# Fractal properties of Aldous–Kendall random metric

G.Guillaume<EMAIL_ADDRESS>[
Université Paris-Saclay, Orsay, Francepresep=, ]e1

###### Abstract

Investigating a model of scale-invariant random spatial network suggested by
Aldous, Kendall constructed a random metric $T$ on $\mathbb{R}^{d}$, for which
the distance between points is given by the optimal connection time, when
travelling on the road network generated by a Poisson process of lines with a
speed limit. In this paper, we look into some fractal properties of that
random metric. In particular, although almost surely the metric space
$\left(\mathbb{R}^{d},T\right)$ is homeomorphic to the usual Euclidean
$\mathbb{R}^{d}$, we prove that its Hausdorff dimension is given by
$(\gamma-1)d/(\gamma-d)>d$, where $\gamma>d$ is a parameter of the model;
which confirms a conjecture of Kahn. We also find that the metric space
$\left(\mathbb{R}^{d},T\right)$ equipped with the Lebesgue measure exhibits a
multifractal property, as some points have untypically big balls around them.

###### Abstract

[language=french] En étudiant un modèle de “scale-invariant random spatial
network” suggéré par Aldous, Kendall a construit une métrique aléatoire $T$
sur $\mathbb{R}^{d}$, pour laquelle la distance entre les points est donnée
par le temps de trajet optimal, lorsqu’on se déplace sur le réseau de routes
engendré par un processus de Poisson de droites avec une limitation de
vitesse. Dans cet article, nous nous intéressons aux propriétés fractales de
cette métrique aléatoire. En particulier, bien que presque sûrement l’espace
métrique $\left(\mathbb{R}^{d},T\right)$ soit homéomorphe à l’espace euclidien
$\mathbb{R}^{d}$, nous montrons que sa dimension de Hausdorff est donnée par
$(\gamma-1)d/(\gamma-d)>d$, où $\gamma>d$ est un paramètre du modèle ; cela
confirme une conjecture de Kahn. Nous montrons par ailleurs que l’espace
métrique $\left(\mathbb{R}^{d},T\right)$ muni de la mesure de Lebesgue est
multifractal, puisque certains points se trouvent être au centre de boules
atypiquement grosses.

60D05,

Random geometry,

Poisson process,

Hausdorff dimension,

###### keywords:

[class=MSC]

###### keywords:

## Introduction and main results

In this paper, we are interested in fractal properties of a self-similar
random metric on $\mathbb{R}^{d}$, which was constructed by Kendall in the
pioneer paper [7]. Investigating a model suggested by Aldous in [1, Subsection
4.1], the author of [7] considers a Poisson random measure $\Pi$ with
intensity proportional to $\mu_{d}\otimes v^{-\gamma}\mathrm{d}v$ on
$\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$, where $\mathbb{L}_{d}$ is the space
of affine lines in $\mathbb{R}^{d}$ and $\mu_{d}$ is its unique (up to
multiplicative constant) invariant measure, and $\gamma>d$ is a parameter of
the model. Viewing each element $(\ell,v)$ of
$\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$ as a road in $\mathbb{R}^{d}$, with
$v$ the speed limit on the line $\ell$, the atoms of the measure $\Pi$ are
seen as the roads of a network. Kendall shows that every pair of points in
$\mathbb{R}^{d}$ can be connected using this random road network, by paths
that respect the speed limits; thus, denoting by $T(x,y)$ the optimal
connection time between points $x,y\in\mathbb{R}^{d}$, the function
$T:(x,y)\mapsto T(x,y)$ defines a metric on $\mathbb{R}^{d}$. By construction,
the random metric $T$ is invariant in distribution under rotations and
translations, and satisfies a scaling property: for every
$x,y\in\mathbb{R}^{d}$, we have the equality in distribution
${T(x,y)\overset{\text{\tiny law}}{=}|x-y|^{(\gamma-d)/(\gamma-1)}\cdot
T(0,e_{d})}$, where $e_{d}=(0,\ldots,0,1)$ denotes the $d$-th vector of the
canonical basis of $\mathbb{R}^{d}$. (The model will be presented in more
detail in Section 1.)

Although the random metric space $\left(\mathbb{R}^{d},T\right)$ is almost
surely homeomorphic to the usual Euclidean $\mathbb{R}^{d}$, in this paper we
prove the following, confirming a conjecture of Kahn (see [5, Section 7]).

######  Theorem 1.

Almost surely, the metric space $\left(\mathbb{R}^{d},T\right)$ has Hausdorff
dimension

$\frac{(\gamma-1)d}{\gamma-d}>d.$

This fractal property is reminiscent of the Brownian sphere, for which the
Hausdorff dimension $4$ is greater than the “topological dimension” $2$ (see,
e.g, [11, Theorem 2 and Theorem 3]).

To compute the above Hausdorff dimension, we use the Lebesgue measure as mass
distribution on $\left(\mathbb{R}^{d},T\right)$. This choice intuitively makes
sense given that $T$ is invariant (in distribution) under rotations and
translations. Implementing the standard “energy method” leads us to estimating
the _quick connection probability_ , i.e, the probability
$\mathbb{P}(T(0,e_{d})\leq t)$ for $0$ and $e_{d}$ — two points at unit
Euclidean distance — to be connected by the road network in time $t$, as $t\to
0^{+}$. We obtain the following sharp estimate.

######  Theorem 2.

There exists constants $c,C>0$ such that

$c\cdot t^{\sigma}\leq\mathbb{P}(T(0,e_{d})\leq t)\leq C\cdot t^{\sigma}$ for
all $t\in[0,1]$, where $\sigma=\frac{(\gamma-1)(d+\gamma-2)}{\gamma-d}$.

Then, we pursue our analysis by investigating the Lebesgue measure of balls
for the metric $T$. Denoting by
${\overline{\Gamma}(x,t)=\left\\{y\in\mathbb{R}^{d}:T(x,y)\leq t\right\\}}$
the closed $T$-ball centered at $x\in\mathbb{R}^{d}$ and with radius $t>0$, we
are interested in the behaviour of the quantity
$\lambda\left(\overline{\Gamma}(x,t)\right)$ as $t\to 0^{+}$. In particular,
we prove the following, where $\mathcal{L}$ denotes the union of the lines
$\ell$ over the roads $(\ell,v)$ of $\Pi$.

######  Theorem 3.

Almost surely, the following holds:

  1. 1.

for $\lambda$-almost every $x\in\mathbb{R}^{d}$, we have

$\lambda\left(\overline{\Gamma}(x,t)\right)=t^{s^{*}+o(1)}$ as $t\to 0^{+}$,
where $s^{*}=\frac{(\gamma-1)d}{\gamma-d}$,

  2. 2.

for every $x\in\mathcal{L}$, we have

$\lambda\left(\overline{\Gamma}(x,t)\right)=t^{s_{*}+o(1)}$ as $t\to 0^{+}$,
with $s_{*}=\frac{(\gamma-1)d}{\gamma-d}-\frac{d-1}{\gamma-d}<s^{*}$,

  3. 3.

for every $x\in\mathbb{R}^{d}$, we have

$t^{s^{*}+o(1)}\leq\lambda\left(\overline{\Gamma}(x,t)\right)\leq
t^{s_{*}-o(1)}\quad\text{as $t\to 0^{+}$.}$

In words, although for _typical_ points $x$ we have
$\lambda\left(\overline{\Gamma}(x,t)\right)\approx t^{s^{*}}$ for small $t$,
with $s^{*}$ the Hausdorff dimension of $\left(\mathbb{R}^{d},T\right)$, there
exists special points $x$ — namely points _on roads_ — for which
$\lambda\left(\overline{\Gamma}(x,t)\right)\approx t^{s_{*}}$ for small $t$,
with a smaller exponent $s_{*}$ (see the opening of Section 3 for a
heuristic). This _multifractal_ property of the measured metric space
$\left(\mathbb{R}^{d},T,\lambda\right)$ distinguishes it for instance from the
Brownian sphere equipped with its volume measure, for which the single
exponent $4$ describes the behaviour of the volume of balls around _all_
points (see [10, Proposition 6.1, equations $(38)$ and $(39)$, and Theorem
7.2]). Although it can be the case in $\left(\mathbb{R}^{d},T,\lambda\right)$
that other points than points on roads exhibit a different behaviour than
typical points (see the results of Section 4 below), the third item of Theorem
3 above tells us that the two behaviours of typical points and points on roads
are extremal.

#### Organisation of the paper.

In a first section, we present the model in detail and recall the construction
of Kendall’s metric $T$. Then, we present the proof of Theorem 1 and Theorem 2
in Section 2. In Section 3, we study the Lebesgue measure of balls for the
metric $T$, and — in particular — prove Theorem 3. We conclude this paper in
Section 4, where we discuss some natural questions our results leave open.

|  |
---|---|---
Figure 1: Simulations by Arvind Singh of the distance-to-the-origin function,
for the metric $T$ induced by the Poisson process $\Pi$, in dimension $d=2$.
In each picture, the origin is the dark (red) point at the centre, and all
points are coloured (from red to blue) according to their distance to the
origin: the isocolour sets correspond to level sets for the distance-to-the-
origin function. This naturally reveals the roads of the network. From left to
right, the parameter $\gamma$ increases (a natural coupling for the speed
limits allows to use the same lines).

##  1 Introduction to Kendall’s metric

In this section, we present the model in detail, recalling the construction of
Kendall’s metric. For more details, we refer to the original paper [7]. See
also Kahn’s subsequent paper [5].

###  1.1 Definition of the Poisson process of roads

We start by introducing the Poisson process of roads $\Pi$. We call _road_ a
couple $(\ell,v)$, with $\ell\subset\mathbb{R}^{d}$ an affine line, and
$v\in\mathbb{R}_{+}^{*}$ the speed limit on $\ell$. Denoting by
$\mathbb{L}_{d}$ the set of affine lines in $\mathbb{R}^{d}$, the process
$\Pi$ will be defined as a Poisson random measure on the space of roads
$\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$. To describe its intensity measure,
let us recall the following.

#### The space of lines.

In this paragraph, we briefly present the usual topology and invariant measure
on the affine Grassmannian $\mathbb{L}_{d}$. We refer to [16, Section 13.2]
for more details, see also [2, Section 8.2]. The space of lines can
conveniently be described as follows. Choosing a reference line, for instance
the line $\ell_{d}=\\{xe_{d}=(0,\ldots,0,x);~{}x\in\mathbb{R}\\}$, and
denoting by ${\ell_{d}^{\perp}=\mathbb{R}^{d-1}\times\\{0\\}}$ its orthogonal
hyperplane, consider the surjective mapping

$\begin{matrix}\Phi:&\ell_{d}^{\perp}\times\mathbf{SO}\left(\mathbb{R}^{d}\right)&\longrightarrow&\mathbb{L}_{d}\\\
&(w,g)&\longmapsto&g(w+\ell_{d}).\end{matrix}$

The space $\mathbb{L}_{d}$ is endowed with the finest topology that makes
$\Phi$ into a continuous map (final topology), and with the corresponding
Borel $\sigma$-algebra.

##### Invariant measure.

On $\mathbb{L}_{d}$ there exists a unique, up to multiplicative constant,
locally finite Borel measure which is invariant under rotations and
translations. It can be described as the pushforward
$\mu_{d}=\Phi_{*}\left[\lambda_{d}^{\perp}\otimes\eta\right]$ of the measure
$\lambda_{d}^{\perp}\otimes\eta$ by the map $\Phi$, where
$\lambda_{d}^{\perp}$ denotes the Lebesgue measure on the $(d-1)$-dimensional
Euclidean space $\ell_{d}^{\perp}$, and $\eta$ is the Haar probability on the
compact group $\mathbf{SO}\left(\mathbb{R}^{d}\right)$. The Borel measure
$\mu_{d}$ satisfies the following invariance and scaling property: for
$x\in\mathbb{R}^{d}$, $r>0$ and $h\in\mathbf{SO}\left(\mathbb{R}^{d}\right)$,
the pushforward of $\mu_{d}$ by the continuous function

$\begin{matrix}f:&\mathbb{L}_{d}&\longrightarrow&\mathbb{L}_{d}\\\
&\ell&\longmapsto&x+r\cdot h(\ell)\\\ \end{matrix}$

is the measure $f_{*}\mu_{d}=r^{-(d-1)}\cdot\mu_{d}$. For a compact subset
$K\subset\mathbb{R}^{d}$, we denote by $[K]=\\{\ell\in\mathbb{L}_{d}:\ell\cap
K\neq\emptyset\\}$ the (closed) subset of lines that hit $K$. If
$L\subset\mathbb{R}^{d}$ is another compact subset, we simply write $[K;L]$
instead of $[K]\cap[L]$. Writing
${\overline{B}(x,r)=\left\\{y\in\mathbb{R}^{d}:|x-y|\leq r\right\\}}$ for the
closed Euclidean ball centered at $x\in\mathbb{R}^{d}$ with radius $r>0$, we
have

$\mu_{d}\left[\overline{B}(x,r)\right]=\upsilon_{d-1}r^{d-1},$ (1)

where the constant $\upsilon_{s}=\frac{\pi^{s/2}}{\Gamma(s/2+1)}$ corresponds
to the Lebesgue measure of the unit Euclidean ball in $\mathbb{R}^{s}$. In
complement to (1), we have the following.

######  Lemma 1.1.

There exists constants $C,c>0$ such that for every $x\neq y\in\mathbb{R}^{d}$:

$\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]\leq
C\cdot\frac{r^{d-1}\cdot s^{d-1}}{|x-y|^{d-1}}\quad\text{for all $r,s>0$,}$

and

$\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]\geq
c\cdot\frac{r^{d-1}\cdot s^{d-1}}{|x-y|^{d-1}}\quad\text{for all
$0<r,s\leq|x-y|$.}$

For the convenience of the reader, a proof is provided in the Appendix.

#### The Poisson process of roads $\Pi$.

Fix a parameter $\gamma>d$, and let $\Pi$ be a Poisson random measure with
intensity ${c\cdot\mu_{d}\otimes v^{-\gamma}\mathrm{d}v}$ on
$\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$, where the normalising constant
$c=\upsilon_{d-1}^{-1}\cdot(\gamma-1)$ is chosen so as to have, in view of
(1):

$c\cdot\mu_{d}\left[\overline{B}(x,r)\right]\cdot\int_{v_{0}}^{\infty}\frac{\mathrm{d}v}{v^{\gamma}}=r^{d-1}\cdot
v_{0}^{-(\gamma-1)}\quad\text{for all $x\in\mathbb{R}^{d}$, $r>0$ and
$v_{0}\in\mathbb{R}_{+}^{*}$.}$ (2)

The atoms of the measure $\Pi$ are seen as the roads of a network: for
$(\ell,v)\in\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$, the positive real number
$v$ represents the speed limit on the line $\ell$, and we say that $(\ell,v)$
is a road of $\Pi$ when $(\ell,v)$ is an atom of the measure $\Pi$. We also
use the notation $\Pi$ for the set of atoms of the measure $\Pi$. For
instance, we write $\mathcal{L}=\bigcup_{(\ell,v)\in\Pi}\ell$ for the union of
the lines $\ell$ over the roads $(\ell,v)$ of $\Pi$. For
$v_{0}\in\mathbb{R}_{+}^{*}$, we denote by $\Pi_{v_{0}}$ the restriction of
the Poisson random measure $\Pi$ to the Borel subset
$\mathbb{L}_{d}\times[v_{0},\infty[$.

##### First properties.

The road network generated by the process almost surely has the following
properties:

  1. (/)

we have $\Pi\left(\mathbb{L}_{d}\times\\{v\\}\right)\leq 1$ for all
$v\in\mathbb{R}_{+}^{*}$,

  2. $(\varnothing)$

for every $R>0$ and $v_{0}\in\mathbb{R}_{+}^{*}$, there are only finitely many
roads of $\Pi_{v_{0}}$ that pass through the ball $\overline{B}(0,R)$,

  3. $(\sharp)$

for every $x\in\mathbb{R}^{d}$ and $r>0$, there are infinitely many roads of
$\Pi$ that pass through the ball $\overline{B}(x,r)$.

In addition,

  1. $(\flat)$

for any given $x\in\mathbb{R}^{d}$, almost surely no road of $\Pi$ passes
through $x$,

  2. $(\flat\flat)$

in dimensions $d\geq 3$, almost surely the roads of $\Pi$ do not intersect,

  3. $(*)$

in dimension $d=2$, almost surely there is no crossing of three roads from
$\Pi$.

For the convenience of the reader, a proof of $(\flat\flat)$ and $(*)$ is
provided in the Appendix.

##### Speed limits.

The speed limits on the lines naturally extend to a (random) speed-limit
function $V:\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}$, defined by:

$V(x)=\sup\\{v;~{}(\ell,v)\in\Pi:\ell\ni x\\}\quad\text{for all
$x\in\mathbb{R}^{d}$,}$

with the convention $\sup\emptyset=0$. If no road of $\Pi$ passes through $x$
then $V(x)=0$, otherwise $x$ lies on at most two roads of $\Pi$, and $V(x)$
corresponds to the maximal speed limit. For $x\in\mathbb{R}^{d}$ and $r>0$, we
denote by $V^{x}_{r}=\sup_{\overline{B}(x,r)}V$ the speed limit of the fastest
road passing through the ball $\overline{B}(x,r)$. For every
$v_{0}\in\mathbb{R}_{+}^{*}$, we have

$\mathbb{P}\left(V^{x}_{r}\geq
v_{0}\right)=\mathbb{P}\left(\Pi\left(\left[\overline{B}(x,r)\right]\times[v_{0},\infty[\right)>0\right)=1-\exp\left[-r^{d-1}\cdot
v_{0}^{-(\gamma-1)}\right].$ (3)

In particular, we have $V^{x}_{r}\overset{\text{\tiny
law}}{=}r^{(d-1)/(\gamma-1)}\cdot V^{0}_{1}$. Let us keep in mind that
$V^{x}_{r}$ is of order $r^{(d-1)/(\gamma-1)}$. For $x,y\in\mathbb{R}^{d}$ and
$r,s>0$, we denote by $V^{x,y}_{r,s}$ the speed limit of the fastest road
passing through both balls $\overline{B}(x,r)$ and $\overline{B}(y,s)$,
writing $V^{x,y}_{r}$ instead of $V^{x,y}_{r,r}$ for short.

###  1.2 Construction of the metric $T$

In this subsection, we recall Kendall’s construction of the metric $T$. The
distance $T(x,y)$ will be given by the optimal connection time between points
$x,y\in\mathbb{R}^{d}$, using the random road network generated by $\Pi$,
respecting the speed limits.

####  $\Pi$-paths.

To travel on the road network generated by the process, we use Kendall’s
$\Pi$-paths. The reason for their definition is that one wants to connect
every pair of points in $\mathbb{R}^{d}$ by continuous paths that come with a
notion of instantaneous speed, and the usual piecewise $C^{1}$ paths fail at
that task because of the properties $(\flat)$ and $(\flat\flat)$ of the road
network. We call _regular path_ a continuous function
$\xi:[0,T]\rightarrow\mathbb{R}^{d}$ for which there exists an integrable
function $\dot{\xi}:[0,T]\rightarrow\mathbb{R}^{d}$ such that

$\xi(t)=\xi(0)+\int_{0}^{t}\dot{\xi}(s)\mathrm{d}s\quad\text{for all
$t\in[0,T]$.}$

Note that by the Lebesgue differentiation theorem (see, e.g, [15, Theorem
7.7]), the instantaneous speed $\dot{\xi}$ of a regular path $\xi$ can be
recovered as ${\dot{\xi}(t)=\lim_{\delta\to
0^{+}}(\xi(t+\delta)-\xi(t-\delta))/(2\delta)}$ for almost every
$t\in{]0,T[}$. Now, given a realisation of $\Pi$, a _$\Pi$ -path_ is a regular
path $\xi:[0,T]\rightarrow\mathbb{R}^{d}$ which respects the speed limits,
i.e, such that

$\left|\dot{\xi}(t)\right|\leq V(\xi(t))\quad\text{for almost every
$t\in[0,T]$.}$

##### A random metric on $\mathbb{R}^{d}$.

Kendall’s [7, Theorem 3.6] shows that almost surely, every pair of points in
$\mathbb{R}^{d}$ can be connected by a $\Pi$-path. This is not trivial in view
of properties $(\flat)$ and $(\flat\flat)$ of the road network, and it gives
meaning to the definition:

$T(x,y)=\inf\left\\{T>0:\text{there exists a $\Pi$-path
$\xi:[0,T]\rightarrow\mathbb{R}^{d}$ connecting $x$ to
$y$}\right\\}\quad\text{for all $x,y\in\mathbb{R}^{d}$.}$ (4)

Then, it is not difficult to check that the (random) function

$\begin{matrix}T:&\mathbb{R}^{d}\times\mathbb{R}^{d}&\longrightarrow&\mathbb{R}_{+}\\\
&(x,y)&\longmapsto&T(x,y)\end{matrix}$

is a metric. At least, the fact that $T(x,x)=0$ for all $x\in\mathbb{R}^{d}$,
the symmetry, and the triangle inequality, are immediate. For the positivity,
consider the following simple argument, which is illustrated in Figure 2. Fix
a realisation of $\Pi$: any $\Pi$-path connecting points $x\neq
y\in\mathbb{R}^{d}$ has to travel an Euclidean distance at least $r=|x-y|$
inside the ball $\overline{B}(x,r)$, where the speed is limited at
$V^{x}_{r}$. Thus, the optimal connection time $T(x,y)$ is bounded from below
by the ratio $r/V^{x}_{r}$, which is positive.

Figure 2: Sketch of a $\Pi$-path connecting points $x\neq y\in\mathbb{R}^{d}$.
For $0<r\leq|x-y|$, let $z\in\partial B(x,r)$ be the first exit point of the
$\Pi$-path from the ball $\overline{B}(x,r)$: between $x$ and $z$, the
$\Pi$-path travels an Euclidean distance $r$ inside the ball
$\overline{B}(x,r)$, at speed at most $V^{x}_{r}$. Thus, we have $T(x,y)\geq
r/V^{x}_{r}$.

(Note that the positivity of $T$ also follows from a more delicate result of
Kendall we mention below, which is the existence of $\Pi$-geodesics.)

######  Remark 1.

Equivalently, the metric $T$ can be seen as a first-passage percolation
metric. With the random field ${W:x\in\mathbb{R}^{d}\mapsto
V(x)^{-1}\in[0,\infty]}$ we have, for all $x,y\in\mathbb{R}^{d}$:

$T(x,y)=\inf\left\\{\int_{0}^{1}W(\varsigma(s))\left|\dot{\varsigma}(s)\right|\mathrm{d}s,\quad\text{$\varsigma:[0,1]\rightarrow\mathbb{R}^{d}$
regular path connecting $x$ to $y$}\right\\}.$

Indeed, given a $\Pi$-path $\xi:[0,T]\rightarrow\mathbb{R}^{d}$ connecting $x$
to $y$, the reparametrised regular path ${\varsigma:s\in[0,1]\mapsto\xi(sT)}$
satisfies
$\int_{0}^{1}W(\varsigma(s))\left|\dot{\varsigma}(s)\right|\mathrm{d}s\leq T$.
In the other direction, let ${\varsigma:[0,1]\rightarrow\mathbb{R}^{d}}$ be a
regular path connecting $x$ to $y$, such that
$\int_{0}^{1}W(\varsigma(s))\left|\dot{\varsigma}(s)\right|\mathrm{d}s<\infty$.
Calling this quantity $T$, consider

$\psi:t\in[0,T]\mapsto\inf\left\\{u\in[0,1]:\int_{0}^{u}W(\varsigma(s))\left|\dot{\varsigma}(s)\right|\mathrm{d}s\geq
t\right\\}.$

The reparametrisation ${\xi:t\in[0,T]\mapsto\varsigma(\psi(t))}$ is a
$\Pi$-path that connects $x$ to $y$ in time $T$.

#####  $\Pi$-geodesics.

_$\Pi$ -geodesics_ are defined as $\Pi$-paths travelling in optimal time: the
$\Pi$-path $\xi:[0,T]\rightarrow\mathbb{R}^{d}$ is a $\Pi$-geodesic when no
$\Pi$-path $\varsigma:[0,S]\rightarrow\mathbb{R}^{d}$ connecting $\xi(0)$ to
$\xi(T)$ satisfies $S<T$. Kendall’s [7, Corollary 3.5] shows that the infimum
in (4) is in fact a minimum: for every $x,y\in\mathbb{R}^{d}$, there exists a
$\Pi$-geodesic connecting $x$ to $y$ in optimal time $T(x,y)$.

####  $\varepsilon$-$\Pi$-paths.

A natural idea to make sense of $\Pi$-paths is to approximate them by simple
sequential paths that nearly respect the speed limits. Following Kendall, for
$\varepsilon>0$ we call _$\varepsilon$ -$\Pi$-path_ a continuous path
$\xi:[0,T]\rightarrow\mathbb{R}^{d}$ for which there exists a subdivision
$0\leq r_{1}<s_{1}\leq\ldots\leq r_{k}<s_{k}\leq T$ of the interval $[0,T]$,
and a collection $(\ell_{1},v_{1});\ldots;(\ell_{k},v_{k})$ of roads of $\Pi$
such that:

  * •

for each $i\in\llbracket 1,k\rrbracket$, the points $\xi(r_{i})$ and
$\xi(s_{i})$ lie on $\ell_{i}$, and are connected linearly by $\xi$ at speed
$v_{i}$ over the time interval $[r_{i},s_{i}]$,

  * •

for each $i\in\llbracket 0,k\rrbracket$, with the conventions $s_{0}=0$ and
$r_{k+1}=T$, the points $\xi(s_{i})$ and $\xi(r_{i+1})$ are connected linearly
by $\xi$ at speed $\varepsilon$ over the time interval $[s_{i},r_{i+1}]$.

This definition is illustrated in Figure 3.

Figure 3: Sketch of an $\varepsilon$-$\Pi$-path connecting points $x$ and $y$
in $\mathbb{R}^{d}$. The big light (cyan) segments are travelled on a road of
$\Pi$ at maximal speed, and the smaller dark (blue) segments are travelled
between roads at speed $\varepsilon$.

In particular, any $\varepsilon$-$\Pi$-path
$\xi:[0,T]\rightarrow\mathbb{R}^{d}$ is a regular path such that

$\left|\dot{\xi}(t)\right|\leq\varepsilon\vee V(\xi(t))\quad\text{for almost
every $t\in[0,T]$.}$

With this cutoff, setting

$T_{\varepsilon}(x,y)=\inf_{\begin{subarray}{c}k\geq 0\\\
(\ell_{1},v_{1});\ldots;(\ell_{k},v_{k})\in\Pi\\\
a_{1},b_{1}\in\ell_{1};\ldots;a_{k},b_{k}\in\ell_{k}\end{subarray}}\left\\{\frac{|x-a_{1}|}{\varepsilon}+\frac{|a_{1}-b_{1}|}{v_{1}}+\ldots+\frac{|a_{k}-b_{k}|}{v_{k}}+\frac{|b_{k}-y|}{\varepsilon}\right\\}\quad\text{for
all $x,y\in\mathbb{R}^{d}$}$

defines a metric $T_{\varepsilon}$ on $\mathbb{R}^{d}$. Given that for each
$x,y\in\mathbb{R}^{d}$, the quantity $T_{\varepsilon}(x,y)$ is nondecreasing
as $\varepsilon$ decreases, one defines an extended metric
${\overline{T}:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow[0,\infty]}$, by
setting:

$\overline{T}(x,y)=\lim_{\varepsilon\downarrow
0}T_{\varepsilon}(x,y)\quad\text{for all $x,y\in\mathbb{R}^{d}$.}$

Kendall’s results [7, Theorem 3.8] and [7, Theorem 3.11] show that in fact
this definition coincides with that of $T$, i.e, for every
$x,y\in\mathbb{R}^{d}$, we have

$T_{\varepsilon}(x,y)\uparrow T(x,y)\quad\text{as $\varepsilon\downarrow 0$}.$
(5)

##### Measurability.

A byproduct of this approximation argument is the measurability of $T$ as a
function of $\omega;x,y$, which justifies every application of Fubini’s
theorem in this paper. For each $\varepsilon>0$, it is possible to write

$T_{\varepsilon}(x,y)=\tau_{\varepsilon}(\Pi;x,y)\quad\text{for all
$x,y\in\mathbb{R}^{d}$,}$ (6)

for some measurable function $\tau_{\varepsilon}$, and it follows that

$T(x,y)=\tau(\Pi;x,y)\quad\text{for all $x,y\in\mathbb{R}^{d}$,}$ (7)

for some measurable function $\tau$.

###  1.3 Fundamental properties of $T$

In this section, we present some fundamental properties of the metric $T$;
namely, its invariance and scaling property, its mixing, and the probabilistic
estimate of Kahn. We do not claim any new results here. In particular, the
estimates below show that $\left(\mathbb{R}^{d},T\right)$ is almost surely
homeomorphic to the usual Euclidean $\mathbb{R}^{d}$, which was already noted
by Kendall (see [7, Section 7]) and Kahn (see [5, Remark 5.1]).

#### Invariance and scaling.

The invariance and scaling properties of the process’s intensity measure
naturally extend to the random metric $T$. Note that $T$ may be viewed as a
random variable with values in the space $\mathbf{F}$ of functions from
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ to $\mathbb{R}$, endowed as usual with
the product $\sigma$-algebra.

######  Proposition 1.1.

For every $x\in\mathbb{R}^{d}$, $r>0$ and
$h\in\mathbf{SO}\left(\mathbb{R}^{d}\right)$, we have the equality in
distribution

$(T(x+r\cdot h(y),x+r\cdot h(z)))_{y,z\in\mathbb{R}^{d}}\overset{\text{\tiny
law}}{=}\left(r^{\frac{\gamma-d}{\gamma-1}}\cdot
T(y,z)\right)_{y,z\in\mathbb{R}^{d}}.$

###### Proof.

Since the intensity measure of $\Pi$ is left invariant by the continuous map

$\begin{matrix}f:&\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}&\longrightarrow&\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}\\\
&(\ell,v)&\longmapsto&\left(x+r\cdot h(\ell),r^{\frac{d-1}{\gamma-1}}\cdot
v\right),\end{matrix}$

the Poisson random measure $\Pi^{\prime}=f_{*}\Pi$ has the same distribution
as $\Pi$. Therefore, denoting by
${T^{\prime}(\cdot,\cdot)=\tau(\Pi^{\prime};\cdot,\cdot)}$ the random metric
induced by the optimal connection time using $\Pi^{\prime}$-paths, where
$\tau$ is the measurable function of (7), we have on the one hand the equality
in distribution

$\left(T^{\prime}(x+r\cdot h(y),x+r\cdot
h(z))\right)_{y,z\in\mathbb{R}^{d}}\overset{\text{\tiny law}}{=}(T(x+r\cdot
h(y),x+r\cdot h(z)))_{y,z\in\mathbb{R}^{d}}.$

On the other hand we have, deterministically:

$\tau\left(\Pi^{\prime};x+r\cdot h(y),x+r\cdot
h(z)\right)=\frac{r}{r^{(d-1)/(\gamma-1)}}\cdot\tau(\Pi;y,z)=r^{\frac{\gamma-d}{\gamma-1}}\cdot\tau(\Pi;y,z)\quad\text{for
all $y,z\in\mathbb{R}^{d}$.}$

This concludes the proof. ∎

#### Mixing.

In the course of the proof above, we have seen that the process $\Pi$ is
stationary with respect to the translations

$\left(\begin{matrix}f_{x}:&\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}&\longrightarrow&\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}\\\
&(\ell,v)&\longmapsto&(x+\ell,v)\end{matrix}\right)_{x\in\mathbb{R}^{d}},$ (8)

in the sense that for every $x\in\mathbb{R}^{d}$, we have
$f_{x}^{*}\Pi\overset{\text{\tiny law}}{=}\Pi$. Furthermore, it is mixing, as
shown in the following proposition. Let us denote by $\mathbb{M}$ the space of
atomic measures on $\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$. The Poisson
random measure $\Pi$ is a random variable with values in $\mathbb{M}$.

######  Proposition 1.2.

For every bounded measurable functions $\varphi$ and $\psi$ from $\mathbb{M}$
to $\mathbb{R}$, we have

$\mathbb{E}[\varphi(\Pi)\cdot\psi(f_{x}^{*}\Pi)]\underset{|x|\to\infty}{\longrightarrow}\mathbb{E}[\varphi(\Pi)]\cdot\mathbb{E}[\psi(\Pi)],$

where $f_{x}$ has been defined in (8).

We are indebted to an anonymous referee for pointing out that the above result
holds in more generality than as we were stating in an earlier version of this
paper, and for providing helpful references.

###### Proof.

The proof uses a classical approximation argument from measure theory,
similarly as in [8, proof of Theorem 3.5]. This proof strategy also appears
elsewhere in the literature, see for instance [4, proof of Proposition 2.3.7].

Let $\varphi$ and $\psi$ be bounded measurable functions from $\mathbb{M}$ to
$\mathbb{R}$. For $n\in\mathbb{N}^{*}$, let us denote by
$\rho_{n}:\mathbb{M}\rightarrow\mathbb{M}$ the restriction map to the Borel
subset
${\left[\overline{B}(0,n)\right]\times[1/n,\infty[}\subset\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$,
and by ${\mathcal{A}_{n}=\sigma(\rho_{n}(\Pi))}$ the $\sigma$-algebra
generated by $\rho_{n}(\Pi)$. Fix $\varepsilon>0$. As $\sigma(\Pi)$ coincides
with the $\sigma$-algebra generated by $\bigcup_{n\geq 1}\mathcal{A}_{n}$, by
a standard measure theoretic argument (see, e.g, [6, Lemma 3.16]), there
exists $n\in\mathbb{N}^{*}$, and bounded measurable functions $\varphi_{n}$
and $\psi_{n}$ from $\mathbb{M}$ to $\mathbb{R}$, such that

$\mathbb{E}[|\varphi(\Pi)-\varphi_{n}(\rho_{n}(\Pi))|]\leq\varepsilon\quad\text{and}\quad\mathbb{E}[|\psi(\Pi)-\psi_{n}(\rho_{n}(\Pi))|]\leq\varepsilon.$
(9)

Denoting by $C>0$ a constant which dominates $\varphi$, $\psi$, $\varphi_{n}$
and $\psi_{n}$, we have on the one hand:

$|\mathbb{E}[\varphi(\Pi)\cdot\psi(f_{x}^{*}\Pi)]-\mathbb{E}[\varphi_{n}(\rho_{n}(\Pi))\cdot\psi_{n}(\rho_{n}(f_{x}^{*}\Pi))]|\leq
2C\varepsilon.$ (10)

On the other hand, the random measures $\rho_{n}(\Pi)$ and
$\rho_{n}(f_{x}^{*}\Pi)$ are “asymptotically independent”. Indeed, denote by
$G_{x}$ the event: “no road of $\Pi_{1/n}$ passes through both balls
$\overline{B}(0,n)$ and $\overline{B}(-x,n)$”. The following holds:

  * •

we have $\mathbb{P}(G_{x})\rightarrow 1$ as $|x|\to\infty$ (indeed, the
integer $n$ is fixed, and we have
$\mu_{d}\left[\overline{B}(0,n)~{};~{}\overline{B}(-x,n)\right]\rightarrow 0$
as $|x|\to\infty$, for instance by Lemma 1.1),

  * •

on $G_{x}$, the random measure $\rho_{n}(\Pi)$ coincides with the restriction
$\Pi_{0\setminus{-x}}$ of the Poisson random measure $\Pi$ to the Borel subset
${\left(\left[\overline{B}(0,n)\right]\middle\backslash\left[\overline{B}(-x,n)\right]\right)\times[1/n,\infty[}$,
and $\rho_{n}(f_{x}^{*}\Pi)$ coincides with the restriction
$\Pi_{{-x}\setminus 0}$ of $\Pi$ to
${\left(\left[\overline{B}(-x,n)\right]\middle\backslash\left[\overline{B}(0,n)\right]\right)\times[1/n,\infty[}$,

  * •

the random measures $\Pi_{0\setminus{-x}}$ and $\Pi_{{-x}\setminus 0}$ are
independent.

With these three ingredients, standard manipulations show that

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

Then, equations (9) and (10) allow to conclude: as $|x|\to\infty$, we have

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

∎

#### Probabilistic estimates.

Finally, we present some fundamental probabilistic estimates on the metric
$T$.

##### Kahn’s estimate.

First, let us state Kahn’s estimate on the $T$-diameter of Euclidean balls.
For $x\in\mathbb{R}^{d}$ and $r>0$, consider the random variable111As a
consequence of Proposition 1.3 below, almost surely the function
$T:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}$ is continuous,
thus the supremum can be reduced to points $y,z$ having rational coordinates.

$T^{*}_{x,r}=\sup_{y,z\in\overline{B}(x,r)}T(y,z).$

By invariance and scaling (Proposition 1.1), we have the equality in
distribution $T^{*}_{x,r}\overset{\text{\tiny law}}{=}T^{*}_{0,1}\cdot
r^{(\gamma-d)/(\gamma-1)}$. Kahn’s [5, Theorem 3.1] shows the existence of
constants $J,\kappa>0$ such that

$\mathbb{P}\left(T^{*}_{0,1}>t\right)\leq J\exp\left[-\kappa
t^{\gamma-1}\right]\quad\text{for all $t\in\mathbb{R}_{+}^{*}$}.$ (11)

##### A uniform control.

Intuitively, for given $x,y\in\mathbb{R}^{d}$, the random variable $T(x,y)$ is
of order $|x-y|^{(\gamma-d)/(\gamma-1)}$, since by invariance and scaling, we
have the equality in distribution ${T(x,y)\overset{\text{\tiny
law}}{=}|x-y|^{(\gamma-d)/(\gamma-1)}\cdot T(0,e_{d})}$. Slightly adapting
Kahn’s proof of [5, Theorem 3.1], one can obtain the following estimate.

######  Proposition 1.3.

Almost surely, for every $R>0$, there exists a constant $C=C(\omega,R)>0$ such
that

$T(x,y)\leq
C\cdot|x-y|^{\frac{\gamma-d}{\gamma-1}}\cdot\ln\left(\frac{4R}{|x-y|}\right)^{\frac{1}{\gamma-1}}\quad\text{for
all $x\neq y\in\overline{B}(0,R)$}.$ (12)

In particular, the map
$\mathrm{id}:\left(\overline{B}(0,R),|\cdot|\right)\rightarrow\left(\overline{B}(0,R),T\right)$
is $\beta$-Hölder, for every $\beta\in{]0,(\gamma-d)/(\gamma-1)[}$.

###### Proof.

The idea (of Kahn) is to combine Kendall’s construction of $\Pi$-paths between
pairs of points ([7, Theorem 3.1]), that we recall for convenience, with a
uniform lower bound on the speed limits of the roads involved (Lemma 1.2
below).

Fix $R\in\mathbb{N}^{*}=\\{1,2,\ldots\\}$, and let $\alpha\in{]0,1[}$ be a
small parameter to be adjusted throughout the proof. Let $(x,y)$ be a pair of
points in $\overline{B}(0,R)$: Kendall constructs a $\Pi$-path connecting $x$
to $y$ in the following recursive way. If $x=y$ then there is nothing to do,
otherwise consider the fastest road $(\ell,v)$ of $\Pi$ that passes through
both balls $\overline{B}(x,\alpha|x-y|)$ and ${\overline{B}(y,\alpha|x-y|)}$,
and denote by $x^{\prime}$ (resp. $y^{\prime}$) the orthogonal projection of
$x$ (resp. $y$) onto the line $\ell$. The road $(\ell,v)$ allows to connect
$x^{\prime}$ to $y^{\prime}$ in time $|x^{\prime}-y^{\prime}|/v$, and it
remains to connect $x$ to $x^{\prime}$, and $y^{\prime}$ to $y$. For this,
recursively apply the procedure to the pairs of points $(x,x^{\prime})$ and
$(y^{\prime},y)$. To encode the construction, let us mark the nodes
$\varnothing;1,2;\ldots$ of the infinite (complete) binary tree $\mathbb{T}$
in the following way. Initially, set
$(x_{\varnothing},y_{\varnothing})=(x,y)$. Then, for every
${n\in\mathbb{N}=\\{0,1,\ldots\\}}$ such that the pairs of points
$((x_{u},y_{u});~{}u\in\mathbb{T}:|u|=n)$ have been constructed, proceed as
follows. For each node $u\in\mathbb{T}$ such that $|u|=n$, consider the
fastest road $(\ell_{u},v_{u})$ of $\Pi$ that passes through both balls
$\overline{B}(x_{u},\alpha|x_{u}-y_{u}|)$ and
${\overline{B}(y_{u},\alpha|x_{u}-y_{u}|)}$, and denote by $x_{u}^{\prime}$
(resp. $y_{u}^{\prime}$) the orthogonal projection of $x_{u}$ (resp. $y_{u}$)
onto the line $\ell_{u}$. Recalling the notation introduced above, note that
$v_{u}=V^{x_{u},y_{u}}_{\alpha|x_{u}-y_{u}|}$. Record the time
$T_{u}=|x_{u}^{\prime}-y_{u}^{\prime}|/v_{u}$ it takes to connect
$x_{u}^{\prime}$ to $y_{u}^{\prime}$ using the road $(\ell_{u},v_{u})$, and
define the pairs of points associated to the children $u1$ and $u2$ of $u$ in
$\mathbb{T}$ as $(x_{u1},y_{u1})=(x_{u},x_{u}^{\prime})$ and
${(x_{u2},y_{u2})=(y_{u}^{\prime},y_{u})}$. From all these “middle”
connections $x_{u}^{\prime}\to y_{u}^{\prime}$, we claim that it is indeed
possible to obtain a $\Pi$-path connecting $x$ to $y$, which travels in time

$\sum_{u\in\mathbb{T}}T_{u}=\sum_{u\in\mathbb{T}}\frac{|x_{u}^{\prime}-y_{u}^{\prime}|}{v_{u}}\leq\sum_{u\in\mathbb{T}}\frac{|x_{u}-y_{u}|}{V^{x_{u},y_{u}}_{\alpha|x_{u}-y_{u}|}}.$
(13)

Therefore, the above right hand side provides an upper bound on $T(x,y)$. Now,
consider the following lemma. Note that, assuming $\alpha\leq 1/3$, we have
$x_{u},y_{u}\in\overline{B}(0,2R)$ for all $u\in\mathbb{T}$.

######  Lemma 1.2.

Almost surely, there exists a constant $c>0$ such that

$V^{x,y}_{\alpha|x-y|}\geq
c\cdot\frac{|x-y|^{\frac{d-1}{\gamma-1}}}{\ln(4R/(\alpha|x-y|))^{\frac{1}{\gamma-1}}}\quad\text{for
all $x\neq y\in\overline{B}(0,2R)$}.$ (14)

Then, equation (12) will be obtained by combining (13) and (14).

###### Proof.

The proof relies on a discretisation argument, together with the Borel-
Cantelli lemma.

For each $n\in\mathbb{N}$, set $r_{n}=\alpha^{n}\cdot 4R$, and denote by
$\left(\overline{B}(z,r_{n}/2)\right)_{z\in\mathcal{Z}_{n}}$ a covering of
$\overline{B}(0,2R)$ by balls of radius $r_{n}/2$, with centres
$z\in\overline{B}(0,2R)$ more than $r_{n}/2$ apart from each other. In
particular, the balls $\overline{B}(z,r_{n}/4)$ for $z\in\mathcal{Z}_{n}$ are
disjoint and included in $\overline{B}(0,2R+r_{n}/4)$, and we have
${\\#\mathcal{Z}_{n}\leq(8R/r_{n}+1)^{d}=\left(2\alpha^{-n}+1\right)^{d}}$.
Let $x\neq y\in\overline{B}(0,2R)$, and let $n\in\mathbb{N}^{*}$ be such that
$r_{n}\leq|x-y|\leq r_{n-1}$. There exists
$x^{\prime},y^{\prime}\in\mathcal{Z}_{n+1}$ such that
$x\in\overline{B}(x^{\prime},r_{n+1}/2)$ and
${y\in\overline{B}(y^{\prime},r_{n+1}/2)}$. Since $r_{n+1}=\alpha
r_{n}\leq\alpha|x-y|$, by the triangle inequality we have
$V^{x,y}_{\alpha|x-y|}\geq V^{x^{\prime},y^{\prime}}_{r_{n+1}/2}$. Now, let us
control the $V^{x^{\prime},y^{\prime}}_{r_{n+1}/2}$ uniformly in
$x^{\prime},y^{\prime}$. First, note that $x^{\prime}$ and $y^{\prime}$ are at
distance at most

$|x^{\prime}-x|+|x-y|+|y-y^{\prime}|\leq\frac{r_{n+1}}{2}+r_{n-1}+\frac{r_{n+1}}{2}=\left(\frac{\alpha}{2}+\frac{1}{\alpha}+\frac{\alpha}{2}\right)\cdot
r_{n}=:C\cdot r_{n}$

from each other. Using Lemma 1.1, we find that there exists a constant $c>0$
such that

$\mathbb{P}\left(V^{x^{\prime},y^{\prime}}_{r_{n+1}/2}<r_{n}^{\frac{d-1}{\gamma-1}}\cdot\zeta_{n}^{-1}\right)\leq\exp\left[-c\cdot
r_{n}^{d-1}\cdot\left(r_{n}^{\frac{d-1}{\gamma-1}}\cdot\zeta_{n}^{-1}\right)^{-(\gamma-1)}\right]=\exp\left[-c\zeta_{n}^{\gamma-1}\right],$

where the $\zeta_{n}$ are positive real numbers which will be adjusted later.
Denoting by $A_{n}$ the event: “there exists
${x^{\prime},y^{\prime}\in\mathcal{Z}_{n+1}}$ with
$|x^{\prime}-y^{\prime}|\leq Cr_{n}$ such that
${V^{x^{\prime},y^{\prime}}_{r_{n+1}/2}<r_{n}^{(d-1)/(\gamma-1)}\cdot\zeta_{n}^{-1}}$”,
by a union bound we get

$\mathbb{P}(A_{n})\leq\\#\mathcal{Z}_{n}^{2}\cdot\exp\left[-c\zeta_{n}^{\gamma-1}\right]\leq\left(2\alpha^{-n}+1\right)^{2d}\cdot\exp\left[-c\zeta_{n}^{\gamma-1}\right].$

Then, setting

$\zeta_{n}=\left(\frac{3d\ln(1/\alpha)}{c}\cdot
n\right)^{\frac{1}{\gamma-1}},\quad\text{so
that}\quad\exp\left[-c\zeta_{n}^{\gamma-1}\right]=\alpha^{3dn},$

we obtain $\sum_{n\geq 1}\mathbb{P}(A_{n})<\infty$. By the Borel-Cantelli
lemma, almost surely the event $A_{n}$ fails to be realised for all
sufficiently large $n$, hence there exists a constant $c^{\prime}>0$ such that
for every $n\in\mathbb{N}^{*}$:

$V^{x^{\prime},y^{\prime}}_{r_{n+1}/2}\geq c^{\prime}\cdot
r_{n}^{\frac{d-1}{\gamma-1}}\cdot n^{-\frac{1}{\gamma-1}}\quad\text{for all
$x^{\prime},y^{\prime}\in\mathcal{Z}_{n+1}$ such that
$|x^{\prime}-y^{\prime}|\leq Cr_{n}$.}$

It follows that

$V^{x,y}_{\alpha|x-y|}\geq
c^{\prime}\cdot(\alpha|x-y|)^{\frac{d-1}{\gamma-1}}\cdot\log_{1/\alpha}\left(\frac{4R}{\alpha|x-y|}\right)^{-\frac{1}{\gamma-1}}\quad\text{for
all $x\neq y\in\overline{B}(0,2R)$,}$

which proves (14). ∎

Back to the proof of the proposition, let us plug (14) into (13): this yields

$T(x,y)\leq\frac{1}{c}\sum_{u\in\mathbb{T}}|x_{u}-y_{u}|^{\frac{\gamma-d}{\gamma-1}}\cdot\ln\left(\frac{4R}{\alpha|x_{u}-y_{u}|}\right)^{\frac{1}{\gamma-1}}=\frac{1}{c}\sum_{u\in\mathbb{T}}\phi(|x_{u}-y_{u}|),$

with $\phi:s\mapsto s^{(\gamma-d)/(\gamma-1)}\cdot\ln(4R/(\alpha
s))^{1/(\gamma-1)}$. Note that $|x_{u}-y_{u}|\leq\alpha^{|u|}\cdot|x-y|\leq
2R$ for all $u\in\mathbb{T}$. Now, a straightforward analysis shows that,
assuming $\alpha\leq 2e^{-1/(\gamma-d)}$, the function $\phi$ is nondecreasing
over the interval $]0,2R]$: using this, we get

$\begin{split}T(x,y)&\leq\frac{1}{c}\sum_{u\in\mathbb{T}}\phi\left(\alpha^{|u|}\cdot|x-y|\right)\\\
&=\frac{1}{c}\sum_{n\geq
0}2^{n}\cdot\phi\left(\alpha^{n}\cdot|x-y|\right)=\frac{1}{c}\sum_{n\geq
0}\left(2\alpha^{\frac{\gamma-d}{\gamma-1}}\right)^{n}\cdot|x-y|^{\frac{\gamma-d}{\gamma-1}}\cdot\ln\left(\frac{4R}{\alpha^{n+1}\cdot|x-y|}\right)^{\frac{1}{\gamma-1}}.\end{split}$

Finally, we have

$\ln\frac{4R}{\alpha^{n+1}\cdot|x-y|}=(n+1)\ln\frac{1}{\alpha}+\ln\frac{4R}{|x-y|}\leq\left((n+1)\cdot\frac{\ln(1/\alpha)}{\ln
2}+1\right)\cdot\ln\frac{4R}{|x-y|}=:a_{n}\cdot\ln\frac{4R}{|x-y|}$

for all $n\in\mathbb{N}$. Thus, we obtain

$T(x,y)\leq\frac{1}{c}\sum_{n\geq
0}\left(2\alpha^{\frac{\gamma-d}{\gamma-1}}\right)^{n}\cdot
a_{n}^{\frac{1}{\gamma-1}}\cdot|x-y|^{\frac{\gamma-d}{\gamma-1}}\cdot\ln\left(\frac{4R}{|x-y|}\right)^{\frac{1}{\gamma-1}}\quad\text{for
all $x\neq y\in\overline{B}(0,R)$.}$

Assuming $\alpha<2^{-(\gamma-1)/(\gamma-d)}$, we see that $\sum_{n\geq
0}\left(2\alpha^{\frac{\gamma-d}{\gamma-1}}\right)^{n}\cdot
a_{n}^{\frac{1}{\gamma-1}}<\infty$, which achieves to prove (12). The second
assertion of the proposition is a straightforward consequence of that
equation. ∎

In particular, almost surely, for every $R>0$, the map
${\mathrm{id}:\left(\overline{B}(0,R),|\cdot|\right)\rightarrow\left(\overline{B}(0,R),T\right)}$
is continuous. By the compactness of $\overline{B}(0,R)$, this map is in fact
a homeomorphism (see, e.g, [13, Theorem 26.6]), and globally so is the map
${\mathrm{id}:\left(\mathbb{R}^{d},|\cdot|\right)\rightarrow\left(\mathbb{R}^{d},T\right)}$.
Therefore, the metric $T$ may be viewed as a random variable with values in
the space $\mathbf{C}$ of continuous functions from
$\mathbb{R}^{d}\times\mathbb{R}^{d}$ to $\mathbb{R}$, endowed as usual with
the topology of uniform convergence on compact subsets.

##  2 Hausdorff dimension and quick connection probability

In this section, we compute the Hausdorff dimension of the random metric space
$\left(\mathbb{R}^{d},T\right)$, and provide sharp estimates on the _quick
connection probability_ , i.e, the probability $\mathbb{P}(T(0,e_{d})\leq t)$
for two points at unit Euclidean distance to be connected by the road network
in time $t$, as $t\to 0^{+}$. Actually, doing the former is quite
straightforward once the latter has been established, so we consider it first.

###  2.1 Hausdorff dimension of $\left(\mathbb{R}^{d},T\right)$

Theorem 1 states that the Hausdorff dimension of
$\left(\mathbb{R}^{d},T\right)$ is given by ${(\gamma-1)d/(\gamma-d)>d}$: this
confirms a conjecture of Kahn, stated in [5, Section 7]. For a reference on
Hausdorff dimension see, e.g, [14, Chapter 4]. The proof below is conditional
on the estimate of Theorem 2, which we prove in the next subsection; namely,
we assume here that the following holds: there exists a constant $C>0$ such
that

$\mathbb{P}(T(0,e_{d})\leq t)\leq C\cdot t^{\sigma}$ for all $t\in[0,1]$,
where $\sigma=\frac{(\gamma-1)(d+\gamma-2)}{\gamma-d}$. (15)

###### Proof of Theorem 1 assuming (15).

Upper bound. By Proposition 1.3, almost surely, for each $R>0$, the map
${\mathrm{id}:\left(\overline{B}(0,R),|\cdot|\right)\rightarrow\left(\overline{B}(0,R),T\right)}$
is $\beta$-Hölder, for every ${\beta\in{]0,(\gamma-d)/(\gamma-1)[}}$.
Therefore, we have

$\dim_{H}\left(\overline{B}(0,R),T\right)\leq\frac{\dim_{H}\left(\overline{B}(0,R),|\cdot|\right)}{\beta}\quad\text{for
every $\beta\in\left]0,\frac{\gamma-d}{\gamma-1}\right[$,}$

hence $\dim_{H}\left(\overline{B}(0,R),T\right)\leq(\gamma-1)/(\gamma-d)\cdot
d$. By the countable stability of Hausdorff dimension, we obtain

$\dim_{H}\left(\mathbb{R}^{d},T\right)\leq\frac{(\gamma-1)d}{\gamma-d}.$

Lower bound. Let us now turn to the converse inequality. The energy method
(see [14, Theorem 4.27]) states that if the energy integral
$\int_{\overline{B}(0,1)}\int_{\overline{B}(0,1)}T(x,y)^{-s}\mathrm{d}y\mathrm{d}x$
is finite, then $\dim_{H}\left(\overline{B}(0,1),T\right)\geq s$. Therefore,
it suffices to show that for every $s\in[0,(\gamma-1)d/(\gamma-d)[$, we have
$\int_{\overline{B}(0,1)}\int_{\overline{B}(0,1)}T(x,y)^{-s}\mathrm{d}y\mathrm{d}x<\infty$
almost surely. Let then ${s\in[0,(\gamma-1)d/(\gamma-d)[}$: by Fubini’s
theorem, and invariance and scaling, we have

$\begin{split}\mathbb{E}\left[\int_{\overline{B}(0,1)}\int_{\overline{B}(0,1)}\frac{\mathrm{d}y}{T(x,y)^{s}}\mathrm{d}x\right]&=\int_{\overline{B}(0,1)}\int_{\overline{B}(0,1)}\mathbb{E}\left[T(x,y)^{-s}\right]\mathrm{d}y\mathrm{d}x\\\
&=\int_{\overline{B}(0,1)}\int_{\overline{B}(0,1)}\frac{\mathrm{d}y}{|x-y|^{(\gamma-d)s/(\gamma-1)}}\mathrm{d}x\cdot\mathbb{E}\left[T(0,e_{d})^{-s}\right].\end{split}$

As $(\gamma-d)s/(\gamma-1)<d$, we have
$\int_{\overline{B}(0,1)}\int_{\overline{B}(0,1)}|x-y|^{-(\gamma-d)s/(\gamma-1)}\mathrm{d}y\mathrm{d}x<\infty$;
and by assumption, the other term

$\mathbb{E}\left[T(0,e_{d})^{-s}\right]=s\int_{0}^{\infty}\mathbb{P}\left(T(0,e_{d})\leq
t\right)\cdot t^{-s-1}\mathrm{d}t\leq s\int_{0}^{1}C\cdot t^{\sigma}\cdot
t^{-s-1}\mathrm{d}t+1$

is finite as well, since $\sigma-s>(\gamma-1)(\gamma-2)/(\gamma-d)>0$. ∎

To actually complete the above proof, we need to prove Theorem 2, which is the
subject of the next subsection.

###  2.2 Sharp estimates for quick connection probability

In this subsection, we prove Theorem 2. The lower bound is Proposition 2.1
below, and the more delicate upper bound is Proposition 2.3.

#### Lower bound.

A simple way of connecting $0$ and $e_{d}$ quickly is to use a fast road
passing close to both points. This basic idea gives the following lower bound.

######  Proposition 2.1.

There exists a constant $c>0$ such that

$\mathbb{P}(T(0,e_{d})\leq t)\geq c\cdot t^{\sigma}$ for all $t\in[0,1]$,
where $\sigma=\frac{(\gamma-1)(d+\gamma-2)}{\gamma-d}$.

###### Proof.

Let $t\in{]0,1]}$, and let $r=r(t)>0$ be a small parameter to be adjusted
later. We have (see Figure 4)

$T(0,e_{d})\leq T^{*}_{0,r}+\frac{1}{V^{0,e_{d}}_{r}}+T^{*}_{e_{d},r},$

hence

$\mathbb{P}(T(0,e_{d})\leq
t)\geq\mathbb{P}\left(T^{*}_{0,r}\leq\frac{t}{3}~{};~{}V^{0,e_{d}}_{r}\geq\frac{3}{t}~{};~{}T^{*}_{e_{d},r}\leq\frac{t}{3}\right).$

Figure 4: (Planted on $\ell$ is a speed limit sign.) Denoting by $(\ell,v)$
the fastest road passing through both balls $\overline{B}(0,r)$ and
$\overline{B}(e_{d},r)$, with $v=V^{0,e_{d}}_{r}$ by definition, let $x$
(resp. $y$) be the orthogonal projection of $0$ (resp. $e_{d}$) onto $\ell$.
The points $x$ and $y$ can be connected in time $|x-y|/v\leq 1/v$, and
connecting points $0$ and $x$ (resp. $y$ and $e_{d}$) takes time at most
$T^{*}_{0,r}$ (resp. $T^{*}_{e_{d},r}$).

Now, the three events in the above intersection are all increasing, in the
sense that if they hold for $\Pi$, then they also hold for any atomic measure
$\Pi^{\prime}$ such that $\Pi^{\prime}(A)\geq\Pi(A)$ for all measurable sets
$A\subset\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$. Therefore, by the FKG
inequality for Poisson processes (see, e.g, [9, Theorem 20.4]), we have

$\mathbb{P}\left(T^{*}_{0,r}\leq\frac{t}{3}~{};~{}V^{0,e_{d}}_{r}\geq\frac{3}{t}~{};~{}T^{*}_{e_{d},r}\leq\frac{t}{3}\right)\geq\mathbb{P}\left(T^{*}_{0,r}\leq\frac{t}{3}\right)\cdot\mathbb{P}\left(V^{0,e_{d}}_{r}\geq\frac{3}{t}\right)\cdot\mathbb{P}\left(T^{*}_{e_{d},r}\leq\frac{t}{3}\right).$

Next, by (11) together with invariance and scaling, there exists a constant
$c>0$ such that

$\mathbb{P}\left(T^{*}_{0,r}>\frac{t}{3}\right)\leq
J\exp\left[-\frac{ct^{\gamma-1}}{r^{\gamma-d}}\right]\quad\text{and}\quad\mathbb{P}\left(T^{*}_{e_{d},r}>\frac{t}{3}\right)\leq
J\exp\left[-\frac{ct^{\gamma-1}}{r^{\gamma-d}}\right]\quad\text{for every
$t\in\mathbb{R}_{+}^{*}$.}$

At this point, let us set

$r(t)=\left(\frac{ct^{\gamma-1}}{\ln(1+2J)}\right)^{\frac{1}{\gamma-d}},\quad\text{so
that}\quad
J\exp\left[-\frac{ct^{\gamma-1}}{r(t)^{\gamma-d}}\right]\leq\frac{1}{2}.$

With that choice, we get

$\mathbb{P}(T(0,e_{d})\leq
t)\geq\frac{1}{4}\cdot\mathbb{P}\left(V^{0,e_{d}}_{r}\geq\frac{3}{t}\right).$

Finally, the quantity $\mathbb{P}\left(V^{0,e_{d}}_{r}\geq 3/t\right)$ is the
probability for a Poisson random variable with mean $p$ proportional to
$\mu_{d}\left[\overline{B}(0,r)~{};~{}\overline{B}(e_{d},r)\right]\cdot(3/t)^{-(\gamma-1)}$
to be positive. By virtue of Lemma 1.1, the parameter $p$ is bounded between
constants times $r^{2(d-1)}\cdot t^{\gamma-1}$: since

$r(t)^{2(d-1)}\cdot t^{\gamma-1}=\frac{c}{\ln(1+2J)}\cdot t^{\sigma}$

goes to $0$ as $t\to 0^{+}$, the probability
$\mathbb{P}\left(V^{0,e_{d}}_{r}\geq 3/t\right)=1-e^{-p(t)}$ is asymptotic to
$p(t)$, and thus we obtain the result of the proposition. ∎

#### Upper bound.

We now turn to upper bounding the quick connection probability. The idea is
that, for $0$ and $e_{d}$ to be connected quickly, there must exist fast roads
at all scales around both points. More precisely, for every $r\in{]0,1]}$, any
$\Pi$-path connecting $0$ and $e_{d}$ has to travel an Euclidean distance $r$
inside the ball $\overline{B}(0,r)$ (resp. $\overline{B}(e_{d},r)$), at speed
at most $V^{0}_{r}$ (resp. $V^{e_{d}}_{r}$). This is illustrated in Figure 2.
Therefore, on the event $(T(0,e_{d})\leq t)$, we must have $V^{0}_{r}\geq r/t$
and $V^{e_{d}}_{r}\geq r/t$ for all $r\in{]0,1]}$. As mentioned in Section 1,
the typical order of $V^{x}_{r}$ is $r^{(d-1)/(\gamma-1)}$, so — intuitively —
the condition $V^{x}_{r}\geq r/t$ is restrictive only when $r$ is of order at
least $t^{(\gamma-1)/(\gamma-d)}$. Discretising scales, we set $r_{k}=2^{-k}$
for all $k\in\mathbb{N}$, and let $K$ be the largest integer $k$ such that
$r_{k}\geq t^{(\gamma-1)/(\gamma-d)}$. According to the previous discussion,
we have

$\mathbb{P}(T(0,e_{d})\leq
t)\leq\mathbb{P}\left(\text{$V^{0}_{r_{k}}\geq\frac{r_{k}}{t}$ and
$V^{e_{d}}_{r_{k}}\geq\frac{r_{k}}{t}$ for all $k\in\llbracket
0,K\rrbracket$}\right).$ (16)

In the following subparagraph, we explain how to control the probability of
such an event.

##### Multiscale lemma.

In (16), the random variables $V^{0}_{r_{k}}$ and $V^{e_{d}}_{r_{k}}$ for
$k\in\llbracket 0,K\rrbracket$ are correlated; still, in order to control the
right hand side, the idea is to exploit some independence. Such independence
is gained by examining the possible configurations of the process leading to
the realisation of the event at stake (the proof of Lemma 2.1 below should
shed light on this sentence). We derive the following “multiscale lemma”.

######  Lemma 2.1.

Let $(r_{k})_{k\in\mathbb{N}}$ and $(v_{k})_{k\in\mathbb{N}}$ be decreasing
sequences of positive real numbers.

  1. 1.

Let $x\in\mathbb{R}^{d}$. For any integers $0\leq n\leq K$, we have

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
n,K\rrbracket$}\right)\leq\sum_{j=0}^{K-n}\sum_{n=k_{0}<\ldots<k_{j+1}=K+1}\prod_{i=0}^{j}\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq
v_{k_{i}}\right).$ (17)

  2. 2.

Let $x\neq y\in\mathbb{R}^{d}$. For any integers $0\leq m,n\leq K$, we have

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
m,K\rrbracket$ and $V^{y}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
n,K\rrbracket$}\right)\\\
\leq\sum_{j=0}^{2K+1-m-n}\sum_{(m,n)=(k_{0},l_{0})\prec\ldots\prec(k_{j+1},l_{j+1})=(K+1,K+1)}\prod_{i=0}^{j}\varpi_{k_{i},l_{i}}^{k_{i+1},l_{i+1}},$
(18)

where $(k_{i+1},l_{i+1})\succ(k_{i},l_{i})$ means $k_{i+1}\geq k_{i}$,
$l_{i+1}\geq l_{i}$, and $(k_{i+1},l_{i+1})\neq(k_{i},l_{i})$; and with

$\varpi_{k_{i},l_{i}}^{k_{i+1},l_{i+1}}=\begin{cases}\mathbb{P}\left(V^{x,y}_{r_{k_{i+1}-1},r_{l_{i+1}-1}}\geq
v_{k_{i}\wedge l_{i}}\right)&\text{if $k_{i+1}>k_{i}$ and $l_{i+1}>l_{i}$},\\\
\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq v_{k_{i}\wedge
l_{i}}\right)&\text{if $k_{i+1}>k_{i}$ and $l_{i+1}=l_{i}$},\\\
\mathbb{P}\left(V^{y}_{r_{l_{i+1}-1}}\geq v_{k_{i}\wedge
l_{i}}\right)&\text{if $k_{i+1}=k_{i}$ and $l_{i+1}>l_{i}$}.\end{cases}$

The first point was developed by Kahn in [5, proof of Theorem 5.1], in a
context where the scales $r_{k}$ are of order at least $|x-y|$, so that
looking at both balls $\overline{B}(x,r_{k})$ and $\overline{B}(y,r_{k})$ is
redundant. Here in (16), we want to look at scales $r_{k}$ below $1$ around
points $0$ and $e_{d}$, and thus we cannot afford to lose the information on
the second point.

###### Proof.

  1. 1.

Fix $K\in\mathbb{N}$, and let $a_{n}=\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq
v_{k}$ for all $k\in\llbracket n,K\rrbracket$}\right)$ for every
$n\in\llbracket 0,K+1\rrbracket$. We claim that the sequence
$(a_{n})_{n\in\llbracket 0,K+1\rrbracket}$ satisfies the following recurrence
relation: for every $n\in\llbracket 0,K\rrbracket$, we have

$a_{n}\leq\sum_{n<p\leq K+1}\mathbb{P}\left(V^{x}_{r_{p-1}}\geq
v_{n}\right)\cdot a_{p}.$ (19)

Since $a_{K+1}=1$, this yields (17). To prove (19), let $n\in\llbracket
0,K\rrbracket$. Since $n\leq K$, on the event $A_{n}$ we have
$V^{x}_{r_{n}}\geq v_{n}$. Looking at the largest integer $p\geq n$ such that
$V^{x}_{r_{p}}\geq v_{n}$, we write

$a_{n}=\sum_{n\leq p<K}\mathbb{P}\left(\text{$V^{x}_{r_{p}}\geq
v_{n}>V^{x}_{r_{p+1}}$ ; $V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
p+1,K\rrbracket$}\right)+\mathbb{P}\left(V^{x}_{r_{K}}\geq v_{n}\right).$

Let us call $B_{p}$ the event

$\left(\text{$V^{x}_{r_{p}}\geq v_{n}>V^{x}_{r_{p+1}}$ ; $V^{x}_{r_{k}}\geq
v_{k}$ for all $k\in\llbracket p+1,K\rrbracket$}\right).$

On $B_{p}$, we have
$X_{p}:=\Pi\left(\left.\left[\overline{B}(x,r_{p})\right]\middle\backslash\left[\overline{B}(x,r_{p+1})\right]\right.\times[v_{n},\infty[\right)>0$,
and $X_{p}$ is independent of the random variables
$\left(V^{x}_{r_{k}},~{}k\in\llbracket p+1,K\rrbracket\right)$: indeed, these
random variables are all measurable with respect to the $\sigma$-algebra
generated by the restriction of the Poisson random measure $\Pi$ to
$\left[\overline{B}(x,r_{p+1})\right]\times\mathbb{R}_{+}^{*}$. Thus, we
obtain

$\mathbb{P}(B_{p})\leq\mathbb{P}(X_{p}>0)\cdot\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq
v_{k}$ for all $k\in\llbracket
p+1,K\rrbracket$}\right)\leq\mathbb{P}\left(V^{x}_{r_{p}}\geq
v_{n}\right)\cdot a_{p+1}.$

Plugging this bound into the previous equation proves (19), upon reindexing
the sum.

  2. 2.

The idea is the same as before, but things are a bit trickier since we are
dealing with two points: at each scale $r_{k}$ around points $x$ and $y$, the
speed $v_{k}$ can come from the same road.

Fix $K\in\mathbb{N}$. For every $m,n\in\llbracket 0,K+1\rrbracket$, denote by
$A_{m,n}$ the event

$\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket m,K\rrbracket$
and $V^{y}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket n,K\rrbracket$}\right),$

and let $a_{m,n}=\mathbb{P}(A_{m,n})$. We claim that the sequence
$(a_{m,n})_{m,n\in\llbracket 0,K+1\rrbracket}$ satisfies the following
recurrence relation: for every $m,n\in\llbracket 0,K+1\rrbracket$ with
$(m,n)\prec(K+1,K+1)$, we have

$a_{m,n}\leq\sum_{(m,n)\prec(p,q)\preceq(K+1,K+1)}\varpi_{m,n}^{p,q}\cdot
a_{p,q}.$ (20)

Recall that $(m,n)\prec(p,q)$ means $(m,n)\preceq(p,q)$ and
${(p,q)\neq(m,n)}$, where $(m,n)\preceq(p,q)$ stands for $p\geq m$ and $q\geq
n$. We also recall the definition of the $\varpi_{m,n}^{p,q}$ terms:

$\varpi_{m,n}^{p,q}=\begin{cases}\mathbb{P}\left(V^{x,y}_{r_{p-1},r_{q-1}}\geq
v_{m\wedge n}\right)&\text{if $p>m$ and $q>n$},\\\
\mathbb{P}\left(V^{x}_{r_{p-1}}\geq v_{m\wedge n}\right)&\text{if $p>m$ and
$q=n$},\\\ \mathbb{P}\left(V^{y}_{r_{q-1}}\geq v_{m\wedge n}\right)&\text{if
$p=m$ and $q>n$}.\end{cases}$

Since $a_{K+1,K+1}=1$, the fact that $(a_{m,n})_{m,n\in\llbracket
0,K+1\rrbracket}$ satisfies the above recurrence relation yields (18). To
prove (20), let $m,n\in\llbracket 0,K+1\rrbracket$ be such that
$(m,n)\prec(K+1,K+1)$. By invariance, we have the symmetry
${a_{m,n}=a_{n,m}}$, so we might as well assume $m\leq n$. In particular, we
have $m\leq K$, and on the event $A_{m,n}$ we have $V^{x}_{r_{m}}\geq v_{m}$.
Looking at the largest integer $p\geq m$ such that $V^{x}_{r_{p}}\geq v_{m}$,
we write

$a_{m,n}=\sum_{m\leq p<K}\mathbb{P}\left(V^{x}_{r_{p}}\geq
v_{m}>V^{x}_{r_{p+1}}~{};~{}A_{p+1,n}\right)+\mathbb{P}\left(V^{x}_{r_{K}}\geq
v_{m}~{};~{}A_{K+1,n}\right).$ (21)

Let us call $B_{p}$ the event

$\left(V^{x}_{r_{p}}\geq v_{m}>V^{x}_{r_{p+1}}~{};~{}A_{p+1,n}\right).$

On $B_{p}$, either (a) a road with speed limit $v\geq v_{m}$ passes trough
both balls $\overline{B}(x,r_{p})$ and $\overline{B}(y,r_{n})$, or (b) the
ball $\overline{B}(x,r_{p})$ is traversed by a road with speed limit $v\geq
v_{m}$ that does not pass through $\overline{B}(y,r_{n})$.

     * •

In case (a), looking at the largest integer $q\geq n$ such that
$V^{x,y}_{r_{p},r_{q}}\geq v_{m}$, we write

$\mathbb{P}\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{n}}\geq
v_{m}\right)\leq\sum_{n\leq
q<K}\mathbb{P}\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{q}}\geq
v_{m}>V^{x,y}_{r_{p},r_{q+1}}\right)+\mathbb{P}\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{K}}\geq
v_{m}\right).$

On the event $\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{q}}\geq
v_{m}>V^{x,y}_{r_{p},r_{q+1}}\right)$, we have

$X_{p,q}:=\Pi\left(\left(\left[\overline{B}(x,r_{p})\right]\middle\backslash\left[\overline{B}(x,r_{p+1})\right]\right)\cap\left(\left[\overline{B}(y,r_{q})\right]\middle\backslash\left[\overline{B}(x,r_{q+1})\right]\right)\times[v_{m},\infty[\right)>0,$

and the event $A_{p+1,q+1}$ is realised. Since $A_{p+1,q+1}$ belongs to the
$\sigma$-algebra generated by the restriction of the Poisson random measure
$\Pi$ to
$\left(\left[\overline{B}(x,r_{p+1})\right]\cup\left[\overline{B}(y,r_{q+1})\right]\right)\times\mathbb{R}_{+}^{*}$,
it is independent of $X_{p,q}$, and thus we obtain

$\begin{split}\mathbb{P}\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{q}}\geq
v_{m}>V^{x,y}_{r_{p},r_{q+1}}\right)&\leq\mathbb{P}(X_{p,q}>0)\cdot\mathbb{P}(A_{p+1,q+1})\\\
&\leq\mathbb{P}\left(V^{x,y}_{r_{p},r_{q}}\geq
v_{m}\right)\cdot\mathbb{P}(A_{p+1,q+1})=\varpi_{m,n}^{p+1,q+1}\cdot
a_{p+1,q+1}.\end{split}$

Similarly, we have

$\mathbb{P}\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{K}}\geq
v_{m}\right)\leq\varpi_{m,n}^{p+1,K+1}\cdot a_{p+1,K+1}.$

We get

$\mathbb{P}\left(B_{p}~{};~{}V^{x,y}_{r_{p},r_{n}}\geq
v_{m}\right)\leq\sum_{n\leq q<K}\varpi_{m,n}^{p+1,q+1}\cdot
a_{p+1,q+1}+\varpi_{m,n}^{p+1,K+1}\cdot a_{p+1,K+1}=\sum_{n<q\leq
K+1}\varpi_{m,n}^{p+1,q}\cdot a_{p+1,q}.$

     * •

In case (b), we have

$Y_{p}:=\Pi\left(\left.\left(\left[\overline{B}(x,r_{p})\right]\middle\backslash\left[\overline{B}(x,r_{p+1})\right]\right)\middle\backslash\left[\overline{B}(y,r_{n})\right]\right.\times[v_{m},\infty[\right)>0,$

and the event $A_{p+1,n}$ is realised. Since $A_{p+1,n}$ belongs to the
$\sigma$-algebra generated by the restriction of the Poisson random measure
$\Pi$ to
$\left(\left[\overline{B}(x,r_{p+1})\right]\cup\left[\overline{B}(y,r_{n})\right]\right)\times\mathbb{R}_{+}^{*}$,
it is independent of $Y_{p}$, and thus we obtain

$\mathbb{P}\left(B_{p}~{};~{}V^{x}_{r_{p}}\geq
v_{m}>V^{y}_{r_{n}}\right)\leq\mathbb{P}(Y_{p}>0)\cdot\mathbb{P}(A_{p+1,n})\leq\mathbb{P}\left(V^{x}_{r_{p}}\geq
v_{m}\right)\cdot\mathbb{P}(A_{p+1,n})=\varpi_{m,n}^{p+1,n}\cdot a_{p+1,n}.$

With the above distinction, we see that

$\mathbb{P}(B_{p})\leq\sum_{n<q\leq K+1}\varpi_{m,n}^{p+1,q}\cdot
a_{p+1,q}+\varpi_{m,n}^{p+1,n}\cdot
a_{p+1,n}=\sum_{q=n}^{K+1}\varpi_{m,n}^{p+1,q}\cdot a_{p+1,q}.$

Similarly, we have

$\mathbb{P}\left(V^{x}_{r_{K}}\geq
v_{m}~{};~{}A_{K+1,n}\right)\leq\sum_{q=n}^{K+1}\varpi_{m,n}^{K+1,q}\cdot
a_{K+1,q}.$

Plugging these bounds into (21), we get

$a_{m,n}\leq\sum_{m\leq p<K}\sum_{q=n}^{K+1}\varpi_{m,n}^{p+1,q}\cdot
a_{p+1,q}+\sum_{q=n}^{K+1}\varpi_{m,n}^{K+1,q}\cdot a_{K+1,q}=\sum_{m<p\leq
K+1}\sum_{q=n}^{K+1}\varpi_{m,n}^{p,q}\cdot a_{p,q},$

which proves (20).

∎

In order to extract some quantitative information from the previous lemma,
consider the following technical proposition, which will be used several times
in the paper.

######  Proposition 2.2.

Let $(r_{k})_{k\in\mathbb{N}}$ and $(v_{k})_{k\in\mathbb{N}}$ be decreasing
sequences of positive real numbers.

  1. 1.

Let $x\in\mathbb{R}^{d}$. For any integers $0\leq n\leq K$, we have

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
n,K\rrbracket$}\right)\leq r_{K}^{d-1}\cdot
v_{n}^{-(\gamma-1)}\cdot\exp\left[\sum_{k=n+1}^{K}r_{k-1}^{d-1}\cdot
v_{k}^{-(\gamma-1)}\right].$

  2. 2.

Let $x\neq y\in\mathbb{R}^{d}$. Assume that $\left(r_{k}=|x-y|\cdot
2^{-k}\right)_{k\in\mathbb{N}}$, and $(v_{k}=r_{k}/t)_{k\in\mathbb{N}}$, for
some $t\in\mathbb{R}_{+}^{*}$. There exists positive constants $C$ and $c$,
which do not depend on $x$, $y$ or $t$, such that for every $K\in\mathbb{N}$:

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ and $V^{y}_{r_{k}}\geq v_{k}$
for all $k\in\llbracket 0,K\rrbracket$}\right)\leq
C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\exp\left[c\cdot r_{K}^{-(\gamma-d)}\cdot
t^{\gamma-1}\right].$

######  Remark 2.

In the other direction, we have the obvious lower bounds

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
n,K\rrbracket$}\right)\geq\mathbb{P}\left(V^{x}_{r_{K}}\geq
v_{n}\right)=1-\exp\left[-r_{K}^{d-1}\cdot v_{n}^{-(\gamma-1)}\right],$

and

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ and $V^{y}_{r_{k}}\geq v_{k}$
for all $k\in\llbracket
0,K\rrbracket$}\right)\geq\mathbb{P}\left(V^{x,y}_{r_{K}}\geq v_{0}\right)\geq
1-\exp\left[c^{\prime}\cdot\frac{r_{K}^{d-1}\cdot
r_{K}^{d-1}}{|x-y|^{d-1}}\cdot v_{0}^{-(\gamma-1)}\right],$

where $c^{\prime}>0$ is a constant obtained from Lemma 1.1. Under mild
assumptions, these lower bounds match the above upper bounds up to
multiplicative constants. Thus, informally, if one asks $\Pi$ to put fast
roads at all scales near one or two points, then the less expensive way for
that to be realised is to have one very fast road passing very close to the
prescribed points, and which alone is responsible for all these events. This
was formulated by Kahn for the one-point case in [5, Remark 5.1].

###### Proof.

Let $K\in\mathbb{N}$.

  1. 1.

By item 1 of the multiscale lemma, we have

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
n,K\rrbracket$}\right)\leq\sum_{j=0}^{K-n}\sum_{n=k_{0}<\ldots<k_{j+1}=K+1}\prod_{i=0}^{j}\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq
v_{k_{i}}\right).$

Using the inequality $\mathbb{P}(\mathrm{Poisson}(\lambda)>0)\leq\lambda$, we
have

$\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq v_{k_{i}}\right)\leq
r_{k_{i+1}-1}^{d-1}\cdot v_{k_{i}}^{-(\gamma-1)}\quad\text{for all integers
$k_{i+1}>k_{i}$,}$

hence

$\prod_{i=0}^{j}\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq v_{k_{i}}\right)\leq
r_{K}^{d-1}\cdot v_{n}^{-(\gamma-1)}\cdot\prod_{i=1}^{j}r_{k_{i}-1}^{d-1}\cdot
v_{k_{i}}^{-(\gamma-1)}.$

It follows that

$\begin{split}\sum_{0=k_{0}<\ldots<k_{j+1}=K+1}\prod_{i=0}^{j}\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq
v_{k_{i}}\right)&\leq r_{K}^{d-1}\cdot
v_{n}^{-(\gamma-1)}\cdot\sum_{n=k_{0}<\ldots<k_{j+1}=K+1}\prod_{i=1}^{j}r_{k_{i}-1}^{d-1}\cdot
v_{k_{i}}^{-(\gamma-1)}\\\ &\leq r_{K}^{d-1}\cdot
v_{n}^{-(\gamma-1)}\cdot\frac{1}{j!}\cdot\sum_{n+1\leq k_{1},\ldots,k_{j}\leq
K}\prod_{i=1}^{j}r_{k_{i}-1}^{d-1}\cdot v_{k_{i}}^{-(\gamma-1)}\\\
&=r_{K}^{d-1}\cdot
v_{n}^{-(\gamma-1)}\cdot\frac{1}{j!}\cdot\left[\sum_{k=n+1}^{K}r_{k-1}^{d-1}\cdot
v_{k}^{-(\gamma-1)}\right]^{j}.\end{split}$

Finally, summing over $j$ we obtain

$\sum_{j=0}^{K-n}\sum_{n=k_{0}<\ldots<k_{j+1}=K+1}\prod_{i=0}^{j}\mathbb{P}\left(V^{x}_{r_{k_{i+1}-1}}\geq
v_{k_{i}}\right)\leq r_{K}^{d-1}\cdot
v_{n}^{-(\gamma-1)}\cdot\exp\left[\sum_{k=n+1}^{K}r_{k-1}^{d-1}\cdot
v_{k}^{-(\gamma-1)}\right].$

  2. 2.

By item 2 of the multiscale lemma, we have (with the notation introduced in
Lemma 2.1):

$\mathbb{P}\left(\text{$V^{x}_{r_{k}}\geq v_{k}$ and $V^{y}_{r_{k}}\geq v_{k}$
for all $k\in\llbracket
0,K\rrbracket$}\right)\leq\sum_{j=0}^{2K+1}\sum_{(0,0)=(k_{0},l_{0})\prec\ldots\prec(k_{j+1},l_{j+1})=(K+1,K+1)}\prod_{i=0}^{j}\varpi_{k_{i},l_{i}}^{k_{i+1},l_{i+1}}.$

To continue, recall the assumptions: we have $r_{k}=|x-y|\cdot 2^{-k}$ for all
$k\in\mathbb{N}$, and $v_{k}=r_{k}/t$ for all $k\in\mathbb{N}$, for some
$t\in\mathbb{R}_{+}^{*}$. We claim that there exists a constant $C>0$, not
depending on $x$, $y$ or $t$, such that

$\varpi_{m,n}^{p,q}\leq C\cdot\frac{r_{p-1}^{d-1}\cdot r_{q-1}^{d-1}\cdot
t^{\gamma-1}}{r_{m}^{(d+\gamma-2)/2}\cdot
r_{n}^{(d+\gamma-2)/2}}\quad\text{for all integers $(p,q)\succ(m,n)$},$ (22)

where for convenience we have set $r_{-1}=2|x-y|$. Indeed, let
$m,n,p,q\in\mathbb{N}$ be such that $(p,q)\succ(m,n)$.

     * •

If $p>m$ and $q>n$, then
$\varpi_{m,n}^{p,q}=\mathbb{P}\left(V^{x,y}_{r_{p-1},r_{q-1}}\geq v_{m\wedge
n}\right)$. Using the inequality
${\mathbb{P}(\mathrm{Poisson}(\lambda)>0)\leq\lambda}$ and Lemma 1.1, we see
that there exists a constant $C>0$ such that

$\varpi_{m,n}^{p,q}\leq C\cdot\frac{r_{p-1}^{d-1}\cdot
r_{q-1}^{d-1}}{|x-y|^{d-1}}\cdot v_{m\wedge
n}^{-(\gamma-1)}=C\cdot\frac{r_{p-1}^{d-1}\cdot r_{q-1}^{d-1}\cdot
t^{\gamma-1}}{|x-y|^{d-1}\cdot r_{m\wedge n}^{\gamma-1}}\leq
C\cdot\frac{r_{p-1}^{d-1}\cdot r_{q-1}^{d-1}\cdot
t^{\gamma-1}}{r_{m}^{(d+\gamma-2)/2}\cdot r_{n}^{(d+\gamma-2)/2}}.$

     * •

If $p>m$ and $q=n$, then
$\varpi_{m,n}^{q,n}=\mathbb{P}\left(V^{x}_{r_{p-1}}\geq v_{m\wedge n}\right)$.
Using the inequality $\mathbb{P}(\mathrm{Poisson}(\lambda)>0)\leq\lambda$, we
see that

$\varpi_{m,n}^{p,q}\leq r_{p-1}^{d-1}\cdot v_{m\wedge
n}^{-(\gamma-1)}=2^{-(d-1)}\cdot\frac{r_{p-1}^{d-1}\cdot r_{q-1}^{d-1}\cdot
t^{\gamma-1}}{r_{n}^{d-1}\cdot r_{m\wedge n}^{\gamma-1}}\leq
2^{-(d-1)}\cdot\frac{r_{p-1}^{d-1}\cdot r_{q-1}^{d-1}\cdot
t^{\gamma-1}}{r_{m}^{(d+\gamma-2)/2}\cdot r_{n}^{(d+\gamma-2)/2}}.$

     * •

If $p=m$ and $q>n$, then
$\varpi_{m,n}^{q,n}=\mathbb{P}\left(V^{x}_{r_{q-1}}\geq v_{m\wedge n}\right)$.
In the same way as above, we get

$\varpi_{m,n}^{p,q}\leq 2^{-(d-1)}\cdot\frac{r_{p-1}^{d-1}\cdot
r_{q-1}^{d-1}\cdot t^{\gamma-1}}{r_{m}^{(d+\gamma-2)/2}\cdot
r_{n}^{(d+\gamma-2)/2}}.$

Now armed with (22), we obtain

$\begin{split}\prod_{i=0}^{j}\varpi_{k_{i},l_{i}}^{k_{i+1},l_{i+1}}&\leq
C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}\cdot
t^{\gamma-1}}{r_{0}^{(d+\gamma-2)/2}\cdot
r_{0}^{(d+\gamma-2)/2}}\cdot\prod_{i=1}^{j}\left(C\cdot\frac{r_{k_{i}-1}^{d-1}\cdot
r_{l_{i}-1}^{d-1}\cdot t^{\gamma-1}}{r_{k_{i}}^{(d+\gamma-2)/2}\cdot
r_{l_{i}}^{(d+\gamma-2)/2}}\right)\\\ &=C\cdot\frac{r_{K}^{d-1}\cdot
r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\prod_{i=1}^{j}\frac{2^{d+\gamma-2}\cdot C\cdot
t^{\gamma-1}}{r_{k_{i}-1}^{(\gamma-d)/2}\cdot
r_{l_{i}-1}^{(\gamma-d)/2}}.\end{split}$

It follows that

$\displaystyle\sum_{(0,0)=(k_{0},l_{0})\prec\ldots\prec(k_{j+1},l_{j+1})=(K+1,K+1)}\prod_{i=0}^{j}\varpi_{k_{i},l_{i}}^{k_{i+1},l_{i+1}}$
$\displaystyle\leq$ $\displaystyle C\cdot\frac{r_{K}^{d-1}\cdot
r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\sum_{(0,0)=(k_{0},l_{0})\prec\ldots\prec(k_{j+1},l_{j+1})=(K+1,K+1)}\prod_{i=1}^{j}\frac{2^{d+\gamma-2}\cdot
C\cdot t^{\gamma-1}}{r_{k_{i}-1}^{(\gamma-d)/2}\cdot
r_{l_{i}-1}^{(\gamma-d)/2}}$ $\displaystyle\leq$ $\displaystyle
C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\frac{1}{j!}\cdot\sum_{(0,0)\prec(k_{1},l_{1});\ldots;(k_{j},l_{j})\prec(K+1,K+1)}\prod_{i=1}^{j}\frac{2^{d+\gamma-2}\cdot
C\cdot t^{\gamma-1}}{r_{k_{i}-1}^{(\gamma-d)/2}\cdot
r_{l_{i}-1}^{(\gamma-d)/2}}$ $\displaystyle=$ $\displaystyle
C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\frac{1}{j!}\cdot\left[\sum_{(0,0)\prec(k,l)\prec(K+1,K+1)}\frac{2^{d+\gamma-2}\cdot
C\cdot t^{\gamma-1}}{r_{k-1}^{(\gamma-d)/2}\cdot
r_{l-1}^{(\gamma-d)/2}}\right]^{j}$ $\displaystyle\leq$ $\displaystyle
C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\frac{1}{j!}\cdot\left[2^{d+\gamma-2}\cdot
C\cdot\left(\sum_{k=0}^{K+1}r_{k-1}^{-(\gamma-d)/2}\right)^{2}\cdot
t^{\gamma-1}\right]^{j}.$

Finally, summing over $j$ we obtain

$\sum_{j=0}^{2K+1}\sum_{(0,0)=(k_{0},l_{0})\prec\ldots\prec(k_{j+1},l_{j+1})=(K+1,K+1)}\prod_{i=0}^{j}\varpi_{k_{i},l_{i}}^{k_{i+1},l_{i+1}}\\\
\leq C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}}{|x-y|^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\exp\left[2^{d+\gamma-2}\cdot
C\cdot\left(\sum_{k=0}^{K+1}r_{k-1}^{-(\gamma-d)/2}\right)^{2}\cdot
t^{\gamma-1}\right].$

The result follows, because $\sum_{k=0}^{K+1}r_{k-1}^{-(\gamma-d)/2}\leq
r_{K}^{-(\gamma-d)/2}\cdot\sum_{l\geq 0}2^{-(\gamma-d)/2\cdot l}$.

∎

##### The upper bound.

With the previous results, we can now proceed with the upper bound for the
quick connection probability. Mercifully, all the work has been done already.

######  Proposition 2.3.

There exists a constant $C>0$ such that

$\mathbb{P}(T(0,e_{d})\leq t)\leq C\cdot t^{\sigma}$ for all $t\in[0,1]$,
where $\sigma=\frac{(\gamma-1)(d+\gamma-2)}{\gamma-d}$.

###### Proof.

Let $t\in{]0,1]}$. Set $r_{k}=2^{-k}$ and $v_{k}=r_{k}/t$ for all
$k\in\mathbb{N}$, and let
${K=\left\lfloor\log_{2}t^{-(\gamma-1)/(\gamma-d)}\right\rfloor}$ be the
largest integer $k$ such that $r_{k}\geq t^{(\gamma-1)/(\gamma-d)}$. By (16),
we have

$\mathbb{P}(T(0,e_{d})\leq t)\leq\mathbb{P}\left(\text{$V^{0}_{r_{k}}\geq
v_{k}$ and $V^{e_{d}}_{r_{k}}\geq v_{k}$ for all $k\in\llbracket
0,K\rrbracket$}\right).$

By Proposition 2.2, there exists positive constants $C$ and $c$, not depending
on $t$, such that

$\mathbb{P}\left(\text{$V^{0}_{r_{k}}\geq v_{k}$ and $V^{e_{d}}_{r_{k}}\geq
v_{k}$ for all $k\in\llbracket 0,K\rrbracket$}\right)\leq
C\cdot\frac{r_{K}^{d-1}\cdot r_{K}^{d-1}}{1^{d-1}}\cdot
v_{0}^{-(\gamma-1)}\cdot\exp\left[c\cdot r_{K}^{-(\gamma-d)}\cdot
t^{\gamma-1}\right].$

The result follows, since $r_{K}\leq 2t^{(\gamma-1)/(\gamma-d)}$, $v_{0}=1/t$,
and $r_{K}^{-(\gamma-d)}\leq t^{-(\gamma-1)}$. ∎

##  3 Lebesgue measure of $T$-balls

We pursue our analysis with the Lebesgue measure as mass distribution on
$\left(\mathbb{R}^{d},T\right)$, by examining the Lebesgue measure of balls
for the metric $T$. Recalling the notation
${\overline{\Gamma}(x,t)=\left\\{y\in\mathbb{R}^{d}:T(x,y)\leq t\right\\}}$
for the $T$-ball centered at $x\in\mathbb{R}^{d}$ with radius $t>0$, we are
interested in the behaviour of the quantity
$\lambda\left(\overline{\Gamma}(x,t)\right)$ as $t\to 0^{+}$.

Note that for fixed $x\in\mathbb{R}^{d}$ and $t>0$, by invariance and scaling,
we have the equality in distribution

$\lambda\left(\overline{\Gamma}(x,t)\right)\overset{\text{\tiny
law}}{=}\lambda\left(\overline{\Gamma}(0,1)\right)\cdot t^{s^{*}},$ (23)

where $s^{*}=(\gamma-1)d/(\gamma-d)$ is the Hausdorff dimension of
$\left(\mathbb{R}^{d},T\right)$. Therefore, for “typical” points $x$ we should
have $\lambda\left(\overline{\Gamma}(x,t)\right)\approx t^{s^{*}}$ for small
$t$. On the other hand, $T$-balls around points on roads should be bigger than
typical balls, since $\Pi$-paths travelling in time $t$ from a point on a road
can first use this road for a time $t/2$, and then still have time $t/2$ left
to move around. Heuristically, as illustrated in Figure 6, those balls should
have Lebesgue measure $\approx t^{s_{*}}$, with
${s_{*}=s^{*}-(d-1)/(\gamma-d)}$.

Figure 5: Fix a realisation of $\Pi$. Let $(\ell,v)$ be a road of $\Pi$, and
let $x\in\ell$. The $T$-ball $\overline{\Gamma}(x,t)$ contains the union
$B_{t}=\bigcup_{y\in\ell:|x-y|\leq vt/2}\overline{\Gamma}(y,t/2)$. On the
picture, a “typical” $T$-ball of radius $t/2$ has been represented around $x$,
and replicated along the line $\ell$. Heuristically, this $T$-ball has
Lebesgue measure $\approx t^{s^{*}}$, and thus spans an Euclidean distance
$\approx t^{s^{*}/d}=t^{(\gamma-1)/(\gamma-d)}$. Therefore, the union $B_{t}$
should have Lebesgue measure ${\approx t\cdot t^{-(\gamma-1)/(\gamma-d)}\cdot
t^{s^{*}}=t^{s_{*}}}$ (the first two factors account for the number of
“disjoint” typical $T$-balls of radius $t/2$ it takes to span an Euclidean
distance $vt/2$, and the third term for the Lebesgue measure of a typical
$T$-ball with radius $t/2$).

|
---|---

Figure 6: Simulations by Arvind Singh. On the left hand side, a typical
$T$-ball. On the right hand side, the origin is conditioned to be traversed by
a road, and we seem to observe a shape similar to the one in Figure 6.

In this section, the above heuristics are made rigorous in two different ways:
in a first subsection we adopt a distributional point of view, before turning
to the proof of Theorem 3 in Subsection 3.2.

###  3.1 Distributional point of view

Here we adopt a distributional point of view, describing the law of
$\lambda\left(\overline{\Gamma}(X,t)\right)$ as $t\to 0^{+}$, when $X$ is a
point picked “uniformly at random”, possibly on a road. The main result of
this subsection is Proposition 3.1

#### Typical points.

Fixing $R>0$, let $X$ be a random variable with uniform distribution on
$\overline{B}(0,R)$, independent of $\Pi$. For every $t>0$, by invariance and
scaling, we have the equality in distribution

$\lambda\left(\overline{\Gamma}(X,t)\right)\overset{\text{\tiny
law}}{=}\lambda\left(\overline{\Gamma}(0,1)\right)\cdot t^{s^{*}}.$

In particular, trivially we have the convergence in distribution

$\frac{\lambda\left(\overline{\Gamma}(X,t)\right)}{t^{s^{*}}}\underset{t\to
0^{+}}{\overset{\text{\tiny
law}}{\longrightarrow}}\lambda\left(\overline{\Gamma}(0,1)\right).$

Note that in the above setting, almost surely $X$ is not on a road of $\Pi$.
Let us now consider the case where $X$ is a point chosen “uniformly at random”
on a road of $\Pi$.

#### Typical points on roads.

Fixing $R>0$ and $v_{0}\in\mathbb{R}_{+}^{*}$, let us write here
$N^{R}_{v_{0}}$ in place of
$\Pi\left(\left[\overline{B}(0,R)\right]\times[v_{0},\infty[\right)$ for
short. Define a random variable $X$ as follows: on the event
$\left(N^{R}_{v_{0}}>0\right)$, pick a road $(\ell,v)$ of $\Pi$ in the set
${\left[\overline{B}(0,R)\right]\times[v_{0},\infty[}$ uniformly at random,
and let $X$ be a uniform point on the line segment
$\ell\cap\overline{B}(0,R)$; on the complementary event
$\left(N^{R}_{v_{0}}=0\right)$, just set $X=0$ (unimportant). What we prove
here is the following.

######  Proposition 3.1.

There exists a constant $\Theta>0$, not depending on $R$ or $v_{0}$, such that
the following holds: conditionally on the event
$\left(N^{R}_{v_{0}}>0\right)$, we have the convergence in probability

$\frac{\lambda\left(\overline{\Gamma}(X,t)\right)}{V(X)\cdot
t^{s_{*}}}\underset{t\to 0^{+}}{\longrightarrow}\Theta,$

where $V(X)$ corresponds to the speed limit of the road on which lies $X$.

As we shall see in the proof, the constant $\Theta$ emerges from a
subadditivity argument.

###### Proof.

It suffices to prove that for each $\varepsilon>0$,

$\mathbb{E}\left[\frac{1}{N^{R}_{v_{0}}}\sum_{\begin{subarray}{c}(\ell,v)\in\Pi\\\
\text{$\ell$ hits $\overline{B}(0,R)$ and $v\geq
v_{0}$}\end{subarray}}\fint_{\ell\cap\overline{B}(0,R)}\mathbf{1}\left(\left|\Theta-\frac{\lambda\left(\overline{\Gamma}(x,t)\right)}{v\cdot
t^{s_{*}}}\right|>\varepsilon\right)\mathrm{d}x~{};~{}N^{R}_{v_{0}}>0\right]\underset{t\to
0^{+}}{\longrightarrow}0,$

where the symbol $\fint_{\ell\cap\overline{B}(0,R)}\mathrm{d}x$ denotes
averaging in $x$ over the line segment $\ell\cap\overline{B}(0,R)$. In the
above expectation, bounding $\mathbf{1}\left(N^{R}_{v_{0}}>0\right)$ by
$N^{R}_{v_{0}}$, we find that it suffices to prove that as $t\to 0^{+}$, we
have

$\mathbb{E}\left[\sum_{\begin{subarray}{c}(\ell,v)\in\Pi\\\ \text{$\ell$ hits
$\overline{B}(0,R)$ and $v\geq
v_{0}$}\end{subarray}}\fint_{\ell\cap\overline{B}(0,R)}\mathbf{1}\left(\left|\Theta-\frac{\lambda\left(\overline{\Gamma}(x,t)\right)}{v\cdot
t^{s_{*}}}\right|>\varepsilon\right)\mathrm{d}x\right]\longrightarrow 0.$

Let us write $\varphi(\ell,v;\Pi)$ for

$\fint_{\ell\cap\overline{B}(0,R)}\mathbf{1}\left(\left|\Theta-\frac{\lambda\left(\overline{\Gamma}(x,t)\right)}{v\cdot
t^{s_{*}}}\right|>\varepsilon\right)\mathrm{d}x\cdot\mathbf{1}\left(\text{$\ell$
hits $\overline{B}(0,R)$ and $v\geq v_{0}$}\right).$

By the Slivnyak–Mecke theorem (see, e.g, [9, Theorem 4.1]), we have

$\mathbb{E}\left[\sum_{(\ell,v)\in\Pi}\varphi(\ell,v;\Pi)\right]=c\int_{\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}}\varphi\left(\ell,v;\Pi+\delta_{(\ell,v)}\right)\mathrm{d}\left[\mu_{d}\otimes
v^{-\gamma}\mathrm{d}v\right](\ell,v),$

where $c=\upsilon_{d-1}^{-1}\cdot(\gamma-1)$ is the normalising constant of
(2). This yields

$\mathbb{E}\left[\sum_{\begin{subarray}{c}(\ell,v)\in\Pi\\\ \text{$\ell$ hits
$\overline{B}(0,R)$ and $v\geq
v_{0}$}\end{subarray}}\fint_{\ell\cap\overline{B}(0,R)}\mathbf{1}\left(\left|\Theta-\frac{\lambda\left(\overline{\Gamma}(x,t)\right)}{v\cdot
t^{s_{*}}}\right|>\varepsilon\right)\mathrm{d}x\right]\\\
=c\int_{\left[\overline{B}(0,R)\right]}\int_{v_{0}}^{\infty}\left\\{\fint_{\ell\cap\overline{B}(0,R)}\mathbb{P}\left(\left|\Theta-\frac{\lambda\left(\overline{\Gamma_{\ell,v}}(x,t)\right)}{v\cdot
t^{s_{*}}}\right|>\varepsilon\right)\mathrm{d}x\right\\}\frac{\mathrm{d}v}{v^{\gamma}}\mathrm{d}\mu_{d}(\ell).$

The notation $\overline{\Gamma_{\ell,v}}(x,t)$ in the last integral denotes
the closed ball of radius $t$ centered at $x$ for the random metric
$T_{\ell,v}$ induced by the process ${\Pi_{\ell,v}=\Pi+\delta_{(\ell,v)}}$,
i.e, obtained using the road $(\ell,v)$ on top of the roads of $\Pi$. By the
bounded convergence theorem, we see that it suffices to prove the following to
complete the proof of the proposition.

######  Lemma 3.1.

Fix $(\ell,v)\in\mathbb{L}_{d}\times\mathbb{R}_{+}^{*}$, and $x\in\ell$.
Setting $\Pi_{\ell,v}=\Pi+\delta_{(\ell,v)}$, consider the random metric
${T_{\ell,v}:\mathbb{R}^{d}\times\mathbb{R}^{d}\rightarrow\mathbb{R}_{+}}$
induced by $\Pi_{\ell,v}$, which is given by
$T_{\ell,v}(y,z)=\tau\left(\Pi_{\ell,v};y,z\right)$ for all
$y,z\in\mathbb{R}^{d}$ — where $\tau$ is the measurable function of (7) —, and
consider the corresponding balls
${\overline{\Gamma_{\ell,v}}(x,t)=\left\\{y\in\mathbb{R}^{d}:T_{\ell,v}(x,y)\leq
t\right\\}}$. There exists a constant $\Theta>0$, not depending on $x$ or
$(\ell,v)$, such that

$\frac{\lambda\left(\overline{\Gamma_{\ell,v}}(x,t)\right)}{v\cdot
t^{s_{*}}}\underset{t\to 0^{+}}{\overset{\mathbb{P}}{\longrightarrow}}\Theta.$

###### Proof.

For ease of writing, we consider the case where $\ell=\ell_{d}$, and $x=0$.
Let $t>0$. On the event $G_{t}=\left(V^{0}_{vt}<v\right)$, we have

$\overline{\Gamma_{\ell_{d},v}}(0,t)=\bigcup_{|y|\leq
vt}\overline{\Gamma}(ye_{d},t-|y|/v).$

Indeed, the converse inclusion is always true; and for the direct inclusion,
let $z\in\overline{\Gamma_{\ell_{d},v}}(0,t)$. Denote by $ye_{d}$ the last
exit point from $\ell_{d}$ of the $\Pi_{\ell_{d},v}$-geodesic $\xi$ connecting
$0$ to $z$, with $y\in\mathbb{R}$. On $G_{t}$, the $\Pi_{\ell_{d},v}$-geodesic
$\xi$ must remain inside the ball $\overline{B}(0,vt)$; in particular, we have
$|y|\leq vt$. Moreover, since on that event $(\ell_{d},v)$ is the fastest road
of $\Pi_{\ell_{d},v}$ passing through $\overline{B}(0,vt)$, the
$\Pi_{\ell_{d},v}$-geodesic $\xi$ must connect $0$ to $ye_{d}$ using the road
$(\ell_{d},v)$: we deduce that $T_{\ell_{d},v}(0,ye_{d})=|y|/v$. Finally,
since $\xi$ does not use the road $(\ell_{d},v)$ between $ye_{d}$ and $z$, we
have $T_{\ell_{d},v}(ye_{d},z)=T(ye_{d},z)$. We conclude that
$T(ye_{d},z)=T_{\ell_{d},v}(0,z)-|y|/v\leq t-|y|/v$.

Since $\mathbb{P}(G_{t})\rightarrow 1$ as $t\to 0^{+}$, to prove the Lemma it
suffices to show that

$\frac{\lambda\left(\bigcup_{|y|\leq
vt}\overline{\Gamma}(ye_{d},t-|y|/v)\right)}{v\cdot t^{s_{*}}}\underset{t\to
0^{+}}{\overset{\mathbb{P}}{\longrightarrow}}\Theta.$

Now, by scaling, we have the equality in distribution

$\lambda\left(\bigcup_{|y|\leq
vt}\overline{\Gamma}(ye_{d},t-|y|/v)\right)\overset{\text{\tiny
law}}{=}\lambda\left(\bigcup_{|y|\leq
u}\overline{\Gamma}\left(ye_{d},1-|y|/u\right)\right)\cdot
t^{s^{*}},\quad\text{where $u=vt^{-(d-1)/(\gamma-d)}$},$

hence

$\frac{\lambda\left(\bigcup_{|y|\leq
vt}\overline{\Gamma}(ye_{d},t-|y|/v)\right)}{v\cdot
t^{s_{*}}}\overset{\text{\tiny law}}{=}\frac{\lambda\left(\bigcup_{|y|\leq
u}\overline{\Gamma}\left(ye_{d},1-|y|/u\right)\right)}{u}.$

To conclude, let us show that

$\frac{\lambda\left(\bigcup_{|y|\leq
u}\overline{\Gamma}(ye_{d},1-|y|/u)\right)}{u}\underset{u\to\infty}{\overset{\mathbb{P}}{\longrightarrow}}\Theta.$

Exploiting subadditivity. For each $t>0$, let

$\Lambda^{t}_{a,b}=\lambda\left(\bigcup_{a\leq y\leq
b}\overline{\Gamma}(ye_{d},t)\right)\quad\text{for all $a\leq
b\in\mathbb{R}$.}$

We claim that the random variables $\left(\Lambda^{t}_{m,n}\right)_{m\leq
n\in\mathbb{N}}$ satisfy the following:

  1. (i)

for each $m\leq n\in\mathbb{N}$, we have
$\Lambda^{t}_{0,n}\leq\Lambda^{t}_{0,m}+\Lambda^{t}_{m,n}$,

  2. (ii)

for each $q\in\mathbb{N}^{*}$, the sequence
$\left(\Lambda^{t}_{qn,q(n+1)}\right)_{n\in\mathbb{N}}$ is stationary and
mixing,

  3. (iii)

the distribution of $\left(\Lambda^{t}_{m,m+n}\right)_{n\in\mathbb{N}}$ does
not depend on the integer $m\in\mathbb{N}$,

  4. (iv)

we have $\mathbb{E}\left[\Lambda^{t}_{0,1}\right]<\infty$, and there exists a
constant $c=c(t)>0$ such that $\mathbb{E}\left[\Lambda^{t}_{0,n}\right]\geq
c\cdot n$ for all $n\in\mathbb{N}^{*}$.

These are the assumptions of the subadditive ergodic theorem as it is stated
in [3, Theorem 6.4.1]. Therefore, by the subadditive ergodic theorem, there
exists a constant $\theta=\theta(t)>0$ such that

$\frac{\Lambda^{t}_{0,u}}{u}\underset{u\to\infty}{\longrightarrow}\theta(t)\quad\text{almost
surely and in $L^{1}$}.$ (24)

The subadditivity (i) is clear. (iii) and the stationarity in (ii) follow from
invariance, and the mixing in (ii) follows from Proposition 1.2. It remains to
check (iv). To ease notation, let us take $t=1$, and simply write
$\Lambda_{a,b}$ instead of $\Lambda^{1}_{a,b}$. We begin with the second part
of (iv): we have

$\begin{split}\mathbb{E}[\Lambda_{0,n}]&=\int_{\mathbb{R}^{d}}\mathbb{P}(T(z;[0,n]e_{d})\leq
1)\mathrm{d}z\\\
&\geq\int_{\mathbb{R}^{d-1}\times[0,n]}\mathbb{P}(T((w,z_{d});(0,z_{d}))\leq
1)\mathrm{d}(w,z_{d})\\\
&=\int_{0}^{n}\int_{\mathbb{R}^{d-1}}\mathbb{P}(T((w,0);0)\leq
1)\mathrm{d}w\mathrm{d}z_{d}=\int_{\mathbb{R}^{d-1}}\mathbb{P}\left(T(0,e_{d})\leq|w|^{-\frac{\gamma-d}{\gamma-1}}\right)\mathrm{d}w\cdot
n,\end{split}$

where we used invariance and scaling for the last two equalities. Finally, for
the first part of (iv), write

$\mathbb{E}[\Lambda_{0,1}]=\int_{\mathbb{R}^{d}}\mathbb{P}(T(z;[0,1]e_{d})\leq
1)\mathrm{d}z\leq\upsilon_{d}2^{d}+\int_{|z|\geq
2}\mathbb{P}(T(z;[0,1]e_{d})\leq 1)\mathrm{d}z.$

To bound the last term, let $|z|\geq 2$. As for upper bounding the quick
connection probability, we argue that if there exists $y\in[0,1]e_{d}$ such
that $T(z,y)\leq 1$, then we must have $V^{y}_{\rho}\geq\rho$ and
$V^{z}_{\rho}\geq\rho$ for all $\rho\in{]0,|y-z|]}$. By the triangle
inequality, it follows that $V^{0}_{2\rho}\geq V^{y}_{\rho}\geq\rho$ and
$V^{z}_{2\rho}\geq V^{z}_{\rho}\geq\rho$ for all $\rho\in[1,|z|/2]$. Thus,
setting $\rho_{k}=|z|/2\cdot 2^{-k}$ for all $k\in\mathbb{N}$, and letting
$K=\left\lfloor\log_{2}(|z|/2)\right\rfloor$ be the largest integer $k$ such
that $\rho_{k}\geq 1$, we have:

$\mathbb{P}(T(z;[0,1]e_{d})\leq
1)\leq\mathbb{P}\left(\text{$V^{0}_{2\rho_{k}}\geq\rho_{k}$ and
$V^{z}_{2\rho_{k}}\geq\rho_{k}$ for all $k\in\llbracket
0,K\rrbracket$}\right).$

Using Proposition 2.2 with $\left(r_{k}=2\rho_{k}=|z|\cdot
2^{-k}\right)_{k\in\mathbb{N}}$ and
$(v_{k}=\rho_{k}=r_{k}/2)_{k\in\mathbb{N}}$, we obtain that there exists a
constant $C>0$ such that

$\mathbb{P}(T(z;[0,1]e_{d})\leq 1)\leq\frac{C}{|z|^{d+\gamma-2}}\quad\text{for
all $|z|\geq 2$,}$

which achieves to show that $\mathbb{E}[\Lambda_{0,1}]<\infty$.

Concluding the proof. We claim that the following is a consequence of (24):

$\frac{\lambda\left(\bigcup_{0\leq y\leq
u}\overline{\Gamma}(ye_{d},1-y/u)\right)}{u}\underset{u\to\infty}{\overset{\mathbb{P}}{\longrightarrow}}\int_{0}^{1}\theta(1-r)\mathrm{d}r.$
(25)

To prove this, fix $n\in\mathbb{N}^{*}$. For every $u>0$, we have

$\bigcup_{0\leq y\leq
u}\overline{\Gamma}\left(ye_{d},1-\frac{y}{u}\right)\subset\bigcup_{0\leq
k<n}\bigcup_{ku/n\leq
y\leq(k+1)u/n}\overline{\Gamma}\left(ye_{d},1-\frac{k}{n}\right)=:\bigcup_{0\leq
k<n}R^{n}_{k}(u)$

and

$\bigcup_{0\leq y\leq
u}\overline{\Gamma}\left(ye_{d},1-\frac{y}{u}\right)\supset\bigcup_{0\leq
k<n}\bigcup_{ku/n\leq
y\leq(k+1)u/n}\overline{\Gamma}\left(ye_{d},1-\frac{k+1}{n}\right)=:\bigcup_{0\leq
k<n}L^{n}_{k}(u).$

Then, the Bonferroni inequalities yield the bounds

$\sum_{0\leq k<n}\lambda(L^{n}_{k}(u))-\sum_{0\leq
i<j<n}\lambda\left(L^{n}_{i}(u)\cap
L^{n}_{j}(u)\right)\leq\lambda\left(\bigcup_{0\leq y\leq
u}\overline{\Gamma}\left(ye_{d},1-\frac{y}{u}\right)\right)\leq\sum_{0\leq
k<n}\lambda\left(R^{n}_{k}(u)\right).$ (26)

Now, let us first control the cross term $\chi_{u}:=\sum_{0\leq
i<j<n}\lambda\left(L^{n}_{i}(u)\cap L^{n}_{j}(u)\right)$ in the left hand
side: we have

$\begin{split}\mathbb{E}\left[\chi_{u}\right]&\leq\sum_{0\leq
i<j<n}\mathbb{E}\left[\lambda\left(\left(\bigcup_{iu/n\leq
y\leq(i+1)u/n}\overline{\Gamma}(ye_{d},1)\right)\cap\left(\bigcup_{ju/n\leq
y\leq(j+1)u/n}\overline{\Gamma}(ye_{d},1)\right)\right)\right]\\\
&\leq\frac{n(n-1)}{2}\cdot\mathbb{E}\left[\lambda\left(\left(\bigcup_{-u\leq
y\leq 0}\overline{\Gamma}(ye_{d},1)\right)\cap\left(\bigcup_{0\leq y\leq
u}\overline{\Gamma}(ye_{d},1)\right)\right)\right],\end{split}$

where we used invariance for the last inequality. By the inclusion-exclusion
formula and by invariance, we can write

$\mathbb{E}\left[\lambda\left(\left(\bigcup_{-u\leq y\leq
0}\overline{\Gamma}(ye_{d},1)\right)\cap\left(\bigcup_{0\leq y\leq
u}\overline{\Gamma}(ye_{d},1)\right)\right)\right]=\mathbb{E}[\Lambda_{-u,u}]-\mathbb{E}[\Lambda_{-u,0}]-\mathbb{E}[\Lambda_{0,u}]=\mathbb{E}[\Lambda_{0,2u}]-2\mathbb{E}[\Lambda_{0,u}].$

By subadditivity, we have $\mathbb{E}[\Lambda_{0,u}]/u\rightarrow\theta(1)$ as
$u\to\infty$, and it follows that the above is $o(u)$. In particular, by
Markov’s inequality, we have $\chi_{u}/u\overset{\mathbb{P}}{\rightarrow}0$ as
$u\to\infty$. Next, for each $k\in\llbracket 0,n\llbracket$, by (24) and
invariance, we have

$\frac{\lambda(R^{n}_{k}(u))}{u}\underset{u\to\infty}{\overset{\mathbb{P}}{\longrightarrow}}\frac{\theta(1-k/n)}{n}\quad\text{and}\quad\frac{\lambda(L^{n}_{k}(u))}{u}\underset{u\to\infty}{\overset{\mathbb{P}}{\longrightarrow}}\frac{\theta(1-(k+1)/n)}{n}.$

Therefore, dividing (26) by $u$, the right and left hand sides respectively
converge, as $u\to\infty$, to

$\sum_{0\leq k<n}\frac{\theta(1-k/n)}{n}\quad\text{and}\quad\sum_{0\leq
k<n}\frac{\theta(1-(k+1)/n)}{n}-0\quad\text{in probability.}$

Since these quantities can be made arbitrarily close to
$\int_{0}^{1}\theta(1-r)\mathrm{d}r$ by picking $n$ large enough, we obtain
(25). By invariance, we get as well

$\frac{\lambda\left(\bigcup_{-u\leq y\leq
0}\overline{\Gamma}(ye_{d},1-|y|/u)\right)}{u}\underset{u\to\infty}{\overset{\mathbb{P}}{\longrightarrow}}\int_{0}^{1}\theta(1-r)\mathrm{d}r=\int_{-1}^{0}\theta(1-|r|)\mathrm{d}r.$

Using one last time the inclusion-exclusion formula, we conclude that

$\frac{\lambda\left(\bigcup_{|y|\leq
u}\overline{\Gamma}(ye_{d},1-|y|/u)\right)}{u}\underset{u\to\infty}{\overset{\mathbb{P}}{\longrightarrow}}\int_{-1}^{0}\theta(1-|r|)\mathrm{d}r+\int_{0}^{1}\theta(1-r)\mathrm{d}r-0=\int_{-1}^{1}\theta(1-|r|)\mathrm{d}r=:\Theta,$

since — as above — the cross term is negligible. ∎

∎

###  3.2 Almost sure results

In this subsection, with a different point of view we establish Theorem 3,
which states that almost surely the following holds:

  1. 1.

for $\lambda$-almost every $x\in\mathbb{R}^{d}$, we have
$\lambda\left(\overline{\Gamma}(x,t)\right)=t^{s^{*}+o(1)}$ as $t\to 0^{+}$,

  2. 2.

for every $x\in\mathcal{L}$, we have
$\lambda\left(\overline{\Gamma}(x,t)\right)=t^{s_{*}+o(1)}$ as $t\to 0^{+}$,

  3. 3.

for every $x\in\mathbb{R}^{d}$, we have
$t^{s^{*}+o(1)}\leq\lambda\left(\overline{\Gamma}(x,t)\right)\leq
t^{s_{*}-o(1)}$ as $t\to 0^{+}$.

We recall here that $s^{*}=(\gamma-1)d/(\gamma-d)$, and
$s_{*}=s^{*}-(d-1)/(\gamma-d)$. To prove Theorem 3, we derive the upper bound
in item 1 (Proposition 3.2), the lower bound in item 2 (Proposition 3.4), and
both bounds of item 3 (Proposition 3.3 for the lower bound, and Proposition
3.5 for the upper bound). Let us start with the following ¡¡ upper bound for
typical points ¿¿.

######  Proposition 3.2.

Almost surely, for $\lambda$-almost every $x\in\mathbb{R}^{d}$, there exists a
constant $C=C(\omega,x)>0$ such that

$\lambda\left(\overline{\Gamma}(x,t)\right)\leq C\cdot
t^{s^{*}}\cdot\ln(1/t)^{2}\quad\text{for all $t\in{]0,1]}$.}$

In particular, for $\lambda$-almost every $x\in\mathbb{R}^{d}$, we have
$\lambda\left(\overline{\Gamma}(x,t)\right)=t^{s^{*}+o(1)}$ as $t\to 0^{+}$.

Such a statement does not come as a surprise, since the Lebesgue measure is a
mass distribution on $\left(\mathbb{R}^{d},T\right)$ for which the energy
integrals $\left(\iint T(x,y)^{-s}\mathrm{d}y\mathrm{d}x\right)_{0\leq
s<s^{*}}$ are finite (see the discussion in [12, Chapter 8]).

###### Proof.

Recall the equality in distribution (23). We claim that the random variable
$\lambda\left(\overline{\Gamma}(0,1)\right)$ is integrable. Indeed, we have
(by Fubini’s theorem and scaling)

$\mathbb{E}\left[\lambda\left(\overline{\Gamma}(0,1)\right)\right]=\int_{\mathbb{R}^{d}}\mathbb{P}(T(0,y)\leq
1)\mathrm{d}y=\int_{\mathbb{R}^{d}}\mathbb{P}\left(T(0,e_{d})\leq|y|^{-\frac{\gamma-d}{\gamma-1}}\right)\mathrm{d}y,$

and by Theorem 2, there exists a constant $C>0$ such that

$\mathbb{P}\left(T(0,e_{d})\leq|y|^{-\frac{\gamma-d}{\gamma-1}}\right)\leq\frac{C}{|y|^{d+\gamma-2}}\quad\text{for
all $|y|\geq 1$.}$

Now, let $x\in\mathbb{R}^{d}$, and set $t_{n}=2^{-n}$ for all
$n\in\mathbb{N}$. By (23) and Markov’s inequality, we have

$\mathbb{P}\left(\lambda\left(\overline{\Gamma}(x,t_{n})\right)\geq
t_{n}^{s^{*}}\cdot
n^{2}\right)=\mathbb{P}\left(\lambda\left(\overline{\Gamma}(0,1)\right)\geq
n^{2}\right)\leq\frac{\mathbb{E}\left[\lambda\left(\overline{\Gamma}(0,1)\right)\right]}{n^{2}}\quad\text{for
all $n\in\mathbb{N}^{*}$.}$

Therefore, by the Borel-Cantelli lemma and by Fubini’s theorem, we obtain

$\displaystyle\mathbb{E}\left[\lambda\left\\{x\in\mathbb{R}^{d}:\text{$\lambda\left(\overline{\Gamma}(x,t_{n})\right)\geq
t_{n}^{s^{*}}\cdot n^{2}$ for infinitely many $n$}\right\\}\right]$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{d}}\mathbb{P}\left(\text{$\lambda\left(\overline{\Gamma}(x,t_{n})\right)\geq
t_{n}^{s^{*}}\cdot n^{2}$ for infinitely many $n$}\right)\mathrm{d}x$
$\displaystyle=$ $\displaystyle 0.$

Therefore, almost surely, for $\lambda$-almost every $x\in\mathbb{R}^{d}$,
there exists a constant $C=C(\omega,x)>0$ such that

$\lambda\left(\overline{\Gamma}(x,t_{n})\right)\leq C\cdot t_{n}^{s^{*}}\cdot
n^{2}\quad\text{for all $n\in\mathbb{N}^{*}$.}$

The result of the proposition readily follows. ∎

In the other direction, we have the following ¡¡ lower bound for all points
¿¿, as a simple consequence of Proposition 1.3.

######  Proposition 3.3.

Almost surely, for every $R>0$, there exists a positive function $r$
(depending on $\omega$ and $R$), defined over some interval $]0,t_{0}[$ and
satisfying $r(t)=t^{(\gamma-1)/(\gamma-d)+o(1)}$ as $t\to 0^{+}$, such that
for every $t\in{]0,t_{0}[}$, we have

$\overline{\Gamma}(x,t)\supset\overline{B}(x,r(t))\quad\text{for all
$x\in\overline{B}(0,R)$.}$

In particular, for every $x\in\mathbb{R}^{d}$, we have
$\lambda\left(\overline{\Gamma}(x,t)\right)\geq t^{s^{*}+o(1)}$ as $t\to
0^{+}$.

###### Proof.

By Proposition 1.3, almost surely, for every $R>0$, there exists a constant
$C=C(\omega,R)>0$ such that

$T(x,y)\leq
C\cdot|x-y|^{\frac{\gamma-d}{\gamma-1}}\cdot\ln\left(\frac{4R}{|x-y|}\right)^{\frac{1}{\gamma-1}}\quad\text{for
all $x\neq y\in\overline{B}(0,R)$.}$

Now, consider the function $\phi:r\in{]0,2R]}\mapsto
r^{(\gamma-d)/(\gamma-1)}\cdot\ln(4R/r)^{1/(\gamma-1)}$. A straightforward
analysis shows that $\phi$ is strictly increasing over the interval
$]0,r_{0}[$, where $r_{0}=4Re^{1/(\gamma-d)}$; and that if we denote by
$\psi:{]0,\phi(r_{0})[}\rightarrow{]0,r_{0}[}$ its inverse, then we have
$\psi(t)=t^{(\gamma-1)/(\gamma-d)+o(1)}$ as $t\to 0^{+}$. Setting
$r(t)=\psi(t/C)$, so that $C\cdot\phi(r(t))=t$, for $t$ small enough so as to
have $r(t)\leq R/2$, we get $T^{*}_{x,r(t)}\leq t$ for every
$x\in\overline{B}(0,R/2)$, i.e.
$\overline{\Gamma}(x,t)\supset\overline{B}(x,r(t))$. This yields the first
assertion; and the second follows by taking the Lebesgue measure. ∎

The previous ¡¡ lower bound for all points ¿¿ effortlessly yields the
following ¡¡ lower bound for points on roads ¿¿, with the argument of Figure
6.

######  Proposition 3.4.

Almost surely, for every $x\in\mathcal{L}$, we have
$\lambda\left(\overline{\Gamma}(x,t)\right)\geq t^{s_{*}+o(1)}$ as $t\to
0^{+}$.

###### Proof.

Fix a realisation of $\Pi$. Let $(\ell,v)$ be a road of $\Pi$, and let
$x\in\ell$. Let $R>|x|$: by Proposition 3.3, there exists a positive function
$r$, defined over some interval $]0,t_{0}[$ and satisfying
$r(t)=t^{(\gamma-1)/(\gamma-d)+o(1)}$ as $t\to 0^{+}$, such that for every
$t\in{]0,t_{0}[}$, we have $\overline{\Gamma}(x,t)\supset\overline{B}(x,r(t))$
for all $x\in\overline{B}(0,R)$. As explained in Figure 6, we have

$\overline{\Gamma}(x,t)\supset\bigcup_{y\in\ell:|x-y|\leq
vt/2}\overline{\Gamma}\left(y,\frac{t}{2}\right),$

thus

$\overline{\Gamma}(x,t)\supset\bigcup_{y\in\ell:|x-y|\leq
vt/2}\overline{B}\left(y,r\left(\frac{t}{2}\right)\right)\quad\text{for all
sufficiently small $t$.}$

Placing points $(y_{i})_{i\in\mathbb{Z}}$ to be $2r(t/2)$-apart on $\ell$ so
that the balls $\left(\overline{B}(y_{i},r(t/2));~{}i\in\mathbb{Z}\right)$
have disjoint interiors, we obtain

$\lambda\left(\overline{\Gamma}(x,t)\right)\geq\\#\left\\{i\in\mathbb{Z}:|x-y_{i}|\leq\frac{vt}{2}\right\\}\cdot\upsilon_{d}r\left(\frac{t}{2}\right)^{d}\geq\left(\frac{vt}{2r(t/2)}-1\right)\cdot\upsilon_{d}r\left(\frac{t}{2}\right)^{d}\quad\text{for
all sufficiently small $t$.}$

This lower bound is of order $t\cdot r(t/2)^{d-1}=t^{s_{*}+o(1)}$ as $t\to
0^{+}$, concluding the proof. ∎

Finally, to complete the proof of Theorem 3, it remains to prove the following
¡¡ upper bound for all points ¿¿.

######  Proposition 3.5.

Almost surely, for every $x\in\mathbb{R}^{d}$, we have
$\lambda\left(\overline{\Gamma}(x,t)\right)\leq t^{s_{*}-o(1)}$ as $t\to
0^{+}$.

###### Proof.

For positive real numbers $R$ and $t$, consider the random variable
$\Lambda^{R}_{t}=\sup_{x\in\overline{B}(0,R)}\lambda\left(\overline{\Gamma}(x,t)\right)$222In
all rigour, work with the actual $\Pi$-measurable random variable
$\overline{\Lambda}^{R}_{t}=\sup_{x\in\overline{B}(0,R)\cap\mathbb{Q}}\lambda\left(\overline{\Gamma}(x,2t)\right)$,
which upper bounds $\Lambda^{R}_{t}$ by virtue of Fatou’s lemma.. To prove the
proposition, we will show that almost surely, for each $s\in[0,s_{*}[$, we
have $\Lambda^{R}_{t}\leq t^{s}$ for all sufficiently small $t$.

Let $s\in[0,s_{*}[$. By scaling, we have the equality in distribution

$\Lambda^{R}_{t}\overset{\text{\tiny law}}{=}\Lambda^{R^{\prime}}_{1}\cdot
t^{s^{*}},\quad\text{where $R^{\prime}=t^{-\frac{\gamma-1}{\gamma-d}}\cdot
R$.}$

Therefore,

$\mathbb{P}\left(\Lambda^{R}_{t}>t^{s}\right)=\mathbb{P}\left(\Lambda^{R^{\prime}}_{1}>(R/R^{\prime})^{(\gamma-d)(s-s^{*})/(\gamma-1)}\right).$

Since we have $(\gamma-d)(s-s^{*})/(\gamma-1)<-(d-1)/(\gamma-1)$, there exists
$\varepsilon>0$ such that

$\mathbb{P}\left(\Lambda^{R^{\prime}}_{1}>(R/R^{\prime})^{(\gamma-d)(s-s^{*})/(\gamma-1)}\right)\leq\mathbb{P}\left(\Lambda^{R^{\prime}}_{1}>R^{\prime\frac{d-1}{\gamma-1}+\varepsilon}\right)\quad\text{for
all sufficiently large $R$}.$

Now, suppose we show that

$\sum_{n\geq
0}\mathbb{P}\left(\Lambda^{R_{n}}_{1}>R_{n}^{\frac{d-1}{\gamma-1}+\varepsilon}\right)<\infty,\quad\text{where
$R_{n}=2^{n}$ for all $n\in\mathbb{N}$.}$ (27)

Then, going back up the previous reasoning, we get

$\sum_{n\geq
0}\mathbb{P}\left(\Lambda^{R}_{t_{n}}>t_{n}^{s}\right)<\infty,\quad\text{with
$t_{n}=(R\cdot 2^{-n})^{\frac{\gamma-d}{\gamma-1}}$ for all
$n\in\mathbb{N}$,}$

and by the Borel-Cantelli lemma we conclude that almost surely, we have
$\Lambda^{R}_{t_{n}}\leq t_{n}^{s}$ for all sufficiently large $n$. The result
of the proposition readily follows.

In view of the above considerations, let us fix a realisation of $\Pi$, and
consider the following deterministic lemma.

######  Lemma 3.2.

Let $\varepsilon\in{]0,(\gamma-d)/(\gamma-1)[}$, set
$\alpha_{i}=(1-\varepsilon)^{i}\cdot((d-1)/(\gamma-1)+\varepsilon)$ for all
$i\in\mathbb{N}$, and let $j\in\mathbb{N}^{*}$ be large enough so that
$\alpha_{j}d\leq\varepsilon$. Assume that there exists
$N_{0},\ldots,N_{j-1}\in\mathbb{N}$ such that for every $R\geq 1$, the
following holds, for each $i\in\llbracket-1,j\llbracket$:

  1. $(i=-1)$

$V^{0}_{2R}\leq R^{(d-1)/(\gamma-1)+\varepsilon}$,

  2. $(i=0)$

for every $x\in\overline{B}(0,R)$, there are at most $N_{0}$ roads of
$\Pi_{R^{\alpha_{1}}}$ that pass through the ball
$\overline{B}(x,R^{\alpha_{0}})$,

  3. $(1\leq i<j)$

for every $x\in\overline{B}(0,R+R^{\alpha_{0}}+\ldots+R^{\alpha_{i-1}})$,
there are at most $N_{i}$ roads of $\Pi_{R^{\alpha_{i+1}}}$ that pass through
the ball $\overline{B}(x,2R^{\alpha_{i}})$.

Then, there exists a constant $C>0$ such that

$\Lambda^{R}_{1}\leq C\cdot
R^{\frac{d-1}{\gamma-1}+2\varepsilon}\quad\text{for every $R\geq 1$}.$

To prove (27), it will then remain to show that the assumptions of the lemma
are satisfied with very high probability.

###### Proof.

Let $R\geq 1$. The idea is that the information of items
$(i,~{}i\in\llbracket-1,j\llbracket)$ bootstraps into the right upper bound
for $\Lambda^{R}_{1}$.

Fix $x\in\overline{B}(0,R)$: we wish to upper bound
$\lambda\left(\overline{\Gamma}(x,1)\right)$.

  1. $(i=-1)$

First, as $V^{0}_{2R}\leq R^{(d-1)/(\gamma-1)+\varepsilon}\leq R$, the
$T$-ball $\overline{\Gamma}(x,1)$ must be included in $\overline{B}(0,2R)$.
Then, since the speed of $\Pi$-paths inside $\overline{B}(0,2R)$ cannot exceed
$R^{(d-1)/(\gamma-1)+\varepsilon}=R^{\alpha_{0}}$, actually
$\overline{\Gamma}(x,1)$ is included in $\overline{B}(x,R^{\alpha_{0}})$.

  2. $(i=0)$

By assumption, there are at most $N_{0}$ roads of $\Pi_{R^{\alpha_{1}}}$ that
pass through the ball $\overline{B}(x,R^{\alpha_{0}})$: let us denote by
$\mathcal{L}_{\varnothing}$ the corresponding set of lines. Note that it could
be the case that no road of $\Pi_{R^{\alpha_{1}}}$ passes through $x$: to
allow for the following argument, let us fix an arbitrary road
$\left(\ell_{0},R^{\alpha_{0}}\right)$ that passes through $x$, and add
$\ell_{0}$ to $\mathcal{L}_{\varnothing}$. Now, for each line
$\ell\in\mathcal{L}_{\varnothing}$, we place points to be $2R^{\alpha_{1}}$
apart on $\ell\cap\overline{B}(x,R^{\alpha_{0}})$, so that the Euclidean balls
with radius $R^{\alpha_{1}}$ around those points cover the line segment. Let
us denote _all_ these points by $x_{1},\ldots,x_{\kappa(\varnothing)}$: we
have

$\kappa(\varnothing)\leq\\#\mathcal{L}_{\varnothing}\cdot\left(\frac{R^{\alpha_{0}}}{R^{\alpha_{1}}}+1\right)\leq(N_{0}+1)\cdot
2R^{\varepsilon\alpha_{0}},$

and the balls $\left(\overline{B}(x_{k},2R^{\alpha_{1}});~{}k\in\llbracket
1,\kappa(\varnothing)\rrbracket\right)$ cover $\overline{\Gamma}(x,1)$.
Indeed, given $y\in\overline{\Gamma}(x,1)$, denote by $z_{0}$ the last exit
point from $\bigcup_{\ell\in\mathcal{L}_{\varnothing}}\ell$ of the
$\Pi$-geodesic $\xi$ that connects $x$ to $y$ (this $z_{0}$ would be ill-
defined if we had not added $\ell_{0}$ to $\mathcal{L}_{\varnothing}$). As an
element of
$\bigcup_{\ell\in\mathcal{L}_{\varnothing}}\ell\cap\overline{B}(x,R^{\alpha_{0}})$,
the point $z_{0}$ lies in the ball $\overline{B}(x_{k_{0}},R^{\alpha_{1}})$
for some $k_{0}\in\llbracket 1,\kappa(\varnothing)\rrbracket$. Since the speed
of $\xi$ cannot exceed $R^{\alpha_{1}}$ between $z_{0}$ and $y$, the point $y$
must lie in
${\overline{B}(z_{0},R^{\alpha_{1}})\subset\overline{B}(x_{k_{0}},2R^{\alpha_{1}})}$.

  3. $(i=1)$

For each $u\in\llbracket 1,\kappa(\varnothing)\rrbracket$, we proceed as
follows. As $x_{u}\in\overline{B}(0,R+R^{\alpha_{0}})$, by assumption there
are at most $N_{1}$ roads with speed limit greater than or equal to
$R^{\alpha_{2}}$ that pass through the ball
$\overline{B}(x_{u},2R^{\alpha_{1}})$: let us denote by $\mathcal{L}_{u}$ the
corresponding set of lines. For each line $\ell\in\mathcal{L}_{u}$, we place
points to be $2R^{\alpha_{2}}$ apart on
$\ell\cap\overline{B}(x_{u},2R^{\alpha_{1}})$, so that the Euclidean balls
with radius $R^{\alpha_{2}}$ around those points cover the line segment. Let
us denote _all_ these points by $x_{u1},\ldots,x_{u\kappa(u)}$ (where $uk$ is
to be understood as the concatenation of the word $u$ with the letter $k$). We
have

$\kappa(u)\leq\\#\mathcal{L}_{u}\cdot\left(\frac{2R^{\alpha_{1}}}{R^{\alpha_{2}}}+1\right)\leq
N_{1}\cdot 3R^{\varepsilon\alpha_{1}},$

and the balls
$\left(\overline{B}\left(x_{uk},2R^{\alpha_{2}}\right);~{}u\in\llbracket
1,\kappa(\varnothing)\rrbracket,k\in\llbracket 1,\kappa(u)\rrbracket\right)$
cover $\overline{\Gamma}(x,1)$. Indeed, let $y\in\overline{\Gamma}(x,1)$. We
know that $z_{0}$, the last exit point from the set
$\bigcup_{\ell\in\mathcal{L}_{\varnothing}}\ell$ of the $\Pi$-geodesic $\xi$
connecting $x$ to $y$, lies in the ball $\overline{B}(x_{u},R^{\alpha_{1}})$
for some $u\in\llbracket 1,\kappa(\varnothing)\rrbracket$. Moreover, by the
definition of $z_{0}$, the speed of $\xi$ cannot exceed $R^{\alpha_{1}}$
between $z_{0}$ and $y$. Then, denote by $z_{1}$ the last exit point from
$\bigcup_{\ell\in\mathcal{L}_{u}}\ell$ of the $\Pi$-geodesic segment between
$z_{0}$ and $y$. As an element of
$\bigcup_{\ell\in\mathcal{L}_{u}}\ell\cap\overline{B}(x_{u},2R^{\alpha_{1}})$,
the point $z_{1}$ lies in the ball $\overline{B}(x_{uk},R^{\alpha_{2}})$ for
some $k\in\llbracket 1,\kappa(u)\rrbracket$. Finally, since the speed of $\xi$
cannot exceed $R^{\alpha_{2}}$ between $z_{1}$ and $y$, the point $y$ must lie
in
${\overline{B}(z_{1},R^{\alpha_{2}})\subset\overline{B}(x_{uk},2R^{\alpha_{2}})}$.

  4. $\ldots$

Continuing like this by induction, we construct a plane tree $\mathbb{T}$ in
which the root $\varnothing$ has $\kappa(\varnothing)$ children
$1,\ldots,\kappa(\varnothing)$, each $u\in\llbracket
1,\kappa(\varnothing)\rrbracket$ has $\kappa(u)$ children
$u1,\ldots,u\kappa(u)$, etc. By construction, for each $i\in\llbracket
1,j\rrbracket$, the balls
$\left(B(x_{u},2R^{\alpha_{i}});~{}u\in\mathbb{T}:|u|=i\right)$ cover
$\overline{\Gamma}(x,1)$.

Since the balls
$\left(B(x_{u},2R^{\alpha_{j}});~{}u\in\mathbb{T}:|u|=j\right)$ cover
$\overline{\Gamma}(x,1)$, we have

$\begin{split}\lambda\left(\overline{\Gamma}(x,1)\right)&\leq\\#\\{u\in\mathbb{T}:|u|=j\\}\cdot\upsilon_{d}(2R^{\alpha_{j}})^{d}\\\
&\leq(N_{0}+1)\cdot 2R^{\varepsilon\alpha_{0}}\cdot N_{1}\cdot
3R^{\varepsilon\alpha_{1}}\cdot\ldots\cdot N_{j-1}\cdot
3R^{\varepsilon\alpha_{j-1}}\cdot\upsilon_{d}(2R^{\alpha_{j}})^{d}\\\
&=:C\cdot
R^{\varepsilon(\alpha_{0}+\ldots+\alpha_{j-1})+\alpha_{j}d}.\end{split}$

This bound is uniform in $x\in\overline{B}(0,R)$, and the constant $C$ does
not depend on $R$. The result of the lemma follows, since

$\varepsilon(\alpha_{0}+\ldots+\alpha_{j-1})+\alpha_{j}d\leq\varepsilon\sum_{i\geq
0}(1-\varepsilon)^{i}\cdot\left(\frac{d-1}{\gamma-1}+\varepsilon\right)+\varepsilon=\frac{d-1}{\gamma-1}+2\varepsilon.$

∎

Let $R_{n}=2^{n}$ for all $n\in\mathbb{N}$. Note that we may assume without
loss of generality that $\varepsilon$ in (27) is less than
${(\gamma-d)/(\gamma-1)}$. To prove (27), in view of the previous lemma, it
remains to find integers $N_{0},\ldots,N_{j-1}$ for which the assumptions of
the lemma are satisfied, on events whose complements have summable
probability. For each $i\in\llbracket 0,j\llbracket$, fix $N_{i}\in\mathbb{N}$
large enough so that
$(1-\alpha_{i})d-\alpha_{i}((1-\varepsilon)(\gamma-1)-(d-1))(N_{i}+1)<0$. We
show below that there exists constants $C_{i},c_{i}>0$ such that item $(i)$ of
Lemma 3.2 holds with probability at least $1-C_{i}2^{-c_{i}n}$. This will
complete the proof of the proposition.

  1. $(i=-1)$

Using the inequality $\mathbb{P}(\mathrm{Poisson}(\lambda)>0)\leq\lambda$, we
get

$\mathbb{P}\left(V^{0}_{2R_{n}}>R_{n}^{\frac{d-1}{\gamma-1}+\varepsilon}\right)\leq(2R_{n})^{d-1}\cdot
R_{n}^{-((d-1)+\varepsilon(\gamma-1))}=2^{d-1}\cdot
2^{-\varepsilon(\gamma-1)n}=:C_{-1}\cdot 2^{-c_{-1}\cdot n}\quad\text{for all
$n\in\mathbb{N}$.}$

  2. $(i=0)$

For each $n\in\mathbb{N}$, denote by
$\left(\overline{B}(x,R_{n}^{\alpha_{0}});~{}x\in\mathcal{X}_{0}\right)$ a
covering of $\overline{B}(0,R_{n})$ by balls of radius $R_{n}^{\alpha_{0}}$,
with centres ${x\in\overline{B}(0,R_{n})}$ more than $R_{n}^{\alpha_{0}}$
apart from each other. We have $\\#\mathcal{X}_{0}\leq\left(2R_{n}\cdot
R_{n}^{-\alpha_{0}}+1\right)^{d}\leq\left(3R_{n}^{1-\alpha_{0}}\right)^{d}$.
Now, suppose that there exists an $x\in\overline{B}(0,R_{n})$ for which more
than $N_{0}$ roads of $\Pi_{R_{n}^{\alpha_{1}}}$ pass through the ball
$\overline{B}(x,R_{n}^{\alpha_{0}})$. The point $x$ lies in the ball
$\overline{B}(x^{\prime},R_{n}^{\alpha_{0}})$ for some
$x^{\prime}\in\mathcal{X}_{0}$, and the Poisson random variable
$\Pi\left(\left[\overline{B}(x^{\prime},2R_{n}^{\alpha_{0}})\right]\times[R_{n}^{\alpha_{1}},\infty[\right)$
then exceeds $N_{0}$. By a union bound we get, using the inequality
${\mathbb{P}\left(\mathrm{Poisson}(\lambda)>N_{0}\right)\leq\frac{\lambda^{N_{0}+1}}{(N_{0}+1)!}}$:

$\begin{split}\mathbb{P}\left(\exists
x^{\prime}\in\mathcal{X}_{0}:\Pi\left(\left[\overline{B}(x^{\prime},2R_{n}^{\alpha_{0}})\right]\times[R_{n}^{\alpha_{1}},\infty[\right)>N_{0}\right)&\leq\left(3R_{n}^{1-\alpha_{0}}\right)^{d}\cdot\frac{\left[(2R_{n}^{\alpha_{0}})^{d-1}\cdot
R_{n}^{-\alpha_{1}(\gamma-1)}\right]^{N_{0}+1}}{(N_{0}+1)!}\\\ &=:C_{0}\cdot
R_{n}^{(1-\alpha_{0})d+(\alpha_{0}(d-1)-\alpha_{1}(\gamma-1))(N_{0}+1)}.\end{split}$

The constant $C_{0}$ does not depend on $n$, and by the definition of $N_{0}$,
the exponent

$(1-\alpha_{0})d+(\alpha_{0}(d-1)-\alpha_{1}(\gamma-1))(N_{0}+1)=(1-\alpha_{0})d-\alpha_{0}((1-\varepsilon)(\gamma-1)-(d-1))(N_{0}+1)=:-c_{0}$

is negative.

  3. $(1\leq i<j)$

For each $n\in\mathbb{N}$, denote by
$\left(\overline{B}(x,R_{n}^{\alpha_{i}});~{}x\in\mathcal{X}_{i}\right)$ a
covering of
$\overline{B}\left(0,R_{n}+R_{n}^{\alpha_{0}}+\ldots+R_{n}^{\alpha_{i-1}}\right)$
by balls of radius $R_{n}^{\alpha_{i}}$, with centres
${x\in\overline{B}\left(0,R_{n}+R_{n}^{\alpha_{0}}+\ldots+R_{n}^{\alpha_{i-1}}\right)}$
more than $R_{n}^{\alpha_{i}}$ apart from each other. We have

$\\#\mathcal{X}_{i}\leq\left(2\cdot\frac{R_{n}+R_{n}^{\alpha_{0}}+\ldots+R_{n}^{\alpha_{i-1}}}{R_{n}^{\alpha_{i}}}+1\right)^{d}\leq\left((2i+3)R_{n}^{1-\alpha_{i}}\right)^{d}.$

Now, suppose that there exists an
$x\in\overline{B}\left(0,R_{n}+R_{n}^{\alpha_{0}}+\ldots+R_{n}^{\alpha_{i-1}}\right)$
for which more than $N_{i}$ roads of $\Pi_{R_{n}^{\alpha_{i+1}}}$ pass through
the ball $\overline{B}(x,2R_{n}^{\alpha_{i}})$. The point $x$ lies in the ball
$\overline{B}(x^{\prime},R_{n}^{\alpha_{i}})$ for some
$x^{\prime}\in\mathcal{X}_{i}$, and the Poisson random variable
$\Pi\left(\left[\overline{B}(x^{\prime},3R_{n}^{\alpha_{i}})\right]\times\left[R_{n}^{\alpha_{i+1}},\infty\right[\right)$
then exceeds $N_{i}$. By a union bound we get, using the inequality
$\mathbb{P}\left(\mathrm{Poisson}(\lambda)>N_{i}\right)\leq\frac{\lambda^{N_{i}+1}}{(N_{i}+1)!}$:

$\begin{split}\mathbb{P}\left(\exists
x^{\prime}\in\mathcal{X}_{i}:\Pi\left(\left[\overline{B}(x,3R_{n}^{\alpha_{i}})\right]\times\left[R_{n}^{\alpha_{i+1}},\infty\right[\right)>N_{i}\right)&\leq\left((2i+3)R_{n}^{1-\alpha_{i}}\right)^{d}\cdot\frac{\left[(3R_{n}^{\alpha_{i}})^{d-1}\cdot
R_{n}^{-\alpha_{i+1}(\gamma-1)}\right]^{N_{i}+1}}{(N_{i}+1)!}\\\ &=:C_{i}\cdot
R_{n}^{(1-\alpha_{i})d+(\alpha_{i}(d-1)-\alpha_{i+1}(\gamma-1))(N_{i}+1)}.\end{split}$

The constant $C_{i}$ does not depend on $n$, and by the definition of $N_{i}$,
the exponent

$(1-\alpha_{i})d+(\alpha_{i}(d-1)-\alpha_{i+1}(\gamma-1))(N_{i}+1)=(1-\alpha_{i})d-\alpha_{i}((1-\varepsilon)(\gamma-1)-(d-1))(N_{i}+1)=:-c_{i}$

is negative.

∎

##  4 Discussion

To conclude this paper, we mention and discuss some natural problems our
results leave open.

#### Gauge function and exact Hausdorff measure.

In our analysis, we used the Lebesgue measure as mass distribution on
$\left(\mathbb{R}^{d},T\right)$. Question: is the Lebesgue measure a Hausdorff
measure for some gauge function?

#### Complete description of the multifractal spectrum.

The results of Theorem 3 are short of describing completely the local
behaviour around all points in the measured metric space
$\left(\mathbb{R}^{d},T,\lambda\right)$. Question: can one describe the
behaviour of $\lambda\left(\overline{\Gamma}(x,t)\right)$ as $t\to 0^{+}$ for
all $x\in\mathbb{R}^{d}$?

In the following, we make a modest first step towards that by considering the
simpler question: ¡¡ are there points $x$ other than points on roads for which
$\lambda\left(\overline{\Gamma}(x,t)\right)=t^{s+o(1)}$ as $t\to 0^{+}$, with
an exponent $s<s^{*}$? ¿¿. Note that if we could answer no to the latter, then
with Theorem 3 we would have the complete description mentioned above.

######  Remark 3.

One of our original motivations with this question was to recover the roads of
the process $\Pi$, given the measured metric space
$\left(\mathbb{R}^{d},T,\lambda\right)$. Because of Proposition 4.2 below, it
turns out that points on roads cannot in general be identified as the only
points having untypically big $T$-balls around them. However, the road network
may be recovered from the metric space $\left(\mathbb{R}^{d},T\right)$ through
the speed-limit function $V$, as follows: almost surely, for every
$x\in\mathbb{R}^{d}$, we have

$\varliminf_{\begin{subarray}{c}y\to x\\\ y\neq
x\end{subarray}}\frac{T(x,y)}{|x-y|}=\frac{1}{V(x)}\in[0,\infty].$

In the right direction for a negative answer to the last question, we have the
following.

######  Proposition 4.1.

Almost surely, the set

$\bigcup_{s<s^{*}}\left\\{x\in\mathbb{R}^{d}:\text{$\lambda\left(\overline{\Gamma}(x,t)\right)\geq
t^{s}$ for all sufficiently small $t$}\right\\}$

has Euclidean Hausdorff dimension $1$.

###### Proof.

By Theorem 3, this set contains $\mathcal{L}$, and thus has Euclidean
Hausdorff dimension at least $1$. Now consider the upper bound. It suffices to
prove that for each $s<s^{*}$ and every $R>0$, almost surely the set

$\left\\{x\in\overline{B}(0,R):\text{$\lambda\left(\overline{\Gamma}(x,t)\right)\geq
t^{s}$ for all sufficiently small $t$}\right\\}$

has Euclidean Hausdorff dimension at most $1$ (note that by Theorem 3, this
set is empty for $s<s_{*}$). The idea is that, if $T$-balls with small radius
around $x\in\mathbb{R}^{d}$ are untypically big, then there must be
untypically fast roads passing near $x$ at all scales, and we can control the
probability of this happening with the multiscale lemma. More precisely now,
let $x\in\mathbb{R}^{d}$, and suppose that there exists an $s\in[s_{*},s^{*}[$
such that $\lambda\left(\overline{\Gamma}(x,t)\right)\geq t^{s}$ for all
sufficiently small $t$. Fixing $s^{\prime}\in{]s,s^{*}[}$, set
$r(t)=t^{s^{\prime}/d}$, so that $\upsilon_{d}r(t)^{d}<t^{s}$ for all
sufficiently small $t$. For those $t$, the $T$-ball $\overline{\Gamma}(x,t)$
cannot be contained in $\overline{B}(x,r(t))$, hence we must have
$V^{x}_{r(t)}\geq r(t)/t=r(t)^{\alpha}$, where
$\alpha=1-d/s^{\prime}\in{]0,(d-1)/(\gamma-1)[}$. To conclude, it suffices to
prove that for each $\alpha\in{]0,(d-1)/(\gamma-1)[}$, almost surely, the
Euclidean Hausdorff dimension of the set

$\left\\{x\in\overline{B}(0,R):\text{$V^{x}_{r}\geq r^{\alpha}$ for all
sufficiently small $r$}\right\\}$

is at most $1$. For each $k\in\mathbb{N}$, set $\rho_{k}=2^{-k}$, and let
$\left(\overline{B}(x,\rho_{k});~{}x\in\mathcal{X}_{k}\right)$ be a covering
of $\overline{B}(0,R)$ by balls of radius $\rho_{k}$, with centres
$x\in\overline{B}(0,R)$ more than $\rho_{k}$ apart from each other. In
particular, we have ${\\#\mathcal{X}_{k}\leq(2R/\rho_{k}+1)^{d}}$. Fix
$n\in\mathbb{N}$. For each $K\geq n$, the balls
$\left(\overline{B}(x,\rho_{K});~{}x\in\mathcal{X}_{K}:\text{$V^{x}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right)$ cover the set

$\left\\{x\in\overline{B}(0,R):\text{$V^{x}_{r}\geq r^{\alpha}$ for all
$r\in{]0,\rho_{n}]}$}\right\\}.$

Indeed, each element $x$ of that set lies in the ball
$\overline{B}(x^{\prime},\rho_{K})$ for some $x^{\prime}\in\mathcal{X}_{K}$,
and for every $k\in\llbracket n,K\rrbracket$ we have
$V^{x^{\prime}}_{2\rho_{k}}\geq V^{x}_{\rho_{k}}\geq\rho_{k}^{\alpha}$.
Therefore, the $1$-dimensional Euclidean Hausdorff measure of the above set is
upper bounded by

$\varliminf_{K\to\infty}\\#\left\\{x\in\mathcal{X}_{K}:\text{$V^{x}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right\\}\cdot 2\rho_{K}=:H.$

We wish to see that $H$ is almost surely finite. By Fatou’s lemma, we have

$\mathbb{E}[H]\leq\varliminf_{K\to\infty}\mathbb{E}\left[\\#\left\\{x\in\mathcal{X}_{K}:\text{$V^{x}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right\\}\cdot 2\rho_{K}\right],$

with

$\mathbb{E}\left[\\#\left\\{x\in\mathcal{X}_{K}:\text{$V^{x}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right\\}\cdot
2\rho_{K}\right]=\\#\mathcal{X}_{K}\cdot\mathbb{P}\left(\text{$V^{0}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right)\cdot 2\rho_{K}$

for each $K\geq n$. Using Proposition 2.2 with
${(r_{k}=2\rho_{k})_{k\in\mathbb{N}}}$ and
${(v_{k}=\rho_{k}^{\alpha})_{k\in\mathbb{N}}}$, we obtain

$\begin{split}\mathbb{P}\left(\text{$V^{0}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket
n,K\rrbracket$}\right)&\leq(2\rho_{K})^{d-1}\cdot\rho_{n}^{-\alpha(\gamma-1)}\cdot\exp\left[\sum_{k=n+1}^{K}(2\rho_{k-1})^{d-1}\cdot\rho_{k}^{-\alpha(\gamma-1)}\right]\\\
&\leq(2\rho_{K})^{d-1}\cdot\rho_{n}^{-\alpha(\gamma-1)}\cdot\exp\left[4^{d-1}\cdot\sum_{k\geq
0}\rho_{k}^{(d-1)-\alpha(\gamma-1)}\right].\end{split}$

As $(d-1)-\alpha(\gamma-1)>0$, we have $\sum_{k\geq
0}\rho_{k}^{(d-1)-\alpha(\gamma-1)}<\infty$, and we get

$\mathbb{P}\left(\text{$V^{0}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$ for all
$k\in\llbracket n,K\rrbracket$}\right)\leq C\cdot\rho_{K}^{d-1},$

where the constant $C$ does not depend on $K$. Thus, we obtain

$\\#\mathcal{X}_{K}\cdot\mathbb{P}\left(\text{$V^{0}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right)\cdot
2\rho_{K}\leq(2R/\rho_{K}+1)^{d}\cdot C\cdot\rho_{K}^{d-1}\cdot
2\rho_{K}\leq(2R+\rho_{K})^{d}\cdot C\cdot 2,$

and it follows that

$\mathbb{E}[H]\leq\varliminf_{K\to\infty}\\#\mathcal{X}_{K}\cdot\mathbb{P}\left(\text{$V^{x}_{2\rho_{k}}\geq\rho_{k}^{\alpha}$
for all $k\in\llbracket n,K\rrbracket$}\right)\cdot 2\rho_{K}\leq(2R)^{d}\cdot
C\cdot 2<\infty.$

∎

The result of the previous proposition does not imply though that the set

$\bigcup_{s<s^{*}}\left\\{x\in\mathbb{R}^{d}:\text{$\lambda\left(\overline{\Gamma}(x,t)\right)\geq
t^{s}$ for all sufficiently small $t$}\right\\}$

is reduced to $\mathcal{L}$. In fact it is not, at least in the planar case.

######  Proposition 4.2.

Almost surely, there exists
$x\in\left.\mathbb{R}^{2}\middle\backslash\mathcal{L}\right.$ such that
$\lambda\left(\overline{\Gamma}(x,t)\right)\geq t^{s_{*}+o(1)}$ as $t\to
0^{+}$.

###### Proof.

It suffices to prove that there exists
$x\in\left.\overline{B}(0,1)\middle\backslash\mathcal{L}\right.$ such that
${V^{x}_{r}\geq\ln(1/r)^{-1}}$ for all sufficiently small $r$. Indeed, suppose
this has been done. By Proposition 3.3, there exists a positive function $r$,
defined over some interval $]0,t_{0}[$ and satisfying
$r(t)=t^{(\gamma-1)/(\gamma-d)+o(1)}$ as $t\to 0^{+}$, such that for every
$t\in{]0,t_{0}[}$, we have

$\overline{\Gamma}(y,t)\supset\overline{B}(y,r(t))\quad\text{for all
$y\in\overline{B}(0,2)$.}$

For all sufficiently small $t$, we have both
$\Gamma(x,t/2)\supset\overline{B}(0,r(t/2))$ and
$V^{x}_{r(t/2)}\geq\ln(1/r(t/2))^{-1}$. In particular, the ball
$\overline{B}(x,r(t/2))$ is traversed by a road $(\ell,v)\in\Pi$ with speed
limit $v\geq\ln(1/r(t/2))^{-1}$. Denoting by
$x^{\prime}\in\overline{B}(x,r(t/2))$ the orthogonal projection of $x$ onto
the line $\ell$, we have $T(x,x^{\prime})\leq t/2$, and thus

$\overline{\Gamma}(x,t)\supset\overline{\Gamma}(x^{\prime},t/2)\supset\bigcup_{y\in\ell:|x^{\prime}-y|\leq
vt/2}\overline{\Gamma}\left(y,\frac{t}{2}\right)\supset\bigcup_{y\in\ell:|x^{\prime}-y|\leq
vt/2}\overline{B}\left(y,r\left(\frac{t}{2}\right)\right)\quad\text{for all
sufficiently small $t$.}$

Placing points $(y_{i})_{i\in\mathbb{Z}}$ to be $2r(t/2)$-apart on $\ell$ so
that the balls $\left(\overline{B}(y_{i},r(t/2));~{}i\in\mathbb{Z}\right)$
have disjoint interiors, we get

$\lambda\left(\bigcup_{y\in\ell:|x^{\prime}-y|\leq
vt/2}\overline{B}\left(y,r\left(\frac{t}{2}\right)\right)\right)\geq\\#\left\\{i\in\mathbb{Z}:|x^{\prime}-y_{i}|\leq\frac{vt}{2}\right\\}\cdot\upsilon_{d}r\left(\frac{t}{2}\right)^{d}\geq\left(\frac{vt}{2r(t/2)}-1\right)\cdot\upsilon_{d}r\left(\frac{t}{2}\right)^{d}.$

We thus obtain

$\lambda\left(\overline{\Gamma}(x,t)\right)\geq\left(\frac{\ln(1/r(t/2))^{-1}\cdot
t}{2r(t/2)}-1\right)\cdot\upsilon_{d}r\left(\frac{t}{2}\right)^{d}\quad\text{for
all sufficiently small $t$,}$

which achieves to show that $\lambda\left(\overline{\Gamma}(x,t)\right)\geq
t^{s_{*}+o(1)}$ as $t\to 0^{+}$.

Now, we turn to the proof of the claim. The idea, to obtain a point
$x\in\left.\overline{B}(0,1)\middle\backslash\mathcal{L}\right.$ such that
${V^{x}_{r}\geq\ln(1/r)^{-1}}$ for all sufficiently small $r$, is to construct
it using a compactness argument, forcing it to be close enough to fast roads,
yet away from them (we are indebted to Jonas Kahn for the birth of this idea).
Denote by $(\ell_{1},v_{1});(\ell_{2},v_{2});\ldots$ the roads of $\Pi$
passing through the ball $\overline{B}(0,1)$, with $v_{1}>v_{2}>\ldots$. The
following steps are represented in Figure 7.

|
---|---
Figure 7: Left: the line segment $s_{1}$, the line $\ell_{n_{2}}$, the
rectangle $R_{1}$ and the line segment $s_{2}$. Right: zooming in on the
rectangle $R_{1}$, we represent the line segment $s_{2}$, the line
$\ell_{n_{3}}$, also $\ell_{2}$ which is potentially there, then the rectangle
$R_{2}$ and finally the line segment $s_{3}$.

  1. $(k=1)$

Let $n_{1}=1$. Almost surely, the line $\ell_{1}$ intersects the interior of
the ball $\overline{B}(0,1)$, and we can fix a segment $s_{1}$ of this line
that is included in that interior.

  2. $(k=2)$

Let $n_{2}=\min\left\\{n>1:\text{$\ell_{n}$ meets $s_{1}$}\right\\}$ be the
index of the next line hitting $s_{1}$, and define $R_{1}$ as a rectangle
along that line segment, with half-side length
$0<\delta_{1}\leq\exp\left[-v_{n_{2}}^{-1}\right]$ small enough so that
$R_{1}\subset\overline{B}(0,1)$. Zooming in on the rectangle $R_{1}$, we can
fix a segment $s_{2}$ of the line $\ell_{n_{2}}$ that: (a) is included in the
interior of $R_{1}$, and (b) avoids $\ell_{1}$.

  3. $(k=3)$

Let $n_{3}=\min\left\\{n>n_{2}:\text{$\ell_{n}$ meets $s_{2}$}\right\\}$ be
the index of the next line hitting $s_{2}$, and define $R_{2}$ as a rectangle
along that line segment, with half-side length
$0<\delta_{2}\leq\exp\left[-v_{n_{3}}^{-1}\right]$ small enough so as to have
$R_{2}$: (a) included in the rectangle $R_{1}$, and (b) avoiding the line
$\ell_{1}$. Zooming in on the rectangle $R_{2}$, fix a segment $s_{3}$ of the
line $\ell_{n_{3}}$ that: (a) is included in the interior of $R_{2}$, and (b)
avoids the lines $\ell_{n_{2}}$ and $\ell_{2}$ (possibly $n_{2}$ differs from
$2$, and we want $s_{3}$ to avoid the line $\ell_{2}$).

  4. $\ldots$

Repeating the procedure inductively, we construct rectangles $R_{1}\supset
R_{2}\supset\ldots$ such that, for each $k\in\mathbb{N}^{*}$, the rectangle
$R_{k}$ avoids the lines $\ell_{1},\ldots,\ell_{k-1}$, and any point $x\in
R_{k}$ is at distance at most
$\delta_{k}\leq\exp\left[-v_{n_{k+1}}^{-1}\right]$ from $\ell_{n_{k}}$.

By compactness, the intersection $\bigcap_{k\geq 1}R_{k}$ is non-empty (see,
e.g, [13, special case after Theorem 26.9]): let $x$ be an element of that
intersection. By construction, the point $x$ belongs to $\overline{B}(0,1)$,
does not lie on any of the lines $\ell_{1},\ell_{2},\ldots$, and satisfies
$d(x,\ell_{n_{k}})\leq\exp\left[-v_{n_{k+1}}^{-1}\right]$ for all
$k\in\mathbb{N}^{*}$. Thus, the point $x$ is not on a road, and we claim that
$V^{x}_{r}\geq\ln(1/r)^{-1}$ for all sufficiently small $r$. Indeed, for
$r\in\left]0,\exp\left[-v_{1}^{-1}\right]\right]$, taking $k\in\mathbb{N}^{*}$
such that $\exp\left[-v_{n_{k+1}}^{-1}\right]\leq
r\leq\exp\left[-v_{n_{k}}^{-1}\right]$, we have
$d(x,\ell_{n_{k}})\leq\exp\left[-v_{n_{k+1}}^{-1}\right]\leq r$, hence
$V^{x}_{r}\geq v_{n_{k}}\geq\ln(1/r)^{-1}$. ∎

## Appendix

###### Proof of Lemma 1.1.

Although the proof we present here is based on covering arguments, the result
can be obtained by direct computations, with the description of $\mu_{d}$ as a
pushforward measure recalled in Section 1. Let $x\neq y\in\mathbb{R}^{d}$ and
$r,s>0$, and without loss of generality assume $r\geq s$.

Lower bound. For $r\leq|x-y|$, let
$\left(\overline{B}(y^{\prime},s)\right)_{y^{\prime}\in\mathcal{Y}}$ be a
covering of $\partial B(x,|x-y|)$ by balls of radius $s$, with centres
$y^{\prime}\in\partial B(x,|x-y|)$ more than $s$ apart from each other. Since
the balls
$\left(\overline{B}(y^{\prime},s/2);~{}y^{\prime}\in\mathcal{Y}\right)$ are
disjoint and included in the annulus
$\left\\{z\in\mathbb{R}^{d}:|x-y|-s/2\leq|x-z|\leq|x-y|+s/2\right\\}$, we have
$\\#\mathcal{Y}\cdot\upsilon_{d}(s/2)^{d}\leq\upsilon_{d}\cdot
d(|x-y|+s/2)^{d-1}\cdot s$, i.e, since $s\leq r\leq|x-y|$:

$\\#\mathcal{Y}\leq\left(\frac{2}{s}\right)^{d}\cdot
d\left(\frac{3|x-y|}{2}\right)^{d-1}\cdot
s=:C\cdot\frac{|x-y|^{d-1}}{s^{d-1}}.$

Integrating

$\mathbf{1}\left(\text{$\ell$ hits
$\overline{B}(x,r)$}\right)\leq\frac{\\#\left\\{y^{\prime}\in\mathcal{Y}:\text{$\ell$
hits $\overline{B}(x,r)$ and
$\overline{B}(y^{\prime},s)$}\right\\}}{2}\quad\text{for all
$\ell\in\mathbb{L}_{d}$}$

with respect to $\mu_{d}$ yields, using invariance:

$\upsilon_{d-1}r^{d-1}\leq\frac{\\#\mathcal{Y}\cdot\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]}{2}.$

We get

$\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]\geq\frac{2\upsilon_{d-1}}{C}\cdot\frac{r^{d-1}\cdot
s^{d-1}}{|x-y|^{d-1}}.$

Upper bound. If $r\geq|x-y|$, then we have

$\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]\leq\mu_{d}\left[\overline{B}(y,s)\right]=\upsilon_{d-1}s^{d-1}\leq\upsilon_{d-1}\cdot\frac{r^{d-1}\cdot
s^{d-1}}{|x-y|^{d-1}}.$

In the case $r\leq|x-y|$, let $\mathcal{Y}$ be a maximal $3s$-separated subset
of $\partial B(x,3|x-y|)$. Since the balls
$\left(\overline{B}(y^{\prime},3s);~{}y^{\prime}\in\mathcal{Y}\right)$ cover
$\partial B(x,3|x-y|)$, the balls
$\left(\overline{B}(y^{\prime},4s);~{}y^{\prime}\in\mathcal{Y}\right)$ cover
the annulus $\left\\{z\in\mathbb{R}^{d}:3|x-y|-s\leq|x-z|\leq
3|x-y|+s\right\\}$, and we have
$\\#\mathcal{Y}\cdot\upsilon_{d}(4s)^{d}\geq\upsilon_{d}\cdot
d(3|x-y|-s)^{d-1}\cdot 2s$, i.e, since $s\leq r\leq|x-y|$:

$\\#\mathcal{Y}\geq(4s)^{-d}\cdot d(2|x-y|)^{d-1}\cdot
2s=:c\cdot\frac{|x-y|^{d-1}}{s^{d-1}}.$

Now, the balls
$\left(\overline{B}(y^{\prime},s);~{}y^{\prime}\in\mathcal{Y}\right)$ are at
distance at least $s$ from one another, and included in the annulus

$\left\\{z\in\mathbb{R}^{d}:3|x-y|-s\leq|x-z|\leq 3|x-y|+s\right\\}.$

The intersection of a line hitting $\overline{B}(x,r)$ with that annulus is
the disjoint union of two line segments, each of length at most
$2\sqrt{3}\cdot s$. Therefore, any line passing through the ball
$\overline{B}(x,r)$ cannot hit more than $2\left(1+2\sqrt{3}\right)$ of the
$\overline{B}(y^{\prime},s)$, and we have

$\mathbf{1}\left(\text{$\ell$ hits
$\overline{B}(x,r)$}\right)\geq\frac{\\#\left\\{y^{\prime}\in\mathcal{Y}:\text{$\ell$
hits $\overline{B}(x,r)$ and
$\overline{B}(y^{\prime},s)$}\right\\}}{2\left(1+2\sqrt{3}\right)}\quad\text{for
all $\ell\in\mathbb{L}_{d}$.}$

Integration with respect to $\mu_{d}$ yields, upon using invariance:

$\upsilon_{d-1}r^{d-1}\geq\frac{\\#\mathcal{Y}\cdot\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]}{2\left(1+2\sqrt{3}\right)},$

and we get

$\mu_{d}\left[\overline{B}(x,r)~{};~{}\overline{B}(y,s)\right]\leq\frac{2\left(1+2\sqrt{3}\right)\upsilon_{d-1}}{c}\cdot\frac{r^{d-1}\cdot
s^{d-1}}{|x-y|^{d-1}}.$

∎

###### Proof of $(\flat\flat)$ and $(*)$.

Fix $R>0$ and $v_{0}\in\mathbb{R}_{+}^{*}$. Let $\delta>0$, and let
$\left(\overline{B}(x,\delta)\right)_{x\in\mathcal{X}}$ be a covering of
$\overline{B}(0,R)$ by balls of radius $\delta$, with centres
$x\in\overline{B}(0,R)$ more than $\delta$ apart so that the
$\left(\overline{B}(x,\delta/2)\right)_{x\in\mathcal{X}}$ are disjoint. In
particular, we have $\\#\mathcal{X}\leq(2R/\delta+1)^{d}$. On the underlying
probability space, we have the following inclusions:

$A_{2}:=\left(\text{there is a crossing of two roads of $\Pi_{v_{0}}$ inside
$\overline{B}(0,R)$}\right)\subset\bigcup_{x\in\mathcal{X}}\left(\Pi\left(\left[\overline{B}(x,\delta)\right]\times[v_{0},\infty[\right)\geq
2\right),$

and

$A_{3}:=\left(\text{there is a crossing of three roads of $\Pi_{v_{0}}$ inside
$\overline{B}(0,R)$}\right)\subset\bigcup_{x\in\mathcal{X}}\left(\Pi\left(\left[\overline{B}(x,\delta)\right]\times[v_{0},\infty[\right)\geq
3\right).$

By union bounds, we get

$\mathbb{P}\left(\bigcup_{x\in\mathcal{X}}\left(\Pi\left(\left[\overline{B}(x,\delta)\right]\times[v_{0},\infty[\right)\geq
2\right)\right)\leq\left(\frac{2R}{\delta}+1\right)^{d}\cdot\frac{\left(\delta^{d-1}\cdot
v_{0}^{-(\gamma-1)}\right)^{2}}{2!}=:a_{2}(\delta)$

and

$\mathbb{P}\left(\bigcup_{x\in\mathcal{X}}\left(\Pi\left(\left[\overline{B}(x,\delta)\right]\times[v_{0},\infty[\right)\geq
3\right)\right)\leq\left(\frac{2R}{\delta}+1\right)^{d}\cdot\frac{\left(\delta^{d-1}\cdot
v_{0}^{-(\gamma-1)}\right)^{3}}{3!}=:a_{3}(\delta).$

As $\delta\to 0^{+}$, we have $a_{2}(\delta)=O\left(\delta^{d-2}\right)$,
hence $A_{2}$ is negligible in dimensions $d\geq 3$; and
$a_{3}(\delta)=O\left(\delta^{2d-3}\right)$, hence $A_{3}$ is negligible in
all dimensions $d\geq 2$. Upon taking countable unions over $R$ and $v_{0}$,
we obtain $(\flat\flat)$ and $(*)$. ∎

[Acknowledgements] I warmly thank Nicolas Curien and Arvind Singh for their
constant support and guidance, and for their helpful comments on draft
versions of this paper. I would also like to thank Jonas Kahn for the
stimulating discussions we had while he was visiting in Orsay, and Wilfrid
Kendall for his feedback on an earlier version of this paper. Finally, I am
thankful to the referees for their careful reading of the paper: their remarks
have helped improve the presentation a great deal. I acknowledge support from
the ERC Advanced Grant 740943 GeoBrown.

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