by

\begin{align*}\phi_x(s)=\lim_{a \downarrow 0}\frac{1}{a}\int_{B}{\textrm d} z\,\textbf{1}\big\{0\le r_s(x+z)-z\le a\big\}\end{align*}

the density of points started at x that are on their way back from their target location, and that are at time s precisely at the origin. Note that a device satisfies Condition (C3) at time t if and only if it passes the crossing on its way to its starting location in the time interval $[t-\varepsilon(2v)^{-1},t]$ . Therefore, we can rewrite

(3) \begin{equation}\int_{A}{\textrm d} x\, \eta_x\int_{B}{\textrm d} z\,\textbf{1}\big\{0\le r_t(x+z)-z \le \varepsilon/2\big\}=\int_{t-\varepsilon/(2v)}^t{\textrm d} s\int_{A}{\textrm d} x\, \eta_x\phi_x(s),\end{equation}

and we will prove that, $\mathbb{P}$ -almost surely, conditioned on S, there exist constants $T_0$ and C that depend on v, ${\varrho_{\rm I,min}}$ , and $\varepsilon$ but not on S, x, or $\kappa^S$ , such that $\inf_{t\ge T_0}\phi_x(t)\gt C\gt 0$ . In order to derive this statement, the main tool is the following identity for $\phi_x(t)$ :

\begin{equation*} \phi_{x}(t) = \sum_{n=1}^\infty \frac{1}{2\varepsilon n} \textbf{1}\big\{x+2b+(n-1)2(b+x)\le vt\lt x+2b+2\varepsilon+(n-1)2(x+b+\varepsilon)\big\}.\end{equation*}

Let us explain this formula. Here, n represents the number of times that a device started at x reaches its destination, which is uniformly distributed in B. With every reflection in B, the density is distributed to an interval of length $2\varepsilon n$ , where $\varepsilon$ is the length of B, again uniformly at random. Now, $x+2b$ , respectively $x+2b+2\varepsilon$ , are the shortest, respectively the longest, distance that a device started at x travels before it revisits the crossing c for the first time. For each consecutive visit to the origin an additional length of $2(b+x)$ , respectively $2(x+b+\varepsilon)$ , has to be added as the length of a complete cycle from the origin to its destination and back.

In other words, each indicator becomes non-zero in $x+2b+(n-1)2(x+b)$ and adds a mass of $(2\varepsilon n)^{-1}$ over a length of $2n\varepsilon$ . For $n\lt(x+b)/\varepsilon$ the indicator functions are disjoined, i.e. for $vt\lt x+2b+((x+b)/\varepsilon-1)2(x+b)$ there is at most one indicator function that is non-zero. Now, for $n\gt(x+b)/\varepsilon$ , the interval for vt in which the indicator function is non-negative is longer than the minimal cycle length. In other words, it is possible that at least two indicator functions are non-negative.

Now, for $(x+b)/\varepsilon\lt n\lt 2(x+b)/\varepsilon$ the indicator functions contribute for more than the length of a full cycle $2(x+b)$ and less than twice that length. Therefore, for each $vt \in [x+2b+(n_1-1)2(b+x),x+2b+(n_2-1)2(b+x)]$ there are either exactly one or two non-negative indicators, where $n_i=\lceil i(x+b)/\varepsilon \rceil$ . As each indicator has a weight of at least $(4\varepsilon(x+b)/\varepsilon)^{-1}$ we can bound $\phi_x(t)\gt(4\varepsilon(x+b)/\varepsilon)^{-1}$ .

More generally, in the interval $vt \in [x+2b+(n_m-1)2(b+x),x+2b+(n_{m+1}-1)2(b+x)]$ there are either m or $m+1$ overlapping indicator functions whose contributions can each be bounded by $(2n_{m+1}\varepsilon)^{-1}$ . Therefore, we can bound

\begin{equation*} \phi_x(t)\gt\frac{m}{2(n_{m+1})\varepsilon} = \frac{m}{2\varepsilon \lceil (m+1) (x+b)/\varepsilon \rceil} \ge \frac{1}{2\varepsilon \lceil 2(x+b)/\varepsilon \rceil}\ge \frac{1}{4(2b+\varepsilon)}=C\gt 0,\end{equation*}

where we used that the bound monotonically increases in m and $x\le b+\varepsilon$ . Therefore, for $T_0=(x+2b+(n_1-1)2(b+x))v^{-1}$ the desired bound is realized and integration of (3) yields

\begin{equation*} p_S(s_1,s_2)\ge \varepsilon(2v)^{-1}(8b+4\varepsilon)^{-1} \int_{A}{\textrm d} x \,\eta_x=\tilde p_S(s_1,s_2)\gt 0.\end{equation*}

Noting that the requirements in Conditions (C4) and (C5) can be considered independently, respectively with probability $p_4={\varrho_{\rm I}}([{\varrho_{\rm I,min}},{\varrho_{\rm I,min}}+\varepsilon/2])$ and $p_5={\varrho_{\rm I}}([{\varrho_{\rm I,min}},{\varrho_{\rm I,min}}+\varepsilon])$ , the number of devices that satisfy the criterion of a bridging device $X^{s_1,s_2}$ can be bounded from below by a Poisson-distributed random variable with expectation $\tilde p_S(s_1,s_2)p_4p_5$ . As c is open if there is at least one bridging device for each combination of open adjacent streets, we can bound

\begin{equation*} \mathbb{P}(c \text{ is open at time } t\mid S)\ge \prod_{\substack{i\neq j, s_i,s_j\, \text{open}\\c \in s_i,c \in s_j}} \big(1-\exp\!(\!-\lambda \tilde p(s_i,s_j)p_4p_5)\big)\quad \text{for all } t\ge T,\end{equation*}

which proves the lemma.

4.3. Proof of Theorem 2

For the proof we adopt a multiscale argument in the spirit of the proof of existence of subcritical regimes for the Poisson–Boolean model with random radii in [Reference Gouéré19]. We note that this approach has also been applied in the setting of Cox–Boolean models with random radii in [Reference Jahnel, Tóbiás and Cali29]. More specifically, we adapt the arguments for absence of percolation in the connectivity graph for sufficiently large velocities presented in [Reference Cali, Hinsen, Jahnel and Wary11]. Roughly speaking, we prove that a Cox–Boolean model in which any two devices that could transmit the infection somewhere on their path and are sufficiently close are connected, becomes subcritical if the white knight intensity is sufficiently large.

First, we call a street $s=(c_1,c_2)$ blocked for (device) x if either $|s|\lt v_{\rm min}{\varrho_{\rm I,min}}/2$ or there exist two white knights $Y_1,Y_2\in Y^{{\lambda_{\rm W}}}$ such that for $i\in \{1,2\}$ we have ${\varrho_{\rm I,min}}\gt{\rho_{\rm W}}(Y_i,x)$ , $Y_{i,t}\in s$ , and $|c_i-Y_{i,t}|\lt r$ for all $t\ge 0$ . Otherwise we call s unblocked for x. Then, we consider the thinned process

\begin{multline*}X^{\lambda,{\rm th}}=\big\{X_i\in X^\lambda\colon \text{there exists }y\in{\textrm{supp}}\big(\kappa^S(X_i,{\textrm d} y)\big) \\ \text{ and } s\in\ell_S(X_i,y)\text{ such that}\, \textit{s}\, \text{ is unblocked for $X_i$}\big\}.\end{multline*}

In words, for devices in $X^\lambda\setminus X^{\lambda,{\rm th}}$ every possible path only contains streets that are either too short to perform any transmission of the infection or contain a white knight that patches the device before it can transmit the infection on the street. Let us note that here we use that $r\gt v_{\rm max}{\varrho_{\rm W,min}}$ is large enough to allow a successful patch. Using these definitions, we note that, if the infection survives globally, then this implies that the geostatistical Cox–Boolean model $\mathcal{C}=\bigcup_{X_i\in X^{\lambda,{\rm th}}}B_{\ell_S(X_i)}(X_i)$ contains an unbounded component. Hence, in order to prove absence of global survival, it suffices to prove the absence of percolation in this Cox–Boolean model, which is also well-defined by the local connectedness assumption. Let $\mathcal{C}_x(V)$ denote the connected component containing x of the geostatistical Cox–Boolean model, based on points in $V\subset\mathbb{R}^2$ . Then, we define, for any $x\in \mathbb{R}^2$ and $\alpha\gt 0$ , the event

\begin{equation*}G(x,\alpha)=\big\{\mathcal{C}_x(B_{10\alpha}(x))\not\subset B_{8\alpha}(x)\big\}\end{equation*}

that the cluster containing x, formed by disks whose centers lie in $B_{10\alpha}(x)$ , reaches beyond $B_{8\alpha}(x)$ . Consider the set of $\alpha$ -local points $A_S(\alpha)=\{x\in S\colon \ell_S(x)\lt\alpha \}$ , let $S^*$ denote the Palm version of S, and recall the stabilization radii $R_y$ of the street system. Recall that $R(V)=\sup_{y\in V\cap\, \mathbb{Q}^2}R_y$ for any $V\subset\mathbb{R}^2$ . Then, we can employ the following key lemma that establishes a scaling relation in the model.

Lemma 2. ([Reference Cali, Hinsen, Jahnel and Wary11, Lemma III.1].) Consider the geostatistical Cox–Boolean model. There exists a constant $c\gt 0$ such that, for all $\alpha\gt 0$ ,

\begin{equation*}\mathbb{P}(G(o,10\alpha)) \le c\mathbb{P} (G(o,\alpha))^2+c\alpha^2\mathbb{P}\big(o\in A^{\rm c}_{S^*}(\alpha)\big)+c\mathbb{P}(R(Q_{10\alpha})\ge \alpha).\end{equation*}

In the next lemma we deviate from [Reference Cali, Hinsen, Jahnel and Wary11, Proof of Theorem II.3]. We show that the local percolation probability becomes zero for large white knight intensities.

Lemma 3. There exists $c\gt 0$ such that

\begin{multline*}\mathbb{P}(G(o,\alpha))\le c \alpha^2\mathbb{P}\big(\textit{there exists } y\in{\textrm{supp}}\big(\kappa^{S^*}(o,{\textrm d} y)\big) \\ \textit{ and at least one unblocked }s\in\ell_{S^*}(o,y)\big),\end{multline*}

and $\mathbb{P}\big(\textit{there exists }y\in{\textrm{supp}}\big(\kappa^{S^*}(o,{\textrm d} y)\big)\textit{ and at least one unblocked }s\in\ell_{S^*}(o,y)\big)$ tends to zero as ${\lambda_{\rm W}}$ tends to infinity.

Proof of Lemma 3. Note that

\begin{align*}\mathbb{P}(G(o,\alpha))&\le \mathbb{P}\big(X^{\lambda,{\rm th}}(B_{10\alpha})\gt 0\big)\le \mathbb{E}\big[X^{\lambda,{\rm th}}(B_{10\alpha})\big]\\&\le \lambda\mathbb{E}\bigg[\int_{S\cap B_{10\alpha}}{\textrm d} x\,\mathbb{P}\big(\text{there exists } y\in{\textrm{supp}}\big(\kappa^S(x,{\textrm d} y)\big)\text{ and } s\in\ell_S(x,y)\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{such that}\, \textit{s}\,\text{ is unblocked for}\, \textit{x}\big)\bigg]\\&\le \lambda 10^2\alpha^2\mathbb{P}\big(\text{there exists } y\in{\textrm{supp}}\big(\kappa^{S^*}(o,{\textrm d} y)\big)\text{ and } s\in\ell_{S^*}(o,y)\\&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\text{such that } \,\textit{s}\, \text{ is unblocked for }\,\textit{o}\big).\end{align*}

But, by dominated convergence,

\begin{equation*}\mathbb{P}\big(\text{there exists } y\in{\textrm{supp}}\big(\kappa^{S^*}(o,{\textrm d} y)\big)\text{ and } s\in\ell_{S^*}(o,y)\text{ such that }\,\textit{s}\, \text{ is unblocked for}\, \textit{o}\big)\end{equation*}

tends to zero as ${\lambda_{\rm W}}$ tends to infinity, and hence we arrive at the desired result.

As a final input for the proof, we recall the following essential result about convergence properties of functions satisfying some scaling inequality.

Now we are in the position to prove the theorem.

Proof of Theorem 2. Recall that S is assumed to be a PVT or PDT. In order to prove that ${\lambda_{\rm W,c}}\lt\infty$ , it suffices to show that $\lim_{\alpha\uparrow\infty}\mathbb{P}\big(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb{R}^2))\gt X^{\lambda,{\rm th}}(B_{8\alpha})\big)=0$ for all sufficiently large ${\lambda_{\rm W}}$ . But this is true if $\lim_{\alpha\uparrow\infty}\mathbb{P}(G(0,\alpha))=0$ since, by [Reference Cali, Hinsen, Jahnel and Wary11][Lemma III.4 and III.5],

\begin{equation*}\mathbb{P}\big(X^{\lambda,{\rm th}}\big(\mathcal{C}_o\big(\mathbb{R}^2\big)\big)\gt X^{\lambda,{\rm th}}(B_{8\alpha})\big) \le \mathbb{P}(G(o,\alpha))+\lambda\mathbb{E}\bigg[\int_{S}{\textrm d} x\,\textbf{1}\big\{10\alpha\le |x|\le 9\alpha+\ell_S(x)\big\}\bigg],\end{equation*}

where $\mathbb{E}\big[\int_{S}{\textrm d} x\,\textbf{1}\{10\alpha\le |x|\le 9\alpha+\ell_S(x)\}\big]$ tends to zero as $\alpha$ tends to infinity. Now, in order to show that $\lim_{\alpha\uparrow\infty}\mathbb{P}(G(0,\alpha))=0$ we apply Lemmas 2, 3, and 4 for proper choices of f and g.

For this, note that, in Lemma 2, since we assume stabilization, $\mathbb{P}(R(Q_{10\alpha})\ge \alpha)$ tends to zero as $\alpha$ tends to infinity, and the same is true for $ \alpha ^2\mathbb{P}(o\in A^{\rm c}_{S^*}(\alpha))$ by [Reference Cali, Hinsen, Jahnel and Wary11, Lemma III.2]. Hence, we can define

\begin{align*}\alpha_1&\,:\!=\inf\Big\{s\ge 1\colon \phi_1(\alpha)=\alpha ^2\mathbb{P}\big(o\in A^{\rm c}_{S^*}\big)\lt\big(8c^2\big)^{-1}\text{ for all }\alpha\ge s\Big\}, \\\alpha_2&\,:\!=\inf\big\{s\ge 1\colon \phi_2(\alpha)=\mathbb{P}(R(Q_{10\alpha})\ge \alpha)\lt\big(8c^2\big)^{-1}\text{ for all }\alpha\ge s\big\},\end{align*}

set $\alpha_{\rm c}=\alpha_1\vee\alpha_2$ , and note that $\alpha_{\rm c}\lt\infty$ . Further, we let

\begin{align*}{\lambda_{\rm W,c}}=\inf\Big\{{\lambda_{\rm W}}\ge 1\colon \mathbb{P}\big(&\text{there exists }y\in{\textrm{supp}}\big(\kappa^{S^*}(o,{\textrm d} y)\big)\\ &\text{ and at least one unblocked }s\in\ell_{S^*}(o,y)\big)\lt\tfrac 1 2 (100c\alpha_{\rm c})^{-2}\Big\},\end{align*}

where ${\lambda_{\rm W,c}}\lt\infty$ by Lemma 3. Then, we write $f(\alpha)=c\mathbb{P}(G(o,10\alpha_{\rm c}\alpha))$ and $g(\alpha)=c^2(\phi_1(\alpha_{\rm c}\alpha)+\phi_2(\alpha_{\rm c}\alpha))$ and hence, again using Lemma 3, we have $f(\alpha)\le \frac12$ for all $1\le \alpha\le10$ and ${\lambda_{\rm W}}\gt{\lambda_{\rm W,c}}$ , and, by definition, $g(\alpha)\le \frac14$ for all $1\le \alpha$ . Finally, using Lemma 2,

\begin{equation*} f(\alpha) \le c^2(\mathbb{P}(G(o,\alpha_{\rm c}\alpha))^2+\phi_1(\alpha_{\rm c}\alpha)+\phi_2(\alpha_{\rm c}\alpha)) = f(\alpha/10)^2+g(\alpha)\quad\text{ for all }\alpha \ge 10.\end{equation*}

Hence, since $\lim_{\alpha\uparrow\infty}g(\alpha)=0$ , an application of Lemma 4 gives the result.

4.4. Proof of Theorem 3

We prove the theorem in two parts.

Proof of Theorem 3 parts (ii) and (iii). Note that part (ii) is the statement of Theorem 1 which puts no restrictions on the white knight intensity. Further, part (iii) follows from [Reference Cali, Hinsen, Jahnel and Wary11][Theorem 3.3] where a subcritical percolation regime is established for sufficiently fast speeds, independent of the device intensities. This in particular implies that global survival is possible only with probability 0.

Hence, it only remains to prove part (i). The main idea is that small velocities make it impossible to transmit the infection through crossings if sufficiently many white knights are available. We say that a street $s=(c_1,c_2)$ is closed if:

1. (A) $|s|\gt 2v{\rho_{\rm W}}$ ;

2. (B) there exist white knights $Y^s\subset Y^{{\lambda_{\rm W}}}$ such that, for all $t\ge 0$ and $x\in s$ , $|x-Y_{i,t}|\lt r/2$ for some $Y_i\in Y^s$ ; and

3. (C) if, at time $t\ge 0$ , there exists an infected $X_i\in X^\lambda$ with $X_{i,t}=c_1$ (respectively $X_{i,t}=c_2$ ), then $\{X_j\in X^\lambda\cap s\colon |X_{j,u}-c_2|\le v{\rho_{\rm W}}\text{ for some }u\in \tau(t)\}=\emptyset$ (respectively $\{X_j\in X^\lambda\cap s\colon |X_{j,u}-c_1|\le v{\rho_{\rm W}}\text{ for some }u\in \tau(t)\}=\emptyset$ ). Here, $\tau(t)$ is the time interval in which the infection reaches $s\cap B_{v{\rho_{\rm W}}+r}(c_2)$ (respectively $s\cap B_{v{\rho_{\rm W}}+r}(c_1)$ ).

A street is called open if it is not closed. The key idea is that a closed street cannot be used for the transmission of the infection. Indeed, Condition (A) guarantees that the street is sufficiently long to allow the white knights to patch, and the probability that the condition is satisfied tends to 1 as v tends to 0. Further, Condition (B) guarantees the existence of white knights at all times close to every point on the street, and in particular close to the crossings. These white knights help to patch any incoming infected device, and the condition becomes likely to be satisfied for large white knight intensities. Now, for Condition (C), note first that a large device intensity $\lambda$ makes it probable that an infection travels through a street unpatched since we assume that ${\rho_{\rm W}}\gt{\rho_{\rm I}}$ . Also, due to the continuous movement of the devices, it is unavoidable that an infected device enters the street at some time. However, in order for an infected device to leave the street unpatched, it needs to receive the infection at the opposite end of the street, sufficiently close to the crossing, i.e. in $s\cap B_{v{\rho_{\rm W}}+r}(c_2)$ (respectively $s\cap B_{v{\rho_{\rm W}}+r}(c_1)$ ). Only in this case can it exit the street unpatched from the blocking white knight that is positioned there. Very roughly, then, again for small v, Condition (C) becomes likely.

What makes the probability that Condition (C) is satisfied challenging to control is the fact that, in principle, a street at any given time could host devices from distant spatial areas. In order to cope with this we again employ a multi-scale argument very similar to the one presented in Section 4.3. Using the definitions for $\ell_S$ from there, we now consider the thinned point cloud

\begin{equation*}X^{\lambda,{\rm th}}=\big\{X_i\in X^\lambda\colon \text{there exists }y\in{\textrm{supp}}\big(\kappa^S(X_i,{\textrm d} y)\big)\text{ and } s\in\ell_S(X_i,y)\text{ such that} \,\textit{s}\, \text{ is open}\big\}.\end{equation*}

Again, the idea is that any device that is not in $X^{\lambda,{\rm th}}$ only finds closed streets on any possible path and hence cannot contribute to the spread of the infection. Thus, it suffices to show subcriticality in the associated geostatistical Cox–Boolean model $\mathcal{C}=\bigcup_{X_i\in X^{\lambda,{\rm th}}}B_{\ell_S(X_i)}(X_i)$ . However, the situation is more complicated compared to the proof of Theorem 2 since the thinning procedure depends on the entire point cloud. Still, we can employ Lemmas 2 and 4, and only need to replace Lemma 3 with the following, which is presented in terms of $\mathbb{P}^*$ , the Palm version of $\mathbb{P}$ .

Lemma 5. There exists $c\gt 0$ such that

\begin{equation*}\mathbb{P}(G(o,\alpha))\le c \alpha^2\mathbb{P}^*\big(\textit{there exists }y\in{\textrm{supp}}\big(\kappa^S(o,{\textrm d} y)\big)\textit{ and } s\in\ell_S(o,y)\text{ such that }\,\textit{s}\,\text{ is open}\big)\end{equation*}

and

\begin{equation*}\limsup_{{\lambda_{\rm W}}\uparrow\infty}\limsup_{v\downarrow0}\mathbb{P}^*\big(\textit{there exists }y\in{\textrm{supp}}\big(\kappa^S(o,{\textrm d} y)\big)\textit{ and } s\in\ell_S(o,y)\textit{ such that s is open}\big)=0.\end{equation*}

Proof of Lemma 5. First note that we can use the Slivnyak–Mecke formula and translation invariance to estimate

\begin{align*}\mathbb{P}(G(o,\alpha))&\le \mathbb{P}\big(X^{\lambda,{\rm th}}(B_{10\alpha})\gt 0\big)\le \mathbb{E}\big[X^{\lambda,{\rm th}}(B_{10\alpha})\big] \\&\le \lambda 10^2\alpha^2\mathbb{P}^*\big(\text{there exists }y\in{\textrm{supp}}\big(\kappa^{S}(o,{\textrm d} y)\big)\text{ and } s\in\ell_{S}(o,y)\text{ such that}\, \textit{s}\, \text{is open}\big).\end{align*}

We highlight here that, differently to Lemma 3, the probability also extends to the devices $X^\lambda$ since openness is defined not only with respect to white knights and the street system.

In order to show the second statement of the lemma, we employ stabilization based on the stabilizing random field $\{R_x\}_{x\in\mathbb{R}^2}$ . For all sufficiently large n, we have, for $\varepsilon^{\prime}\gt 0$ ,

\begin{align*}&\mathbb{P}^*\big(\text{there exists }y\in{\textrm{supp}}\big(\kappa^{S}(o,{\textrm d} y)\big)\text{ and } s\in\ell_{S}(o,y)\text{ such that}\, \textit{s}\, \text{is open}\big) \\&\le \mathbb{P}^*\big(R(Q_{2n})\lt n/2,\,\text{there exists }y\in{\textrm{supp}}\big(\kappa^{S}(o,{\textrm d} y)\big)\text{ and } s\in\ell_{S}(o,y)\text{ such that}\, \textit{s} \, \text{is open}\big)+\varepsilon^{\prime}\\&\le \mathbb{P}^*(\text{there exists an open $s\in Q_{2n}$})+\varepsilon^{\prime},\end{align*}

where we used that $\mathbb{P}(R(Q_{2n})\ge n/2)$ tends to zero as n tends to infinity, as well as the fact that under the event $R(Q_{2n})\lt n/2$ any shortest path starting in o must be contained in $Q_{2n}$ . Now, by the Markov inequality,

\begin{align*}\mathbb{P}^*(\text{there exists an open $s\in Q_{2n}$})\le \mathbb{E}^*\Bigg[\sum_{s\in S\cap Q_{2n}}\mathbb{P}^*(s\text{ is open}\mid S)\Bigg]\end{align*}

and, using dominated convergence and the fact that $\mathbb{E}^*[\#\{s\in S\cap Q_{2n}\}]\lt\infty$ for PVT and PDT, it suffices to show that, for all $s\in S\cap Q_{2n}$ , $\limsup_{{\lambda_{\rm W}}\uparrow\infty}\limsup_{v\downarrow0}\mathbb{P}^*(s\text{ is open}\mid S)=0$ almost surely with respect to S. In view of the definition of closed streets, let C(v) denote the event that Condition (A) is satisfied for s, and similarly $D({\lambda_{\rm W}})$ and $E(v,\lambda,{\lambda_{\rm W}})$ for the events associated with Conditions (B) and (C). Then,

\begin{align*}\mathbb{P}^*(s\text{ is open}\mid S)\le \mathbb{P}(C^c(v)\mid S)+\mathbb{P}(D^c({\lambda_{\rm W}})\mid S)+\mathbb{P}(E^c(v,\lambda,{\lambda_{\rm W}})\mid S),\end{align*}

where the first and third summands tend to zero as v tends to zero, and then the second summand also tends to zero as ${\lambda_{\rm W}}$ tends to infinity. This completes the proof.

Proof of Theorem 3 part (i). For the proof we can follow the same line of argument as used for the proof of Theorem 2, replacing Lemma 3 with Lemma 5 and adjusting the subsequent arguments. We need to prove existence of ${\lambda_{\rm W,c}}\lt\infty$ such that, for all ${\lambda_{\rm W}}\gt{\lambda_{\rm W,c}}$ , there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that $\lim_{\alpha\uparrow\infty}\mathbb{P}(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb{R}^2))\gt X^{\lambda,{\rm th}}(B_{8\alpha}))=0$ for all $v\lt v_{c}(\lambda,{\lambda_{\rm W}})$ . Referencing [Reference Cali, Hinsen, Jahnel and Wary11][Lemmas III.2, III.4, and III.5], as well as Lemma 2 and the initial argument presented in Theorem 2, it suffices to note that, with the help of Lemma 5, there exists ${\lambda_{\rm W,c}}\lt\infty$ such that, for all ${\lambda_{\rm W}}\gt{\lambda_{\rm W,c}}$ , there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that, for all $v\lt v_{c}(\lambda,{\lambda_{\rm W}})$ ,

\begin{equation*}\mathbb{P}^*\big(\text{there exists }y\in{\textrm{supp}}\big(\kappa^S(o,{\textrm d} y)\big)\text{ and } s\in\ell_S(o,y)\text{ such that} \, \textit{s} \, \text{is open}\big)\lt\tfrac 1 2 (100c\alpha_{\rm c})^{-2},\end{equation*}

where $\alpha_{\rm c}$ is defined as in the proof of Theorem 2. Then, using the same definitions for f and g, the result follows by an application of Lemma 4.

4.5. Proof of Proposition 2

Proof of Proposition 2. First note that, by the assumption of supercriticality, there is a positive probability that the typical device is connected to infinity within $g_{\rho,r}(S,X^\lambda)$ via at least one path of susceptible devices, i.e. in the absence of white knights. However, in the case that $T_{\rm I}\le T_{\rm W}$ , any white knight that is connected to the path can only eliminate the infection after the infected device has already passed the infection to the subsequent susceptible device, leading to an unstoppable sequence of infections along the path, independent of the choice of ${\lambda_{\rm W}}$ .

On the other hand, in the case $T_{\rm I}\gt T_{\rm W}$ , it suffices to consider the model conditioned on the event $\{X_o\leftrightsquigarrow\infty\}$ that $X_o$ is in the infinite cluster of $g_{\rho,r}(S,X^\lambda)$ . Let us denote by D the graph distance of $X_o$ to the set of white knights, and note that $D\lt\infty$ almost surely as ${\lambda_{\rm W}}\gt 0$ . Then, the closest white knight $Y_i$ experiences an infection attempt at time $DT_{\rm I}$ . Note that, at this time, any infected device has a distance to $Y_i$ given by at most 2D. This comes from the fact that the transmission times are deterministic and there is a one-to-one correspondence between the graph distance to the typical device and elapsed time. However, there exists $N\in \mathbb{N}$ such that $NT_{\rm I}\gt(N+2D)T_{\rm W}$ , and therefore the infection can reach at most a graph distance of $N+D$ . Indeed, any device $X_i$ at graph distance $N+D$ from $X_o$ is in contact with $Y_i$ via $N+2D$ edges connecting infected or patched devices. But, since the patching is faster than the infection, for this N, on the joint path between $X_o$ , $Y_i$ , and $X_i$ , the patch will have caught up with the infection before time $DT_{\rm I}+(2D+N)T_{\rm W}$ , which is strictly smaller than $(N+D)T_{\rm I}$ .

4.6. Proof of Theorem 4

Proof of Theorem 4. As ${\varrho_{\rm W,min}} \gt 0$ , we can choose b large enough that ${\varrho^b_{\rm I,min}}\lt {\varrho_{\rm W,min}}$ . Furthermore, we can choose $\mu^{\prime}$ with $v^{\prime}_{\textrm{min}}$ such that $v_{\textrm{min}}\rho\lt v^{\prime}_{\textrm{min}}{\varrho^b_{\rm I,min}}\lt\min(a_{\rm c}/2,r,c/2)$ . Now $v^{\prime}_{\textrm{min}}$ and ${\varrho^b_{\rm I,min}}$ satisfy the conditions of Theorem 1. A close inspection of the proof of Theorem 1 reveals that, under the conditions in this theorem, there exists an infinite sequence of crossing devices, boundary devices, and devices connecting boundary devices $\{X_{n_i}\}_{i\in \mathbb{N}}$ that connect the typical device to infinity in the dynamical model considered in Theorem 1. Let us argue that this given sequence is also connected in $g_{\rho,r}(S, X^\lambda)$ . Indeed, as the chain of devices between boundary devices is strongly localized, they are connected for all $t\ge 0$ , especially for a continuous time window of length $\rho$ . Now, the crossing devices are localized in such a way that they travel at least a distance of $v^{\prime}_{\textrm{min}}{\varrho^b_{\rm I,min}}$ on their respective streets. As $v_{\textrm{min}}\rho\lt v^{\prime}_{\textrm{min}}{\varrho^b_{\rm I,min}}$ the bridging devices can connect to their respective boundary devices. Therefore, the infection can propagate in $g_{\rho,r}(S, X^\lambda)$ along the given sequence to infinity, which completes the proof.

4.7. Proof of Theorem 5

Proof of Theorem 5. Using similar arguments to the analysis of chase–escape dynamics on the Gilbert graph in [Reference Hinsen, Jahnel, Cali and Wary23], we call a device $X_i\in V^{M,p}$ good if, in the case of an infection, the infection is transmitted to all neighbors of $X_i$ before any neighbor of $X_i$ is able to patch it. If there is an infinite path of good devices, every device on this path is guaranteed to become infected and hence, if there exists an infinite component of good devices with positive probability, this implies survival of the infection also with positive probability. Now, if a device $X_i$ has at most M neighbors, the probability of being good is bounded by

(4) \begin{align} &\mathbb{P}\Bigg(\min_{X_j \in N(X_i)} {\rho_{\rm W}} (X_j,X_i) \gt \max_{X_j \in X^\lambda \cap N(X_i)} {\rho_{\rm I}}(X_i,X_j)\Bigg) \nonumber \\ & \quad \ge \mathbb{P}\Bigg(\min_{i \in \{1,\dots,M\}} \rho_{{\rm W},i} \gt \max_{i \in \{1,\dots, M\}} \rho_{{\rm I},i}^b\Bigg) = \mathbb{P}\Bigg(\min_{i \in \{1,\dots,M\}} \rho_{{\rm W},i} \gt b^{-1}\max_{i \in \{1,\dots, M\}} \rho_{{\rm I},i}\Bigg),\end{align}

independently of $X_i$ , where the $\rho_{{\rm I},i}^b$ , and respectively the $\rho_{{\rm W},i}$ , are i.i.d. random variables with distribution ${\varrho^b_{\rm I}}$ and $\varrho_{\rm W}$ .

Now, by assumption, we can choose M and p such that $g^{M,p}_{\rho,r}(S,X^\lambda)$ percolates with positive probability. Hence, as ${\varrho_{\rm W}}$ had no atom at zero we can choose $b=b(M)$ sufficiently large such that the bound in (4) is larger than p.

Note that the bound is uniform over the devices, since the infection and patch times are independent by definition, and therefore we can couple the connectivity graph to an independent thinning with parameter p that dominates the process of good devices. As a consequence, $g^{M,p}_{\rho,r}(S,X^\lambda)$ is a subgraph of the thinned graph of good devices with at most M neighbors. Hence, we have established percolation of good devices with positive probability as desired.

4.8. Proof of Theorem 6

Proof of Theorem 6. Our strategy is again to employ the arguments from Sections 4.3 and 4.4. That is, we realize that it suffices to show absence of percolation in the geostatistical Cox–Boolean model $\mathcal{C}=\bigcup_{X_i\in X^{\lambda,{\rm th}}}B_{\ell_S(X_i)}(X_i)$ , where

\begin{equation*}X^{\lambda,{\rm th}}=\Bigg\{X_i\in X^\lambda\colon \min_{X_j \in N_\lambda(X_i)} {\rho_{\rm I}}(X_i,X_j) \lt \min_{X_j \in N_{\lambda_{\rm W}}(X_i)} {\rho_{\rm W}}(X_j,X_i)\Bigg\},\end{equation*}

with $N_\lambda(X_i)$ the set of neighbors of $X_i$ in $g_{\rho,r}(S,X^\lambda)$ and $N_{{\lambda_{\rm W}}}(X_i)$ the set of neighbors of $X_i$ in $g_{\rho,r}(S,X_i\cup Y^{\lambda_{\rm W}})$ . Indeed, devices in $X^\lambda\setminus X^{\lambda,{\rm th}}$ are patched before they can transmit the infection and can therefore not contribute to the dissipation of the infection.

Again, the thinning procedure depends on the entire point cloud, but we can still employ Lemmas 2 and 4 and only need to replace Lemma 3 by the following, which is presented in terms of $\mathbb{P}^*$ , the Palm version of $\mathbb{P}$ .

Lemma 6. There exists $c\gt 0$ such that

\begin{equation*}\mathbb{P}(G(o,\alpha))\le c \alpha^2\mathbb{P}^*\Bigg(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) \lt \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\Bigg)\end{equation*}

and

\begin{equation*}\limsup_{{\lambda_{\rm W}}\uparrow\infty}\mathbb{P}^*\Bigg(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) \lt \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\Bigg)=0.\end{equation*}

Proof of Lemma 6. Again, by the Slivnyak–Mecke formula and translation invariance we can estimate

\begin{align*}\mathbb{P}(G(o,\alpha))&\le \mathbb{P}(X^{\lambda,{\rm th}}(B_{10\alpha})\gt 0)\le \mathbb{E}[X^{\lambda,{\rm th}}(B_{10\alpha})] \\&\le \lambda 10^2\alpha^2\mathbb{P}^*\Bigg(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) \lt \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\Bigg).\end{align*}

Now,

(5) \begin{align}\mathbb{P}^*\Bigg(\min_{X_j \in N_\lambda(o)} {\rho_{\rm I}}(o,X_j) \lt \min_{X_j \in N_{\lambda_{\rm W}}(o)} {\rho_{\rm W}}(X_j,o)\Bigg)\le \mathbb{E}^*\big[\mathbb{P}({\rho_{\rm W}}\gt{\varrho_{\rm I,min}})^{N_{\lambda_{\rm W}}(o)}\big],\end{align}

where ${\rho_{\rm W}}$ is a random variable distributed according to ${\varrho_{\rm W}}$ . Note that $\mathbb{P}({\rho_{\rm W}}\gt{\varrho_{\rm I,min}})\lt 1$ by our assumptions on ${\varrho_{\rm W,min}}$ . Further, the expression on the right-hand side of (5) can be further bounded from above by considering only those white knights that mimic the behavior of the typical device at the origin in the sense that they start close to the origin and have a target location close to the target location of the typical device. Then, by dominated convergence, the right-hand side of (5) tends to zero as ${\lambda_{\rm W}}$ tends to infinity, and this completes the proof.

As the lemma is now proved, this concludes the proof of Theorem 6.

Proof of Theorem 3 part (i). We can follow the same line of argument as used for the proof of Theorem 2, replacing Lemma 3 by Lemma 5 and adjusting the subsequent arguments. We need to prove the existence of ${\lambda_{\rm W,c}}\lt\infty$ such that, for all ${\lambda_{\rm W}}\gt{\lambda_{\rm W,c}}$ , there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that $\lim_{\alpha\uparrow\infty}\mathbb{P}\big(X^{\lambda,{\rm th}}(\mathcal{C}_o(\mathbb{R}^2))\gt X^{\lambda,{\rm th}}(B_{8\alpha})\big)=0$ for all $v\lt v_{c}(\lambda,{\lambda_{\rm W}})$ . Referencing [Reference Cali, Hinsen, Jahnel and Wary11][Lemmas III.2, III.4, and III.5], as well as Lemma 2 and the initial argument presented in Theorem 2, it suffices to note that, with the help of Lemma 5, there exists ${\lambda_{\rm W,c}}\lt\infty$ such that, for all ${\lambda_{\rm W}}\gt{\lambda_{\rm W,c}}$ , there exists $v_{c}(\lambda,{\lambda_{\rm W}})$ such that, for all $v\lt v_{c}(\lambda,{\lambda_{\rm W}})$ ,

\begin{equation*}\mathbb{P}^*\big(\text{there exists }y\in{\textrm{supp}}\big(\kappa^S(o,{\textrm d} y)\big)\text{ and } s\in\ell_S(o,y)\text{ such that}\, \textit{s} \,{is open}\big)\lt\tfrac 1 2 (100c\alpha_{\rm c})^{-2},\end{equation*}

where $\alpha_{\rm c}$ is defined as in the proof of Theorem 2. Then, using the same definitions for f and g, the result follows by an application of Lemma 4.