# Almost sharp sharpness for Poisson Boolean percolation

Barbara Dembin  Vincent Tassion

###### Abstract

We consider Poisson Boolean percolation on $\mathbb{R}^{d}$ with power-law
distribution on the radius with a finite $d$-moment for $d\geq 2$. We prove
that subcritical sharpness occurs for all but a countable number of power-law
distributions. This extends the results of Duminil-Copin–Raoufi–Tassion [5]
where subcritical sharpness is proved under the assumption that the radii
distribution has a $5d-3$ finite moment. Our proofs techniques are different
from [5]: we do not use randomized algorithm and rely on specific independence
properties of Boolean percolation, inherited from the underlying Poisson
process.

We also prove supercritical sharpness for any distribution with a finite
$d$-moment and the continuity of the critical parameter for the truncated
distribution when the truncation goes to infinity.

##  1 Introduction

#### Overview

Boolean percolation was introduced by Gilbert in [8] as a continuous version
of Bernoulli percolation, introduced by Broadbent and Hammersley [3]. We
consider a Poisson point process of intensity $\lambda>0$ on $\mathbb{R}^{d}$
and on each point, we center a ball of potentially random radius. In Boolean
percolation we are interested in the connectivity properties of the occupied
set: it is defined as the subset of $\mathbb{R}^{d}$ consisting of all the
points covered by at least one ball.

This model undergoes a phase transition in $\lambda$ for the existence of an
unbounded connected component of balls. For $\lambda<\lambda_{c}$, all the
connected components are bounded, and for $\lambda>\lambda_{c}$, there exists
at least one unbounded connected component.

When the radii of the balls are bounded, Boolean percolation exhibits a very
similar behaviour as standard Bernoulli percolation. In contrast, when the law
of the radii has fat tails, the existence of large balls creates long-range
dependencies in the occupied set, leading to different behaviours, and proving
them often requires new methods. This is particularly true for the subcritical
phase: When the radii are bounded, the connection probability between two
fixed points of $\mathbb{R}^{d}$ decays exponentially fast in the distance
between the two points: there is subcritical sharpness. But this cannot be the
case when the radii have subexponential tails, because the connection
probability is always larger than the probability that the two points are
covered by a large ball. Nevertheless, a notion of sharpness can be defined in
this case. This was established in [5] under a moment assumption. In this
paper, we extend the result to a minimal moment assumptions, but we restrict
to a particular family of laws for the radii. We also establish supercritical
sharpness, by relying on standard methods for Bernoulli percolation.

#### Definition of the model

Let us more formally define Boolean percolation. Let $d\geq 2$. Denote by
$\|\cdot\|$ the $\ell_{2}$-norm on $\mathbb{R}^{d}$. For $r>0$ and
$x\in\mathbb{R}^{d}$, set

$\mathrm{B}^{x}_{r}:=\left\\{y\in\mathbb{R}^{d}:\,\|y-x\|\leq
r\right\\}\qquad\text{and}\qquad\partial\mathrm{B}^{x}_{r}:=\left\\{y\in\mathbb{R}^{d}:\,\|y-x\|=r\right\\}$

for the closed ball of radius $r$ centered at $x$ and its boundary. For short,
we will write $\mathrm{B}_{r}$ for $\mathrm{B}_{r}^{0}$. For a subset $\eta$
of $\mathbb{R}^{d}\times\mathbb{R}_{+}$, we define the occupied set associated
to $\eta$ by

$\mathcal{O}(\eta):=\bigcup_{(z,r)\in\eta}\mathrm{B}_{r}^{z}.$

Let $\mu$ be a measure on $\mathbb{R}_{+}$ and $\lambda>0$. Let $\eta$ be a
Poisson point process of intensity $\lambda\,dz\otimes\mu$ where $dz$ is the
Lebesgue measure on $\mathbb{R}^{d}$. Write $\mathbb{P}_{\lambda,\mu}$ for the
law of $\eta$ and $\mathbb{E}_{\lambda,\mu}$ for the expectation under the law
$\mathbb{P}_{\lambda,\mu}$. We will work with measures $\mu$ such that

$\displaystyle\int_{\mathbb{R}_{+}}t^{d}d\mu(t)<\infty.$ (1.1)

This hypothesis is natural for the study of percolation properties of the
occupied set $\mathcal{O}(\eta)$. Indeed, Hall proved in [12] that this
condition is necessary and sufficient to avoid that all the space is covered.
We say that two points $x$ and $y$ in $\mathbb{R}^{d}$ are connected by $\eta$
if there exists a continuous path in $\mathcal{O}(\eta)$ that joins $x$ to
$y$. We say that two sets $A$ and $B$ are connected if there exists $x\in A$
and $y\in B$ such that $x$ and $y$ are connected by $\eta$. We denote by
$\\{A\longleftrightarrow B\\}$ this event.

The phase transition of Boolean percolation is defined as follows. Define the
critical parameter by

$\lambda_{c}(\mu):=\sup\left\\{\lambda\geq
0:\lim_{r\rightarrow\infty}\mathbb{P}_{\lambda,\mu}\left(0\longleftrightarrow\partial\mathrm{B}_{r}\right)=0\right\\}.$

Under the minimal assumption (1.1), Gouéré proved in [9] that
$0<\lambda_{c}(\mu)<\infty$.

#### Subcritical sharpness

For Bernoulli percolation, the terminology "subcritical sharpness" refers to
the exponential decay of the connection probabilities in the subcritical
regime, see [16, 2, 6]. For Boolean percolation with subexponential radii
distribution, such exponential decay does not hold for any $\lambda>0$. In
order to define subcritical sharpness, we introduce the following critical
point (following Gouéré–Théret [10])

$\widehat{\lambda}_{c}(\mu):=\inf\left\\{\lambda\geq
0:\,\inf_{r>0}\mathbb{P}_{\lambda,\mu}(\mathrm{B}_{r}\longleftrightarrow\partial\mathrm{B}_{2r})>0\right\\}.$

In the regime $\lambda<\widehat{\lambda}_{c}(\mu)$, renormalization arguments
apply and provide an accurate description of the model, see
Gouéré,Gouéré–Théret [9, 10]. For example, when $\mu$ has a subexponential
tail, it was proved in [5] that the probability that $0$ is connected to
distance $n$ is equivalent to the probability that the origin is covered by
one ball intersecting $\partial B_{n}$. _Subcritical sharpness_ corresponds to
coincidence of the two critical points
$\widehat{\lambda}_{c}(\mu)=\lambda_{c}(\mu)$

In the case of bounded radius, Boolean percolation behaves at a high level in
a similar way as standard percolation. The subcritical sharpness is known, and
there is exponential decay in the subcritical regime [19, 5, 7]. In the case
of unbounded radius, the subcritical sharpness has been proved by Duminil-
Copin–Raoufi–Tassion [5], when the distribution has a $5d-3$ finite moment,
using randomized algorithms and OSSS inequality (they made a discretization of
the space-radius to be able to use the OSSS inequality). Later
Last–Pecatti–Yogeshwaran [14] proved a continuous version of the OSSS
inequality [17] and obtained sharpness for a general class of models, but
their result does not include all the Boolean percolation with the minimal
assumption (1.1). A key idea in the OSSS approach is to control the
probability that a given ball is revealed when certain exploration algorithms
are run. This method fails for fat tails, because the revealment of large
balls by the randomized algorithm is too large, and new ideas are needed to
bypass this obstacle.

In the special case of $d=2$, the subcritical sharpness has been proved by
Ahlberg–Tassion–Teixeira [1] under minimal assumption (1.1) using crucially
the planarity (the proofs relies on Russo–Seymour–Welsh theory).

In this paper, we investigate the subcritical sharpness for distribution $\mu$
under the minimal assumption (1.1). We restrict our study to the following
family of power-law distributions. For $\delta>0$, set

$\mu_{\delta}:=\frac{1}{r^{d+1+\delta}}\mathbf{1}_{r\geq 1}dr.$

The main result of this paper is the following.

######  Theorem 1.1 (Almost sharp subcritical sharpness).

There exists $\mathcal{D}\subset(0,\infty)$ at most countable such that for
all $\delta\in(0,\infty)\setminus\mathcal{D}$, we have

$\lambda_{c}(\mu_{\delta})=\widehat{\lambda}_{c}(\mu_{\delta}).$ (1.2)

######  Remark 1.2.

The set $\mathcal{D}$ is defined as the set of discontinuity points of
$\delta\mapsto\lambda_{c}(\mu_{\delta})$. By monotonicity, we know that
$\mathcal{D}$ is at most countable, but we expect that $\lambda_{c}(\delta)$
is actually continuous in $\delta$, namely that $\mathcal{D}=\emptyset$, but
we were not able to prove this.

######  Remark 1.3.

We call this result almost sharp sharpness, because it states that for the
laws $\\{\mu_{\delta},\delta\\}$, Boolean percolation satisfies subcritical
sharpness under the minimal assumption (1.1) (except possibly for countably
many $\delta$). In particular, it implies that for every $d_{0}>d$, there
exists a distribution with infinite $d_{0}$-moment for which subcritical
sharpness holds.

Our proof techniques are very different from [5], in particular we do not use
randomized algorithm. We prove a Talagrand formula [18] for Boolean
percolation to prove a sharp threshold. There is a real difficulty in applying
Talagrand formula in this context. The big balls play the role of a dictator
so we need to increase their intensity a lot to prove a sharp threshold. Doing
a small sprinkling in $\lambda$ is not sufficient to obtain a sharp threshold
in that context.

#### Supercritical sharpness

In the supercritical regime of bond percolation, the key property to start a
renormalization is the so-called local uniqueness: with high probability a
large box contains a unique macroscopic cluster. If this property is true for
every $p>p_{c}$, we say that the model has supercritical sharpness. The main
step to prove local uniqueness is to prove percolation in slabs using
Grimmett–Marstrand argument [11]. They proved that for any $p>p_{c}$, for $k$
large enough there exists an infinite connected component in the slab
$S_{k}:=\mathbb{R}^{2}\times[-k,k]^{d-2}$. For Bernoulli bond percolation on
$\mathbb{Z}^{d}$, supercritical sharpness is more involved and delicate than
its subcritical counterpart. In contrast, for Boolean percolation, subcritical
appears more challenging than supercritical percolation: as discussed in the
previous section, establishing subcritical sharpness requires new ideas, while
for supercritical sharpness, the proof follows essentially the same lines as
in the case of bond percolation. Some small adaptations are needed, but the
proof is rather simpler for Boolean than the lattice version, thanks to
rotational symmetries specific to a continuous model. In this paper, we prove
that supercritical sharpness holds for any distribution satisfying (1.1). We
define

$\lambda_{c}^{S_{k}}(\mu):=\sup\left\\{\lambda\geq
0:\lim_{r\rightarrow\infty}\mathbb{P}_{\lambda,\mu}\left(0\stackrel{{\scriptstyle\mathcal{O}(\eta\cap(S_{k}\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\partial\mathrm{B}_{r}\right)=0\right\\}.$

The following theorem is proved using the seedless renormalization scheme of
[4], which is a refinement of the original argument [11].

######  Theorem 1.4.

Let $\mu$ be a distribution that satisfies (1.1). We have

$\inf_{n\geq 1}\lambda_{c}^{S_{n}}(\mu|_{[0,n]})=\lambda_{c}(\mu)\,$

where $\mu|_{[0,n]}$ is the restriction of the distribution on the interval
$[0,n]$.

We deduce from this result the two following straightforward corollaries. We
have supercritical sharpness for any distribution $\mu$.

######  Corollary 1.5 (Supercritical sharpness).

Let $\mu$ be a distribution that satisfies (1.1). We have

$\lim_{k\rightarrow\infty}\lambda_{c}^{S_{k}}(\mu)=\lambda_{c}(\mu).$

The following corollary states that the function $\lambda_{c}$ is continuous
with respect to the truncated distribution.

######  Corollary 1.6 (Continuity of $\lambda_{c}$ for truncated radius).

Let $\mu$ be a distribution that satisfies (1.1). We have

$\lim_{n\rightarrow\infty}\lambda_{c}(\mu|_{[0,n]})=\lambda_{c}(\mu).$

#### Key steps and inputs for the proof Theorem 1.1.

We proceed by contradiction and assume that there exists a non empty regime
$\widehat{\lambda}_{c}(\delta)<\lambda_{c}(\delta)$. In other words, we assume
the existence of a regime
$\lambda\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$ for which
there is not infinite cluster but the crossing probability
$\mathbb{P}_{\lambda}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{2n})$
does not go to $0$ (along an infinite subsequence of $n$). We consider
$\lambda\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$, and we prove
that for arbitrarily small $\varepsilon>0$, the process percolates at
parameter $\lambda+\varepsilon$, which is a contradiction to
$\lambda<\lambda_{c}(\delta)$. We proceed in 3 steps:

Step 1: sharpness for truncated radii.

If $\lambda>\widehat{\lambda}_{c}$, the cluster of $\mathrm{B}_{n}$ cannot be
too small: at least for some scales, it must intersect
$\partial\mathrm{B}_{2n}$ with positive probability. By using a union bound,
we can deduce a good lower bound on the probability of
$\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{N}$ for $N\gg n$. On the
other hand, if $\lambda<\lambda_{c}$ we know that the set of balls of radius
$\leq n$ do not percolate. Furthermore, by the known sharpness for bounded
radii, the connection probabilities for the truncated process (obtained be
keeping only the balls of radius $\leq n$) decay exponentially fast. Hence
$\mathrm{B}_{n}$ cannot be connected too far without using large balls,
therefore the cluster of $\mathrm{B}_{n}$ must contain balls of radius
$\mathrm{B}_{r}$ for $r\geq n$. We can make this idea quantitative by using
the $\varphi_{p}(S)$ argument of [6] (developed in [19] for Boolean
percolation): For arbitrary $\alpha>0$, we obtain,

$\mathbb{P}_{\lambda}(\mathcal{E}_{n})\geq\frac{1}{n^{\alpha}},$ (1.3)

where $\mathcal{E}_{n}:=\left\\{\text{$\mathrm{B}_{n}$ is connected to an open
ball of radius larger than $n$}\right\\}$ (see figure 1).

Step 2: sharp threshold in $\delta$ via hypercontractivity.

The event $\mathcal{E}_{n}$ is rotation invariant, and we expect such event to
satisfy a sharp threshold. (A small increase in the parameter is sufficient to
increase the probability close to one. For a long time in the project, we did
not think that this part would be difficult, but we faced two obstacles. The
first one is conceptual: there is no general sharp threshold in $\lambda$ (see
Section 4.1 for an example that illustrates this fact). We overcome this
difficulty by proving a sharp threshold in $\delta$ instead: when varying
$\delta$, the density of large balls increases much faster than the density of
the small ones, which prevents the large balls to play a "dictator" role on a
full range of parameters. The second difficulty comes from the fact that we
want to prove a sharp threshold for rotational invariant events. A natural
strategy would be to discretize the space and the radius and apply a discrete
Talagrand formula. However, there is a theoretical obstacle in pursuing this
strategy: for dimensions $d\geq 3$, there is no rotational invariant way to
discretize the space. This is a consequence of the fact that the group of
orthogonal transformations is not abelian in dimension $d\geq 3$. Hence,
discretizing the model implies losing all the symmetries of the continuous
one. For this reason, we had to prove a continuous version of Talagrand’s
formula, by using a suitable encoding and limiting procedure. By applying this
formula to the event $\mathcal{E}_{n}$, we obtain that

$\mathbb{P}_{\lambda,\delta-\eta}(\mathcal{E}_{n})\geq
1-\frac{1}{n^{\alpha}}.$ (1.4)

Step 3: Grimmett–Marstrand dynamic exploration:

thanks to the previous step we know that the event $\mathcal{E}_{n}$ occurs
with high probability for the distribution $\mu_{\delta-\varepsilon}$. The
open ball in the definition of the event $\mathcal{E}_{n}$ plays the role of
the seed in Grimmett–Marstrand dynamic renormalization. Up to a small
sprinkling $\varepsilon$ in $\lambda$, we can prove that there exists an
infinite connected component with positive probability. It follows that

$\lambda+\varepsilon>\lambda_{c}(\mu_{\delta-\eta}).$ (1.5)

Conclusion.

Equation (1.5) is not exactly the contradiction we were aiming for because we
had to reduce $\delta$, nevertheless we still reach a contradiction if
$\delta$ is a continuity point of $\lambda_{c}$.

Figure 1: Illustration of the event $\mathcal{E}_{n}$

#### Organization of the paper

In the next section, we state the three key propositions corresponding to the
three steps and prove Theorem 1.1. In Section 3, we prove that in the
fidicious regime, with not too small probability, balls are connected to
larger open balls (this corresponds to Step 1). In Section 4, we prove a
Talagrand formula for Boolean percolation and prove a sharp threshold result
for rotational invariant events (this corresponds to Step 2 of the sketch of
the proof). In Section 5, we prove that there is percolation up to a
sprinkling in the intensity using a Grimmett–Marstrand dynamic exploration
using big balls as seeds (this corresponds to Step 3). Finally, in Section 6,
we prove Theorem 1.4.

#### Acknowledgements

We thank Tobias Hartnick and Matthis Lehmkühler for useful exchanges regarding
discretization of the sphere. We thank Jean-Baptiste Gouéré for his careful
reading and his useful comments. This project has received funding from the
European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation program (grant agreement No 851565).

##  2 Proof of Theorem 1.1

In this section, we state the main propositions corresponding to the three
steps of the sketch of proof and prove Theorem 1.1.

For short, we will write $\lambda_{c}(\delta),\widehat{\lambda}_{c}(\delta)$
instead of $\lambda_{c}(\mu_{\delta}),\widehat{\lambda}_{c}(\mu_{\delta})$ and
$\mathbb{P}_{\lambda,\delta}$ instead of $\mathbb{P}_{\lambda,\mu_{\delta}}$.

For $\rho\geq 1$, we define

$\mathcal{E}_{n,N}(\rho):=\left\\{\exists(x,r)\in\eta\cap((\mathrm{B}_{(\rho+1/2)N}\setminus\mathrm{B}_{\rho
N})\times[n,+\infty)):\mathrm{B}_{n}\stackrel{{\scriptstyle\mathcal{O}(\eta\cap(\mathrm{B}_{(\rho+1/2)N}\times\mathbb{R}))}}{{\longleftrightarrow}}\mathrm{B}_{r}^{x}\right\\}.$

For technical reasons arising in the proof of Grimmett–Marstrand, we need this
extra parameter $\rho$. In the regime
$(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$, we can prove that there
exists $n\leq N$ such that for any $\rho\in[1,2d]$, the event
$\mathcal{E}_{n,N}(\rho)$ occurs with a not too small probability. This
corresponds to Step 1 of the sketch of the proof (it is proved in Section 3).

######  Proposition 2.1.

Let $\delta>0$. Assume that
$\widehat{\lambda}_{c}(\delta)<\lambda_{c}(\delta)$. Let $\varepsilon>0$ and
$\lambda\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$. For any
$n_{0}\geq 1$, there exists $n\geq n_{0}$ and $N\geq 2n$ such that

$\forall\rho\in[1,2d]\qquad\mathbb{P}_{\lambda,\delta}(\mathcal{E}_{n,N}(\rho))\geq\frac{1}{n^{\varepsilon}}.$

In the second step of the proof, we prove a sharp threshold in $\delta$. In
Section 4, we prove an inhomogeneous Talagrand formula for Boolean percolation
in Proposition 4.3. The following proposition is proved in greater generality
in Section 4.

######  Proposition 2.2.

Let $\lambda,\delta>0$. There exists $\kappa>0$ such that the following holds.
For every $\varepsilon\in(0,\delta/2\kappa)$, $n\geq\kappa$, $N\geq 2n$ and
$\rho\in[1,2d]$,

$\mathbb{P}_{\lambda,{\delta}}(\mathcal{E}_{n,N}(\rho))\geq\frac{1}{n^{\varepsilon}}\implies\mathbb{P}_{\lambda,{\delta-\kappa\varepsilon}}(\mathcal{E}_{n,N}(\rho))\geq
1-\frac{1}{n^{\varepsilon}}.$

Once we know that a big ball is connected to another open big ball with a very
good probability, this big ball will play the role of the seed in
Grimmett–Marstrand proof using dynamic renormalization. The following
proposition (proved in Section 5) is a Grimmett–Marstrand type proposition
that corresponds to the Step 3 of the sketch of proof.

######  Proposition 2.3.

Let $\mu$ be a distribution with a finite $d$-moment. There exists
$\varepsilon_{0}>0$ depending only on $d$ such that for all
$\varepsilon\leq\varepsilon_{0}$, $n\leq N$, if we have for every
$\rho\in[1,2d]$

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n,N}(\rho))\geq 1-\varepsilon$

then $\lambda>\lambda_{c}(\mu)-\frac{\lambda K_{d}}{|\log\varepsilon|}$ where
$K_{d}>0$ depends only on $d$.

Now we have all the ingredients to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let $\delta>0$ such that $\widehat{\lambda}_{c}(\delta)<\lambda_{c}(\delta)$.
Let us prove that $\delta\in\mathcal{D}$. Let
$\lambda\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$. Let $\kappa$
be as in Proposition 2.2. Let $\varepsilon\in(0,\delta/2\kappa)$ and
$n_{0}\geq\kappa$. By Proposition 2.1, there exist $n\geq n_{0}$ and $N\geq
2n$ such

$\forall\rho\in[1,2d]\qquad\mathbb{P}_{\lambda,\delta}(\mathcal{E}_{n,N}(\rho))\geq\frac{1}{n^{\varepsilon}}.$

By Proposition 2.2, we obtain

$\forall\rho\in[1,2d]\qquad\mathbb{P}_{\lambda,\delta-\kappa\varepsilon}(\mathcal{E}_{n,N}(\rho))\geq
1-\frac{1}{n^{\varepsilon}}.$

By Proposition 2.3, it follows that

$\lambda\geq\lambda_{c}(\delta-\kappa\varepsilon)-\frac{{\lambda}_{c}(\delta)K_{d}}{\varepsilon\log
n}.$

There exists an infinite number of $n$ such that the previous inequality
holds. By letting first $n$ go to infinity and then $\varepsilon$ go to $0$,
we obtain that

$\lambda\geq\lambda_{c}(\delta^{-})\,$

where $\lambda_{c}(\delta^{-})$ is the left limit of the function
$\delta\mapsto\lambda_{c}(\delta)$ at $\delta$. We recall that
$\lambda<\lambda_{c}(\delta)$, it yields that

$\lambda_{c}(\delta^{-})<\lambda_{c}(\delta)$

and $\delta\in\mathcal{D}$.

∎

##  3 Connection to large balls in the fidicious regime

In this section, we prove Proposition 2.1.

###  3.1 Correlation length for truncated radii

In this section we consider the law of radii

$\mu=\mu_{\delta}|_{[0,n]},$ (3.1)

and we write $\mathbb{P}_{\lambda}^{n}=\mathbb{P}_{\lambda,\mu}$ for the
corresponding Boolean percolation measure, and $\mathbb{E}_{\lambda}^{n}$ for
the corresponding expectation. For bounded radii, sharpness of the phase
transition is known. In particular, one can define a correlation length in the
subcritical regime, and we collect here some quantitative statements that can
be deduced from previously known arguments.

Denote by $\mathcal{B}(\mathbb{R}^{d})$ the Borel $\sigma$-algebra of
$\mathbb{R}^{d}$. Following Ziesche [19] (see also [6]), define for every
$\lambda\geq 0$, $n\geq 1$ and $S\in\mathcal{B}(\mathbb{R}^{d})$ such that
$\mathrm{B}_{n}\subset S$

$\varphi_{\lambda}^{n}(S):=\mathbb{E}_{\lambda}^{n}\big{[}\sum_{(z,r)\in\eta}\mathbf{1}[\mathrm{B}_{r}^{z}\cap\partial
S\neq\emptyset,\mathrm{B}_{n}\stackrel{{\scriptstyle\\{(w,s)\in\eta:\mathrm{B}_{s}^{w}\subset
S\\})}}{{\longleftrightarrow}}\mathrm{B}_{r}^{z}\big{]}$ (3.2)

This functional is instrumental in the proof of sharpness, and we will use the
following properties. Let $\lambda\geq 0$, $N\geq n\geq 1$.

  1. 1

If there exists $S\in\mathcal{B}(\mathbb{R}^{d})$ such that
$\mathrm{B}_{n}\subset S\subset\mathrm{B}_{N}$ and
$\varphi^{n}_{\lambda}(S)\leq 1/e$ then

$\forall\ell\geq
N\qquad\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{\ell})\leq\exp\left(-\left\lfloor{\ell}/{N}\right\rfloor\right).$

  2. 2

If for all $S\in\mathcal{B}(\mathbb{R}^{d})$ such that $\mathrm{B}_{n}\subset
S\subset\mathrm{B}_{N}$ we have $\varphi^{n}_{\lambda}(S)\geq 1/e$ then

$\forall\lambda^{\prime}>\lambda\qquad\mathbb{P}_{\lambda^{\prime}}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{N})\geq\frac{\lambda^{\prime}-\lambda}{e\lambda^{\prime}}.$

These properties are not exactly stated in [19] but they can be obtained from
the proof of the first and third item of Theorem 3.1 in [19]. These properties
give rise to a natural definition of a correlation length in the subcritical
regime: For every $n\geq 1$ and every $\lambda\geq 0$, let

$\mathrm{L}_{\lambda}(n):=\inf\left\\{\ell\in[n,+\infty):\exists
S\in\mathcal{B}(\mathbb{R}^{d}),\,\mathrm{B}_{n}\subset
S\subset\mathrm{B}_{\ell},\,\varphi_{\lambda}^{n}(S)\leq\frac{1}{e}\right\\}.$

where we use the convention $\inf\emptyset=+\infty$. For fixed $\lambda>0$ and
$n\geq 1$, by using Properties 1 and 2 and continuity, we have

$\displaystyle\forall\ell\geq\mathrm{L}_{\lambda}(n)$
$\displaystyle\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{\ell})\leq\exp\left(-\left\lfloor{\ell}/{\mathrm{L}_{\lambda}(n)}\right\rfloor\right)$
(3.3) $\displaystyle\forall\lambda^{\prime}>\lambda$
$\displaystyle\mathbb{P}_{\lambda^{\prime}}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{L_{\lambda}(n)})\geq\frac{\lambda^{\prime}-\lambda}{e\lambda^{\prime}}.$
(3.4)

In the subcritical regime and up to logarithmic corrections, the quantity
$\mathrm{L}_{\lambda}(n)$ can be related to the more standards definition of
correlation length for the truncated model (defined as the exponential rate of
decay of the the connection probabilities).

###  3.2 Behaviour of the correlation length in the fidicious regime

By definition, the correlation length satisfies
$\mathrm{L}_{\lambda}(n)\in[n,\infty]$. In order to prove Proposition 2.1, we
will use that the correlation length is not degenerated in the fidicious
regime, which is the content of the following lemma.

######  Lemma 3.1.

For any $\lambda\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$, there
exists $n_{0}\geq 1$ such that

$\forall n\geq n_{0}\qquad 3n\leq\mathrm{L}_{\lambda}(n)<+\infty.$ (3.5)

###### Proof.

Let $\lambda>0$. We prove the following two implications, from which the lemma
follows.

$\displaystyle\left(\exists n\geq
1\qquad\mathrm{L}_{\lambda}(n)=+\infty\right)\implies\lambda\geq\lambda_{c}(\delta),$
$\displaystyle\left(\mathrm{L}_{\lambda}(n)\leq 3n\text{ for infinitely many
}n\geq 1\right)\implies\lambda\leq\widehat{\lambda}_{c}(\delta).$

The first implication is a direct consequence of Equation (3.4). For the
second implication, let $n$ such that $\mathrm{L}_{\lambda}(n)\leq 3n$ and set
$N:=n^{1+\frac{\delta}{2d}}$. If $\mathrm{B}_{N}$ is connected to
$\partial\mathrm{B}_{2N}$, either the connection uses only balls of radius
smaller than or equal to $n$, or there must exist a ball of radius at least
$n$ intersecting $\mathrm{B}_{2N}$. Therefore, by Markov inequality

$\mathbb{P}_{\lambda,\mu_{\delta}}(\mathrm{B}_{N}\leftrightarrow\partial\mathrm{B}_{2N})\leq\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{N}\leftrightarrow\partial\mathrm{B}_{2N})+\mathbb{E}_{\lambda,\mu}(|\eta\cap
E|),$ (3.6)

where $E:=\left\\{(x,r)\in\mathbb{R}^{d}\times[n,+\infty[:\|x\|_{2}\leq
2N+r\right\\}$. Consider a covering of $\mathrm{B}_{N}$ with at most
$C(N/n)^{d}$ balls of radius $n$ with center on $\mathrm{B}_{N}$ (where $C$ is
a constant depending only on $d$). If $\mathrm{B}_{N}$ is connected to
$\partial\mathrm{B}_{2N}$, then at least one of the balls of the covering must
be connected to $\partial\mathrm{B}_{2N}$. By the union bound, translation
invariance, we obtain

$\displaystyle\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{N}\leftrightarrow\partial\mathrm{B}_{2N})\leq
C\left(\tfrac{N}{n}\right)^{d}\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{N})\overset{\eqref{eq:1}}{\leq}C\left(\tfrac{N}{n}\right)^{d}\exp\big{(}-\lfloor
N/3n\rfloor\big{)},$ (3.7)

where we used the hypothesis $\mathrm{L}_{\lambda}(n)\leq 3n$ on the second
inequality.

Let us now bound the second term in (3.6). The quantity $|\eta\cap E|$ is a
Poisson random variable with parameter $(dz\otimes\mu_{\delta})(E)$, hence

$\mathbb{E}_{\lambda,\mu}(|\eta\cap E|)=(dz\otimes\mu_{\delta})(E)\leq
C^{\prime}\int_{n}^{\infty}(N+t)^{d}d\mu_{\delta}(t)\leq
C^{\prime\prime}\frac{N^{d}}{n^{d+\delta}}+C^{\prime\prime}\int_{N}^{\infty}t^{d}d\mu_{\delta}(t),$
(3.8)

where $C^{\prime},C^{\prime\prime}$ are constants depending only on $d$.

Combining the previous inequalities that hold for an infinite number of $n$,
we get that

$\inf_{N\geq
1}\mathbb{P}_{\lambda,\delta}(\mathrm{B}_{N}\longleftrightarrow\partial\mathrm{B}_{N})=0,$

which concludes that $\lambda\leq\widehat{\lambda}_{c}(\delta)$. ∎

######  Lemma 3.2.

Let $\varepsilon>0$,
$\lambda_{0}\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$. For any
$n_{0}\geq 1$, there exist $n\geq n_{0}$ and
$\lambda\in\left(\frac{\widehat{\lambda}_{c}(\delta)+\lambda_{0}}{2},\lambda_{0}\right)$
such that

$\mathrm{L}_{\lambda-\frac{1}{\log
n}}(n^{1+\varepsilon})\geq\mathrm{L}_{\lambda}(n)^{1+\frac{\varepsilon}{2}}.$
(3.9)

###### Proof.

For contradiction, let $n_{0}\geq 1$, and assume that for every $n\geq n_{0}$
and
$\lambda\in\left(\frac{\widehat{\lambda}_{c}(\delta)+\lambda_{0}}{2},\lambda_{0}\right)$

$\mathrm{L}_{\lambda-\frac{1}{\log
n}}(n^{1+\varepsilon})\leq\mathrm{L}_{\lambda}(n)^{1+\frac{\varepsilon}{2}}.$
(3.10)

Let $N_{0}\geq n_{0}$, and set $N_{k}:=N_{0}^{(1+\varepsilon)^{k}}$ and
$\lambda_{k}:=\lambda_{0}-\sum_{0\leq i\leq k-1}\frac{1}{\log(N_{i})}$ for
every $k\geq 1$. By choosing $N_{0}$ large enough we can ensure that
$\lambda_{k}>\frac{\widehat{\lambda}_{c}(\delta)+\lambda_{0}}{2}$ for every
$k$. Thanks to Lemma 3.1, we have that $L_{\lambda_{0}}(N_{0})<+\infty$. By
induction, Inequality (3.10) implies

$\forall k\geq
0\qquad\mathrm{L}_{\lambda_{k}}(N_{k})\leq\mathrm{L}_{\lambda_{0}}(N_{0})^{(1+\frac{\varepsilon}{2})^{k}},$
(3.11)

which contradicts the lower bound $L_{\lambda}(n)\geq n$.

∎

###### Proof of Proposition 2.1.

Let $\delta>0$. Assume that
$\widehat{\lambda}_{c}(\delta)<\lambda_{c}(\delta)$. Let $\varepsilon>0$,
$\lambda_{0}\in(\widehat{\lambda}_{c}(\delta),\lambda_{c}(\delta))$. By Lemma
3.1, there exists $n_{0}\geq 1$ such that

$\forall n\geq
n_{0}\quad\forall\lambda\geq\frac{\widehat{\lambda}_{c}(\delta)+\lambda_{0}}{2}\qquad\mathrm{L}_{\lambda}(n)\geq\mathrm{L}_{\frac{\widehat{\lambda}_{c}(\delta)+\lambda_{0}}{2}}(n)\geq
3n.$ (3.12)

By Lemma 3.2, there exist $n\geq n_{0}$ and
$\lambda\in(\frac{\widehat{\lambda}_{c}(\delta)+\lambda_{0}}{2},\lambda_{0})$
such that

$\mathrm{L}_{\lambda-\frac{1}{\log
n}}(n^{1+\varepsilon})\geq\mathrm{L}_{\lambda}(n)^{1+\frac{\varepsilon}{2}}.$
(3.13)

Set

$N:=\frac{1}{9d}\cdot{\mathrm{L}_{\lambda-\frac{1}{\log
n}}(n^{1+\varepsilon})}.$

Thanks to inequalities (3.3) and (3.4), we have

$\forall\ell\geq\mathrm{L}_{\lambda}(n)\qquad\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{\ell})\leq\exp\left(-\left\lfloor{\ell}/{\mathrm{L}_{\lambda}(n)}\right\rfloor\right)$
(3.14)

and

$\mathbb{P}^{n^{1+\varepsilon}}_{\lambda}(\mathrm{B}_{n^{1+\varepsilon}}\longleftrightarrow\partial\mathrm{B}_{9dN})\geq\frac{1}{\log
n}.$ (3.15)

By the union bound, there exists a constant $c_{d}$ depending only on $d$ such
that

$\mathbb{P}^{n^{1+\varepsilon}}_{\lambda}(\mathrm{B}_{n^{1+\varepsilon}}\longleftrightarrow\partial\mathrm{B}_{9dN})\leq
c_{d}n^{(d-1)\varepsilon}\,\mathbb{P}_{\lambda}^{n^{1+\varepsilon}}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{3dN})$

where we use that by inequality (3.12), we have $\mathrm{L}_{\lambda-1/\log
n}(n^{1+\varepsilon})=9dN\geq 3n^{1+\varepsilon}$. Hence, we obtain using
inequality (3.15)

$\mathbb{P}_{\lambda}^{n^{1+\varepsilon}}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{3dN})\geq\frac{1}{c_{d}n^{(d-1)\varepsilon}\log
n}\geq\frac{3}{n^{\varepsilon^{\prime}}}$ (3.16)

where $\varepsilon^{\prime}=d\varepsilon$ (up to choosing a larger $n_{0}$
depending on $\varepsilon$, we can assume that $n$ is large enough such that
the last inequality holds). Let $\rho\in[1,2d]$. By union bound, we have

$\begin{split}\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{(\rho+1/4)N}\longleftrightarrow\partial\mathrm{B}_{(\rho+1/2)N})&\leq
c_{d}\left(\frac{\rho
N}{n}\right)^{d-1}\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{N/2})\\\
&\leq
c_{d}\left(\frac{2dN}{n}\right)^{d-1}\exp\left(-\frac{N^{\frac{\varepsilon}{2+\varepsilon}}}{(9d)^{2}}\right)\end{split}$
(3.17)

where we used inequalities (3.13) and (3.14) in the last inequality. By Lemma
3.1, we have $3dN\geq n^{1+\varepsilon}$, up to choosing a larger $n_{0}$
depending on $\varepsilon$, we can assume that $n$ is large enough such that

$c_{d}\left(\frac{2dN}{n}\right)^{d-1}\exp\left(-\frac{N^{\frac{\varepsilon}{2+\varepsilon}}}{(9d)^{2}}\right)\leq\frac{1}{n^{\varepsilon^{\prime}}}$

Combining inequalities (3.16) and (3.17), it follows that for any
$\rho\in[1,2d]$

$\begin{split}\mathbb{P}_{\lambda,\mu_{\delta}}(\mathcal{E}_{n,N}(\rho))&\geq\mathbb{P}_{\lambda}^{n^{1+\varepsilon}}(\mathrm{B}_{n}\longleftrightarrow\partial\mathrm{B}_{3dN})-\mathbb{P}_{\lambda}^{n}(\mathrm{B}_{(\rho+1/4)N}\longleftrightarrow\partial\mathrm{B}_{(\rho+1/2)N})\\\
&-\mathbb{P}_{\lambda,\mu_{\delta}}(\exists(x,r)\in\mathrm{B}_{\rho
N}\times\mathbb{R}_{+}:\mathrm{B}_{r}^{x}\cap\partial\mathrm{B}_{(\rho+1/4)N})\\\
&\geq\frac{2}{n^{\varepsilon^{\prime}}}-\frac{c_{d}}{n^{\delta}}\geq\frac{1}{n^{\varepsilon^{\prime}}}.\end{split}$

The result follows from the fact that
$\mathbb{P}_{\lambda,\mu_{\delta}}(\mathcal{E}_{n,N}(\rho))\leq\mathbb{P}_{\lambda_{0},\mu_{\delta}}(\mathcal{E}_{n,N}(\rho))$.

∎

##  4 Sharp threshold

In this section, we prove a Talagrand formula for inhomogeneous weights. By a
proper encoding of a Poisson point process, we prove a Talagrand formula for
Boolean percolation. We apply this result to prove Proposition 2.2.

###  4.1 Why do we need to sprinkle in $\delta$?

A sprinkling in $\lambda$ is not sufficient to prove a sharp threshold in full
generality. Let us illustrate this fact by a simple example. Denote by
$\mathcal{F}$ the event that there exists an open ball of radius at least
$n^{d/(d+\delta)}$ centered inside the ball $\mathrm{B}_{n}$. We have for
every $\lambda>0$,

$\mathbb{P}_{\lambda,\mu_{\delta}}(\mathcal{F})=1-\exp(-\alpha_{d}\lambda),$

where $\alpha_{d}$ is the volume of the unit ball in $\mathbb{R}^{d}$. Hence,
even if the event $\mathcal{F}$ has a lot of symmetries (the event is
invariant under rotations), there is no sharp threshold in $\lambda$. The big
balls, that are present with a very small probability, play a dictator role in
the sense that the presence of one ball is enough for the event to occur. In
the event $\mathcal{E}_{n,N}(\rho)$ (defined in Section 2), the big balls may
also play a dictator role. As for the event $\mathcal{F}$, we cannot hope to
prove a sharp threshold in $\lambda$ for the event $\mathcal{E}_{n,N}(\rho)$.

By slightly reducing $\delta$, the density of big balls increase faster
compare to the density of small balls. Going back to the event $\mathcal{F}$,
we have for $\varepsilon>0$

$\mathbb{P}_{\lambda,\mu_{\delta-\varepsilon}}(\mathcal{F})=1-\exp(-\lambda
n^{d\varepsilon/(d+\delta)}).$

Hence, the event $\mathcal{F}$ has a sharp threshold in $\delta$.

###  4.2 Sharp threshold for inhomogeneous percolation

In this section, we state general results on sharp threshold for inhomogeneous
weights. We will need the following result that is a Talagrand formula that
holds for inhomogeneous Bernoulli percolation. Let $N\geq 1$, and
$p_{1},\dots,p_{N}\in[0,1]$, set

$\mathrm{P}_{p_{1},\dots,p_{N}}:=\otimes_{i=1}^{N}\mathrm{Ber}(p_{i})$

where $\mathrm{Ber}(p)$ is the distribution on $\\{0,1\\}$ of Bernoulli of
parameter $p$. For $i\in\\{1,\dots,N\\}$ define $\tau_{i}$ the function on
$\\{0,1\\}^{N}$ that switches the value of the $i$-th bit.

######  Proposition 4.1 (Inhomogeneous Talagrand).

There exists a universal constant $C>0$ such that for any family
$(p_{i})_{1\leq i\leq N}\in(0,1/2]^{N}$ for any
$f:\\{0,1\\}^{N}\rightarrow\\{0,1\\}$. We have

$\sum_{i=1}^{N}p_{i}|\log p_{i}|\operatorname{Inf}_{i}(f)\geq
C\operatorname{Var}(f)\log\left(\frac{1}{\max_{1\leq i\leq
N}p_{i}\operatorname{Inf}_{i}(f)}\right)\,$

where

$\operatorname{Inf}_{i}(f):=\mathrm{P}_{p_{1},\dots,p_{N}}(f\circ\tau_{i}\neq
f).$

######  Remark 4.2.

For simplicity, we prove a version of inhomogeneous Talagrand for $p$’s that
are close to $0$. We did not try to optimize the proof to obtain a more
symmetric version.

We were not able to find a proof of this inhomogeneous version of Talagrand
formula in the literature. For completeness, we include a proof of Proposition
4.1 in the appendix.

###  4.3 Inhomogeneous Talagrand for Boolean percolation

In this section, we prove a Talagrand formula for Boolean percolation. For
$x\in\mathbb{Z}^{d}$ and $n\geq 1$, define

$\mathrm{S}_{n}^{x}:=\left(x+\left[-\frac{1}{2},\frac{1}{2}\right)^{d}\right)\times[n,n+1)\,.$

Let $\delta>0$ and $\lambda>0$. For short, we will write $\mathbb{P}_{\delta}$
for $\mathbb{P}_{\lambda,\mu_{\delta}}$. The parameter $\lambda$ will remain
fixed in all this section.

For $\mathcal{E}$ a finite increasing event, denote

$\mathrm{Piv}_{n}^{x}(\mathcal{E}):=\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}_{\delta}(\eta\notin\mathcal{E},\eta\cup\\{(z,r)\\}\in\mathcal{E})dzd\mu_{\delta}(r).$

Set $\Omega$ the set of countable subsets of
$\mathbb{R}^{d}\times\mathbb{R}_{+}$. We define the following distance between
points in $\Omega$: for any $\eta,\eta^{\prime}\in\Omega$

$\mathrm{d}(\eta,\eta^{\prime}):=\left\\{\begin{array}[]{ll}+\infty&\mbox{ if
$|\eta|\neq|\eta^{\prime}|$}\\\ \inf_{\pi:\eta\mapsto\eta^{\prime}\text{
bijection}}\sup\\{\|x-x^{\prime}\|+|r-r^{\prime}|:(x^{\prime},r^{\prime})=\pi((x,r))\\}&\mbox{otherwise}\end{array}\right.$

where we use the convention that $|\eta|=|\eta^{\prime}|$ if both sets are
infinite. We will denote by $\partial\mathcal{E}$ the boundary of
$\mathcal{E}$ for the topology associated with the distance $\mathrm{d}$ in
the space $\Omega$. For $\eta\in\Omega$ and $N\geq 1$, we define the
neighborhood $\mathcal{N}_{N}(\eta,\varepsilon)$ at distance $\varepsilon$
restricted to $\mathrm{B}_{N}$

$\mathcal{N}_{N}(\eta,\varepsilon):=\left\\{\eta^{\prime}\in\Omega:\,\mathrm{d}(\eta\cap(\mathrm{B}_{N}\times\mathbb{R}_{+}),\eta^{\prime}\cap(\mathrm{B}_{N}\times\mathbb{R}_{+}))\leq\varepsilon\right\\}$

Let $N$ be such that the event $\mathcal{E}$ only depends on balls centered in
$\mathrm{B}_{N}$. We will need the following notion of approximate pivotality
at level $\varepsilon>0$

$\mathrm{Piv}_{n}^{x,\varepsilon}(\mathcal{E}):=\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}_{\delta}(\exists\eta^{\prime}\in\mathcal{N}_{N}(\eta,\varepsilon):\eta^{\prime}\notin\mathcal{E},\eta^{\prime}\cup\\{(z,r)\\}\in\mathcal{E})dzd\mu_{\delta}(r).$

The following proposition is a Talagrand formula for increasing event in
Boolean percolation.

######  Proposition 4.3.

Let $\lambda>0$, $\delta_{0},\delta_{1}>0$. There exists $C_{0}>0$ such that
the following holds. For any $\delta\in(\delta_{0},\delta_{1})$, For any
finite increasing event $\mathcal{E}$ such that
$\mathbb{P}_{\delta_{0}}(\partial\mathcal{E})=0$, we have

$\sum_{x\in\mathbb{Z}^{d},n\geq 1}\log
n\,\mathrm{Piv}_{n}^{x}(\mathcal{E})\geq
C_{0}\mathbb{P}_{\delta}(\mathcal{E})(1-\mathbb{P}_{\delta}(\mathcal{E}))\log\left(\frac{1}{\max_{x\in\mathbb{Z}^{d},n\geq
1}\mathrm{Piv}_{n}^{x}(\mathcal{E})}\right).$

######  Remark 4.4.

This proposition holds for increasing events for the two following partial
orders. An event $\mathcal{E}$ is coloring increasing if for any
$\eta\in\mathcal{E}$ for any $\eta^{\prime}$ such that
$\mathcal{O}(\eta)\subset\mathcal{O}(\eta^{\prime})$ then
$\eta^{\prime}\in\mathcal{E}$.

We also define the partial order $\prec$ on
$\mathbb{R}^{d}\times\mathbb{R}_{+}$ as follows. We say that
$\eta\preceq\eta^{\prime}$ if for any $(x,r)\in\eta$ there exists
$(y,r^{\prime})\in\eta^{\prime}$ such that
$\mathrm{B}_{r}^{x}\subset\mathrm{B}_{r^{\prime}}^{y}$. We say that an event
$\mathcal{E}$ is increasing for $\prec$ if $\eta\in\mathcal{E}$ and
$\eta\preceq\eta^{\prime}$ then $\eta^{\prime}\in\mathcal{E}$.

######  Remark 4.5.

We believe that the hypothesis $\mathbb{P}_{\delta}(\partial\mathcal{E})=0$
holds for any increasing event. Note that we will only work on connection
events. For connection events, if $\eta\in\partial\mathcal{E}$ then there
exists $(z,r),(z^{\prime},r^{\prime})\in\eta$ such that
$\|z-z^{\prime}\|=r+r^{\prime}$. It follows that
$\mathbb{P}_{\delta}(\partial\mathcal{E})=0$.

######  Remark 4.6.

At a first look, it may seem that the formula in Proposition 4.3 involves an
underlying discretization of the space. We would like to emphasize that the
pivotal quantities $\mathrm{Piv}_{n}^{x}(\mathcal{E})$ are defined in terms of
the continuous model itself and not a discretized version of it. One can check
that the formula is equivalent to the existence of a constant $C>0$ such that

$\int_{(z,r)\in\mathbb{R}^{d}\times\mathbb{R}_{+}}I_{z,r}(\mathcal{E})\log
r\,dzd\mu(r)\geq
C\mathbb{P}(\mathcal{E})(1-\mathbb{P}(\mathcal{E}))\log\left(\frac{1}{\sup_{(z,r)\in\mathbb{R}^{d}\times\mathbb{R}_{+}}I_{z,r}(\mathcal{E})}\right)$

where
$I_{z,r}(\mathcal{E})=\mathbb{P}(\eta\notin\mathcal{E},\eta\cup\\{(z,r)\\}\in\mathcal{E})$.

We will need the following lemma to prove Proposition 4.3.

######  Lemma 4.7.

Let $\mathcal{E}$ be a finite increasing event such that for any
$\lambda,\delta>0$, $\mathbb{P}_{\delta}(\partial\mathcal{E})=0$. We have for
any $x\in\mathbb{Z}^{d}$ and $n\geq 1$

$\lim_{\varepsilon\rightarrow
0}\mathrm{Piv}_{n}^{x,\varepsilon}(\mathcal{E})=\mathrm{Piv}_{n}^{x}(\mathcal{E}).$

###### Proof.

Let $N$ be such that $\mathcal{E}$ only depends on points centered in
$\mathrm{B}_{N}\times\mathbb{R}_{+}$. Let $\varepsilon>0$ and
$(z,r)\in\mathrm{S}_{n}^{x}$, we have

$\begin{split}\mathbb{P}_{\delta}&(\exists\eta^{\prime}\in\mathcal{N}_{N}(\eta,\varepsilon):\eta^{\prime}\notin\mathcal{E},\eta^{\prime}\cup\\{(z,r)\\}\in\mathcal{E})\\\
&\leq\mathbb{P}_{\delta}(\eta\in\mathcal{N}_{N}(\partial\mathcal{E},\varepsilon))+\mathbb{P}_{\delta}(\eta\cup\\{(z,r)\\}\in\mathcal{N}_{N}(\partial\mathcal{E},\varepsilon))+\mathbb{P}_{\delta}(\eta\notin\mathcal{E},\eta\cup\\{(z,r)\\}\in\mathcal{E}).\end{split}$

It follows that

$\begin{split}\mathrm{Piv}_{n}^{x,\varepsilon}(\mathcal{E})\leq&\mathrm{Piv}_{n}^{x}(\mathcal{E})+\mathbb{P}_{\delta}(\mathcal{N}_{N}(\partial\mathcal{E},\varepsilon))+\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}_{\delta}(\eta\cup\\{(z,r)\\}\in\mathcal{N}_{N}(\partial\mathcal{E},\varepsilon))dzd\mu_{\delta}(r).\end{split}$

We have

$\lim_{\varepsilon\rightarrow
0}\mathbb{P}_{\delta}(\mathcal{N}_{N}(\partial\mathcal{E},\varepsilon))=\mathbb{P}_{\delta}(\partial\mathcal{E})=0.$

Similarly, by dominated convergence

$\lim_{\varepsilon\rightarrow
0}\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}_{\delta}(\eta\cup\\{(z,r)\\}\in\mathcal{N}_{N}(\partial\mathcal{E},\varepsilon))dzd\mu_{\delta}(r)=\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}_{\delta}(\eta\cup\\{(z,r)\\}\in\partial\mathcal{E})dzd\mu_{\delta}(r).$

Using Proposition 2.1 in [13], we have

$\begin{split}\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}_{\delta}(\eta\cup\\{(z,r)\\}\in\partial\mathcal{E})dzd\mu_{\delta}(r)&\leq\int_{(z,r)\in\mathbb{R}^{d}\times\mathbb{R}_{+}}\mathbb{P}_{\delta}(\eta\cup\\{(z,r)\\}\in\partial\mathcal{E},\eta\notin\mathcal{E})dzd\mu_{\delta}(r)\\\
&=\frac{\partial}{\partial\lambda}\mathbb{P}_{\delta}(\partial\mathcal{E})=0\end{split}$

The result follows.

∎

Denote by

$F(t):=\int_{1}^{t}\frac{d+\delta}{r^{d+1+\delta}}dr\,.$

For $n\geq 1$ define

$\forall t\in[0,1]\qquad F_{n}^{-1}(t):=\inf\left\\{s\geq
n:\frac{F(s)-F(n)}{F(n+1)-F(n)}\geq t\right\\}\,.$

We will frequently use the following claim in the proof of Proposition 4.3.
The proof follows from an easy computation.

######  Claim 4.8.

There exists a constant $c$ depending on $\delta$ and $d$ such that the
following holds. Let $n\geq 1$, $i\geq 1$. Let $t_{0}<t_{1}\in[n,n+1)$. We
have

$F_{n}^{-1}(t_{1})-F_{n}^{-1}(t_{0})\leq c(t_{1}-t_{0}),$

Moreover, if

$F_{n}^{-1}(t_{1})-F_{n}^{-1}(t_{0})\leq\frac{1}{2^{i}},$

then

$t_{1}-t_{0}\leq\frac{1}{2^{i}}.$

###### Proof of Proposition 4.3.

Fix $\lambda$ and $\delta_{0}<\delta_{1}$. Set

$g(n):=\frac{F(n+1)-F(n)}{d+\delta}\,$

and

$p(k,t):=\mathbb{P}(\mathcal{P}(t)\geq k+1|\mathcal{P}(t)\geq k)$

where $\mathcal{P}(t)$ is a Poisson random variable of parameter $t$.

#### Encoding the Poisson point process.

For each $\mathrm{S}_{n}^{x}$, we associate a Bernoulli random variable
$\alpha_{0}$ of parameter $p(0,\lambda g(n))$. This random variable will
encode the presence of at least one point in the cell $\mathrm{S}_{n}^{x}$.
For each $(\mathrm{S}_{n}^{x},k)$ with $k\geq 1$, we associate a Bernoulli
random variable $\alpha_{k}$ of parameter $p(k,\lambda g(n))$ (that will
encode the presence of a $k$-th ball in the cell $\mathrm{S}_{n}^{x}$), an
i.i.d. family $(\beta_{i})_{i\geq 1}$ of Bernoulli random variables of
parameter $1/2$ and an i.i.d. family
$(\gamma_{i}^{(1)},\dots,\gamma_{i}^{(d)})_{i\geq 1}$ of Bernoulli random
variables of parameter $1/2$ (that will encode respectively the radius and the
position of the $k$-th ball of the cell). Write $\mathbb{P}$ for the joint law
of these families of Bernoulli random variable. Finally, define for
$m\in\mathbb{N}\cup\\{+\infty\\}$

$z_{k}^{m}(\mathrm{S}_{n}^{x})=z_{k}^{m}:=x+\left(\sum_{i=1}^{m}\frac{\gamma_{i}^{(1)}}{2^{i}},\dots,\sum_{i=1}^{m}\frac{\gamma_{i}^{(d)}}{2^{i}}\right)-\left(\frac{1}{2},\dots,\frac{1}{2}\right)$

and

$r_{k}^{m}(\mathrm{S}_{n}^{x})=r_{k}^{m}:=F_{n}^{-1}\left(\sum_{i=1}^{m}\frac{\beta_{i}}{2^{i}}\right)\,.$

When $m=+\infty$, we write $z_{k}$ and $r_{k}$ instead of $z_{k}^{+\infty}$
and $r_{k}^{+\infty}$. It is easy to check that $z_{k}$ is distributed
uniformly on $x+[-1/2,1/2)^{d}$ and $r_{k}$ is distributed given the
distribution $(d+\delta)\mu_{\delta}$ conditioned on being in the interval
$[n,n+1]$. Denote by $\eta_{\mathrm{S}_{n}^{x}}$ the following random set:

$\eta_{\mathrm{S}_{n}^{x}}:=\left\\{(z_{k},r_{k}):\prod_{i=0}^{k-1}\alpha_{i}=1\right\\}.$

We have

$\mathbb{P}(|\eta_{\mathrm{S}_{n}^{x}}|=k)=\mathbb{P}\left(\alpha_{k}=0,\prod_{i=0}^{k-1}\alpha_{i}=1\right)=(1-p(k,\lambda
g(n)))\prod_{i=0}^{k-1}p(i,\lambda g(n))=\mathbb{P}(\mathcal{P}(\lambda
g(n))=k).$

Hence, the set $\eta_{\mathrm{S}_{n}^{x}}$ is a Poisson point process of
intensity $\lambda dz\otimes\mu_{\delta}$ restricted to the cell
$\mathrm{S}_{n}^{x}$. It yields that the set
$\eta=\cup_{x,n}\eta_{\mathrm{S}_{n}^{x}}$ is a Poisson point process of
intensity $\lambda dz\otimes\mu_{\delta}$. In particular, we have

$\mathbb{P}(\eta\in\mathcal{E})=\mathbb{P}_{\delta}(\mathcal{E}).$ (4.1)

#### Approximation of the Poisson point process using a finite encoding.

In order to apply Talagrand formula, we need to approximate the Poisson point
process with a finite encoding. First, since the event $\mathcal{E}$ is
finite, there exists $N$ such that the event only depends on balls in
$\cup_{x\in\mathrm{B}_{N}\cap\mathbb{Z}^{d},n\geq 1}\mathrm{S}_{n}^{x}$. Let
$K\geq 1$. Denote by $\mathcal{F}_{K}$ the event where there exists a cell
with more than $K$ balls or a ball whose radius is larger than $KN$:

$\mathcal{F}_{K}:=\left\\{\exists(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[0,KN]:\,|\eta_{\mathrm{S}_{n}^{x}}|\geq
K\right\\}\cup\left\\{\eta\cap(\mathrm{B}_{N}\times[KN,+\infty))\neq\emptyset\right\\}.$

It is easy to check that

$\lim_{K\rightarrow\infty}\mathbb{P}(\mathcal{F}_{K})=0\,.$

Write $\overline{z}_{k}$ and $\overline{r}_{k}$ for $z_{k}^{K}$ and
$r_{k}^{K}$. We now define $\overline{\eta}$ the approximate version of $\eta$
defined by truncating up to level $K$:

$\overline{\eta}_{\mathrm{S}_{n}^{x}}:=\left\\{(\overline{z}_{k},\overline{r}_{k}):\prod_{i=0}^{k-1}\alpha_{i}=1,\,k\leq
K\right\\}$

and

$\overline{\eta}:=\bigcup_{(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[0,KN]}\overline{\eta}_{\mathrm{S}_{n}^{x}}.$

Note that on the event $\mathcal{F}_{K}^{c}$, we have

$\overline{\eta}\in\mathcal{N}_{N}(\eta,\varepsilon_{K})$ (4.2)

where $\varepsilon_{K}$ depends on $K$ and goes to $0$ when $K$ goes to
infinity. Fix $(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[0,KN]$. For
$(z,r)\in\mathrm{S}_{n}^{x}$, we define its projection $(\pi(z),\pi_{n}(r))$
as follows

$(\pi(z),\pi_{n}(r)):=\left(2^{-K}\lfloor
2^{K}z\rfloor,F_{n}^{-1}\left(2^{-K}\left\lfloor
2^{K}\frac{F(r)-F(n)}{F(n+1)-F(n)}\right\rfloor\right)\right)\,.$

In particular we have

$(\overline{z}_{k},\overline{r}_{k})=(\pi(z_{k}),\pi_{n}(r_{k}))\,.$

Thanks to Claim 4.8, we have for $r\in[n,n+1)$

$0\leq r-\pi_{n}(r)\leq c2^{-K}.$ (4.3)

We need to compute the influence of all the variables on the event
$\mathbf{1}_{\overline{\eta}\in\mathcal{E}}$. We define the influence as
follows

$\operatorname{Inf}_{\alpha_{0}}^{(x,n)}(\mathcal{E}):=\mathbb{P}(\mathbf{1}_{\overline{\eta}\in\mathcal{E}}\circ\tau_{\alpha_{0}}^{(x,n)}\neq\mathbf{1}_{\overline{\eta}\in\mathcal{E}})$

where $\tau_{\alpha_{0}}^{(x,n)}$ is the function that switches the value of
the bit corresponding to the variable $\alpha_{0}$ of the cell
$\mathrm{S}_{n}^{x}$. The other influences are defined similarly.

#### Influence of $\alpha_{0}$.

The variable $\alpha_{0}$ dictates if there is at least one point in
$\mathrm{S}_{n}^{x}$. If the variable is influential then the cell
$\mathrm{S}_{n}^{x}$ is pivotal. We say that the cell is pivotal if
$\overline{\eta}\setminus\mathrm{S}_{n}^{x}\notin\mathcal{E}$ and
$\overline{\eta}\setminus\mathrm{S}_{n}^{x}\cup\\{(x,n+1+\sqrt{d})\\}\in\mathcal{E}$.
It yields that

$\begin{split}\operatorname{Inf}_{\alpha_{0}}^{(x,n)}(\mathcal{E})&\leq\mathbb{P}(\overline{\eta}\setminus\mathrm{S}_{n}^{x}\notin\mathcal{E},\overline{\eta}\setminus\mathrm{S}_{n}^{x}\cup\\{(x,n+1+\sqrt{d})\\}\in\mathcal{E})\\\
&=\frac{\mathbb{P}(\overline{\eta}\notin\mathcal{E},\overline{\eta}\cup\\{(x,n+1+\sqrt{d})\\}\in\mathcal{E})}{\mathbb{P}(\overline{\eta}\cap\mathrm{S}_{n}^{x}=\emptyset)}.\end{split}$

We have

$\begin{split}\mathbb{P}&(\overline{\eta}\notin\mathcal{E},\overline{\eta}\cup\\{(x,n+1+\sqrt{d})\\}\in\mathcal{E})\\\
&=\frac{1}{g(n+2d)}\int_{(z,r)\in\mathrm{S}_{n+2d}^{x}}\mathbb{P}(\overline{\eta}\notin\mathcal{E},\overline{\eta}\cup\\{(x,n+1+\sqrt{d})\\}\in\mathcal{E})dzd\mu_{\delta}(r)\\\
&\leq\frac{1}{g(n+2d)}\int_{(z,r)\in\mathrm{S}_{n+2d}^{x}}\mathbb{P}(\overline{\eta}\notin\mathcal{E},\overline{\eta}\cup\\{(z,r)\\}\in\mathcal{E})dzd\mu_{\delta}(r)\end{split}$

and using (4.2)

$\frac{1}{g(n+2d)}\int_{(z,r)\in\mathrm{S}_{n+2d}^{x}}\mathbb{P}(\overline{\eta}\notin\mathcal{E},\overline{\eta}\cup\\{(z,r)\\}\in\mathcal{E})dzd\mu_{\delta}(r)\leq\mathbb{P}(\mathcal{F}_{K})+\frac{\mathrm{Piv}_{n+2d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+2d)}.$

Besides, we have

$\mathbb{P}(\overline{\eta}\cap\mathrm{S}_{n}^{x}=\emptyset)=\exp(-\lambda
g(n))\geq\exp(-\lambda)\,.$

Finally, we have

$\operatorname{Inf}_{\alpha_{0}}^{(x,n)}(\mathcal{E})\leq\left(\mathbb{P}(\mathcal{F}_{K})+\frac{\mathrm{Piv}_{n+2d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+2d)}\right)\exp(\lambda).$

#### Influence of $\alpha_{k}$ for $k\leq K-1$.

For $\alpha_{k}$ to be influential, we must have that
$\prod_{i=0}^{k-1}\alpha_{i}=1$. This occurs with probability
$\mathbb{P}(\mathcal{P}(\lambda g(n))\geq k)$. Moreover, the cell
$\mathrm{S}_{n}^{x}$ must be pivotal. By the same computations as above, we
have

$\operatorname{Inf}_{\alpha_{k}}^{(x,n)}(\mathcal{E})\leq\mathbb{P}(\mathcal{P}(\lambda
g(n))\geq
k)\left(\mathbb{P}(\mathcal{F}_{K})+\frac{\mathrm{Piv}_{n+2d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+2d)}\right)\exp(\lambda).$

Fix $1\leq k\leq K$. We now consider the influence of variables encoding the
position and the radius of the $k$-th ball of the cell $\mathrm{S}_{n}^{x}$.

#### Influence of $\gamma_{j}^{(1)}$ for $j\leq K$.

For $\gamma_{j}^{(1)}$ to be influential, the $k$-th ball of the cell
$\mathrm{S}_{n}^{x}$ has to be present, that is
$\prod_{i=0}^{k-1}\alpha_{i}=1$. This occurs with probability
$\mathbb{P}(\mathcal{P}(\lambda g(n))\geq k)$.

Denote by $\Lambda$ the random box of side length $2^{-j+1}$ centered at a
point in $2^{-j+1}\mathbb{Z}^{d}$ that contains $\overline{z}_{k}$. Set
$\overline{\eta}^{(k)}=\overline{\eta}\setminus\\{(\overline{z}_{k},\overline{r}_{k})\\}$.
For $\gamma_{j}^{(1)}$ to be pivotal the function
$z\mapsto\mathbf{1}_{\overline{\eta}^{(k)}\cup\\{(z,\overline{r}_{k})\\}\in\mathcal{E}}$
must be non-constant on $\Lambda$. Let us assume that the function is non-
constant on $\Lambda$. Set

$a:=\inf\\{t\geq n:\exists
z\in\Lambda\quad\overline{\eta}^{(k)}\cup\\{(z,t)\\}\in\mathcal{E}\\}$

and

$b:=\sup\\{t\leq n+1:\exists
z\in\Lambda\quad\overline{\eta}^{(k)}\cup\\{(z,t)\\}\notin\mathcal{E}\\}.$

It is easy to check that
$\overline{\eta}^{(k)}\cup\\{(w,a+d2^{-j+1})\\}\in\mathcal{E}$ for all
$w\in\Lambda$. Hence, we have $a\leq b\leq a+d2^{-j+1}$. If $\gamma_{j}^{(1)}$
is pivotal, then $\overline{r}_{k}=\pi_{n}(r_{k})\in[a,b]$ and by inequality
(4.3), we have $\overline{r}_{k}\in[a,b+c2^{-K}]$. Using Claim 4.8, we have

$\begin{split}&\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}\left(\overline{\eta}^{(k)}\notin\mathcal{E},\overline{\eta}^{(k)}\cup\\{(\pi(z),\pi_{n}(r))\\}\in\mathcal{E},\pi_{n}(r)\in[a,b]\right)dzd\mu_{\delta}(r)\\\
&\quad\leq\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}\left(\overline{\eta}\setminus\mathrm{S}_{n}^{x}\notin\mathcal{E},\overline{\eta}\setminus\mathrm{S}_{n}^{x}\cup\\{(z,n+d)\\}\in\mathcal{E},r\in[a,b+c2^{-K}]\right)dzd\mu_{\delta}(r)\\\
&\quad\leq
2c^{2}d2^{-j+1}g(n)\int_{z\in(x+[-1/2,1/2)^{d})}\mathbb{P}\left(\overline{\eta}\setminus\mathrm{S}_{n}^{x}\notin\mathcal{E},\overline{\eta}\setminus\mathrm{S}_{n}^{x}\cup\\{(z,n+d)\\}\in\mathcal{E}\right)dzd\mu_{\delta}(r)\\\
&\quad\leq
c^{2}d2^{-j+2}\frac{g(n)}{g(n+d)}\int_{(z,r)\in\mathrm{S}_{n+d}^{x}}\mathbb{P}\left(\overline{\eta}\setminus\mathrm{S}_{n}^{x}\notin\mathcal{E},\overline{\eta}\setminus\mathrm{S}_{n}^{x}\cup\\{(z,n+d)\\}\in\mathcal{E}\right)dz\\\
&\quad\leq
c^{2}d2^{-j+2}\frac{g(n)}{g(n+d)}(\mathrm{Piv}_{n+d}^{x,\varepsilon_{K}}(\mathcal{E})+g(n+d)\mathbb{P}(\mathcal{F}_{K}))\exp(\lambda).\end{split}$

Finally, we have

$\operatorname{Inf}_{\gamma_{j}^{(1)}}^{(x,n,k)}(\mathcal{E})\leq\mathbb{P}(\mathcal{P}(\lambda
g(n))\geq
k)c^{2}d2^{-j+2}\left(\frac{\mathrm{Piv}_{n+d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+d)}+\mathbb{P}(\mathcal{F}_{K})\right)\exp(\lambda).$

The same computations also hold for $\gamma_{j}^{(l)}$ for
$l\in\\{2,\dots,d\\}$.

#### Influence of $\beta_{j}$ for $j\leq K$.

Again for $\beta_{j}$ to be influential, the $k$-th ball has to be present,
that is, we have $\prod_{i=0}^{k-1}\alpha_{i}=1$. This event occurs with
probability $\mathbb{P}(\mathcal{P}(\lambda g(n))\geq k)$. Set
$\overline{\eta}^{(k)}=\overline{\eta}\setminus\\{(\overline{z}_{k},\overline{r}_{k})\\}$.
For $\beta_{j}$ to be influential the function
$r\mapsto\mathbf{1}_{\overline{\eta}^{(k)}\cup\\{(\overline{z}_{k},r)\\}\in\mathcal{E}}$
must be non-constant on $[n,n+1]$. Set for $z\in(x+[-1/2,1/2)^{d})$

$\mathfrak{r}(z):=\inf\\{t\geq
n:\overline{\eta}^{(k)}\cup\\{(z,t)\\}\in\mathcal{E}\\}.$

By Claim 4.8, it is easy to check that if
$\overline{r}_{k}\notin[\mathfrak{r}(\overline{z}_{k})-2^{-j},\mathfrak{r}(\overline{z}_{k})+2^{-j}]$
then $\beta_{j}$ is not pivotal. By similar computations as in the previous
case, we have

$\begin{split}\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}&\left(\begin{array}[]{c}\overline{\eta}^{(k)}\notin\mathcal{E},\overline{\eta}^{(k)}\cup\\{(\pi(z),\pi_{n}(r))\\}\in\mathcal{E},\\\
\pi_{n}(r)\in[\mathfrak{r}(\pi(z))-2^{-j},\mathfrak{r}(\pi(z))+2^{-j}]\end{array}\right)dzd\mu_{\delta}(r)\\\
&\leq\int_{(z,r)\in\mathrm{S}_{n}^{x}}\mathbb{P}\left(\begin{array}[]{c}\overline{\eta}\setminus\mathrm{S}_{n}^{x}\notin\mathcal{E},\overline{\eta}\setminus\mathrm{S}_{n}^{x}\cup\\{(z,n+d)\\}\in\mathcal{E},\\\
r\in[\mathfrak{r}(\pi(z))-2^{-j},\mathfrak{r}(\pi(z))+2^{-j}+c2^{-K}]\end{array}\right)dzd\mu_{\delta}(r)\\\
&\leq
c^{2}2^{-j+2}\frac{g(n)}{g(n+d)}(\mathrm{Piv}_{n+d}^{x,\varepsilon_{K}}(\mathcal{E})+g(n+d)\mathbb{P}(\mathcal{F}_{K}))\exp(\lambda).\end{split}$

Finally,

$\operatorname{Inf}_{\beta_{j}}^{(x,n,k)}(\mathcal{E})\leq\mathbb{P}(\mathcal{P}(\lambda
g(n))\geq
k)c^{2}2^{-j+2}\left(\frac{\mathrm{Piv}_{n+d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+d)}+\mathbb{P}(\mathcal{F}_{K})\right)\exp(\lambda).$

#### Application of inhomogeneous Talagrand.

First, it is easy to check that $p(k,g(n))$ is increasing in $k$ and

$\forall k\geq 0\qquad|\log p(k,\lambda g(n))|\leq|\log p(0,\lambda g(n))|.$

Besides, there exists a constant $C_{0}$ depending only on
$\delta_{0},\delta_{1}$, $d$ and $\lambda$ such that for any $n\geq 1$ and
$\delta\in[\delta_{0},\delta_{1}]$

$|\log p(0,\lambda
g(n))|=-\log\left(1-\exp\left(-\frac{\lambda}{d+\delta}\left(\frac{1}{n^{d+\delta}}-\frac{1}{(n+1)^{d+\delta}}\right)\right)\right)\leq
C_{0}\log(n+1).$

Using the equality

$\sum_{k\geq 1}\mathbb{P}(\mathcal{P}(\lambda g(n))\geq k)=\lambda g(n),$

we get

$\begin{split}\sum_{(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[0,N]}&\Bigg{(}\sum_{k=0}^{K-1}\Big{(}p(k,\lambda
g(n))|\log p(k,\lambda
g(n))|\operatorname{Inf}_{\alpha_{k}}^{(x,n)}(\mathcal{E})\\\ &\hskip
85.35826pt+\frac{\log
2}{2}\sum_{j=1}^{K}\Big{(}\sum_{l=1}^{d}\operatorname{Inf}_{\gamma_{j}^{(l)}}^{(x,n,k+1)}(\mathcal{E})+\operatorname{Inf}_{\beta_{j}}^{(x,n,k+1)}(\mathcal{E})\Big{)}\Big{)}\Bigg{)}\\\
&\leq\lambda
e^{\lambda}\sum_{(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[0,N]}g(n)\Bigg{(}C_{0}\log(n+1)\,\left(\frac{\mathrm{Piv}_{n+2d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+2d)}+\mathbb{P}(\mathcal{F}_{K})\right)\\\
&\hskip 85.35826pt+4c^{2}(d^{2}+1)\log
2\left(\frac{\mathrm{Piv}_{n+d}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+d)}+\mathbb{P}(\mathcal{F}_{K})\right)\Bigg{)}.\end{split}$

We can now apply Proposition 4.1 and the previous inequality

$\begin{split}&\lambda
e^{\lambda}C^{\prime}\sum_{(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[0,N]}\log(n+1)\,(\mathrm{Piv}_{n+2d}^{x,\varepsilon_{K}}(\mathcal{E})+g(n)\mathbb{P}(\mathcal{F}_{K}))\\\
&\geq
C\mathbb{P}(\overline{\eta}\in\mathcal{E})(1-\mathbb{P}(\overline{\eta}\in\mathcal{E}))\left|\log\left(\max_{(x,n)\in(\mathrm{B}_{N}\cap\mathbb{Z}^{d})\times[d,N]}2g(n)\lambda
e^{\lambda}c^{2}\left(\frac{\mathrm{Piv}_{n}^{x,\varepsilon_{K}}(\mathcal{E})}{g(n+d)}+\mathbb{P}(\mathcal{F}_{K})\right)\right)\right|\end{split}$
(4.4)

where $C^{\prime}$ is a constant depending only on $\delta_{0},\delta_{1}$,
$\lambda$ and $d$. Recall that we have the equality in law (4.1). Besides, we
have

$|\mathbb{P}(\overline{\eta}\in\mathcal{E})-\mathbb{P}(\eta\in\mathcal{E})|\leq\mathbb{P}(\mathcal{F}_{K})+\mathbb{P}(\mathrm{B}(\partial\mathcal{E},\varepsilon_{K})).$

Since $\mathbb{P}(\eta\in\partial\mathcal{E})=0$, we have

$\lim_{K\rightarrow\infty}\mathbb{P}(\overline{\eta}\in\mathcal{E})=\mathbb{P}(\eta\in\mathcal{E})\,.$

The result follows by letting $K$ goes to infinity in inequality (4.4) and by
Lemma 4.7.

∎

###  4.4 Sharp threshold for Boolean percolation

In this section, we prove a more general version of Proposition 2.2 using the
result of the previous section. We say that $\mathcal{E}$ is rotational
invariant if for any rotation $\rho$ of center $0$, we have

$\rho(\mathcal{E}):=\\{\\{(\rho(x),r):(x,r)\in\eta\\}:\eta\in
E\\}=\mathcal{E}.$

The following proposition proves a sharp threshold for finite rotational
invariant events not depending on the balls centered in $\mathrm{B}_{n}$.

######  Proposition 4.9.

Let $\lambda,\delta>0$. There exists $\kappa>0$ such that the following holds.
For every $\varepsilon\in(0,\delta/2\kappa)$, $n\geq\kappa$ and
$\mathcal{E}\subset\mathbb{R}^{d}\times\mathbb{R}_{+}$ finite increasing event
that does not depend on balls centered in $\mathrm{B}_{n}$, such that
$\mathcal{E}$ is rotational invariant and
$\mathbb{P}_{\delta_{0}}(\partial\mathcal{E})=0$ for any $\delta_{0}>0$, we
have

$\mathbb{P}_{{\delta}}(\mathcal{E})\geq\frac{1}{n^{\varepsilon}}\implies\mathbb{P}_{{\delta-\kappa\varepsilon}}(\mathcal{E})\geq
1-\frac{1}{n^{\varepsilon}}.$

###### Proof.

Let $\lambda,\delta_{1}>0$. In this proof $\lambda$ is omitted from the
notations as it remains fixed. Set $\delta_{0}=\delta_{1}/2$. In what follows,
we will work with $\delta\in[\delta_{0},\delta_{1}]$. Let $\kappa$ that will
be defined later. Let $n\geq\kappa$ and $\mathcal{E}$ as in the statement of
the proposition.

Let us first compute the derivative of
$\delta\mapsto\mathbb{P}_{\delta}(\mathcal{E})$. We have

$\begin{split}\mu_{\delta-t}=\frac{1}{r^{d+1+\delta-t}}\mathbf{1}_{r\geq
1}dr=r^{t}\mu_{\delta}&=\left(1+\exp(t\log r)-1\right)\mu_{\delta}\\\
&=\left(1+t\log r+tF_{r}(t)\right)\mu_{\delta}\end{split}$

where $F_{r}(t):=(\exp(t\log r)-1)/t-\log r$. We have

$\lim_{t\rightarrow 0}\sup_{r\in[0,N]}|F_{r}(t)|=0\,.$

We can apply Theorem 6.6 of [13] to compute the derivative of
$\delta\mapsto\mathbb{P}_{\delta}(\mathcal{E})$, we get

$\frac{\partial}{\partial\delta}\mathbb{P}_{\delta}(\mathcal{E})=-\lambda\int_{(z,r)\in\mathbb{R}^{d}\times\mathbb{R}_{+}}\mathbb{P}_{\delta}(\eta\notin\mathcal{E},\eta\cup\\{(z,r)\\}\in\mathcal{E})\log
r\,dzd\mu_{\delta}(r).$

We now distinguish two cases.

#### First case

We assume that there exists $x\in\mathbb{Z}^{d}$ and $m\geq 1$ such that
$\mathrm{Piv}_{m}^{x}(\mathcal{E})\geq 1/\sqrt{n}$. Without loss of
generality, we can assume that $m\geq 2$ since
$\mathrm{Piv}_{2}^{x}(\mathcal{E})\geq c\mathrm{Piv}_{1}^{x}(\mathcal{E})$ for
some constant $c$ depending on $\delta_{1}$. Since $\mathcal{E}$ does not
depend on balls centered in $\mathrm{B}_{n}$, we have
$x\notin\mathrm{B}_{n-d}$. It is easy to check that there exist $c_{d}>0$
depending only on $d$,
$\mathrm{S}_{1},\dots,\mathrm{S}_{K}\subset\mathbb{R}^{d}$ with $K\geq
c_{d}n^{d-1}$ such that all the $\mathrm{S}_{i}$ are disjoint and for every
$i\in\\{1,\dots,K\\}$, there exists a rotation sending $x+[-1/2,1/2)^{d}$ on
$\mathrm{S}_{i}$. By rotation invariance, it follows that

$\int_{(z,r)\in\mathrm{S}_{i}\times[m,m+1)}\mathbb{P}_{\delta}(\eta\notin\mathcal{E},\eta\cup\\{(z,r)\\}\in\mathcal{E})dzd\mu_{\delta}(r)\geq\frac{1}{\sqrt{n}}$

and

$-\frac{\partial}{\partial\delta}\mathbb{P}_{\delta}(\mathcal{E})=\lambda\int_{(z,r)}\mathbb{P}_{\delta}(\eta\notin\mathcal{E},\eta\cup\\{(z,r)\\}\in\mathcal{E})\log
r\,dzd\mu_{\delta}(r)\geq c_{d}\lambda\log d\,n^{d-3/2}\,.$

#### Second case

We assume that for every $x\in\mathbb{Z}^{d}$ and $m\geq d$, we have
$\mathrm{Piv}_{m}^{x}(\mathcal{E})\leq 1/\sqrt{n}$. By Proposition 4.3, there
exists $C_{0}$ (depending on $\lambda$ and $\delta_{1}$) such that

$\sum_{x\in\mathbb{Z}^{d},k\geq 1}\log
k\,\mathrm{Piv}_{k}^{x}(\mathcal{E})\geq
C_{0}\,\mathbb{P}_{\delta}(\mathcal{E})(1-\mathbb{P}_{\delta}(\mathcal{E}))\log\left(\frac{1}{\max_{x\in\mathbb{Z}^{d},k\geq
1}\mathrm{Piv}_{k}^{x}(\mathcal{E})}\right)\,.$ (4.5)

We have

$\sum_{x\in\mathbb{Z}^{d},k\geq 1}\log
k\,\mathrm{Piv}_{k}^{x}(\mathcal{E})\leq-\frac{1}{\lambda}\frac{\partial}{\partial\delta}\mathbb{P}_{\delta}(\mathcal{E}).$

By using inequality (4.5):

$-\frac{\partial}{\partial\delta}\mathbb{P}_{\delta}(\mathcal{E})\geq\frac{2}{\kappa}\mathbb{P}_{\delta}(\mathcal{E})(1-\mathbb{P}_{\delta}(\mathcal{E}))\log
n$ (4.6)

where $\kappa$ depends on $\lambda$ and $\delta_{1}$.

#### Conclusion

Finally, set for $t\in[0,\delta_{1}/2\kappa]$

$\delta(t):=\delta_{1}-t\kappa$

and

$f(t):=\mathbb{P}_{\delta(t)}(\mathcal{E}).$

We have for $t\in[0,\delta_{1}/2\kappa]$,
$\delta(t)\in[\delta_{0},\delta_{1}]$. We have for
$t\in[0,\delta_{1}/2\kappa]$ using inequality (4.6) for the second case

$\begin{split}f^{\prime}(t)\geq-\kappa\frac{\partial}{\partial\delta}\mathbb{P}_{\delta(t)}(\mathcal{E})\geq
2f(t)(1-f(t))\log n\end{split}$

Hence, we get

$f^{\prime}(t)\geq\min(C_{d}\lambda n^{d-3/2},2f(t)(1-f(t))\log n)$

where $C_{d}$ depends on $d$. Up to increasing $\kappa$, we can assume that
for $n\geq\kappa$, we have $C_{d}\lambda n^{d-3/2}\geq\log n$. Hence, for
$n\geq\kappa$, we have

$f^{\prime}(t)\geq 2f(t)(1-f(t))\log n.$

By integrating between $0$ and $t_{0}\in[0,\delta_{1}/2\kappa]$, we obtain

$\log\left(\frac{f(t_{0})}{1-f(t_{0})}\frac{1-f(0)}{f(0)}\right)\geq
2t_{0}\log n.$

It yields

$\frac{1}{(1-f(t_{0}))f(0)}\geq\exp\left(2t_{0}\log n\right)$

and

$f(t_{0})\geq 1-n^{\varepsilon}\exp\left(-2t_{0}\log n\right).$

Hence, for $n\geq\kappa$, we have

$\mathbb{P}_{\delta_{1}-\varepsilon\kappa}(\mathcal{E})\geq
1-\frac{1}{n^{\varepsilon}}.$

The result follows.

∎

##  5 Grimmett–Marstrand dynamic renormalization

In this section $\mu$ is a general distribution with a finite $d$-moment. We
aim to prove Proposition 2.3. For $B,E,F\subset\mathbb{R}^{d}$ and $n\geq 1$,
denote

$\mathcal{E}_{n}(B,E,F):=\left\\{\exists(x,r)\in\eta\cap(E\times[n,+\infty)):B\stackrel{{\scriptstyle\mathcal{O}(\eta\cap(F\times\mathbb{R}))}}{{\longleftrightarrow}}\mathrm{B}_{r}^{x}\right\\}.$

For short, we will write $\mathcal{E}_{n}(E,F)$ instead of
$\mathcal{E}_{n}(\mathrm{B}_{n},E,F)$.

For $n\geq 1$, we define $\Lambda_{n}:=[-n,n]^{d}$. We will first need the
following lemma.

######  Lemma 5.1.

There exists a positive constant $c^{\prime}_{d}$ depending only on $d$ such
that for $N\geq n\geq 1$, $\varepsilon>0$ such that for any
$\rho\in[1,\sqrt{d}+2]$ we have

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(\mathrm{B}_{(\rho+1/2)N}\setminus\mathrm{B}_{\rho
N},B_{(\rho+1/2)N})\geq 1-\varepsilon,$

then for any $z\in\mathbb{R}^{d}$ such that $N\leq\|z\|_{2}\leq(\sqrt{d}+2)N$,
we have

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(z+\Lambda_{N},\mathrm{B}_{3\sqrt{d}N}))\geq
1-\varepsilon^{1/c^{\prime}_{d}}.$

###### Proof.

Let $\rho\in[1,\sqrt{d}+2]$. There exists a constant $c^{\prime}_{d}$
depending only on $d$ and a family of balls $(\mathrm{B}_{N}^{x_{i}})_{1\leq
i\leq m_{0}}$ with $m_{0}\leq c^{\prime}_{d}$, such that for all $i\leq
m_{0}$, $x_{i}\in\partial\mathrm{B}_{\rho N}$ and

$\mathrm{B}_{(\rho+1/2)N}\setminus\mathrm{B}_{\rho N}\subset\bigcup_{1\leq
i\leq m_{0}}\mathrm{B}_{N}^{x_{i}}.$

It follows that

$\mathbb{P}_{\lambda,\mu}\left(\bigcup_{1\leq i\leq
m_{0}}\mathcal{E}_{n}(\mathrm{B}_{N}^{x_{i}},\mathrm{B}_{(\rho+1/2)N})\right)\geq
1-\varepsilon.$

By invariance under rotation around $0$, and using the square root trick, we
have

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(\mathrm{B}_{N}+\rho
N\textbf{e}_{1},\mathrm{B}_{(\rho+1/2)N}))\geq
1-\varepsilon^{1/c^{\prime}_{d}}$

where $\textbf{e}_{1}=(1,0,\dots,0)$. Finally, we get for any
$z\in\mathbb{\mathbb{R}}^{d}$ such that $N\leq\|z\|_{2}\leq(\sqrt{d}+2)N$

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(z+\Lambda_{N},\mathrm{B}_{3\sqrt{d}N}))\geq\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(\mathrm{B}_{N}+\|z\|_{2}\textbf{e}_{1},\mathrm{B}_{\|z\|_{2}+N/2}))\geq
1-\varepsilon^{1/c^{\prime}_{d}}.$

∎

###  5.1 Sprinkling

Let $n\geq N$ and $\lambda>0$. For $A\subset\mathbb{R}^{d}$, we denote by
$\Delta A$ the set of all balls intersecting the boundary of $A$, that is,

$\Delta
A:=\left\\{(x,r)\in\mathbb{R}^{d}\times\mathbb{R}_{+}:\mathrm{B}_{r}^{x}\cap\partial
A\neq\emptyset\right\\}.$

In the following sprinkling lemma we consider both the connection with a
deterministic set (that correspond to the set $C$) and with a random set
(balls in the set $B$).

######  Lemma 5.2 (Sprinkling lemma).

Let $\xi>0$, $\beta>0$ and $\lambda>0$. Let $A\subset R\subset\mathbb{R}^{d}$.
Let $B\subset(R\times\mathbb{R}_{+})\setminus\Delta A$. Let $C\subset
R\setminus A$. Let us assume that

$\mathbb{P}_{\lambda,\mu}(A\stackrel{{\scriptstyle
R}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta)\cup C)\geq
1-\exp(-3\lambda/\xi)$ (5.1)

then we have

$\mathbb{P}(A\stackrel{{\scriptstyle\mathcal{O}((\eta\cup\eta^{\prime})\cap(R\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta)\cup
C\,|\,\eta\cap\Delta A=\emptyset)\geq 1-\exp(-\beta)-\exp(-\lambda/\xi)$

where $\eta$ (respectively $\eta^{\prime}$) is a Poisson point process of
intensity $\lambda dz\otimes\mu$ (respectively $\beta\xi dz\otimes\mu$).

###### Proof of Lemma 5.2.

To prove this lemma, it is more convenient to work in a discrete setting
instead of the continuous setting. For that purpose, we introduce a very dense
set of points in an enlarged space in a similar way as in [1]. Set
$\overline{\Omega}=\mathbb{R}^{d}\times\mathbb{R}_{+}\times[0,1]$. Let
$\overline{\eta}$ be a Poisson point process of intensity
$m\lambda_{1}dz\otimes\mu\otimes du$ where $\lambda_{1}=2(\lambda+\beta\xi)$.
On top of every point $z$ of $\overline{\eta}$, we add independently a uniform
random variable $U_{z}$ on $[0,1]$. Set

$\eta_{\lambda}:=\left\\{(x,r):\exists
u\leq\frac{1}{m}\quad(x,r,u)\in\overline{\eta},\quad
U_{(x,r,u)}\leq\frac{\lambda}{\lambda_{1}}\right\\}.$

It is easy to check that the set $\eta_{\lambda}$ is a Poisson point process
of intensity $\lambda dz\otimes\mu$. It follows that

$\mathbb{E}\left[\mathbb{P}\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda}\cap(R\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\overline{\eta}\Big{)}\right]=\mathbb{P}_{\lambda,\mu}(A\stackrel{{\scriptstyle
R}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta)\cup C).$

Denote $\pi$ the projection from $\overline{\Omega}$ to
$\mathbb{R}^{d}\times\mathbb{R}_{+}$

$\pi(\overline{\eta}):=\\{(x,r):(x,r,u)\in\overline{\eta}\\}.$

By similar argument than in [1], we have

$\operatorname{Var}\left(\mathbb{P}\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda}\cap(R\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\pi(\overline{\eta})\Big{)}\right)\leq\frac{1}{m}.$ (5.2)

Let $W$ be the sites in $(\Delta A\times[0,1/m])\cap\overline{\eta}$ that are
connected to $\mathcal{O}(B\cap\eta_{\lambda})\cup C$ in
$\mathcal{O}((\eta_{\lambda}\setminus\Delta A)\cap(R\times\mathbb{R}_{+}))$,
that is

$W:=\left\\{(x,r,u)\in\left(\Delta
A\times\left[0,\frac{1}{m}\right]\right)\cap\overline{\eta}:\mathrm{B}_{r}^{x}\stackrel{{\scriptstyle\mathcal{O}((\eta_{\lambda}\setminus\Delta
A)\cap(R\times\mathbb{R}_{+})))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\right\\}$

Conditionally on $\pi(\overline{\eta})$, the sites in $W$ are independent of
$(U_{z})_{z\in W}$. We have

$\mathbb{P}\left(A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda}\cap(R\times\mathbb{R}_{+}))}}{{\not\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,\Big{|}\,\pi(\overline{\eta})\right)\geq(1-\lambda/\lambda_{1})^{t-1}\mathbb{P}(|W|<t\,|\,\pi(\overline{\eta})).$

Let $\mathcal{E}_{0}$ be the following event

$\mathcal{E}_{0}:=\left\\{\eta_{\lambda}\cap\Delta A=\emptyset\right\\}.$

It yields that

$\begin{split}\mathbb{P}&\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda+\beta\xi}\cap(R\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\mathcal{E}_{0},\pi(\overline{\eta})\Big{)}\\\ &\hskip
85.35826pt\geq\mathbb{P}(\exists z\in W\quad
U_{z}\in(\lambda/\lambda_{1},(\lambda+\beta\xi)/\lambda_{1})],\,|W|\geq
t\,|\,\pi(\overline{\eta}))\\\ &\hskip
85.35826pt\geq(1-(1-\beta\xi/\lambda_{1})^{t})\mathbb{P}(|W|\geq
t\,|\,\pi(\overline{\eta}))\\\ &\hskip
85.35826pt\geq(1-(1-\beta\xi/\lambda_{1})^{t})\left(1-\frac{\mathbb{P}\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda}\cap(R\times\mathbb{R}_{+}))}}{{\not\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\pi(\overline{\eta}))\Big{)}}{(1-\lambda/\lambda_{1})^{t-1}}\right).\end{split}$

Finally by choosing $t=\lambda_{1}/\xi$ and using $-2x\leq\log(1-x)\leq-x$ for
$x\in[0,1/2)$, we get

$\begin{split}\mathbb{P}&\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda+\beta\xi}\cap(R\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\mathcal{E}_{0},\pi(\overline{\eta})\Big{)}\\\ &\hskip
56.9055pt\geq(1-\exp(-\beta))\Big{(}1-\exp(2\lambda/\xi)\mathbb{P}\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda}\cap(R\times\mathbb{R}_{+}))}}{{\not\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\pi(\overline{\eta})\Big{)}\Big{)}\\\ &\hskip 56.9055pt\geq
1-\exp(-\beta)-\exp(2\lambda/\xi)\mathbb{P}\Big{(}A\stackrel{{\scriptstyle\mathcal{O}(\eta_{\lambda}\cap(R\times\mathbb{R}_{+}))}}{{\not\longleftrightarrow}}\mathcal{O}(B\cap\eta_{\lambda})\cup
C\,|\,\pi(\overline{\eta})\Big{)}.\end{split}$

Using the assumption (5.1) and using inequality (5.2), for every $m$ large
enough such that $\mathbb{P}_{\lambda,\mu}(\eta\cap\Delta A=\emptyset)\geq
4/\sqrt{m}$, we have with positive probability using Markov inequality

$\begin{split}&\frac{\mathbb{P}\Big{(}A\stackrel{{\scriptstyle\mathcal{O}((\eta\cup\eta^{\prime})\cap(R\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathcal{O}(B\cap\eta)\cup
C,\eta\cap\Delta A=\emptyset\Big{)}+4/\sqrt{m}}{\mathbb{P}(\eta\cap\Delta
A=\emptyset)-4/\sqrt{m}}\\\ &\hskip 199.16928pt\geq
1-\exp(-\beta)-\exp(-\lambda/\xi)-\frac{4\exp(2\lambda/\xi)}{\sqrt{m}}\end{split}$

where $\eta$ (respectively $\eta^{\prime}$) is a Poisson point process of
intensity $\lambda dz\otimes\mu$ (respectively $\beta\xi dz\otimes\mu$). The
result follows by letting $m$ go to infinity. ∎

###  5.2 Dynamic exploration

In this section, we implement a Grimmett–Marstrand dynamic exploration to
prove Proposition 2.3. This section is inspired from [4]. Fix $\beta\geq 0$
large enough such that

$\exp(-\beta)\leq\frac{1-p_{c}^{site}}{3}$

and $\varepsilon_{0}>0$ small enough such that

$\varepsilon_{0}^{1/3}\leq\frac{1-p_{c}^{site}}{3}$

where $p_{c}^{site}$ is the critical parameter of Bernoulli site percolation
on $\mathbb{Z}^{2}$. Thanks to this choice, we have

$1-\exp(-\beta)-\varepsilon_{0}^{1/3}>p_{c}^{site}.$ (5.3)

Let $c^{\prime}_{d}$ be the constant of Lemma 5.1. Let us assume that there
exists $\varepsilon\leq\varepsilon_{0}^{c^{\prime}_{d}}$ and $n\leq N$ such
that we have for every $\rho\in[1,\sqrt{d}+2]$

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(\mathrm{B}_{(\rho+1/2)N}\setminus\mathrm{B}_{\rho
N},\mathrm{B}_{(\rho+1/2)N})\geq 1-\varepsilon.$

From now on, $\varepsilon,n,N$ are fixed such that the previous inequality
holds. By Lemma 5.1, for all $z\in\mathbb{\mathbb{R}}^{d}$ such that
$N\leq\|z\|_{2}\leq(\sqrt{d}+2)N$, we have

$\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n}(z+\Lambda_{N},\mathrm{B}_{3\sqrt{d}N}))\geq
1-\varepsilon^{1/c^{\prime}_{d}}\geq 1-\varepsilon_{0}.$ (5.4)

For every $x\in\mathbb{Z}^{2}$, set

$\Lambda_{x}:=2Nx+\Lambda_{N}\subset\mathbb{R}^{d}\qquad\text{and}\qquad\widetilde{B}_{x}:=2Nx+\mathrm{B}_{4\sqrt{d}N}\subset\mathbb{R}^{d}$

where we identify $x=(x_{1},x_{2})$ with
$(x_{1},x_{2},0,\dots,0)\in\mathbb{Z}^{d}$. Let $\eta$ be a Poisson point
process of intensity $\lambda dz\otimes\mu$. Let
$(\eta^{x},x\in\mathbb{Z}^{2})$ be a family of independent Poisson point
process of intensity $\beta\xi dz\textbf{1}_{\widetilde{B}_{x}}\otimes\mu$
where $\xi$ is such that $\varepsilon^{1/c^{\prime}_{d}}=\exp(-3\lambda/\xi)$.
We aim to prove that the box $\Lambda_{0}$ intersects an infinite connected
component of balls with positive probability in
$\mathcal{O}(\eta\cup_{x\in\mathbb{Z}^{2}}\eta^{x})$. For a configuration
$\eta$ we denote by $\mathscr{C}(\eta)$ the set of balls in $\eta$ that are
connected to $\mathrm{B}_{n}$ in $\mathcal{O}(\eta)$, that is,

$\mathscr{C}(\eta):=\bigcup\left\\{\mathrm{B}_{r}^{x}:(x,r)\in\eta,\mathrm{B}_{n}\stackrel{{\scriptstyle\mathcal{O}(\eta)}}{{\longleftrightarrow}}\mathrm{B}_{r}^{x}\right\\}.$

Set $X_{0}:=(A_{0},B_{0})=(\\{0\\},\emptyset)$,
$\mathcal{B}_{0}:=\mathrm{B}_{n}$ and $\eta_{0}:=\eta$. Let $t\geq 0$. We
build $\eta_{t+1}$, $\mathcal{B}_{t+1}$ and $X_{t+1}$ from $\eta_{t}$,
$\mathcal{B}_{t}$ and $X_{t}$ as follows. If there is no edge in
$(\mathbb{Z}^{2},\mathbb{E}^{2})$ connecting $A_{t}$ to $(A_{t}\cup
B_{t})^{c}$, then set $X_{t+1}:=X_{t}$, $\mathcal{B}_{t+1}:=\mathcal{B}_{t}$
$\eta_{t+1}:=\eta_{t}$. Otherwise, let us denote by $x_{t}$ the extremity in
$(A_{t}\cup B_{t})^{c}$ of this edge (if there are several such edges, we
choose one using a deterministic rule) and let $t_{0}<t$ be such that
$x_{t_{0}}$ is the other extremity of the edge. By definition, we have
$x_{t_{0}}\in A_{t}$. Define

$\eta_{t+1}:=\eta_{t}\cup\eta^{x_{t}}$

and

$(A_{t+1},B_{t+1}):=\left\\{\begin{array}[]{ll}(A_{t}\cup\\{x_{t}\\},B_{t})&\mbox{if
the event
$\mathcal{E}_{n}(\mathcal{B}_{t_{0}},\Lambda_{x_{t}},\widetilde{B}_{x_{t}})$
occurs for $\eta_{t+1}$}\\\
(A_{t},B_{t}\cup\\{x_{t}\\})&\mbox{otherwise.}\end{array}\right.$

If $x_{t}\in A_{t+1}$, to define $\mathcal{B}_{t}$, we pick according to some
deterministic rule a ball in

$\left\\{\mathrm{B}_{r}^{x}:\,(x,r)\in\eta_{t+1}\cap(\Lambda_{x_{t}}\times[n,+\infty)),\,\mathcal{B}_{t_{0}}\stackrel{{\scriptstyle\mathcal{O}(\eta_{t+1}\cap(\widetilde{B}_{x_{t}}\times\mathbb{R}_{+}))}}{{\longleftrightarrow}}\mathrm{B}_{r}^{x}\right\\}.$

Note that

$\eta_{\infty}:=\bigcup_{t\geq
0}\eta_{t}\subset\left(\eta\cup\bigcup_{x\in\mathbb{Z}^{2}}\eta^{x}\right)$

and if the exploration never ends then the ball $\mathrm{B}_{n}$ is connected
to infinity in $\mathcal{O}(\eta_{\infty})$. We aim to prove that

$\mathbb{P}(B_{t+1}=B_{t}|X_{0},\dots,X_{t})\geq
1-\exp(-\beta)-\varepsilon^{1/(3c^{\prime}_{d})}>p_{c}^{site}.$

Let us condition on $\mathscr{C}(\eta_{0}),\dots,\mathscr{C}(\eta_{t})$. Let
$x_{t}$ be as defined and $x_{t_{0}}$ corresponding to the other endpoint. By
construction $x_{t_{0}}\in A_{t}$ and $\mathcal{B}_{t_{0}}$ is connected to
$\mathrm{B}_{n}$ in $\mathcal{O}(\eta_{t})$, that is
$\mathcal{B}_{t_{0}}\subset\mathscr{C}(\eta_{t})$. It follows that

$\begin{split}\mathbb{P}&(\mathcal{E}_{n}(\mathcal{B}_{t_{0}},\Lambda_{x_{t}},\widetilde{B}_{x_{t}})\text{
occurs for $\eta_{t+1}$}|\mathscr{C}(\eta_{0}),\dots,\mathscr{C}(\eta_{t}))\\\
&\hskip
142.26378pt=\mathbb{P}(\mathcal{E}_{n}(\mathscr{C}(\eta_{t}),\Lambda_{x_{t}},\widetilde{B}_{x_{t}})\text{
occurs for $\eta_{t+1}$}|\mathscr{C}(\eta_{t})).\end{split}$

If
$\mathcal{E}_{n}(\mathcal{B}_{t_{0}},\Lambda_{x_{t}},\widetilde{B}_{x_{t}})$
occurs for $\eta_{t}$ then we are done and $x_{t}\in\mathcal{A}_{t+1}$.
Otherwise, we want to do a sprinkling to get another chance for this event to
occur. Let $z\in\Lambda_{x_{t_{0}}}$. Since $\|x_{t_{0}}-x_{t}\|_{\infty}=1$,
it follows that

$N\leq\|z-2Nx_{t}\|_{2}\leq(\sqrt{d}+2)N.$

Note that $\mathscr{C}(\eta_{t})$ contains a ball of radius at least $n$
centered in $\Lambda_{x_{t_{0}}}$, thanks to inequality (5.4), it follows that

$\mathbb{P}(\mathcal{E}_{n}(\mathscr{C}(\eta_{t}),\Lambda_{x_{t}},\widetilde{B}_{x_{t}})\text{
occurs for $\eta^{\prime}$}|\mathscr{C}(\eta_{t}))\geq
1-\varepsilon^{1/c^{\prime}_{d}}$

where $\eta^{\prime}$ is an independent Poisson point process of intensity
$\lambda dz\otimes\mu$. Note that when we condition on $\mathscr{C}(\eta_{t})$
the points in $\eta_{t}$ such that the corresponding ball does not intersect
$\mathscr{C}(\eta_{t})$ are unexplored and
$\eta_{t}\cap\Delta\mathscr{C}(\eta_{t})=\emptyset$.

$\begin{split}\mathbb{P}&(\mathcal{E}_{n}(\mathscr{C}(\eta_{t}),\Lambda_{x_{t}},\widetilde{B}_{x_{t}})\text{
occurs for $\eta_{t+1}$}|\mathscr{C}(\eta_{t}))\\\
&\qquad=\mathbb{P}(\mathcal{E}_{n}(\mathscr{C}(\eta_{t}),\Lambda_{x_{t}},\widetilde{B}_{x_{t}})\text{
occurs for
$\eta^{\prime}\cup\eta^{x_{t}}$}|\,\eta^{\prime}\cap\Delta\mathscr{C}(\eta_{t})=\emptyset,\mathscr{C}(\eta_{t}))\end{split}$

where $\eta^{\prime}$ is an independent Poisson point process of intensity
$\lambda dz\otimes\mu$. By using the Lemma 5.2 conditioning on
$\mathscr{C}(\eta_{t})$ (for $A=\mathscr{C}(\eta_{t})$,
$B=(\Lambda_{x_{t}}\times[n,+\infty))\setminus\Delta\mathscr{C}(\eta_{t})$ and
$C=\emptyset$) and inequality (5.3), it follows that

$\mathbb{P}(\mathcal{E}_{n}(\mathcal{B}_{t_{0}},\Lambda_{x_{t}},\widetilde{B}_{x_{t}})\text{
occurs for
$\eta_{t+1}$}|\mathscr{C}(\eta_{0}),\dots,\mathscr{C}(\eta_{t}))\geq
1-\exp(-\beta)-\varepsilon^{1/(3c^{\prime}_{d})}>p_{c}^{site}.$

We need the following lemma to conclude.

######  Lemma 5.3 (Lemma 9 in [4]).

Let $X_{t}=(A_{t},B_{t})$ be a random exploration sequence and assume that
there exists some $q>p_{c}^{site}$ such that for every $t\geq 0$

$\mathbb{P}(B_{t+1}=B_{t}|X_{0},\dots,X_{t})\geq q\quad\text{a.s.,}$

then the process X percolates with probability larger than a constant
$c=c(q)>0$ that can be taken arbitrarily close to 1 provided that q is close
enough to 1.

We can deduce that with positive probability, there exists an infinite
connected component in $\mathcal{O}(\eta_{\infty})$. Note that there exists a
constant $\kappa_{d}$ depending only on the dimension $d$ such that
$\eta_{\infty}$ is stochastically dominated by a Poisson point process of
intensity $(\lambda+\kappa_{d}\beta\xi)dz\otimes\mu$. Hence,

$\lambda\left(1+\frac{3\kappa_{d}c^{\prime}_{d}\beta}{-\log\varepsilon}\right)=\lambda+\kappa_{d}\beta\xi\geq\lambda_{c}(\mu).$

The result follows.

##  6 Supercritical sharpness: proof of Theorem 1.4

In this section, we prove Theorem 1.4 by adapting the seedless
Grimmett–Marstrand strategy in [4] to the Boolean context. We refer to [4] as
most of the arguments will only be sketched. In what follows, we fix $\mu$ a
distribution on the radius that has a finite $d$-moment. Let $\eta$ be a
Poisson point process of intensity $\lambda dz\otimes\mu$. Let $k\leq K$.
Denote by $A_{2}(k,K)$ the event where there exists at least two disjoint
connected components in
$\mathcal{O}(\eta\setminus\mathcal{B}(\Lambda_{K}))\cap\Lambda_{K}$ that
intersect both $\Lambda_{k}$ and $\partial\Lambda_{K}$ where

$\mathcal{B}(\Lambda_{K}):=\left\\{(x,r)\in(\mathbb{R}^{d}\times\mathbb{R}_{+})\setminus(\Lambda_{2K}\times[0,K]):\mathrm{B}_{r}^{x}\cap\Lambda_{K}\neq\emptyset\right\\}.$

The set $\mathcal{B}(\Lambda_{K})$ may be thought as a set of bad balls that
are typically not present when $K$ is large. We will need the following claim
that follows from similar computations as in the inequality (3.8).

######  Claim 6.1.

For all $\lambda>0$. We have

$\lim_{K\rightarrow\infty}\mathbb{P}_{\lambda,\mu}(\eta\cap\mathcal{B}(\Lambda_{K})\neq\emptyset)=0.$

######  Proposition 6.2.

There exists some $C>0$ and $\varepsilon_{0}>0$ (depending on $d$ only) such
that the following holds. Let $\varepsilon\in(0,\varepsilon_{0})$ and let
$1\leq k\leq K\leq n\leq N$ such that $K\leq\varepsilon^{2}n$ and

  1. (a)

$\mathbb{P}_{\lambda,\mu}\left(0\stackrel{{\scriptstyle\mathcal{O}(\eta\setminus(\mathcal{B}(\Lambda_{K})\cup\mathcal{B}(\Lambda_{N})))}}{{\longleftrightarrow}}\partial\Lambda_{N}\right)\geq\varepsilon$
and
$\mathbb{P}_{\lambda,\mu}\left(0\stackrel{{\scriptstyle\mathcal{O}(\eta\setminus\mathcal{B}(\Lambda_{K}))}}{{\longleftrightarrow}}\partial\Lambda_{K}\right)\geq\varepsilon$,

  2. (b)

$\mathbb{P}_{\lambda,\mu}\left(\Lambda_{k}\stackrel{{\scriptstyle\mathcal{O}(\eta\setminus(\mathcal{B}(\Lambda_{K})\cup\mathcal{B}(\Lambda_{N})))}}{{\longleftrightarrow}}\partial\Lambda_{N}\right)\geq
1-\exp\left(-\frac{1}{\varepsilon}\right)$,

  3. (c)

$\mathbb{P}_{\lambda,\mu}(A_{2}(k,K))\leq\exp\left(-\frac{1}{\varepsilon}\right)$
and
$\mathbb{P}_{\lambda,\mu}(A_{2}(n,N))\leq\exp\left(-\frac{1}{\varepsilon}\right)$.

Then, we have

$\lambda+C\varepsilon\geq\lambda_{c}^{S_{4dN}}(\mu|_{[0,N]}).$

Let us first explain how we can deduce Theorem 1.4 from Proposition 6.2. We
will need the following lemma which is a consequence of the work of Meester
and Roy [15] who proved uniqueness of the infinite cluster in Boolean
percolation.

######  Lemma 6.3.

Let $\lambda>\lambda_{c}(\mu)$. We have

$\lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty}\mathbb{P}_{\lambda,\mu}(A_{2}(n,N))=0.$

###### Proof of Theorem 1.4.

Let $\lambda>\lambda_{c}(\mu)$. Let $\varepsilon_{0}$ and $C$ be as in the
statement of Proposition 6.2. There exists $\varepsilon\in(0,\varepsilon_{0})$
such that

$\inf_{n\geq
1}\mathbb{P}_{\lambda,\mu}(0\longleftrightarrow\partial\Lambda_{n})\geq
2\varepsilon.$

Moreover, thanks to Claim 6.1, Lemma 6.3 and the existence of an infinite
cluster in the supercritical regime, there exists $n_{0}\geq 1$ large enough
such that

$\forall n\geq
n_{0}\qquad\mathbb{P}_{\lambda,\mu}(\eta\cap\mathcal{B}(\Lambda_{n})\neq\emptyset)\leq\frac{1}{4}\exp\left(-\frac{1}{\varepsilon}\right),$
$\forall n\geq n_{0}\qquad\inf_{N\geq
n_{0}}\mathbb{P}_{\lambda,\mu}(A_{2}(n,N))\leq\frac{1}{2}\exp\left(-\frac{1}{\varepsilon}\right)$

and

$\forall n\geq n_{0}\qquad\inf_{N\geq
n}\mathbb{P}_{\lambda,\mu}(\Lambda_{n}\longleftrightarrow\partial\Lambda_{N})\geq
1-\frac{1}{2}\exp\left(-\frac{1}{\varepsilon}\right).$

Let $k=n_{0}$, $K$ be the first integer such that

$\mathbb{P}_{\lambda,\mu}(A_{2}(k,K))\leq\exp\left(-\frac{1}{\varepsilon}\right),$

let $n$ be such that $n\geq 2K/\varepsilon^{2}$ and $N$ be the first integer
such that

$\mathbb{P}_{\lambda,\mu}(A_{2}(n,N))\leq\exp\left(-\frac{1}{\varepsilon}\right).$

It follows that the conditions a, b and c hold and by Proposition 6.2, there
exists $C$ depending only on $d$ such that

$\lambda+C\varepsilon\geq\lambda_{c}^{S_{4dN}}(\mu|_{[0,N]}).$

It follows that

$\lambda+C\varepsilon\geq\inf_{N\geq
1}\lambda_{c}^{S_{N}}(\mu|_{[0,N]})\geq\lambda_{c}(\mu).$

The result follows by letting $\varepsilon$ go to $0$ and $\lambda$ go to
$\lambda_{c}(\mu)$. ∎

Let us now sketch the proof of Proposition 6.2. The following lemma is proved
in the context of percolation in [4] it is a key part to prove Proposition
6.2.

######  Lemma 6.4.

Assume that the conditions a, b and c hold. Then there exists some $c>0$
(depending on $d$ only) such that for every connected set $S\ni 0$ with a
diameter larger than $n$,

$\mathbb{P}_{\lambda,\mu|_{[0,N]}}(S\longleftrightarrow F(N))\geq
1-2\exp\left(-\frac{c}{\varepsilon}\right)$

where $F(N):=\\{(x_{1},\dots,x_{d})\in\partial\Lambda_{N}:x_{1}=N,x_{2}\geq
0,\dots,x_{d}\geq 0\\}$.

###### Sketch of the proof of Lemma 6.4.

We build $x_{1},\dots,x_{l}\in S$ with $l\geq c_{1}/\varepsilon^{2}$ in the
same way as in the proof of Lemma 9 in [4] with the additional condition that
the boxes $x_{i}+\Lambda_{2K}$ are disjoint. For an event $E$, $x+E$ denote
the event translated by $x$. Set

$E_{i}:=\left\\{x_{i}\stackrel{{\scriptstyle\mathcal{O}(\eta\setminus(x_{i}+\mathcal{B}(\Lambda_{K})))}}{{\longleftrightarrow}}x_{i}+\partial\Lambda_{K}\right\\}\cap(x_{i}+A_{2}(k,K))^{c}$

and

$B_{i}:=\left\\{x_{i}+\Lambda_{k}\stackrel{{\scriptstyle\mathcal{O}(\eta\setminus(x_{i}+\mathcal{B}(\Lambda_{K})\cup\mathcal{B}(\Lambda_{N})))}}{{\longleftrightarrow}}\partial\Lambda_{N}\right\\}^{c}.$

The events $E_{i}$ are independent since the event $E_{i}$ only depends on the
Poisson point process inside $x_{i}+\Lambda_{2K}$. If the event
$E_{i}\setminus B_{i}$ occurs for some $i$, then there exists a path from
$x_{i}$ to $x_{i}+\partial\Lambda_{K}$ and a path from $x_{i}+\Lambda_{k}$ to
$\partial\Lambda_{N}$ and these two paths do not use balls in
$x_{i}+\mathcal{B}(\Lambda_{K})$. On the event $(x_{i}+A_{2}(k,K))^{c}$, they
must intersect inside $x_{i}+\Lambda_{K}$. This creates a path from $x_{i}$ to
$\partial\Lambda_{N}$. Note that the square root-trick holds for Boolean
percolation because we have FKG and the required symmetry. We conclude
similarly. ∎

###### Proof of Proposition 6.2.

The proof uses a renormalization and a sprinkling argument (similar to Section
5, at the exception that here we do not use a big ball as a seed but the path
itself).

Let $q$ be large enough such that $c(q)\geq 1/2$ where $c(q)$ is defined in
Lemma 5.3. Fix $\beta>0$ large enough such that

$\exp(-\beta)\leq\frac{1-q}{3}.$

Fix $\varepsilon_{0}$ small enough depending on $d$ and $q$ such that

$2\exp\left(-\frac{c}{3\varepsilon_{0}}\right)\leq\frac{1-q}{3}$

where $c$ depends only on $d$ is as in the statement of Lemma 6.4. Thanks to
this choice, we have

$1-\exp(-\beta)-2\exp\left(-\frac{c}{3\varepsilon_{0}}\right)\geq q.$ (6.1)

Let $\varepsilon\in(0,\varepsilon_{0})$. Let $1\leq k\leq K\leq n\leq N$ such
that $K\leq\varepsilon^{2}n$ and the conditions a, b and c hold.

For every $x\in\mathbb{Z}^{2}$, set

$\Lambda_{x}=2Nx+\Lambda_{N}\subset\mathbb{R}^{d}\qquad\text{and}\qquad\widetilde{B}_{x}=2Nx+\mathrm{B}_{4\sqrt{d}N}\subset\mathbb{R}^{d}$

where we identify $x=(x_{1},x_{2})$ with
$(x_{1},x_{2},0,\dots,0)\in\mathbb{Z}^{d}$.

Let $\eta$ be a Poisson point process of intensity
$\lambda\mathbf{1}_{S_{4dN}}dz\otimes\mu$. Let $(\eta^{x},x\in\mathbb{Z}^{2})$
be a family of independent Poisson point process of intensity
$\beta\varepsilon dz\textbf{1}_{\widetilde{B}_{x}}\otimes\mu$. We aim to prove
that the box $\Lambda_{0}$ intersects an infinite component of balls with
positive probability in $\mathcal{O}(\eta\cup_{x\in\mathbb{Z}^{2}}\eta^{x})$.
For a configuration $\eta$ we denote by $\mathscr{C}(\eta)$ the cluster of $0$
in $\eta$, that is,

$\mathscr{C}(\eta):=\bigcup\left\\{\mathrm{B}_{r}^{x}:(x,r)\in\eta,\,0\stackrel{{\scriptstyle\mathcal{O}(\eta)}}{{\longleftrightarrow}}\mathrm{B}_{r}^{x}\right\\}.$

Set $X_{0}:=(A_{0},B_{0})=(\\{0\\},0)$ and
$\eta_{0}:=\eta\cap(S_{4dN}\times[0,N])$. Let $t\geq 0$. We build $\eta_{t+1}$
and $X_{t+1}$ from $\eta_{t}$ and $X_{t}$ as follows. If there is no edge in
$(\mathbb{Z}^{2},\mathbb{E}^{2})$ connecting $A_{t}$ to $(A_{t}\cup
B_{t})^{c}$, then set $X_{t+1}:=X_{t}$ and $\eta_{t+1}:=\eta_{t}$. Otherwise,
let us denote by $x_{t}$ the extremity in $(A_{t}\cup B_{t})^{c}$ of this edge
(if there are several such edges, we choose one using a deterministic rule).
Define

$\eta_{t+1}:=\eta_{t}\cup\eta^{x_{t}}$

and

$(A_{t+1},B_{t+1}):=\left\\{\begin{array}[]{ll}(A_{t}\cup\\{x_{t}\\},B_{t})&\mbox{if
$0\stackrel{{\scriptstyle\mathcal{O}(\eta_{t+1})}}{{\longleftrightarrow}}\Lambda_{x_{t}}$}\\\
(A_{t},B_{t}\cup\\{x_{t}\\})&\mbox{otherwise.}\end{array}\right.$

Note that

$\eta_{\infty}:=\bigcup_{t\geq
0}\eta_{t}\subset\left(\eta_{0}\cup\bigcup_{x\in\mathbb{Z}^{2}}\eta^{x}\right)$

and if the exploration never ends then $0$ is connected to infinity in
$\mathcal{O}(\eta_{\infty}\cap S_{4dN})$. We aim to prove that

$\mathbb{P}\left(X\text{ percolates
}|\,0\stackrel{{\scriptstyle\mathcal{O}(\eta_{0})}}{{\longleftrightarrow}}\partial\Lambda_{0}\right)\geq\frac{1}{2}.$
(6.2)

The conditioning ensures that $\mathscr{C}(\eta_{0})$ has a diameter at least
$N$. Note that there exists a constant $C$ depending only on the dimension $d$
such that $\eta_{\infty}$ is stochastically dominated by a Poisson point
process of intensity $(\lambda+C\varepsilon)dz\otimes\mu$. If inequality (6.2)
is true then $\lambda+C\varepsilon\geq\lambda_{c}^{S_{4dN}}(\mu|_{[0,N]})$ and
the result follows. Let us now prove that inequality (6.2) holds. Let us
condition on $\mathscr{C}(\eta_{0}),\dots,\mathscr{C}(\eta_{t})$. We have

$\begin{split}\mathbb{P}&\left(\Lambda_{0}\stackrel{{\scriptstyle\mathcal{O}(\eta_{t+1})}}{{\longleftrightarrow}}\Lambda_{x_{t}}|\,\mathscr{C}(\eta_{0}),\dots,\mathscr{C}(\eta_{t}),\partial\Lambda_{0}\cap\mathscr{C}(\eta_{0})\neq\emptyset\right)\\\
&\hskip
142.26378pt=\mathbb{P}\left(\Lambda_{0}\stackrel{{\scriptstyle\mathcal{O}(\eta_{t+1})}}{{\longleftrightarrow}}\Lambda_{x_{t}}|\,\mathscr{C}(\eta_{t}),\partial\Lambda_{0}\cap\mathscr{C}(\eta_{t})\neq\emptyset\right).\end{split}$

On the conditioning by
$\\{\partial\Lambda_{0}\cap\mathscr{C}(\eta_{t})\neq\emptyset\\}$, the cluster
$\mathscr{C}(\eta_{t})$ has a diameter at least $N$. Using Lemma 6.4, we have

$\mathbb{P}(\mathscr{C}(\eta_{t})\longleftrightarrow\Lambda_{x_{t}}\text{
occurs for
$\eta^{\prime}$}|\,\mathscr{C}(\eta_{t}),\partial\Lambda_{0}\cap\mathscr{C}(\eta_{t})\neq\emptyset)\geq
1-2\exp\left(-\frac{c}{\varepsilon}\right)\,.$

where $\eta^{\prime}$ is an independent Poisson point process of intensity
$\lambda dz\otimes\mu$. We conclude that inequality (6.2) holds using the
sprinkling Lemma 5.2, Lemma 5.3 and inequality (6.1) (see the proof of
Proposition 2.3 in Section 5). ∎

###### Proof of Lemma 6.3.

Define $\mathcal{E}_{n,N}$ to be the event that there exists at least two
disjoint connected components in $\mathcal{O}(\eta)\cap\Lambda_{N}$ that
intersects both $\Lambda_{n}$ and $\partial\Lambda_{N}$. We have

$\mathbb{P}_{\lambda,\mu}(A_{2}(n,N))\leq\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n,N})+\mathbb{P}_{\lambda,\mu}(\eta\cap\mathcal{B}(\Lambda_{N})\neq\emptyset).$
(6.3)

Denote by $\mathcal{F}$ the event that there exists at least two infinite
clusters. By [15], this event has probability $0$. We have

$\mathcal{F}=\bigcup_{n\geq 1}\bigcap_{N\geq n}\mathcal{E}_{n,N}$

It follows that

$\lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty}\mathbb{P}_{\lambda,\mu}(\mathcal{E}_{n,N})=0\,.$

By Claim 6.1 and inequality (6.3), it follows that

$\lim_{n\rightarrow\infty}\lim_{N\rightarrow\infty}\mathbb{P}_{\lambda,\mu}(A_{2}(n,N))=0\,.$

This yields the result. ∎

##  Appendix A Inhomogeneous Talagrand formula

In this section, we prove the inhomogeneous Talagrand formula (Proposition
4.1), We will use the original Talagrand formula.

######  Theorem A.1 (Talagrand formula [18]).

There exists a universal constant $C>0$ such that for any $p\in(0,1)$ for any
function $f:\\{0,1\\}^{N}\rightarrow\\{0,1\\}$. We have

$\sum_{i=1}^{N}\operatorname{Inf}_{i}(f)\geq\frac{C}{\log(\frac{2}{p(1-p)})}\operatorname{Var}(f)\log\left(\frac{1}{\max_{1\leq
i\leq N}\operatorname{Inf}_{i}(f)}\right)\,$

where

$\operatorname{Inf}_{i}(f):=\mathrm{P}_{p}^{\otimes n}(f\circ\tau_{i}\neq f).$

###### Proof of Proposition 4.1.

Fix $N\geq 1$. Fix $f:\\{0,1\\}^{N}\rightarrow\\{0,1\\}$ a boolean function.
It is easy to check that the function $\operatorname{Inf}_{i}(f)$ is a
polynomial function of the family $(p_{1},\dots,p_{N})$. As a result, the map
$(p_{1},\dots,p_{N})\mapsto(\operatorname{Inf}_{1}^{p_{1},\dots,p_{N}}(f),\dots,\operatorname{Inf}_{N}^{p_{1},\dots,p_{N}}(f))$
is continuous. Hence, it is enough to check that the inequality holds only for
$p_{1},\dots,p_{N}$ that belong to the set

$E:=\\{m2^{-\ell}:\ell\geq 1,\,m\leq 2^{\ell}-1\\}$

which is dense in $[0,1]$.

From now on, we will assume that $p_{1},\dots,p_{N}\in E$ and write
$p_{i}=m_{i}2^{-\ell_{i}}$ for all $1\leq i\leq N$. Let $1\leq i\leq N$. We
set

$\mathrm{P}:=\otimes_{i=1}^{N}\mathrm{P}_{1/2}^{\otimes\ell_{i}}.$

Let $(X_{i,1},\dots,X_{i,\ell_{i}})$ be $\ell_{i}$ independent Bernoulli of
parameter $1/2$. Let us define

$\pi(X_{i,1},\dots,X_{i,\ell_{i}}):=\sum_{k=1}^{\ell_{i}}\frac{X_{i,k}}{2^{k}}.$

It is easy to check that $\pi(X_{i,1},\dots,X_{i,\ell_{i}})$ is uniform on
$\\{k2^{-\ell_{i}}:0\leq k\leq 2^{\ell_{i}}-1\\}$ and

$\mathrm{P}(\pi(X_{i,1},\dots,X_{i,\ell_{i}})\geq
1-m_{i}2^{-\ell_{i}})=m_{i}2^{-\ell_{i}}.$

Set $Y_{i}:=\mathbf{1}_{\pi(X_{i,1},\dots,X_{i,\ell_{i}})\geq 1-p_{i}}$. It
follows that $(Y_{i})_{1\leq i\leq N}$ has the distribution of
$\otimes_{i=1}^{N}\mathrm{Ber}(p_{i})$. Set

$\widetilde{f}(X_{1,1},\dots,X_{1,\ell_{1}},\dots,X_{N,1},\dots,X_{N,\ell_{N}}):=f(Y_{1},\dots,Y_{N}).$

It is easy to check that
$\operatorname{Var}(f)=\operatorname{Var}(\widetilde{f})$. Let us now compute
the influence of $X_{i,j}$ on $Y_{i}$ for $1\leq j\leq\ell_{i}$. We define
$\sigma_{j}^{1}$ (respectively $\sigma_{j}^{0}$ the function that for
$x=(x_{1},\dots,x_{\ell_{i}})\in\\{0,1\\}^{\ell_{i}}$ changes $x_{j}$ into $1$
(respectively into $0$). Set

$\Delta_{j}\pi=\pi\circ\sigma_{j}^{1}-\pi\circ\sigma_{j}^{0}.$

We have

$\operatorname{Inf}_{i,j}(\widetilde{f})=\mathrm{P}(\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0)\operatorname{Inf}_{i}(f).$

Let us now compute $\mathrm{P}(\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0)$. We have

$\left\\{\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0\right\\}=\left\\{\sum_{k=1,k\neq j}^{\ell_{i}}\frac{X_{i,k}}{2^{k}}\leq
1-p_{i},\,\sum_{k=1,k\neq
j}^{\ell_{i}}\frac{X_{i,k}}{2^{k}}+\frac{1}{2^{j}}\geq 1-p_{i}\right\\}.$

Let $j_{i}\geq 1$ be the largest integer such that

$2^{-j_{i}}\geq p_{i}.$

We distinguish two cases. Let us first assume that $j\geq j_{i}$.

$\\{\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0\\}\subset\left\\{\sum_{k=1}^{j-1}\frac{X_{i,k}}{2^{k}}\leq
1-p_{i},\,\sum_{k=1}^{j-1}\frac{X_{i,k}}{2^{k}}\geq
1-p_{i}-\frac{1}{2^{j-1}}\right\\}$

It follows that

$\mathrm{P}(\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0)\leq\frac{1}{2^{j-1}}.$

This rough upper-bound also holds for $j<j_{i}$, but we aim here at obtaining
a better upper-bound in that case. Let us now assume that $j<j_{i}$. We have

$\\{\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0\\}=\left\\{\sum_{k=1}^{j-1}\frac{X_{i,k}}{2^{k}}\leq
1-p_{i},\,\sum_{k=1}^{j-1}\frac{X_{i,k}}{2^{k}}\geq
1-p_{i}-\frac{1}{2^{j}}-\frac{1}{2^{j}}\sum_{k=1}^{\ell_{i}-j}\frac{X_{i,k+j}}{2^{k}}\right\\}.$
(A.1)

Note that we have almost surely

$\sum_{k=1}^{j-1}\frac{X_{i,k}}{2^{k}}\leq 1-\frac{1}{2^{j-1}}.$

Hence, for the event on the right hand side of (A.1) to be non-empty, we need
that

$1-p_{i}-\frac{1}{2^{j}}-\frac{1}{2^{j}}\sum_{k=1}^{\ell_{i}-j}\frac{X_{i,k+j}}{2^{k}}\leq
1-\frac{1}{2^{j-1}}.$

That is

$\sum_{k=1}^{\ell_{i}-j}\frac{X_{i,k+j}}{2^{k}}\geq 1-2^{j}p_{i}.$

It follows that

$\begin{split}\mathrm{P}&(\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0)\\\
&\qquad\leq\mathrm{P}\left(1-p_{i}-\frac{1}{2^{j-1}}\leq\sum_{k=1}^{j-1}\frac{X_{i,k}}{2^{k}}\leq
1-p_{i}\right)\mathrm{P}\left(\sum_{k=1}^{\ell_{i}-j}\frac{X_{i,k+j}}{2^{k}}\geq
1-2^{j}p_{i}\right)\\\ &\qquad\leq\frac{1}{2^{j-1}}2^{j}p_{i}\leq
2p_{i}.\end{split}$

Finally, we have

$\sum_{j=1}^{\ell_{i}}\mathrm{P}(\Delta_{j}\pi(X_{i,1},\dots,X_{i,\ell_{i}})\neq
0)\leq\sum_{j=1}^{j_{i}-1}2p_{i}+\sum_{j=j_{i}}^{\ell_{i}}\frac{1}{2^{j-1}}\leq
4p_{i}|\log p_{i}|$

and

$\sum_{j=1}^{\ell_{i}}\operatorname{Inf}_{i,j}(\widetilde{f})\leq 4p_{i}|\log
p_{i}|\operatorname{Inf}_{i}(f).$

Let us now apply Talagrand formula (Theorem A.1) to $\widetilde{f}$, we obtain

$\sum_{i=1}^{N}\sum_{j=1}^{\ell_{i}}\operatorname{Inf}_{i,j}(\widetilde{f})\geq
C\operatorname{Var}(\widetilde{f})\log\left(\frac{1}{\max\\{\operatorname{Inf}_{i,j}(\widetilde{f}):1\leq
i\leq N,1\leq j\leq\ell_{i}\\}}\right)$

and

$\sum_{i=1}^{N}p_{i}|\log p_{i}|\operatorname{Inf}_{i}(f)\geq
C\operatorname{Var}(f)\log\left(\frac{1}{\max\\{p_{i}\operatorname{Inf}_{i}(f):1\leq
i\leq N\\}}\right).$

The result follows.

∎

## References

  * [1] Daniel Ahlberg, Vincent Tassion, and Augusto Teixeira.  Sharpness of the phase transition for continuum percolation in $\mathbb{R}^{2}$.  Probab. Theory Related Fields, 172(1-2):525–581, 2018.
  * [2] Michael Aizenman and David J. Barsky.  Sharpness of the phase transition in percolation models.  Comm. Math. Phys., 108(3):489–526, 1987.
  * [3] S. R. Broadbent and J. M. Hammersley.  Percolation processes. I. Crystals and mazes.  Proc. Cambridge Philos. Soc., 53:629–641, 1957.
  * [4] Hugo Duminil-Copin, Gady Kozma, and Vincent Tassion.  Upper Bounds on the Percolation Correlation Length, volume 77, pages 347 – 369.  Birkhäuser, Basel, 2021.
  * [5] Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion.  Subcritical phase of $d$-dimensional Poisson–Boolean percolation and its vacant set.  Annales Henri Lebesgue, 3:677–700, 2020.
  * [6] Hugo Duminil-Copin and Vincent Tassion.  A new proof of the sharpness of the phase transition for Bernoulli percolation and the Ising model.  Comm. Math. Phys., 343(2):725–745, 2016.
  * [7] Alessandra Faggionato and Hlafo Mimun.  Connection probabilities in the poisson miller-abrahams random resistor network and other poisson random graphs with bounded edges.  ALEA Lat. Am. J. Probab. Math. Stat., 16:463–486, 2019.
  * [8] E. N. Gilbert.  Random plane networks.  Journal of the Society for Industrial and Applied Mathematics, 9(4):533–543, 1961.
  * [9] Jean-Baptiste Gouéré.  Subcritical regimes in the Poisson Boolean model of continuum percolation.  The Annals of Probability, 36(4):1209 – 1220, 2008.
  * [10] Jean-Baptiste Gouéré and Marie Théret.  Equivalence of some subcritical properties in continuum percolation.  Bernoulli, 25(4B):3714 – 3733, 2019.
  * [11] G. R. Grimmett and J. M. Marstrand.  The supercritical phase of percolation is well behaved.  Proceedings: Mathematical and Physical Sciences, 430(1879):439–457, 1990.
  * [12] Peter Hall.  On Continuum Percolation.  The Annals of Probability, 13(4):1250 – 1266, 1985.
  * [13] Günter Last.  Perturbation analysis of Poisson processes.  Bernoulli, 20(2):486 – 513, 2014.
  * [14] Günter Last, Giovanni Peccati, and D. Yogeshwaran.  Phase transitions and noise sensitivity on the poisson space via stopping sets and decision trees, 2021.
  * [15] Ronald Meester and Rahul Roy.  Uniqueness of Unbounded Occupied and Vacant Components in Boolean Models.  The Annals of Applied Probability, 4(3):933 – 951, 1994.
  * [16] M. V. Menshikov.  Coincidence of critical points in percolation problems.  Dokl. Akad. Nauk SSSR, 288(6):1308–1311, 1986.
  * [17] R. O’Donnell, M. Saks, O. Schramm, and R.A. Servedio.  Every decision tree has an influential variable.  In 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS’05), pages 31–39, 2005.
  * [18] Michel Talagrand.  On russo’s approximate zero-one law.  The Annals of Probability, 22(3):1576–1587, 1994.
  * [19] Sebastian Ziesche.  Sharpness of the phase transition and lower bounds for the critical intensity in continuum percolation on $\mathbb{R}^{d}$.  Annales de l’Institut Henri Poincaré, Probabilités et Statistiques, 54(2):866 – 878, 2018.