$\int_{\psi(\ell^{\prime})\leq
1/r_{n}}\psi(\ell^{\prime})\cdot\mathbf{1}(\text{$[\ell^{\prime}]_{r_{n+1}}$
meets $\ell_{0}\cap[\ell]_{r_{n}}$})\mathrm{d}\mu(\ell^{\prime})\leq
C\cdot\psi(\ell)\cdot r_{n}\cdot\ln(1/r_{n}).$ (28)

Provided that this holds, let us conclude the argument. On the complement of
the bad event $B_{n}$, we have

$\Psi_{n}=\sum_{\begin{subarray}{c}(\ell^{\prime},v^{\prime})\in\Pi_{n}^{\prime}\\\
\psi(\ell^{\prime})\leq
1/r_{n}\end{subarray}}\sum_{(\ell,v)\in\Pi_{n}}\chi(\ell,v;\Pi_{n})\cdot\mathbf{1}\left(\text{$[\ell^{\prime}]_{r_{n+1}}$
meets
$\ell_{0}\cap[\ell]_{r_{n}}$}\right)\cdot\psi(\ell^{\prime})=:\Psi_{n}^{\prime},$

where

$\displaystyle\mathbb{E}[\Psi_{n}^{\prime}\,|\,\mathcal{F}_{n}]$
$\displaystyle=$ $\displaystyle c\int_{\psi(\ell^{\prime})\leq
1/r_{n}}\int_{r_{n+1}^{\alpha}}^{r_{n}^{\alpha}}\sum_{(\ell,v)\in\Pi_{n}}\chi(\ell,v;\Pi_{n})\cdot\mathbf{1}\left(\text{$[\ell^{\prime}]_{r_{n+1}}$
meets $\ell_{0}\cap[\ell]_{r_{n}}$}\right)\cdot\psi(\ell^{\prime})\cdot
v^{-\beta}\mathrm{d}v\mathrm{d}\mu(\ell^{\prime}).$

By (28), we obtain that

$\mathbb{E}[\Psi_{n}^{\prime}\,|\,\mathcal{F}_{n}]\leq C^{\prime}\cdot
r_{n}\cdot\ln(1/r_{n})\cdot
r_{n}^{-\alpha(\beta-1)}\cdot\sum_{(\ell,v)\in\Pi_{n}}\chi(\ell,v;\Pi_{n})\cdot\psi(\ell),$

for some constant $C^{\prime}$. Now, by the definition of $\varepsilon$, we
have

$r_{n}\cdot\ln(1/r_{n})\cdot
r_{n}^{-\alpha(\beta-1)}=o\left(r_{n}^{2\varepsilon/3}\right)\quad\text{as
$n\to\infty$.}$

In particular, there exists a constant $C^{\prime\prime}$ such that
${r_{n}\cdot\ln(1/r_{n})\cdot r_{n}^{-\alpha(\beta-1)}\leq
C^{\prime\prime}\cdot r_{n}^{2\varepsilon/3}}$ for all $n\geq n_{0}$, and we
obtain

$\mathbb{E}[\Psi_{n}^{\prime}\,|\,\mathcal{F}_{n}]\leq C^{\prime\prime}\cdot
r_{n}^{2\varepsilon/3}\cdot\Phi_{n}.$ (29)

Finally, we write

$\mathbb{P}\left(\Phi_{n+1}-\Phi_{n}>r_{n}^{\varepsilon/3}\cdot\Phi_{n}\right)\leq\mathbb{P}(B_{n})+\mathbb{P}\left(\Psi_{n}^{\prime}>r_{n}^{\varepsilon/3}\cdot\Phi_{n}\right).$

The first term is controlled by (27), and for the second term, by (29) and by
Lemma 12 below, we have

$\mathbb{P}\left(\Psi_{n}^{\prime}>r_{n}^{\varepsilon/3}\cdot\Phi_{n}\right)\leq
C^{\prime\prime}\cdot r_{n}^{\varepsilon/3}.$

We deduce that

$\sum_{n\geq
n_{0}}\mathbb{P}\left(\Phi_{n+1}-\Phi_{n}>r_{n}^{\varepsilon/3}\cdot\Phi_{n}\right)<\infty,$

which proves (24) by the Borel–Cantelli lemma.

######  Lemma 12.

Let $X$ and $Y$ be nonnegative random variables. Suppose that $X$ is
measurable with respect to some sub-$\sigma$-algebra $\mathcal{F}$, and that
there exists a constant $C$ such that

$\mathbb{E}[Y\,|\,\mathcal{F}]\leq C\cdot X\quad\text{almost surely.}$

Then, we have

$\mathbb{P}(Y>\lambda\cdot X)\leq\frac{C}{\lambda}\quad\text{for all
$\lambda\in\mathbb{R}_{+}^{*}$.}$

###### Proof of the lemma.

Let $\lambda\in\mathbb{R}_{+}^{*}$. As $\mathbb{P}(Y>\lambda\cdot
X)=\lim_{\eta\downarrow 0}\mathbb{P}(Y\geq\eta+\lambda\cdot X)$, it suffices
to prove that for each $\eta>0$, we have $\mathbb{P}(Y\geq\eta+\lambda\cdot
X)\leq C/\lambda$. In fact, for each $\eta>0$, we have
$\mathbb{P}(Y\geq\eta+\lambda\cdot
X)=\mathbb{E}[\mathbb{P}(Y\geq\eta+\lambda\cdot X\,|\,\mathcal{F})]$, with

$\mathbb{P}(Y\geq\eta+\lambda\cdot
X\,|\,\mathcal{F})\leq\frac{\mathbb{E}[Y\,|\,\mathcal{F}]}{\eta+\lambda\cdot
X}\leq\frac{C\cdot X}{\eta+\lambda\cdot
X}\leq\frac{C}{\lambda}\quad\text{almost surely,}$

by the conditional Markov inequality. ∎

Finally, it remains to check the key property of $\psi$, namely that (27) and
(28) hold. Let us start with (27). We have

$\mathbb{P}(B_{n})\leq
c\cdot\mu\left\\{\ell^{\prime}\in\mathbb{L}:\text{$\psi(\ell^{\prime})\geq
1/r_{n}$ and $[\ell^{\prime}]_{r_{n+1}}$ meets
$\ell_{0}\cap\overline{B}(0,R)$}\right\\}\cdot\int_{r_{n+1}^{\alpha}}^{r_{n}^{\alpha}}v^{-\beta}\mathrm{d}v,$

where $c=2^{-1}\cdot(\beta-1)$ is the multiplicative constant in the intensity
measure of $\Pi$. To compute the right hand side, recall the definition of the
measure $\mu$ from Section 1. In the planar case $d=2$, the group
$\mathbf{SO}\left(\mathbb{R}^{2}\right)$ consists of the rotations

$\left(g_{\theta}:z\in\mathbb{R}^{2}=\mathbb{C}\mapsto e^{i\theta}\cdot
z,\,\theta\in[-\pi,\pi]\right).$

Moreover, we have

$\int_{\mathbf{SO}\left(\mathbb{R}^{2}\right)}\varphi(g)\mathrm{d}g=\frac{1}{2\pi}\int_{-\pi}^{\pi}\varphi(g_{\theta})\mathrm{d}\theta\quad\text{for
all Borel functions
$\varphi:\mathbf{SO}\left(\mathbb{R}^{2}\right)\rightarrow\mathbb{R}_{+}$.}$

Note that when $\ell^{\prime}=g_{\theta}(w+\ell_{0})$, we have

$\ell_{0}\cap[\ell^{\prime}]_{r}=\left\\{\left(-\frac{y}{\sin\theta},0\right)\,;\,y\in[w-r,w+r]\right\\}\quad\text{and}\quad\psi(\ell^{\prime})=\frac{1}{|\sin\theta|}.$

It follows that

$\displaystyle\mu\left\\{\ell^{\prime}\in\mathbb{L}:\text{$\psi(\ell^{\prime})\geq
1/r_{n}$ and $[\ell^{\prime}]_{r_{n+1}}$ meets
$\ell_{0}\cap\overline{B}(0,R)$}\right\\}$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{1/|\sin\theta|\geq
1/r_{n}}\int_{-\infty}^{\infty}\mathbf{1}\left(\exists
y\in[w-r_{n+1},w+r_{n+1}]:\left|\frac{y}{\sin\theta}\right|\leq
R\right)\mathrm{d}w\mathrm{d}\theta$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{|\sin\theta|\leq
r_{n}}\int_{-\infty}^{\infty}\mathbf{1}\left(|w|\leq
R\cdot|\sin\theta|+r_{n+1}\right)\mathrm{d}w\mathrm{d}\theta$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int_{|\sin\theta|\leq
r_{n}}2\cdot(R\cdot|\sin\theta|+r_{n+1})\mathrm{d}\theta$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{|\sin\theta|\leq r_{n}}2\cdot(R\cdot
r_{n}+r_{n})\mathrm{d}\theta=\frac{R+1}{\pi}\cdot
r_{n}\cdot\int_{|\sin\theta|\leq r_{n}}\mathrm{d}\theta.$

In particular, there exists a constant $C=C(R)$ such that

$\mu\left\\{\ell^{\prime}\in\mathbb{L}:\text{$\psi(\ell^{\prime})\geq 1/r_{n}$
and $[\ell^{\prime}]_{r_{n+1}}$ meets
$\ell_{0}\cap\overline{B}(0,R)$}\right\\}\leq C\cdot r_{n}^{2}.$

We deduce that there exists a constant $C^{\prime}$ such that
$\mathbb{P}(B_{n})\leq C^{\prime}\cdot r_{n}^{2}\cdot
r_{n}^{-\alpha(\beta-1)}$, which yields (27). Finally, let us check (28). Fix
$\ell\in\mathbb{L}$, and without loss of generality by the invariance of $\mu$
under translations, assume that $\ell_{0}\cap\ell=\\{0\\}$. In particular, we
have

$\ell_{0}\cap[\ell]_{r_{n}}=\\{(x,0)\,;\,x\in[-\psi(\ell)\cdot
r_{n},\psi(\ell)\cdot r_{n}]\\}.$

It follows that

$\displaystyle\int_{\psi(\ell^{\prime})\leq
1/r_{n}}\psi(\ell^{\prime})\cdot\mathbf{1}\left(\text{$[\ell^{\prime}]_{r_{n+1}}$
meets $\ell_{0}\cap[\ell]_{r_{n}}$}\right)\mathrm{d}\mu(\ell^{\prime})$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{1/|\sin\theta|\leq
1/r_{n}}\int_{-\infty}^{\infty}\frac{1}{|\sin\theta|}\cdot\mathbf{1}\left(\exists
y\in[w-r_{n+1},w+r_{n+1}]:\left|\frac{y}{\sin\theta}\right|\leq\psi(\ell)\cdot
r_{n}\right)\mathrm{d}w\mathrm{d}\theta$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{|\sin\theta|\geq
r_{n}}\int_{-\infty}^{\infty}\frac{1}{|\sin\theta|}\cdot\mathbf{1}\left(|w|\leq\psi(\ell)\cdot
r_{n}\cdot|\sin\theta|+r_{n+1}\right)\mathrm{d}w\mathrm{d}\theta$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi}\int_{|\sin\theta|\geq
r_{n}}\frac{1}{|\sin\theta|}\cdot 2\cdot(\psi(\ell)\cdot
r_{n}\cdot|\sin\theta|+r_{n+1})\mathrm{d}\theta$ $\displaystyle\leq$
$\displaystyle\frac{1}{2\pi}\int_{|\sin\theta|\geq
r_{n}}\frac{1}{|\sin\theta|}\cdot 2\cdot(\psi(\ell)\cdot r_{n}+\psi(\ell)\cdot
r_{n})\mathrm{d}\theta=\frac{2}{\pi}\cdot\psi(\ell)\cdot
r_{n}\cdot\int_{|\sin\theta|\geq r_{n}}\frac{\mathrm{d}\theta}{|\sin\theta|}.$

Equation (28) readily follows, which completes the proof. ∎

Now armed with Lemma 11, we can come to the proof of Lemma 10.

###### Proof of Lemma 10.

Without loss of generality, we assume that $\ell_{0}=\mathbb{R}\times\\{0\\}$
and $v_{0}=1$. Set ${r_{n}=5^{-n}}$ for all $n\in\mathbb{N}$. Moreover, fix a
parameter $\zeta\in{]0,1[}$ to be adjusted, and set ${v_{n}=\zeta\cdot
r_{n}^{1/(\beta-1)}}$ for all $n\in\mathbb{N}$. Finally, fix parameters
$\alpha_{1}<\alpha_{2}<\alpha_{3}\in{]0,1[}$ to be adjusted, and consider the
following claim.

_Claim._ There exists $\varepsilon>0$ such that for all sufficiently large
$n$, on the complement of a bad event $B_{n}$ that has probability at most
$r_{n}^{1+\varepsilon}$, the following holds: if there exists a geodesic
$V^{\prime}$-path ${\gamma:[0,\tau]\rightarrow\mathbb{R}^{2}}$ from a point
outside $\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)$ to a point on
$\ell_{0}\cap\overline{B}(0,r_{n})$ such that

$\gamma[0,\tau]\cap\ell_{0}\subset\overline{B}(0,r_{n}),$

then there must exist a road of $\Pi$ with speed at least $v_{n}^{\alpha_{3}}$
that passes through $\overline{B}\left(0,2r_{n}\right)$.

First, let us explain how this claim yields the result of the lemma. Fix
$\delta>0$, and let $\mathcal{X}$ be the set of points $x\in\ell_{0}$ for
which there exists a geodesic $V^{\prime}$-path
$\gamma:[0,\tau]\rightarrow\mathbb{R}^{2}$ from $x$ to a point outside
$\overline{B}(x,\delta)$ such that $\gamma(t)\notin\ell_{0}$ for all
$t\in{]0,\tau]}$. Let us show that for each $R>0$, almost surely, we have

$\mathcal{X}\cap\overline{B}(0,R)\subset\bigcup_{(\ell,v)\in\Pi}\ell_{0}\cap\ell.$
(30)

For every $n\in\mathbb{N}$, let $\left(B(z,r_{n})\,;\,z\in Z_{n}\right)$ be a
covering of $\ell_{0}\cap\overline{B}(0,R)$ by balls of radius $r_{n}$, with
centres $z\in\ell_{0}\cap\overline{B}(0,R)$ at least $r_{n}$ apart from each
other. By a measure argument, we have $\\#Z_{n}\leq C\cdot r_{n}^{-1}$ for
some constant $C$. Now, for all $n$ large enough so that
$2r_{n}^{\alpha_{1}}\leq\delta$, by the triangle inequality, we have

$\mathcal{X}\cap\overline{B}(0,R)\subset\bigcup_{z\in\mathcal{Z}_{n}}\ell_{0}\cap
B(z,r_{n}),$

where $\mathcal{Z}_{n}$ denotes the set of points $z\in Z_{n}$ for which there
exists a geodesic $V^{\prime}$-path
${\gamma:[0,\tau]\rightarrow\mathbb{R}^{2}}$ from a point outside
$\overline{B}\left(z,r_{n}^{\alpha_{1}}\right)$ to a point on
${\ell_{0}\cap\overline{B}(z,r_{n})}$ such that

$\gamma[0,\tau]\cap\ell_{0}\subset\overline{B}(z,r_{n}).$

Then, for every $z\in Z_{n}$, let $B_{n}^{z}$ be the bad event $B_{n}$, but
for the pushforward $\Pi\circ f_{-z,1}^{-1}$ in place of $\Pi$, where
$f_{x,r}$ is the map of (4). By (3), we have
$\mathbb{P}\left(B_{n}^{z}\right)=\mathbb{P}(B_{n})$. Now, by the union bound,
we have

$\mathbb{P}\left(\exists z\in Z_{n}:\text{$B_{n}^{z}$ is
realised}\right)\leq\sum_{z\in
Z_{n}}\mathbb{P}(B_{n}^{z})=\\#Z_{n}\cdot\mathbb{P}(B_{n})\leq C\cdot
r_{n}^{-1}\cdot r_{n}^{1+\varepsilon}=C\cdot r_{n}^{\varepsilon}.$

By the Borel–Cantelli lemma, we deduce that almost surely, for all
sufficiently large $n$, there is no point $z\in Z_{n}$ for which the bad event
$B_{n}^{z}$ is realised. Thus, for all sufficiently large $n$, we have

$\mathcal{Z}_{n}\subset\left\\{z\in Z_{n}:V^{z}_{2r_{n}}\geq
v_{n}^{\alpha_{3}}\right\\}.$

It follows that

$\mathcal{X}\cap\overline{B}(0,R)\subset\left\\{x\in\ell_{0}:\text{$V^{x}_{3r_{n}}\geq
v_{n}^{\alpha_{3}}$ for all sufficiently large $n$}\right\\},$

which yields (30) by Lemma 11.

Now, let us dive into the proof of the claim. We introduce the “vertical
speed” function
${\phi:\mathbb{L}\times\mathbb{R}_{+}^{*}\rightarrow\mathbb{R}_{+}}$, defined
by
$\phi(\ell,v)=\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|\cdot
v$ for all ${(\ell,v)\in\mathbb{L}\times\mathbb{R}_{+}^{*}}$. Note that we
have $\phi(\ell,v)\leq v$ for all
$(\ell,v)\in\mathbb{L}\times\mathbb{R}_{+}^{*}$. In particular, for every
$\phi_{0}>0$, there are finitely many roads $(\ell,v)\in\Pi$ with
$\phi(\ell,v)\geq\phi_{0}$ that pass through
$\overline{B}(0,r_{n}^{\alpha_{1}})$. Then, for each $k\in\mathbb{N}$, we
define $R_{k}^{1},R_{k}^{2},R_{k}^{3},R_{k}^{4}$ as the four rectangles that
are represented in Figure 11, and we denote by $\Pi_{k}$ the restriction of
$\Pi$ to the set of roads $(\ell,v)$ with speed $v\geq v_{k}$ for which there
exists $i\in\llbracket 1,4\rrbracket$ such that $\ell$ crosses $R_{k}^{i}$ in
the length direction (i.e, passes through both shorter sides of the
rectangle). Note that there exists a constant $c$ such that

$\phi(\ell,v)\geq c\cdot v_{k}\quad\text{for all $(\ell,v)\in\Pi_{k}$.}$ (31)

Finally, let $A_{k}$ be the good event: “for each $i\in\llbracket
1,4\rrbracket$, there is a road of $\Pi$ with speed at least $v_{k}$ that
crosses $R_{k}^{i}$ in the length direction”. We call $k$ a “good scale” if
the event $A_{k}$ is realised.

Figure 11: Definition of the rectangles
$R_{k}^{1},R_{k}^{2},R_{k}^{3},R_{k}^{4}$, and illustration of the good event
$A_{k}$. In this example, the event $A_{k}$ is realised, hence $k$ is a good
scale. Note that there exists a constant $c$ such that for every line
$\ell\in\mathbb{L}$ that crosses $R_{k}^{i}$ in the length direction for some
$i\in\llbracket 1,4\rrbracket$, we have
$\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|\geq c$. Moreover,
note that every line $\ell\in\mathbb{L}$ that crosses $R_{k}^{i}$ in the
length direction for some $i\in\llbracket 1,4\rrbracket$ passes through
$\overline{B}(0,r_{k})$ but does not pass through $\overline{B}(0,r_{k+1})$.

At this point, let us deal with the technical matter of ajudsting our
parameters. First, we choose $\alpha_{1}<\alpha_{2}<\alpha_{3}$ close enough
to $1$ so that

$\alpha_{1}-\frac{\alpha_{3}}{\beta-1}\cdot(\beta-2)+(\alpha_{1}-1)>0$ (32)

and

$\alpha_{1}>1-\frac{\alpha_{3}}{\beta-1}.$ (33)

Next, we set $\phi_{n}^{0}=c\cdot v_{n}^{\alpha_{2}}$, where $c$ is the
constant of (31). Then, using (32), we fix $j\in\mathbb{N}^{*}$ large enough
so that

$\left(\alpha_{1}-\frac{\alpha_{3}}{\beta-1}\cdot(\beta-2)+(\alpha_{1}-1)\right)\cdot
j>1.$ (34)

Now that $j$ has been fixed, note that by (33), the following holds for all
sufficiently large $n$:

$\frac{r_{n}}{\phi_{n}^{0}}+j\cdot
2r_{n}^{\alpha_{1}}\leq\frac{r_{n}}{v_{n}^{\alpha_{3}}}.$ (35)

Finally, we adjust $\zeta$ so that the following bad event, say $D_{n}$, has
probability at most $r_{n}^{2}$ for all sufficiently large $n$: “there is no
good scale between $\alpha_{1}n$ and $\alpha_{2}n$”. To this end, note that
for each $k\in\mathbb{N}$, the event $A_{k}$ is measurable with respect to the
restriction of $\Pi$ to the set of roads that pass through
$\overline{B}(0,r_{k})$ but do not pass through $\overline{B}(0,r_{k+1})$, by
the definition of the $R_{k}^{i}$ (see Figure 11). Moreover, by the definition
of $v_{k}$, we have ${\mathbb{P}(A_{k})=\mathbb{P}(A_{0})=:p(\zeta)}$. It
follows that

$\mathbb{P}(D_{n})=p(\zeta)^{\lfloor\alpha_{2}n\rfloor-\lceil\alpha_{1}n\rceil+1}.$

Since ${\lim_{\zeta\downarrow 0}p(\zeta)=1}$, we can fix $\zeta$ small enough
so that $p(\zeta)^{\alpha_{2}-\alpha_{1}}<1/10$. We deduce that for all
sufficiently large $n$, we have

$\mathbb{P}(D_{n})\leq 10^{-n}=r_{n}^{2}.$

From now on, we assume that $n$ is large enough so that (35) holds. With this
setup, consider the following implication.

_Implication._ On the complement of the bad event $D_{n}$, if there exists a
geodesic $V^{\prime}$-path ${\gamma:[0,\tau]\rightarrow\mathbb{R}^{2}}$ from a
point outside $\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)$ to a point on
$\ell_{0}\cap\overline{B}(0,r_{n})$ such that

$\gamma[0,\tau]\cap\ell_{0}\subset\overline{B}(0,r_{n}),$

and if there is no road of $\Pi$ with speed at least $v_{n}^{\alpha_{3}}$ that
passes through $\overline{B}(0,2r_{n})$, then the following bad event, say
$E_{n}$, must be realised: “there exists $j$ roads $(\ell,v)\in\Pi$ that pass
through $\overline{B}(0,r_{n}^{\alpha_{1}})$ such that
$1/\phi(\ell,v)\in\left[\left.1\middle/\phi_{n}^{0}\right.,\left.1\middle/\phi_{n}^{0}\right.+j\cdot\left.2r_{n}^{\alpha_{1}}\middle/r_{n}\right.\right]$”.

Provided that this implication holds, suppose furthermore that there exists
$\varepsilon>0$ such that

$\mathbb{P}(E_{n})\leq r_{n}^{1+\varepsilon}\quad\text{for all sufficiently
large $n$.}$ (36)

Then, the claim we are after readily follows, with $B_{n}=D_{n}\cup E_{n}$.
Therefore, to complete the proof of the lemma, it suffices to prove the
implication, and check that (36) holds.

We start with the implication. Suppose that there exists a good scale $k$
between $\alpha_{1}n$ and $\alpha_{2}n$, and that there exists a geodesic
$V^{\prime}$-path ${\gamma:[0,\tau]\rightarrow\mathbb{R}^{2}}$ from a point
outside $\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)$ to a point on
$\ell_{0}\cap\overline{B}(0,r_{n})$ such that
${\gamma[0,\tau]\cap\ell_{0}\subset\overline{B}(0,r_{n})}$. Moreover, assume
that there is no road of $\Pi$ with speed at least $v_{n}^{\alpha_{3}}$ that
passes through $\overline{B}(0,2r_{n})$, and let us prove that then, the bad
event $E_{n}$ must be realised. Let
$\phi_{n}^{1}>\ldots>\phi_{n}^{j}\in\left]0,\phi_{n}^{0}\right[$ be defined by

$\frac{1}{\phi_{n}^{i}}=\frac{1}{\phi_{n}^{0}}+i\cdot\frac{2r_{n}^{\alpha_{1}}}{r_{n}}\quad\text{for
all $i\in\llbracket 1,j\rrbracket$.}$

We will prove that for each $i\in\llbracket 1,j\rrbracket$, there must exist a
road $(\ell,v)\in\Pi$ that passes through $\overline{B}(0,r_{n}^{\alpha_{1}})$
such that

$\frac{1}{\phi(\ell,v)}\in\left]\frac{1}{\phi_{n}^{i-1}},\frac{1}{\phi_{n}^{i}}\right].$

Fix $i\in\llbracket 1,j\rrbracket$. We denote by $\mathcal{R}$ be the set of
roads $(\ell,v)\in\Pi$ that pass through $\overline{B}(0,r_{n}^{\alpha_{1}})$
and have vertical speed $\phi(\ell,v)\geq\phi_{n}^{i-1}$. Note that
$\mathcal{R}$ is finite, and contains the roads of $\Pi_{k}$. Indeed, since
$k$ is between $\alpha_{1}n$ and $\alpha_{2}n$, each road $(\ell,v)\in\Pi_{k}$
passes through $\overline{B}(0,r_{n}^{\alpha_{1}})$, and has vertical speed
$\phi(\ell,v)\geq c\cdot v_{n}^{\alpha_{2}}=\phi_{n}^{0}\geq\phi_{n}^{i-1}$,
by (31). Since $k$ is a good scale, it follows that the roads of $\mathcal{R}$
induce a convex polygon $P\subset\overline{B}(0,r_{n}^{\alpha_{1}})$, which
formally is defined as the intersection of the half-spaces
$\left\\{x-\lambda\cdot\mathrm{proj}_{\ell}(0)\,;\,\text{$x\in\ell$ and
$\lambda\geq 0$}\right\\}$, over the roads $(\ell,v)\in\mathcal{R}$. This
polygon $P$ has the following key property: for any point $x$ on its boundary
$\partial P$, and for any point $y\in\ell_{0}\cap\overline{B}(0,r_{n})$, one
can use the roads of $\mathcal{R}$ to drive from $x$ to a point $z$ on
$\ell_{0}\cap\overline{B}(0,r_{n}^{\alpha_{1}})$, in time at most
$\left.d(x,\ell_{0})\middle/\phi_{n}^{i-1}\right.$, and then use the road
$(\ell_{0},1)$ to drive from $z$ to $y$, in time at most $2r_{n}^{\alpha_{1}}$
by the triangle inequality. See Figure 12 for an illustration.

Figure 12: We reproduce Figure 11, assuming furthermore that $k$ is between
$\alpha_{1}n$ and $\alpha_{2}n$. Then, we consider all the roads
$(\ell,v)\in\Pi$ with $\phi(\ell,v)\geq\phi_{n}^{i-1}$ that pass through
$\overline{B}(0,r_{n}^{\alpha_{1}})$, that is, the roads of $\mathcal{R}$. In
this example, this adds exactly two more roads to the picture. The convex
polygon $P$ containing $0$ induced by the roads of $\mathcal{R}$ is
represented here in gray. Let us prove its key property on the example
illustrated here. With the notation introduced on the picture, one can use
$(\ell_{1},v_{1})$ to drive from $x$ to $z_{1}$, in time
$\delta_{1}/\phi(\ell_{1},v_{1})\leq\left.\delta_{1}\middle/\phi_{n}^{i-1}\right.$,
then use $(\ell_{2},v_{2})$ to drive from $z_{1}$ to $z_{2}$, in time
${\delta_{2}/\phi(\ell_{2},v_{2})\leq\left.\delta_{2}\middle/\phi_{n}^{i-1}\right.}$,
and finally use $(\ell_{0},1)$ to drive from $z_{2}$ to $y$, in time
${|z_{2}-y|\leq 2r_{n}^{\alpha_{1}}}$. The total driving time of this
$V^{\prime}$-path is at most
${\left.(\delta_{1}+\delta_{2})\middle/\phi_{n}^{0}\right.+2r_{n}^{\alpha_{1}}}=d(x,\ell_{0})+2r_{n}^{\alpha_{1}}$.

By (35), we have

$\frac{r_{n}}{\phi_{n}^{i-1}}=\frac{r_{n}}{\phi_{n}^{0}}+(i-1)\cdot
2r_{n}^{\alpha_{1}}\leq\frac{r_{n}}{v_{n}^{\alpha_{3}}},$

hence $\phi_{n}^{i-1}\geq v_{n}^{\alpha_{3}}$. In particular, since $\partial
P$ consists of roads of $\mathcal{R}$, that have speed at least
$\phi_{n}^{i-1}$, the circuit $\partial P$ must lie outside
$\overline{B}(0,2r_{n})$. Let us denote by $t\in{]0,\tau[}$ the last time at
which $\gamma$ crosses $\partial P$. As $\gamma$ is geodesic, the key property
of $P$ (see Figure 12) implies that

$\tau-t\leq\frac{d(\gamma(t),\ell_{0})}{\phi_{n}^{i-1}}+2r_{n}^{\alpha_{1}}.$
(37)

On the other hand, observe that over the time interval $[t,\tau]$, the
$V^{\prime}$-path $\gamma$ must travel a distance at least $r_{n}$ by using
the roads of $\Pi$ within the ball $\overline{B}(0,2r_{n})$. This yields

$\tau-t\geq\frac{r_{n}}{V^{0}_{2r_{n}}}>\frac{r_{n}}{v_{n}^{\alpha_{3}}}.$
(38)

By (35), we have

$\frac{r_{n}}{\phi_{n}^{i-1}}+2r_{n}^{\alpha_{1}}=\frac{r_{n}}{\phi_{n}^{0}}+i\cdot
2r_{n}^{\alpha_{1}}\leq\frac{r_{n}}{v_{n}^{\alpha_{3}}}.$

In particular, combining this with (37) and (38), we must have
${d(\gamma(t),\ell_{0})>r_{n}}$. Then, we complement (37) with the bound

$\tau-t\geq\frac{d(\gamma(t),\ell_{0})}{\phi(\ell,v)},$ (39)

where $(\ell,v)$ denotes the road of $\Pi$ with maximal vertical speed used by
$\gamma$ over the time interval $[t,\tau]$. By the definition of $t$, the road
$(\ell,v)$ passes through $\overline{B}(0,r_{n}^{\alpha_{1}})$, and has
vertical speed $\phi(\ell,v)<\phi_{n}^{i-1}$. Moreover, combining (37) and
(39), we obtain that

$\frac{d(\gamma(t),\ell_{0})}{\phi(\ell,v)}\leq\frac{d(\gamma(t),\ell_{0})}{\phi_{n}^{i-1}}+2r_{n}^{\alpha_{1}},$

hence

$\frac{1}{\phi(\ell,v)}\leq\frac{1}{\phi_{n}^{i-1}}+\frac{2r_{n}^{\alpha_{1}}}{d(\gamma(t),\ell_{0})}\leq\frac{1}{\phi_{n}^{i-1}}+\frac{2r_{n}^{\alpha_{1}}}{r_{n}}=\frac{1}{\phi_{n}^{i}}.$

This concludes the proof of the implication.

Finally, to complete the proof of the lemma, it remains to check (36). By
definition, we have

$E_{n}=\left(\Psi_{n}\left[\frac{1}{\phi_{n}^{0}},\frac{1}{\phi_{n}^{0}}+j\cdot\frac{2r_{n}^{\alpha_{1}}}{r_{n}}\right]\geq
j\right),$

where $\Psi_{n}$ denotes the pushforward by $1/\phi$ of the restriction of
$\Pi$ to the set of roads that pass through
$\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)$. By the mapping theorem, the
random measure $\Psi_{n}$ is a Poisson process with intensity measure
$c^{\prime}\cdot r_{n}^{\alpha_{1}}\cdot w^{\beta-2}\mathrm{d}w$ on
$\mathbb{R}_{+}^{*}$, for some constant $c^{\prime}$. Indeed, for every Borel
function ${\varphi:\mathbb{R}_{+}^{*}\rightarrow\mathbb{R}_{+}}$, we have

$\displaystyle\int_{\left\langle\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)\right\rangle}\int_{\mathbb{R}_{+}^{*}}\varphi\left(\frac{1}{\phi(\ell,v)}\right)\cdot
v^{-\beta}\mathrm{d}v\mathrm{d}\mu(\ell)$ $\displaystyle=$
$\displaystyle\int_{\left\langle\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)\right\rangle}\int_{0}^{\infty}\varphi\left(\frac{1}{\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|\cdot
v}\right)\cdot v^{-\beta}\mathrm{d}v\mathrm{d}\mu(\ell)$ $\displaystyle=$
$\displaystyle\int_{\left\langle\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)\right\rangle}\int_{0}^{\infty}\varphi(w)\cdot\left(\frac{1}{\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|\cdot
w}\right)^{-\beta}\cdot\frac{\mathrm{d}w}{\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|\cdot
w^{2}}\mathrm{d}\mu(\ell)$ $\displaystyle=$
$\displaystyle\int_{\left\langle\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)\right\rangle}\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|^{\beta-1}\mathrm{d}\mu(\ell)\cdot\int_{0}^{\infty}\varphi(w)\cdot
w^{\beta-2}\mathrm{d}w,$

with, by the self-similarity of $\mu$ under scaling:

$\int_{\left\langle\overline{B}\left(0,r_{n}^{\alpha_{1}}\right)\right\rangle}\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|^{\beta-1}\mathrm{d}\mu(\ell)=\int_{\left\langle\overline{B}(0,1)\right\rangle}\left|\left\langle\ell,\ell_{0}^{\perp}\right\rangle\right|^{\beta-1}\mathrm{d}\mu(\ell)\cdot
r_{n}^{\alpha_{1}}.$

Using the inequality $\mathbb{P}(\mathrm{Poisson}(\lambda)\geq
j)\leq\lambda^{j}$, it follows that

$\mathbb{P}(E_{n})\leq\left(c^{\prime}\cdot
r_{n}^{\alpha_{1}}\cdot\int_{\left.1\middle/\phi_{n}^{0}\right.}^{\left.1\middle/\phi_{n}^{0}\right.+j\cdot\left.2r_{n}^{\alpha_{1}}\middle/r_{n}\right.}w^{\beta-2}\mathrm{d}w\right)^{j}.$

Finally, since

$\int_{\left.1\middle/\phi_{n}^{0}\right.}^{\left.1\middle/\phi_{n}^{0}\right.+j\cdot\left.2r_{n}^{\alpha_{1}}\middle/r_{n}\right.}w^{\beta-2}\mathrm{d}w\leq\left(\frac{1}{\phi_{n}^{0}}+j\cdot\frac{2r_{n}^{\alpha_{1}}}{r_{n}}\right)^{\beta-2}\cdot
j\cdot\frac{2r_{n}^{\alpha_{1}}}{r_{n}}\leq\left(\frac{1}{v_{n}^{\alpha_{3}}}\right)^{\beta-2}\cdot
j\cdot\frac{2r_{n}^{\alpha_{1}}}{r_{n}}$

by (35), we obtain

$\mathbb{P}(E_{n})\leq\left(c^{\prime}\cdot
r_{n}^{\alpha_{1}}\cdot\left(\frac{1}{v_{n}^{\alpha_{3}}}\right)^{\beta-2}\cdot
j\cdot\frac{2r_{n}^{\alpha_{1}}}{r_{n}}\right)^{j},$

which yields (36), by (34). The proof of the lemma is complete. ∎

##  6 On the cut locus

We conclude this paper with a short discussion about the cut locus. The notion
of cut locus was introduced by Poincaré in the context of smooth manifolds. In
our setting of rough random metric, we follow the definition of Angel,
Kolesnik and Miermont [3]: for a fixed point $x\in\mathbb{R}^{2}$, the weak
cut locus $\mathcal{C}_{x}$ of $\left(\mathbb{R}^{2},T\right)$ with respect to
$x$ is the set of points $y\in\mathbb{R}^{2}$ from which there exists multiple
geodesics to $x$. Actually, by Lemma 2, for each $x$, almost surely
$\mathcal{C}_{x}$ agrees with the strong cut locus, that is the set of points
$y$ from which there exists multiple geodesics to $x$ which are required to be
disjoint in a neighborhood of $y$. In [3], the authors study the stability of
$\mathcal{C}_{x}$ when the base point $x$ is moved over $\mathbb{R}^{2}$, and
the two notions could differ for exceptional points $x$. We shall here focus
on more basic properties, and will henceforth suppose that $x=0$ is fixed. We
use the term cut locus for the weak or strong (indifferently, since the two
are equal almost surely) cut locus $\mathcal{C}_{0}$ with respect to $0$. Let
us make a couple of remarks on $\mathcal{C}_{0}$:

  * •

Almost surely, we have $\mathcal{I}\cap\mathcal{C}_{0}=\emptyset$. Indeed, if
there exist two disjoint geodesics $\gamma_{1},\gamma_{2}$ from an
intersection point $x\in\mathcal{I}$ to $0$, then $\gamma_{1}$ and
$\gamma_{2}$ coalesce at a point $y$ by before hitting $0$, by item 1
Proposition 1. See Figure 13. Since geodesics do not pause en route (Theorem
1), the confluence point $y$ must also belong to $\mathcal{I}$, and the
geodesic segments of $\gamma_{1}$ and $\gamma_{2}$ between $x$ and $y$ are
what we called simple $V$-paths, with the same driving time. But by the
multivariate Mecke formula, almost surely, it is not possible to find two
distinct simple $V$-paths between two points of $\mathcal{I}$ with the same
driving time.

Figure 13: Illustration of the proof of
$\mathcal{I}\cap\mathcal{C}_{0}=\emptyset$.

  * •

Almost surely, for each road $(\ell_{0},v_{0})\in\Pi$, the set
$\ell_{0}\cap\mathcal{C}_{0}$ is countable. Indeed, let $x\in\ell$ be a point
such that there exist two distinct geodesics $\gamma_{1}$ and $\gamma_{2}$
from $x$ to $0$. First, by the previous point, we have
$x\in\mathcal{L}\setminus\mathcal{I}$. Then, by the results of the previous
section, the point $x$ is not a $3^{+}$-star, and there must exist roads
$(\ell_{1},v_{1})\neq(\ell_{2},v_{2})\in\Pi\setminus\\{(\ell_{0},v_{0})\\}$
such that $\gamma_{i}$ leaves $(\ell_{0},v_{0})$ by using the road
$(\ell_{i},v_{i})$, for each $i$ (since $\gamma_{1}$ and $\gamma_{2}$ are
disjoint locally around $x$, these two roads must be distinct). In turn, the
point $x\in\ell$ can be recovered from the two roads $(\ell_{i},v_{i})$: if
$x_{i}$ denotes the intersection point of $\ell_{i}$ with $\ell_{0}$, then $x$
is the unique point $z$ in $\ell$ such that

$\frac{|z-x_{1}|}{v_{0}}+T(x_{1},0)=\frac{|z-x_{2}|}{v_{0}}+T(x_{2},0).$

Therefore, since there are countably many pairs of roads of $\Pi$, the set
$\ell_{0}\cap\mathcal{C}_{0}$ must be countable.

Recall that in smooth contexts (complete, analytic Riemannian surface
homeomorphic to the sphere), the cut locus is known to be a tree with finitely
many branches [24]. In our setting of fractal random metric, we believe that
the cut locus $\mathcal{C}_{0}$ is dense, path-connected and does not contain
any non-trivial cycle. More precisely, the degree of $x\in\mathcal{C}_{0}$,
i.e, the number of connected components of $\mathcal{C}_{0}\setminus\\{x\\}$,
should be equal to the maximal number of distinct geodesics from $x$ to $0$,
and furthermore points with degree at least $3$ should form a countable and
dense subset. This last assertion, and the fact that the cut locus contains a
non-trivial connected set, have been proved for the Lqg random metric by
Gwynne [12]. The kind of topological arguments he used should also work out in
our setting (and in fact the above assertions should hold under relatively
weak assumptions on the planar random metric), and to convince the reader of
this fact we give the following example: the argument of Gwynne [12, Lemma
3.2] shows that

######  Proposition 6.

Almost surely, the set of points
$x\in\left.\mathbb{R}^{2}\middle\backslash\\{0\\}\right.$ such that there
exists (at least) three distinct geodesics from $x$ to $0$ is countable.

###### Proof.

Fix $0<r<R$, and let $A=\left.\overline{B}(0,R)\middle\backslash
B(0,r)\right.$ be the annulus between radii $r$ and $R$ around $0$. Fix
$\delta\in{]0,r[}$, and let us show that almost surely, the set
$\mathcal{X}_{\delta}$ of points $x\in A$ for which there exists three
geodesics $\gamma_{i}:[0,T(x,0)]\rightarrow\mathbb{R}^{2}$ from $x$ to $0$
such that

$\gamma_{i}{]0,T(x,0)]}\cap\gamma_{j}{]0,T(x,0)]}\cap\overline{B}(x,\delta)=\emptyset\quad\text{for
all $i\neq j$}$

is finite. By Proposition 2, there exists $m\in\mathbb{N}^{*}$ such that
almost surely, the following holds: for every $y\in A$, there exists
$\eta_{y}\in{]0,\delta/2[}$ and a cut set
${\mathcal{Z}_{y}\subset\left.\overline{B}(y,\delta/2)\middle\backslash\overline{B}(y,\eta)\right.}$
with cardinality $\\#\mathcal{Z}_{y}\leq m$, such that any geodesic $V$-path
from a point inside $\overline{B}(y,\eta_{y})$ to a point outside
$\overline{B}(y,\delta/2)$ must pass through $z$ for some
$z\in\mathcal{Z}_{y}$. Then, since the open balls $(B(y,\eta_{y})\,;\,y\in A)$
cover $A$, by compactness, there exists a finite subset $Y\subset A$ such that
the open balls $(B(y,\eta_{y})\,;\,y\in Y)$ cover $A$. Now, we claim that
almost surely, we have

$\\#\mathcal{X}_{\delta}\leq\\#Y\cdot\binom{m}{3}<\infty.$

This follows from the argument of Gwynne [12, Lemma 3.2], which shows that for
each $y\in Y$, we have

$\\#\mathcal{X}_{\delta}\cap\overline{B}(y,\eta_{y})\leq\binom{m}{3}.$ (40)

Indeed, for each triple of distinct points
$z_{1},z_{2},z_{3}\in\mathcal{Z}_{y}$, let
$\mathcal{X}_{\delta}^{y;z_{1},z_{2},z_{3}}$ be the set of points
${x\in\overline{B}(y,\eta_{y})}$ for which there exists three geodesics
$\gamma_{i}:[0,T(x,0)]\rightarrow\mathbb{R}^{2}$ from $x$ to $0$ with

$\gamma_{i}{]0,T(x,0)]}\cap\gamma_{j}{]0,T(x,0)]}\cap\overline{B}(x,\delta)=\emptyset\quad\text{for
all $i\neq j$},$

such that $\gamma_{i}$ passes through $z_{i}$ for each $i$. First, since
$\overline{B}(y,\delta/2)\subset\overline{B}(x,\delta)$ for all
$x\in\overline{B}(y,\eta_{y})$, we must have

$\mathcal{X}_{\delta}\cap\overline{B}(y,\eta_{y})\subset\bigcup_{\text{$z_{1},z_{2},z_{3}\in\mathcal{Z}_{y}$
distinct}}\mathcal{X}_{\delta}^{y;z_{1},z_{2},z_{3}}.$

Then, for each triple of distinct points
$z_{1},z_{2},z_{3}\in\mathcal{Z}_{y}$, the argument of Gwynne [12, Lemma 3.2]
shows that $\\#\mathcal{X}_{\delta}^{y;z_{1},z_{2},z_{3}}\leq 1$. See Figure
14 for a visual proof, and [12, Lemma 3.2] for a detailed proof in the
framework of Lqg random metrics. Thus, we obtain (40), which completes the
proof.

Figure 14: Illustration of the argument of Gwynne [12, Lemma 3.2]. Fix
$x\in\mathcal{X}_{\delta}^{y;z_{1},z_{2},z_{3}}$, and draw the corresponding
three geodesics $\gamma_{i}$ from $x$ to $0$, where $\gamma_{i}$ passes
through $z_{i}$ for each $i$. By Lemma 2, their newtork must have a similar
form to the one illustrated here. Then, try to draw three geodesics
$\gamma^{\prime}_{i}$ from a point
$x^{\prime}\in\left.\overline{B}(y,\eta_{y})\middle\backslash\\{x\\}\right.$
to $0$ such that $\gamma^{\prime}_{i}$ passes through $z_{i}$ for each $i$.
Let $z_{i}^{\prime}$ be the point at which $\gamma^{\prime}_{i}$ meets
$\gamma_{i}$, which is necessarily between $x$ and $z_{i}$, and observe that
we cannove have $z_{i}^{\prime}=x$, again by Lemma 2. The existence of such a
point $x^{\prime}$ is simply not possible, and we conclude that
$\\#\mathcal{X}_{\delta}^{y;z_{1},z_{2},z_{3}}\leq 1$.

∎

Back to our discussion, notice that any point of $\mathcal{C}_{0}$ with degree
at least $3$ is in particular a $3^{+}$-star, and thus cannot belong to
$\mathcal{L}\setminus\mathcal{I}$ by Theorem 2. Also, such a point cannot
belong to $\mathcal{I}$ since $\mathcal{I}\cap\mathcal{C}_{0}=\emptyset$, and
so the existence of such points furnishes examples of a $3^{+}$-star which is
not on the geodesic frame (a canonical example of a $3^{+}$-star is given by a
confluence point of two distinct geodesics to a typical point, hence lies on
the geodesic frame). Furthermore, provided that $\mathcal{C}_{0}$ is path-
connected and non-trivial, it must be uncountable, and so most of
$\mathcal{C}_{0}$ lies outside $\mathcal{L}$, since we saw that almost surely
$\mathcal{L}\cap\mathcal{C}_{0}$ is countable. We also believe that the
“branches” of $\mathcal{C}_{0}$ are of fractal nature: almost surely, for
every $x\neq y\in\mathcal{C}_{0}$, and for any path
$\gamma:[0,1]\rightarrow\mathcal{C}_{0}$ from $x$ to $y$, the Hausdorff
dimension of $\gamma[0,1]$ should be strictly greater than $1$ with respect to
the Euclidean metric. See Open problem 1 at the end of the Introduction.

##  7 Appendix

In this section, we present the folklore results that we have deferred to the
appendix. The first one is the following mixing property of $\Pi$.

######  Lemma 13.

Let $\mathbb{M}$ be the space of counting measures on
$\mathbb{L}\times\mathbb{R}_{+}^{*}$, endowed as usual with the
$\sigma$-algebra generated by the maps
$\left(\pi\in\mathbb{M}\mapsto\pi(B),\,\text{$B$ Borel subset of
$\mathbb{L}\times\mathbb{R}_{+}^{*}$}\right)$. For any bounded measurable
functions $\varphi$ and $\psi$ from $\mathbb{M}$ to $\mathbb{R}$, we have

$\mathbb{E}\left[\varphi(\Pi)\cdot\psi\left(f_{k}^{*}\Pi\right)\right]\longrightarrow\mathbb{E}[\varphi(\Pi)]\cdot\mathbb{E}[\psi(\Pi)]\quad\text{as
$k\to\infty$.}$

###### Proof of Lemma 13.

For each ${n\in\mathbb{N}^{*}}$, let
$\rho_{n}:\mathbb{M}\rightarrow\mathbb{M}$ be the restriction map to the set
of roads with speed at least $1/n$ that pass through $\overline{B}(0,n)$, and
let ${\mathcal{A}_{n}=\sigma(\rho_{n}(\Pi))}$ be the $\sigma$-algebra
generated by the restriction $\rho_{n}(\Pi)$ of $\Pi$. Fix bounded measurable
functions $\varphi,\psi:\mathbb{M}\rightarrow\mathbb{R}$, and fix
$\varepsilon>0$. First, we use the following classical measure-theoretic
approximation argument [16, Lemma 3.16]: since the $\sigma$-algebra generated
by $\Pi$ agrees with the $\sigma$-algebra generated by $\bigcup_{n\geq
1}\mathcal{A}_{n}$, there exists $n\in\mathbb{N}^{*}$ and bounded measurable
functions $\varphi_{n},\psi_{n}:\mathbb{M}\rightarrow\mathbb{R}$ such that

$\mathbb{E}[|\varphi(\Pi)-\varphi_{n}(\rho_{n}(\Pi))|]\leq\varepsilon\quad\text{and}\quad\mathbb{E}[|\psi(\Pi)-\psi_{n}(\rho_{n}(\Pi))|]\leq\varepsilon.$
(41)

Now, let $C$ be a constant that dominates $\varphi$, $\psi$, $\varphi_{n}$ and
$\psi_{n}$. On the one hand, for every $k\in\mathbb{N}$, we have

$|\mathbb{E}[\varphi(\Pi)\cdot\psi(f_{k}^{*}\Pi)]-\mathbb{E}[\varphi_{n}(\rho_{n}(\Pi))\cdot\psi_{n}(\rho_{n}(f_{k}^{*}\Pi))]|\leq
2C\varepsilon.$ (42)

On the other hand, the random measures $\rho_{n}(\Pi)$ and
$\rho_{n}\left(f_{k}^{*}\Pi\right)$ are “asymptotically independent”, as
$k\to\infty$. Indeed, let $G_{k}$ be the event: “no road of $\Pi$ with speed
at least $1/n$ passes through $\overline{B}(0,n\cdot r_{k})$”. The following
holds:

  * •

we have $\mathbb{P}(G_{k})\rightarrow 1$ as $k\to\infty$,

  * •

on $G_{k}$, the random measure $\rho_{n}(\Pi)$ agrees with the restriction
$\Pi_{k}^{1}$ of $\Pi$ to the set of roads with speed at least $1/n$ that pass
through $\overline{B}(0,n)$ but do not pass through $\overline{B}(0,n\cdot
r_{k})$, and $\rho_{n}\left(f_{k}^{*}\Pi\right)$ agrees with the restriction
$\Pi_{k}^{2}$ of $\Pi$ to the set of roads with speed at least $1/n\cdot
r_{k}^{(d-1)/(\beta-1)}$ that pass through $\overline{B}(0,n\cdot r_{k})$,

  * •

the random measures $\Pi_{k}^{1}$ and $\Pi_{k}^{2}$ are independent.

With these three ingredients, standard manipulations show that

$\mathbb{E}[\varphi_{n}(\rho_{n}(\Pi))\cdot\psi_{n}(\rho_{n}(f_{k}^{*}\Pi))]\underset{k\to\infty}{\longrightarrow}\mathbb{E}[\varphi_{n}(\rho_{n}(\Pi))]\cdot\mathbb{E}[\psi_{n}(\rho_{n}(\Pi))].$

Then, we conclude with (41) and (42): as $k\to\infty$, we have

$\mathbb{E}[\varphi(\Pi)\cdot\psi(f_{k}^{*}\Pi)]\longrightarrow\mathbb{E}[\varphi(\Pi)]\cdot\mathbb{E}[\psi(\Pi)].$

∎

The second one is the proof of Lemma 3, which is essentially a mix between
[15, Theorem 3.1] of the third author and [6, Proposition 1.3] of the first
author.

###### Proof of Lemma 3.

Beofre diving into the proof, let us recall the following estimate on the
invariant measure $\mu$ on the space of lines. For two (or more) compact
subsets $K_{1},K_{2}\subset\mathbb{R}^{d}$, we use $\langle
K_{1}\,;\,K_{2}\rangle$ as shorthand for $\langle K_{1}\rangle\cap\langle
K_{2}\rangle$. In complement to (1), we have the following estimate: there
exists constants $c$ and $C$ such that for every $x\neq y\in\mathbb{R}^{d}$,
and for every $0<r,s\leq|x-y|$,

$c\cdot\frac{r^{d-1}\cdot
s^{d-1}}{|x-y|^{d-1}}\leq\mu\left\langle\overline{B}(x,r)\,;\,\overline{B}(y,s)\right\rangle\leq
C\cdot\frac{r^{d-1}\cdot s^{d-1}}{|x-y|^{d-1}}.$ (43)

Now, let us dive into the proof. Fix a parameter $\rho\in{]0,r[}$ to be
adjusted, let $\Pi_{\rho}$ be the restriction of $\Pi$ to the set of roads
with speed less than $1$ that do not pass through $\overline{B}(z,\rho)$, and
let $V_{\rho}$ be “speed limits” function induced by $\Pi_{\rho}$. For
$x,y\in\mathbb{R}^{d}$ and $s>0$, we denote by $V_{\rho}(x,y,s)$ the speed of
the fastest road of $\Pi_{\rho}$ that passes through both balls
$\overline{B}(x,s)$ and $\overline{B}(y,s)$. Let us recall Kendall’s
construction of $V$-paths from [17, Theorem 3.1], and use it to construct a
$V_{\rho}$-path between every pair of points $(x,y)$ with $x,y\in A$. Fix a
parameter $\alpha\in{]0,1[}$ to be adjusted, and fix $x,y\in A$. To construct
a $V_{\rho}$-path from $x$ to $y$, we proceed as follows. Assuming that $x\neq
y$ (for otherwise there is nothing to do), consider the fastest road
$(\ell,v)$ of $\Pi_{\rho}$ that passes through both balls
$\overline{B}(x,\alpha|x-y|)$ and ${\overline{B}(y,\alpha|x-y|)}$333In
particular, we will adjust $\alpha$ and $\rho$ to ensure that such a road
exists., and let $x^{\prime}$ (resp. $y^{\prime}$) be the orthogonal
projection of $x$ (resp. $y$) on $\ell$. The road $(\ell,v)$ allows to drive
from $x^{\prime}$ to $y^{\prime}$ in time $|x^{\prime}-y^{\prime}|/v$, and it
remains to find routes from $x$ to $x^{\prime}$ and from $y^{\prime}$ to $y$.
For this purpose, we recursively apply the procedure to the pairs of points
$(x,x^{\prime})$ and $(y^{\prime},y)$. To formalise this, let us encode the
construction in the nodes $\varnothing,1,2,\ldots$ of the infinite binary tree
$\mathbb{B}$ as follows. Initially, we set
$(x_{\varnothing},y_{\varnothing})=(x,y)$. Then, by induction, for every
${n\in\mathbb{N}}$ such that $((x_{u},y_{u})\,;\,u\in\mathbb{B}:|u|\leq n)$
have been constructed, we proceed as follows. For each node $u\in\mathbb{B}$
at height $n$, there is the following alternative.

  * •

If $x_{u}=y_{u}$, then “there is nothing to do”. We let $x_{u}^{\prime}=x_{u}$
and $y_{u}^{\prime}=y_{u}$, and we let $T_{u}=0$ be the driving time of the
trivial $V$-path from $x^{\prime}_{u}$ to $y^{\prime}_{u}$. Then, we define
the pairs of points associated with the children $u1$ and $u2$ of $u$ in
$\mathbb{B}$ as $(x_{u1},y_{u1})=(x_{u},x_{u}^{\prime})$ and
$(x_{u2},y_{u2})=(y_{u}^{\prime},y_{u})$.

  * •

Otherwise, consider the fastest road $(\ell_{u},v_{u})$ of $\Pi_{\rho}$ that
passes through both balls

$\overline{B}(x_{u},\alpha|x_{u}-y_{u}|)\quad\text{and}\quad\overline{B}(y_{u},\alpha|x_{u}-y_{u}|).$

Note that by definition, we have
$v_{u}=V_{\rho}(x_{u},y_{u},\alpha|x_{u}-y_{u}|)$. We let $x_{u}^{\prime}$
(resp. $y_{u}^{\prime}$) be the orthogonal projection of $x_{u}$ (resp.
$y_{u}$) on $\ell_{u}$, and we let
$T_{u}=|x_{u}^{\prime}-y_{u}^{\prime}|/v_{u}$ be the driving time of the
$V$-path that drives from $x_{u}^{\prime}$ to $y_{u}^{\prime}$ using the road
$(\ell_{u},v_{u})$. Then, we define the pairs of points associated with the
children $u1$ and $u2$ of $u$ in $\mathbb{B}$ as
$(x_{u1},y_{u1})=(x_{u},x_{u}^{\prime})$ and
${(x_{u2},y_{u2})=(y_{u}^{\prime},y_{u})}$.

Taking for granted that $\sum_{u\in\mathbb{B}}T_{u}<\infty$, it is a neither
immediate nor difficult deterministic consequence of the construction that all
the $V$-paths from $x_{u}^{\prime}$ to $y_{u}^{\prime}$ for $u\in\mathbb{B}$
yield a $V$-path $\gamma:[0,T]\rightarrow\mathbb{R}^{d}$ from $x$ to $y$, with
driving time $T=\sum_{u\in\mathbb{B}}T_{u}$. The rest of the proof consists in
showing that $\alpha$ and $\rho$ can be adjusted in order to obtain the
desired bound on the driving times $\sum_{u\in\mathbb{B}}T_{u}$ provided by
this construction. Note that for every $u\in\mathbb{B}$ such that $x_{u}\neq
y_{u}$, we have

$T_{u}=\frac{|x_{u}^{\prime}-y_{u}^{\prime}|}{v_{u}}\leq\frac{|x_{u}-y_{u}|}{V_{\rho}(x_{u},y_{u},\alpha|x_{u}-y_{u}|)}.$

Moreover, note that we have $|x_{u}-y_{u}|\leq\alpha^{|u|}\cdot|x-y|$ for all
$u\in\mathbb{B}$, and

$x_{u},y_{u}\in\overline{B}\left(x,\frac{\alpha}{1-\alpha}\right)\cup\overline{B}\left(y,\frac{\alpha}{1-\alpha}\right)\subset(A)_{\alpha/(1-\alpha)}\quad\text{for
all $u\in\mathbb{B}$,}$

where
$(A)_{\varepsilon}=\left\\{x\in\mathbb{R}^{d}:d(x,A)\leq\varepsilon\right\\}$
denotes the $\varepsilon$-neighbourhood of $A$. Now, consider the following
lemma, which provides a uniform lower bound on the speeds

$\left(V_{\rho}(x,y,\alpha|x-y|)\,;\,x\neq
y\in(A)_{\alpha/(1-\alpha)}\right).$

######  Lemma 14.

For all sufficiently small $\alpha$, there exists
$\rho=\rho(\alpha)\in{]0,r[}$ and a positive random variable $\zeta$,
measurable with respect to the restriction of $\Pi_{\rho}$ to the set of roads
that pass through $\overline{B}(0,2)$, such that

$V_{\rho}(x,y,\alpha|x-y|)\geq\zeta\cdot|x-y|^{(d-1)/(\beta-1)}\cdot\ln\left(\frac{4}{\alpha|x-y|}\right)^{-1/(\beta-1)}\quad\text{for
all $x\neq y\in(A)_{\alpha/(1-\alpha)}$.}$

Moreover, there exists constants $C=C(\alpha)$ and $c=(\alpha)$ such that

$\mathbb{P}\left(\zeta<\frac{1}{t}\right)\leq C\cdot\exp\left[-c\cdot
t^{\beta-1}\right]\quad\text{for all $t\in\mathbb{R}_{+}^{*}$.}$

Taking this lemma for granted, we deduce that

$\begin{split}\sum_{u\in\mathbb{B}}T_{u}&\leq\sum_{u\in\mathbb{B}}\frac{|x_{u}-y_{u}|}{\zeta\cdot|x_{u}-y_{u}|^{(d-1)/(\beta-1)}\cdot\ln(4/(\alpha|x_{u}-y_{u}|))^{-1/(\beta-1)}}\\\
&=\zeta^{-1}\cdot\sum_{u\in\mathbb{B}}|x_{u}-y_{u}|^{(\beta-d)/(\beta-1)}\cdot\ln\left(\frac{4}{\alpha|x_{u}-y_{u}|}\right)^{1/(\beta-1)}=\zeta^{-1}\cdot\sum_{u\in\mathbb{B}}\phi(|x_{u}-y_{u}|),\end{split}$

where $\phi:s\mapsto s^{(\beta-d)/(\beta-1)}\cdot\ln(4/(\alpha
s))^{1/(\beta-1)}$. Now, recall that
${|x_{u}-y_{u}|\leq\alpha^{|u|}\cdot|x-y|\leq 2}$ for all $u\in\mathbb{B}$.
Since the function ${t\mapsto t^{-(\beta-d)}\cdot\ln(t)}$ is nonincreasing
over the interval $\left[e^{1/(\beta-d)},\infty\right[$, we deduce that for
$\alpha$ small enough so that $4/(\alpha\cdot 2)\geq e^{1/(\beta-d)}$, the
function $\phi$ is nondecreasing over the interval $]0,2]$. Then, summing over
the height $n$ of $u$ in $\mathbb{B}$, we get

$\begin{split}\sum_{u\in\mathbb{B}}\phi(|x_{u}-y_{u}|)&\leq\sum_{u\in\mathbb{B}}\phi\left(\alpha^{|u|}\cdot|x-y|\right)\\\
&=\sum_{n\geq 0}2^{n}\cdot\phi\left(\alpha^{n}\cdot|x-y|\right)\\\
&=\sum_{n\geq
0}2^{n}\cdot\left(\alpha^{n}\cdot|x-y|\right)^{(\beta-d)/(\beta-1)}\cdot\ln\left(\frac{4}{\alpha^{n+1}\cdot|x-y|}\right)^{1/(\beta-1)}.\end{split}$

Next, using that $|x-y|\geq 2$ we bound, for all $n\in\mathbb{N}$:

$\begin{split}\ln\left(\frac{4}{\alpha^{n+1}\cdot|x-y|}\right)&=(n+1)\ln\left(\frac{1}{\alpha}\right)+\ln\left(\frac{4}{|x-y|}\right)\\\
&\leq(n+1)\cdot\frac{\ln(1/\alpha)}{\ln
2}\cdot\ln\left(\frac{4}{|x-y|}\right)+(n+1)\ln\left(\frac{4}{|x-y|}\right)\\\
&=\left(\frac{\ln(1/\alpha)}{\ln
2}+1\right)\cdot(n+1)\cdot\ln\left(\frac{4}{|x-y|}\right)\\\
&=:C\cdot(n+1)\cdot\ln\left(\frac{4}{|x-y|}\right).\end{split}$

Plugging this bound in the sum above, we obtain

$\displaystyle\sum_{u\in\mathbb{B}}\phi(|x_{u}-y_{u}|)$ $\displaystyle\leq$
$\displaystyle\sum_{n\geq
0}2^{n}\cdot\left(\alpha^{n}\cdot|x-y|\right)^{(\beta-d)/(\beta-1)}\cdot\left(C\cdot(n+1)\cdot\ln\left(\frac{4}{|x-y|}\right)\right)^{1/(\beta-1)}$
$\displaystyle=$ $\displaystyle C^{1/(\beta-1)}\cdot\sum_{n\geq
0}\left(2\cdot\alpha^{(\beta-d)/(\beta-1)}\right)^{n}\cdot(n+1)^{1/(\beta-1)}\cdot|x-y|^{(\beta-d)/(\beta-1)}\cdot\ln\left(\frac{4}{|x-y|}\right)^{1/(\beta-1)}.$

Assuming further that $\alpha$ is small enough so that
$2\cdot\alpha^{(\beta-d)/(\beta-1)}<1$, we get

$\sum_{n\geq
0}\left(2\cdot\alpha^{(\beta-d)/(\beta-1)}\right)^{n}\cdot(n+1)^{1/(\beta-1)}<\infty,$

and we conclude that

$\sum_{u\in\mathbb{B}}\phi(|x_{u}-y_{u}|)\leq
C^{\prime}\cdot|x-y|^{(\beta-d)/(\beta-1)}\cdot\ln\left(\frac{4}{|x-y|}\right)^{1/(\beta-1)}$

for some constant $C^{\prime}=C^{\prime}(\alpha)$. This yields the result of
the lemma, with $\Gamma=C^{\prime}\cdot\zeta^{-1}$ (the tail estimate for
$\Gamma$ comes from the tail estimate for $\zeta$). To actually complete the
proof, it remains to prove Lemma 14.

###### Proof of Lemma 14.

The proof combines a discretisation argument together with tail estimates on
the random variables $V_{\rho}(x,y,r)$. Let $\Delta=2(1+\alpha)/(1-\alpha)$ be
the diameter of $(A)_{\alpha/(1-\alpha)}$, and set
${r_{n}=\alpha^{n}\cdot\Delta}$ for all $n\in\mathbb{N}$. For each
$n\in\mathbb{N}$, let $\left(B(w,r_{n}/2)\,;\,w\in W_{n}\right)$ be a covering
of $(A)_{\alpha/(1-\alpha)}$ by balls of radius $r_{n}/2$, with centres
$w\in(A)_{\alpha/(1-\alpha)}$ at least $r_{n}/2$ apart from each other. By a
measure argument, we have $\\#W_{n}\leq C\cdot\alpha^{-dn}$ for some constant
$C$. Now, we claim that for every $x\neq y\in(A)_{\alpha/(1-\alpha)}$, we have

$V_{\rho}(x,y,\alpha|x-y|)\geq\min\left\\{V_{\rho}\left(w,w^{\prime},\frac{r_{n+1}}{2}\right)\,;\,w\neq
w^{\prime}\in W_{n+1}:|w-w^{\prime}|\leq\left(1+\alpha^{2}\right)\cdot
r_{n-1}\right\\},$ (44)

where $n\in\mathbb{N}^{*}$ is such that $r_{n}\leq|x-y|\leq r_{n-1}$. Indeed,
let ${w,w^{\prime}\in W_{n+1}}$ be such that ${x\in B(w,r_{n+1}/2)}$ and
${y\in B(w^{\prime},r_{n+1}/2)}$. By the $1$-Lipschitz continuity of
$|\cdot|$, we have

$||w-w^{\prime}|-|x-y||\leq|w-x|+|w^{\prime}-y|\leq\frac{r_{n+1}}{2}+\frac{r_{n+1}}{2}=\alpha^{2}\cdot
r_{n-1}.$

In particular, it follows that $w\neq w^{\prime}$, and
$|w-w^{\prime}|\leq\left(1+\alpha^{2}\right)\cdot r_{n-1}$. Next, by the
triangle inequality, we have
${\overline{B}(w,r_{n+1}/2)\subset\overline{B}(x,r_{n+1})}$ and
${\overline{B}(w^{\prime},r_{n+1}/2)\subset\overline{B}(y,r_{n+1})}$. Since
${r_{n+1}=\alpha\cdot r_{n}\leq\alpha|x-y|}$, it follows that
$V_{\rho}(x,y,\alpha|x-y|)\geq V_{\rho}(w,w^{\prime},r_{n+1}/2)$, which
completes the proof of the claim. Now, let us introduce the random variables

$\zeta_{n}=\frac{n^{1/(\beta-1)}}{r_{n-1}^{(d-1)/(\beta-1)}}\cdot\min\left\\{V_{\rho}\left(w,w^{\prime},\frac{r_{n+1}}{2}\right)\,;\,w\neq
w^{\prime}\in W_{n+1}:|w-w^{\prime}|\leq\left(1+\alpha^{2}\right)\cdot
r_{n-1}\right\\},$

and set $\zeta_{*}=\inf_{n\geq 1}\zeta_{n}$. Note that these random variables
are measurable with respect to the restriction of $\Pi_{\rho}$ to the set of
roads that pass through $(A)_{\alpha/(1-\alpha)+r_{2}/2}$. Moreover, we claim
that for every $x\neq y\in(A)_{\alpha/(1-\alpha)}$, we have

$V_{\rho}(x,y,\alpha|x-y|)\geq
c\cdot\zeta_{*}\cdot|x-y|^{(d-1)/(\beta-1)}\cdot\ln\left(\frac{\Delta}{\alpha|x-y|}\right)^{-1/(\beta-1)},$
(45)

where $c=\ln(1/\alpha)^{1/(\beta-1)}$. Indeed, given $x\neq
y\in(A)_{\alpha/(1-\alpha)}$, fix an integer $n\in\mathbb{N}^{*}$ such that
${r_{n}\leq|x-y|\leq r_{n-1}}$. By (44), we have

$V_{\rho}(x,y,\alpha|x-y|)\geq\frac{r_{n-1}^{(d-1)/(\beta-1)}}{n^{1/(\beta-1)}}\cdot\zeta_{n}\geq\frac{|x-y|^{(d-1)/(\beta-1)}}{\log_{1/\alpha}(\Delta/(\alpha|x-y|))^{1/(\beta-1)}}\cdot\zeta_{*},$

and (45) readily follows. From now on, we assume that $\alpha$ is small enough
so that

$1+\frac{\alpha}{1-\alpha}+\frac{r_{2}}{2}=\frac{1+\alpha^{2}\cdot(1+\alpha)}{1-\alpha}\leq
2\quad\text{and}\quad\Delta=\frac{2(1+\alpha)}{1-\alpha}\leq 4.$

It follows that the random variables $\zeta_{n}$ and $\zeta$ are measurable
with the restriction of $\Pi_{\rho}$ to the set of roads that pass through
$\overline{B}(0,2)$, and that

$V_{\rho}(x,y,\alpha|x-y|)\geq
c\cdot\zeta_{*}\cdot|x-y|^{(d-1)/(\beta-1)}\cdot\ln\left(\frac{4}{\alpha|x-y|}\right)^{-1/(\beta-1)}\quad\text{for
all $x\neq y\in(A)_{\alpha/(1-\alpha)}$.}$

To complete the proof of the lemma, it remains to show that $\alpha$ and
$\rho$ can be adjusted in order to obtain the desired tail estimate for
$\zeta=c\cdot\zeta_{*}$. It suffices to show that there exists constants
$C^{\prime}=C^{\prime}(\alpha)$ and $c^{\prime}=c^{\prime}(\alpha)$ such that

$\mathbb{P}\left(\zeta_{*}<\frac{1}{t}\right)\leq
C^{\prime}\cdot\exp\left[-c^{\prime}\cdot t^{\beta-1}\right]\quad\text{for all
sufficiently large $t$.}$

Fix $t\in[2,\infty[$. By the union bound, we have

$\mathbb{P}\left(\zeta_{*}<\frac{1}{t}\right)\leq\sum_{n\geq
1}\mathbb{P}\left(\zeta_{n}<\frac{1}{t}\right).$

Now, recall the definition of $\zeta_{n}$: the quantity
$\mathbb{P}(\zeta_{n}<1/t)$ is the probability that there exists $w\neq
w^{\prime}\in W_{n+1}$ with $|w-w^{\prime}|\leq\left(1+\alpha^{2}\right)\cdot
r_{n-1}$ such that

$V_{\rho}\left(w,w^{\prime},\frac{r_{n+1}}{2}\right)<\frac{1}{t}\cdot\frac{r_{n-1}^{(d-1)/(\beta-1)}}{n^{1/(\beta-1)}}=:v_{n}.$

For fixed $w,w^{\prime}$ as above, the quantity
$\mathbb{P}(V_{\rho}(w,w^{\prime},r_{n+1}/2)<v_{n})$ is the probability that
there is no road of $\Pi_{\rho}$ with speed at least $v_{n}$ that passes
through $\overline{B}(w,r_{n+1}/2)$ and $\overline{B}(w^{\prime},r_{n+1}/2)$.
Therefore, we have
$\mathbb{P}(V_{\rho}(w,w^{\prime},r_{n+1}/2)<v_{n})=e^{-\lambda}$, where

$\lambda=\upsilon_{d-1}^{-1}\cdot\mu\left(\left\langle\overline{B}\left(w,\frac{r_{n+1}}{2}\right)\,;\,\overline{B}\left(w^{\prime},\frac{r_{n+1}}{2}\right)\right\rangle\middle\backslash\overline{B}(z,\rho)\right)\cdot(\beta-1)\cdot\int_{v_{n}}^{1}v^{-\beta}\mathrm{d}v.$
(46)

The constants $\upsilon_{d-1}^{-1}$ and $(\beta-1)$ come from the precise
definition of the intensity measure of $\Pi$, see (2) for a reminder. Since
$v_{n}\leq 1/t\leq 1/2$, we have

$(\beta-1)\cdot\int_{v_{n}}^{1}v^{-\beta}\mathrm{d}v=v_{n}^{-(\beta-1)}-1\geq
v_{n}^{-(\beta-1)}-(2v_{n})^{-(\beta-1)}=\left(1-2^{-(\beta-1)}\right)\cdot
v_{n}^{-(\beta-1)}.$

Now, recall the estimate (43). On the one hand, we have

$\begin{split}\mu\left\langle\overline{B}\left(w,\frac{r_{n+1}}{2}\right)\,;\,\overline{B}\left(w^{\prime},\frac{r_{n+1}}{2}\right)\right\rangle&\geq
c\cdot\frac{(r_{n+1}/2)^{d-1}\cdot(r_{n+1}/2)^{d-1}}{|w-w^{\prime}|^{d-1}}\\\
&\geq
c\cdot\frac{(r_{n+1}/2)^{d-1}\cdot(r_{n+1}/2)^{d-1}}{\left(\left(1+\alpha^{2}\right)\cdot
r_{n-1}\right)^{d-1}}\\\
&=c\cdot\frac{\left(\alpha^{2}\middle/2\right)^{d-1}\cdot\left(\alpha^{2}\middle/2\right)^{d-1}}{\left(1+\alpha^{2}\right)^{d-1}}\cdot
r_{n-1}^{d-1}=:c^{\prime}\cdot r_{n-1}^{d-1}.\end{split}$

On the other hand, we have

$\begin{split}\mu\left\langle\overline{B}\left(w,\frac{r_{n+1}}{2}\right)\,;\,\overline{B}\left(w^{\prime},\frac{r_{n+1}}{2}\right)\,;\,\overline{B}(z,\rho)\right\rangle&\leq\mu\left\langle\overline{B}\left(w,\frac{r_{n+1}}{2}\right)\,;\,\overline{B}(z,\rho)\right\rangle\\\
&\leq C\cdot\frac{(r_{n+1}/2)^{d-1}\cdot\rho^{d-1}}{|w-z|^{d-1}}\\\
&=C\cdot\frac{\left(\alpha^{2}\middle/2\right)^{d-1}\cdot\rho^{d-1}}{|w-z|^{d-1}}\cdot
r_{n-1}^{d-1}.\end{split}$

Now, we want to adjust $\alpha$ and $\rho$ so as to obtain

$\mu\left(\left\langle\overline{B}\left(w,\frac{r_{n+1}}{2}\right)\,;\,\overline{B}\left(w^{\prime},\frac{r_{n+1}}{2}\right)\right\rangle\middle\backslash\overline{B}(z,\rho)\right)\geq\frac{c^{\prime}}{2}\cdot
r_{n-1}^{d-1}$

uniformly in $w,w^{\prime}$ as above. It suffices to ensure that

$C\cdot\frac{\left(\alpha^{2}\middle/2\right)^{d-1}\cdot\rho^{d-1}}{|w-z|^{d-1}}\leq\frac{c^{\prime}}{2}$

uniformly in $w\in(A)_{\alpha/(1-\alpha)}$. First, we assume that $\alpha$ is
small enough so that

$\frac{\alpha}{1-\alpha}\leq\frac{r}{2}.$

This way, we have $|w-z|\geq r/2$ for all $w\in(A)_{\alpha/(1-\alpha)}$. Now
that $\alpha$ has been fixed, we choose $\rho$ small enough so that

$C\cdot\frac{\left(\alpha^{2}\middle/2\right)^{d-1}\cdot\rho^{d-1}}{(r/2)^{d-1}}\leq\frac{c^{\prime}}{2}.$

We can finally come back to our tail estimate: for the parameter $\lambda$ of
(46), we obtain the bound

$\lambda\geq\upsilon_{d-1}^{-1}\cdot\frac{c^{\prime}}{2}\cdot
r_{n-1}^{d-1}\cdot\left(1-2^{-(\beta-1)}\right)\cdot
v_{n}^{-(\beta-1)}=\upsilon_{d-1}^{-1}\cdot\frac{c^{\prime}}{2}\cdot\left(1-2^{-(\beta-1)}\right)\cdot
t^{\beta-1}\cdot n=:c^{\prime\prime}\cdot t^{\beta-1}\cdot n,$

uniformly in $w,w^{\prime}$ as above. By the union bound, we deduce that

$\mathbb{P}\left(\zeta_{n}<\frac{1}{t}\right)\leq\\#W_{n+1}^{2}\cdot\exp\left[-c^{\prime\prime}\cdot
t^{\beta-1}\cdot n\right].$

Finally, recall that $\\#W_{n+1}\leq C\cdot\alpha^{-d(n+1)}$, hence

$\mathbb{P}\left(\zeta_{n}<\frac{1}{t}\right)\leq
C^{2}\cdot\alpha^{-2d(n+1)}\cdot\exp\left[-c^{\prime\prime}\cdot
t^{\beta-1}\cdot
n\right]=C^{2}\cdot\exp\left[2d\ln(1/\alpha)\cdot(n+1)-c^{\prime\prime}\cdot
t^{\beta-1}\cdot n\right].$

It follows that

$\mathbb{P}\left(\zeta_{n}<\frac{1}{t}\right)\leq
C^{2}\cdot\exp\left[-c^{\prime\prime}/2\cdot t^{\beta-1}\cdot
n\right]\quad\text{for all sufficiently large $t$,}$

uniformly in $n\in\mathbb{N}^{*}$. Summing this over $n$, we obtain the
desired tail estimate: we get

$\mathbb{P}\left(\zeta_{*}<\frac{1}{t}\right)\leq\sum_{n\geq
1}C^{2}\cdot\exp\left[-c^{\prime\prime}/2\cdot t^{\beta-1}\cdot
n\right]=C^{2}\cdot\frac{\exp\left[-c^{\prime\prime}/2\cdot
t^{\beta-1}\right]}{1-\exp\left[-c^{\prime\prime}/2\cdot t^{\beta-1}\right]}$

for all sufficiently large $t$. ∎

∎

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