by $\tilde{\eta}\cap(B_{n+r}(0)\times{\mathcal{K}}_{r})$ and hence we restrict
to this set. Define $Z^{s}_{n}=(S\cup\partial B_{s}(0))\oplus B_{2r}(0)$ and
$A_{r}(x):=\\{x\leftrightarrow\partial B_{2r}(x)\\}\cap\\{\partial
B_{2r}(x)\subset\mathcal{O}(\gamma)\\}$ similar to $A^{\infty}_{r}(x)$ defined
in Remark 8.4(i). Further, reasoning as in Remark 8.4(i), we multiply and
divide by $\mathbb{P}(A_{r}(x))$ in the second term, then use the Harris-FKG
inequality and stationarity of the Poisson point process to derive that

$\displaystyle\mathbb{P}(x\in Z^{s}_{n})$
$\displaystyle\leq\mathbf{1}\\{||x|-s|\leq
2r\\}+\mathbf{1}\\{||x|-s|>2r\\}\mathbb{P}(B_{2r}(x)\leftrightarrow\partial
B_{s}(0))$ $\displaystyle\leq\mathbf{1}\\{||x|-s|\leq
2r\\}+\mathbf{1}\\{||x|-s|>2r\\}\mathbb{P}(A_{r}(0))^{-1}\mathbb{P}(x\leftrightarrow\partial
B_{s}(0))$ $\displaystyle\leq\mathbf{1}\\{||x|-s|\leq
2r\\}+\mathbf{1}\\{||x|-s|>2r\\}\mathbb{P}(A_{r}(0))^{-1}\mathbb{P}(x\leftrightarrow\partial
B_{|s-|x||}(x)).$ (8.9)

Denote $C^{\prime}_{\gamma}=\mathbb{P}(A_{r}(0))$ and since
$C^{\prime}_{\gamma}$ is increasing in $\gamma$, $\inf_{\gamma\geq
a}C^{\prime}_{\gamma}=C^{\prime}_{a}>0$. Also, we have that $\inf_{\gamma\geq
a}\inf_{s\leq 2r}\theta_{s}(\gamma)\geq\theta_{2r}(a)>0$ and so from (8.9), we
obtain that for some constant $C_{a}$

$\displaystyle\mathbb{P}(x\in Z_{n}^{s})$
$\displaystyle\leq\mathbf{1}\\{||x|-s|\leq
2r\\}\theta_{2r}(a)^{-1}\theta_{|s-|x||}(\gamma)+\mathbf{1}\\{||x|-s|>2r\\}(C^{\prime}_{a})^{-1}\theta_{|s-|x||}(\gamma)$
$\displaystyle\leq C_{a}\theta_{|s-|x||}(\gamma).$

Thus, choosing $Y$ uniformly at random in $(0,n)$ and setting
$Z_{n}=Z_{n}^{Y}$, we obtain that

$\mathbb{P}(x\in Z_{n})\leq
C_{a}n^{-1}\int_{0}^{n}\theta_{|s-|x||}(\gamma)\mathrm{d}s\leq
2C_{a}n^{-1}\int_{0}^{n}\theta_{s}(\gamma)\mathrm{d}s.$

This shows that our randomized stopping set has the revealment probability as
required in (8.8). Now, we only need to show that it can be constructed as a
randomized CTDT satisfying the assumptions in Theorem 4.2.

We will first describe a sequence of stopping sets in ${\mathbb{R}}^{d}$ and
then describe how to build a CTDT on ${\mathbb{R}}^{d}$ from the same. Let
$s\in(0,n)$.

  * •

Set $S_{0}=\emptyset,E_{0}=\partial B_{s}(0)$.

  * •

Set $E_{1}=(S_{0}\cup E_{0})\oplus B_{2r}(0)$ and $S_{1}$ is the union of
components of
$\mathcal{O}_{k}(\tilde{\eta}\cap(E_{1}\times{\mathcal{K}}_{r}))\cap
B_{n+r}(0)$ intersecting $E_{0}$.

  * •

Given $(S_{i},E_{i})$ for all $1\leq i\leq m\in{\mathbb{N}}$, we construct
$(S_{m+1},E_{m+1})$ as follows :

    * –

If $S_{m}=S_{m-1}$, then the algorithm terminates. Also,
$\mathbf{1}\\{0\leftrightarrow\partial B_{n}(0)]\\}=1$ if there is a path in
$S_{m}$ from $0$ to $\partial B_{n}(0)$ else
$\mathbf{1}\\{0\leftrightarrow\partial B_{n}(0)]\\}=0$. Further, set
$E_{\infty}=E_{m}$.

    * –

If $S_{m}\neq S_{m-1}$. Set $E_{m+1}=(S_{m}\cup E_{0})\oplus B_{2r}(0)$ and
$S_{m+1}$ be the union of components of
$\mathcal{O}_{k}(\tilde{\eta}\cap(E_{m+1}\times{\mathcal{K}}_{r}))\cap
B_{n+r}(0)$ intersecting $S_{m}$.

Observe that if the algorithm terminates after $m$ steps, then $S_{m}=S$ where
$S$ is the union of connected components of $\mathcal{O}(\gamma)\cap
B_{n+r}(0)$ that intersect $\partial B_{s}(0)$ if there exists any and else
$S=\emptyset$. Thus, by definition of $E_{\infty}$, we also obtain that
$E_{\infty}=(S\cup\partial B_{s}(0))\oplus B_{2r}(0)$.

Now we construct the CTDT. For $t\in(m,m+1)$ such that $S_{m}\neq S_{m-1}$,
set $Z^{s}_{n,t}=(S_{m}\cup\partial B_{s}(0))\oplus B_{2r(t-m)}(0)$. Thus, we
have that $Z^{s}_{n}=Z^{s}_{n,\infty}=E_{\infty}$ as required. The graph
measurability and stopping set property (see Remark A.7) of $Z^{s}_{n,t},t\geq
0$ can be argued as in the proof of Theorem 8.3. We will now verify the other
properties of CTDT.

Observe that for all $t\in(m,m+1]$ and any $\epsilon\in(0,t-m)$,

$Z^{s}_{n,t}\setminus
Z^{s}_{n,t-\epsilon}\subset\\{x\in{\mathbb{R}}^{d}:d(x,S_{m}\cup\partial
B_{s}(0))\in(2r(t-\epsilon),2r(t-m)]\\}.$

Also note that $Z^{s}_{n,0}=\partial B_{s}(0)$ has zero measure and further
using the above observation, other properties of CTDT (namely (2.1), (2.2))
can be verified for $Z^{s}_{n,t}$. Next, (4.4) holds trivially by the above
observation and since the intensity measure of $\tilde{\eta}$ is
$\gamma\,\mathrm{d}x\,\mathbb{Q}(\mathrm{d}K)$, a diffuse measure. Finally,
because the intensity measure of $\tilde{\eta}$ is diffuse and the observation
on $Z^{s}_{n,t}\ Z^{s}_{n,t-\epsilon}$, we have that $\\{|X_{i}|\\}_{i\geq 1}$
is a simple point process and this verifies (4.5). Since our function
$f_{n}(\tilde{\eta})$ is a function of
$f_{n}(\tilde{\eta}\cap(B_{n+2r}(0)\times{\mathcal{K}}_{R}))$, assumption
(3.3) holds because of Remark 3.1.

This completes the proof as we have constructed a CTDT satisfying the
assumptions of Theorem 4.2 and having revealment probability as given in
(8.8). ∎

###  9 Confetti Percolation

The confetti percolation model has its origins in the dead leaves model
introduced by Matheron [50]. Various geometric properties of the dead leaves
model have been studied. For example, see [69, 44, 14]. The percolation-
theoretic version of the model called confetti percolation was introduced by
Benjamini and Schramm in [10]. Since then, this model has been investigated in
many works [40, 56, 4, 31]. Although the dead leaves model is defined with
general grains and in arbitrary dimensions, studies of confetti percolation
model are focussed on random balls or squares (i.e., $\ell_{\infty}$ balls) in
two dimensions. We shall consider the former framework of general grains in
arbitrary dimensions and prove a sharp phase transition therein.

Consider a Poisson point process $\tilde{\eta}$ on
$\tilde{{\mathbb{X}}}={\mathbb{R}}^{d}\times{\mathbb{R}}_{+}\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0,1\\}$
with intensity measure
$\tilde{\gamma}(\mathrm{d}(x,t,K_{1},K_{2},a))=\mathrm{d}x\times\mathrm{d}t\times\mathbb{Q}_{1}(\mathrm{d}K_{1})\times\mathbb{Q}_{2}(\mathrm{d}K_{2})\times\nu_{p}(a)$
where
$x\in{\mathbb{R}}^{d},t\in{\mathbb{R}}_{+},K_{1},K_{2}\in{\mathcal{K}}_{r},a\in\\{0,1\\}$
and $\nu_{p}=p\delta_{0}+(1-p)\delta_{1}$ is the standard Bernoulli($p$)
distribution on $\\{0,1\\}$. The interpretation is that $x$ denotes the
location of the particle, $t$ the arrival time, $a$ will determine the colour
of the particle (black if $a=0$ and white otherwise) and $K_{1}$ is the grain
attached to black particles and $K_{2}$ is the grain attached to white
particles. The grains fall on $x$ at time $t$ with colour determined by $a$
and accordingly the grain. Each point in the plane is coloured according to
the first grain that covers it and the question of interest is percolation of
the black region. We will now define this more formally using the framework of
[40, Section 2]. We will assume throughout the section that
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ satisfy assumptions in (8.0).

We define the black region as

$\mathcal{O}_{p}:=\mathcal{O}_{p}(\tilde{\eta})=\mathcal{O}_{p}(\mathbb{Q}_{1},\mathbb{Q}_{2})=\bigcup_{(x,t,K_{1},K_{2},0)\in\tilde{\eta}}(x+K_{1})\setminus\\{\bigcup_{(x^{\prime},t^{\prime},K^{\prime}_{1},K^{\prime}_{2},1)\in\tilde{\eta},t^{\prime}<t}(x^{\prime}+K^{\prime}_{2})\\}.$
(9.1)

The white region $\mathcal{V}_{p}$ can be defined analogously or because
$\tilde{\eta}$ is a space-time Poisson point process, we can observe that
every point has to be coloured black or white because
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ satisfy assumptions in (8.0). Hence
$\mathcal{V}_{p}={\mathbb{R}}^{d}\setminus\mathcal{O}_{p}$ is the white
region.

We now define the percolation events and the corresponding probabilities as in
(8.2). For connected subsets $A,B,R\subset{\mathbb{R}}^{d}$ such that $A\cup
B\subset R$, define

$\displaystyle\\{A\overset{R}{\longleftrightarrow}B\\}$
$\displaystyle:=\\{\mbox{there exists a path in $\mathcal{O}_{p}\cap R$ from
$x$ to $y$ for some $x\in A,y\in B$}\\},$
$\displaystyle\\{A\longleftrightarrow B\\}$
$\displaystyle:=\\{A\overset{{\mathbb{R}}^{d}}{\longleftrightarrow}B\\},$
$\displaystyle\\{0\longleftrightarrow\infty\\}$ $\displaystyle:=\\{\mbox{there
is an unbounded path from $0$ in $\mathcal{O}_{p}$}\\}$
$\displaystyle\theta_{s}(p)$
$\displaystyle:=\theta_{n}(p,\mathbb{Q}_{1},\mathbb{Q}_{2})=\mathbb{P}(0\longleftrightarrow\partial
B_{s}(0)),s>0$ $\displaystyle\theta(p)$
$\displaystyle:=\theta(p,\mathbb{Q}_{1},\mathbb{Q}_{2})=\mathbb{P}(0\longleftrightarrow\infty)$
$\displaystyle Arm_{s,t}(p)$
$\displaystyle:=\\{B^{\infty}_{s}(0)\longleftrightarrow\partial
B^{\infty}_{t}(0)\\},\,\,0<s<t,$ $\displaystyle Cross_{s,t}(p)$
$\displaystyle:=\\{\\{0\\}\times[0,t]^{d-1}\overset{[0,s]\times[0,t]^{d-1}}{\longleftrightarrow}\\{s\\}\times[0,t]^{d-1}\\},\,\,\kappa,t>0.$
(9.2)

Again, by monotonicity (in $n$) of $\theta_{n}(p)$, we have that
$\theta(p)=\lim_{n\to\infty}\theta_{n}(p)$. We suppress the dependence on
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ as they are often fixed. Analogously to
$\mathcal{O}_{p}$, we can define the above events with respect to
$\mathcal{V}_{p}$ and in this case, we will denote the events and
probabilities with a $*$ superscript i.e.,
$\longleftrightarrow^{*},\theta^{*}_{s}(p),\theta^{*}(p),Arm^{*}_{s,t}(p),\theta^{*}(p),Cross^{*}_{s,t}(p)$
and so on.

Throughout this section, we have chosen the intensity to be $1$ but instead we
could have also chosen the intensity to be $\gamma$ for some
$\gamma\in(0,\infty)$ i.e.,
$\tilde{\gamma}(\mathrm{d}(x,t,K_{1},K_{2},a))=\gamma\mathrm{d}x\times\mathrm{d}t\times\mathbb{Q}_{1}(\mathrm{d}K_{1})\times\mathbb{Q}_{2}(\mathrm{d}K_{2})\times\nu_{p}(a)$.
Because of the scale-invariance of the Poisson point process in the time
direction, the probabilities defined above do not depend on the value of
$\gamma$ and hence we have chosen $\gamma=1$ for convenience.

A measurable mapping $f:{\mathbf{N}}(\tilde{{\mathbb{X}}})\to{\mathbb{R}}$ is
said to be black-increasing if for all $\mu\in{\mathbf{N}}$ and
$x^{\prime}=(x,t,K_{1},K_{2})\in{\mathbb{R}}^{d}\times{\mathbb{R}}_{+}\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}$,
we have that

$f_{n}(\mu+\delta_{(x^{\prime},1)})\leq f_{n}(\mu)\leq
f_{n}(\mu+\delta_{(x^{\prime},0)}).$ (9.3)

As usual an event is said to be black-increasing if its indicator function is.
Trivially, if $f$ is a black-increasing function,
$\mathbb{E}[f(\tilde{\eta})]$ is an increasing function in $p$. Since
$\\{0\longleftrightarrow\infty\\}$ is a black-increasing event, $\theta(p)$ is
increasing in $p$ and $\theta^{*}(p)$ is decreasing in $p$. Hence, we can
define the critical probabilities as

$\displaystyle p_{c}$
$\displaystyle:=p_{c}(\mathbb{Q}_{1},\mathbb{Q}_{2})=\inf\\{p:\theta(p,\mathbb{Q}_{1},\mathbb{Q}_{2})>0\\},$
$\displaystyle p_{c}^{*}$
$\displaystyle:=p^{*}_{c}(\mathbb{Q}_{1},\mathbb{Q}_{2})=\sup\\{p:\theta^{*}(p,\mathbb{Q}_{1},\mathbb{Q}_{2})>0\\}.$
(9.4)

Out first theorem shows a sharp phase transition for $\theta_{n}(p)$.

######  Theorem 9.1 (Sharp threshold for confetti percolation).

Consider the black occupied region $\mathcal{O}_{p}$ in the confetti
percolation model as defined in (9.1) with grain distributions
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ satisfying assumptions in (8.0). Then, for
$p_{c}$ as defined above, the following statements hold.

  1. 1.

For all $p<p_{c}$, there exists a constant $\alpha(p)\in(0,\infty)$ such that
$\theta_{n}(p)\leq\exp\\{-\alpha(p)n\\}$ for all $n\geq 1$.

  2. 2.

For all $p^{\prime}<1$, there exists a constant $\alpha\in(0,\infty)$ such
that for all $p\in[p_{c},p^{\prime})$, we have that
$\theta_{n}(p)\geq\alpha(p-p_{c})$ for all $n\geq 1$.

######  Remark 9.2.

  1. 1.

Note that we do not say anything about non-triviality of $p_{c}$ i.e.,
$0<p_{c}<1$. Though we would expect this to hold under the assumptions of
Theorem 9.1, this is beyond the scope of our project. In the planar case
(i.e., $d=2$) non-triviality is known in the the case when
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ are supported on balls (see [4, Theorem 8.3]).
For the general case, one may use a suitable coupling argument similar to that
in Theorem 8.2.

  2. 2.

The above theorem in the case of $d=2$ and when
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ are supported on boxes was shown in [31,
Proposition 1.1]. The proof therein uses the discrete OSSS inequality whereas
we use the continuum version and thereby enabling us to provide a simpler
proof that holds in greater generality. It was remarked in [31, Remark 1.1]
that their method can be adapted for other shapes but our proof allows one to
treat different shapes at once without making any shape-specific argument. The
same remark applies to the Theorem 9.3 as well.

###### Proof.

We shall prove the sharp threshold result using the same proof strategy as in
Theorem 8.2 but with some additional technicalities due to the non-compactness
of $\tilde{{\mathbb{X}}}$.

We fix a $n$ large and $\epsilon\in(0,1/2)$. Suppose we show that for
$p\in[\epsilon,1-\epsilon]$,

$\frac{\mathrm{d}\theta_{n}(p)}{\mathrm{d}p}\geq\frac{b_{r}(\epsilon)n}{2\int_{0}^{n}\theta_{s}(p)\mathrm{d}s}\theta_{n}(p)(1-\theta_{n}(p)),$
(9.5)

where $b_{r}(\epsilon)>0$ will be defined explicitly soon. Now observe that

$1-\theta_{n}(p)\geq
1-\theta_{n}(1-\epsilon)\geq\mathbb{P}(B_{1}(0)\subset\mathcal{V}_{1-\epsilon})>0,$

i.e., there cannot be a path from $0$ to $\partial B_{n}(0)$ if the unit ball
around $0$ is covered by the white region. Thus, we obtain from (9.5) and the
above inequality that

$\frac{\mathrm{d}\theta_{n}(p)}{\mathrm{d}p}\geq\frac{b_{r}(\epsilon)n}{2\int_{0}^{n}\theta_{s}(p)\mathrm{d}s}\theta_{n}(p)(1-\theta_{n}(1-\epsilon)).$

Now using [24, Lemma 1], we obtain that there exists
$p_{0}:=p_{0}(\epsilon)\in[\epsilon,1-\epsilon]$ such that

  1. 1.

For all $p\in[\epsilon,p_{0})$, there exists a constant
$\alpha(p)\in(0,\infty)$ such that $\theta_{n}(p)\leq\exp\\{-\alpha(p)n\\}$
for all $n\geq 1$.

  2. 2.

There exists a constant $\alpha(\epsilon)\in(0,\infty)$ such that for all
$p\in(p_{0},1-\epsilon]$, $\theta_{n}(p)\geq\alpha(\epsilon)(p-p_{0})$ for all
$n\geq 1$.

By definition of $p_{c}$, we have that $\theta(p)=0$ for $p<p_{c}$ and
$\theta(p)>0$ for $p>p_{c}$. From this and the above two statements, the proof
of the two statements in the theorem is complete.

Now, we are left to prove (9.5). Fix $s\in(0,n)$. For $h>0$, let
$\tilde{\eta}_{h}=\tilde{\eta}\cap\tilde{{\mathbb{X}}}_{h}$ where
$\tilde{{\mathbb{X}}}_{h}={\mathbb{R}}^{d}\times[0,h]\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0,1\\}$.
When we want to refer to events in (9.2) but with respect to
$\mathcal{O}_{p}(\tilde{\eta}_{h})$, we shall use $\longleftrightarrow^{h}$
for existence of corresponding paths in $\mathcal{O}(\tilde{\eta}_{h})$ and
$\theta_{s}^{h},\theta_{h}$ for the corresponding percolation probabilities.
We define

$f^{h}_{n}:=\mathbf{1}\\{0\longleftrightarrow^{h}\partial
B_{n}(0)\\},f_{n}:=\mathbf{1}\\{0\longleftrightarrow\partial B_{n}(0)\\}.$

Observe that
$\theta^{h}_{n}(p)=\mathbb{E}[f_{n}^{h}],\theta_{n}(p)=\mathbb{E}[f_{n}].$ Our
proof strategy now is to first use Theorem 4.2 for $f_{n}^{h}$ and derive a
version of the differential inequality (9.5) for $\theta_{n}^{h}$. Then we
will complete the proof by showing that all the terms in the differential
inequality (9.5) are well-approximated by the corresponding truncated versions
for $f_{n}^{h}$. Since $f_{n}^{h}$ is dependent on the point process in a
compact set, we can use ideas similar to those in the proof of Theorem 8.2 to
derive the differential inequality (9.5) for $\theta_{n}^{h}$.

Now, we construct a CTDT satisfying assumptions of Theorem 4.2 for
$f_{n}^{h}$. Let $S$ be the union of connected components of
$\mathcal{O}_{p}(\tilde{\eta}_{h})$ that intersect $\partial B_{s}(0)$. Note
that $S=\emptyset$ if no such component exists. Define
$Z_{n}^{s}:=(S\cup\partial B_{s}(0))\oplus B_{2r}(0)$. We will skip the
construction of $Z_{n}^{s}$ via a CTDT as it is similar to that in Theorem
8.2. We note that $Z_{n}^{s}$ determines $f_{n}$ and further it is a stopping
set as in Remark A.7. Now again as in the derivation of (8.9), using Harris-
FKG inequality (see [31, Definition 2.1 and below]) and translation invariance
of the Poisson point process, we obtain that for $x\in{\mathbb{R}}^{d}$,

$\displaystyle\mathbb{P}(x\in Z_{n}^{s})$
$\displaystyle\leq\mathbb{P}(B_{2r}(x)\cap(S\cup\partial
B_{s}(0))\neq\emptyset)$ $\displaystyle=\mathbf{1}\\{||x|-s|\leq
2r\\}+\mathbf{1}\\{||x|-s|>2r\\}\mathbb{P}(B_{2r}(x)\longleftrightarrow^{h}\partial
B_{s}(0))$ $\displaystyle\leq\mathbf{1}\\{||x|-s|\leq
2r\\}+(b^{h}_{r}(p))^{-1}\mathbf{1}\\{||x|-s|>2r\\}\mathbb{P}(x\longleftrightarrow^{h}\partial
B_{s}(0))$ $\displaystyle\leq\mathbf{1}\\{||x|-s|\leq
2r\\}+(b^{h}_{r}(p))^{-1}\mathbf{1}\\{||x|-s|>2r\\}\theta^{h}_{|s-|x||}(p),$
(9.6)

where $b^{h}_{r}(p)=\mathbb{P}(\\{0\longleftrightarrow^{h}\partial
B_{2r}(0)\\}\cap\\{\partial
B_{2r}(0)\subset\mathcal{O}_{p}(\tilde{\eta}_{h})\\})$ and the usage of
Harris-FKG inequality is justified as the events are black-increasing as
defined in (9.3). Further $b^{h}_{r}(p)\geq
b^{h}_{r}(\epsilon)\geq\mathbb{P}(B_{2r}(0)\subset\mathcal{O}_{\epsilon}(\tilde{\eta}_{h}))$
and the positivity of the latter guarantees the positivity of
$b^{h}_{r}(\epsilon)$. Now we randomize over $s$ as before i.e., set
$Z_{n}:=Z_{n}^{Y}$ where $Y$ is a uniform $(0,n)$-valued random variable. Then
we obtain that

$\displaystyle\delta_{n}$ $\displaystyle:=\sup_{x\in
B_{n+2r}(0)}\mathbb{P}(x\in
Z_{n})\leq(b^{h}_{r}(\epsilon))^{-1}n^{-1}\sup_{x\in
B_{n+2r}(0)}\int_{0}^{n}\theta^{h}_{|s-|x||}(p)\mathrm{d}s$ $\displaystyle\leq
2(b^{h}_{r}(\epsilon))^{-1}n^{-1}\int_{0}^{n}\theta^{h}_{s}(p)\mathrm{d}s,$

where in the first inequality follows from (9.6) and from arguments analogous
to those rehearsed in the proof of Theorem 8.2.

Having constructed a suitable CTDT, we now derive the differential inequality
for $\theta_{n}^{h}$. By $x^{\prime}$ we denote $(x,t,K_{1},K_{2})$. Using
that $f^{h}_{n}$ is black-increasing as in (9.3) and $f^{h}_{n}\in\\{0,1\\}$,
we have that

$|D_{(x^{\prime},0)}f^{h}_{n}|=f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f^{h}_{n}(\tilde{\eta})\leq
f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})\geq|D_{(x^{\prime},1)}f^{h}_{n}|.$

Let $\mathbb{X}$ be such that
$\tilde{{\mathbb{X}}}={\mathbb{X}}\times\\{0,1\\}$. Thus from Theorem 4.2, the
definition of $\delta_{n}$, the above inequalities and a version of the Russo-
Margulis formula in [49, Exercise 19.8], we derive that

$\displaystyle\theta^{h}_{n}(p)(1-\theta^{h}_{n}(p))$
$\displaystyle\leq\delta_{n}\left(p\int_{{\mathbb{X}}}\mathbb{E}[|D_{(x^{\prime},0)}f^{h}_{n}(\tilde{\eta})|]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x\right.$
$\displaystyle\quad+\left.(1-p)\int_{{\mathbb{X}}}\mathbb{E}[|D_{(x^{\prime},1)}f^{h}_{n}(\tilde{\eta})|]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x\right)$
$\displaystyle\leq\delta_{n}\int_{{\mathbb{X}}}\mathbb{E}[f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x$
$\displaystyle=\delta_{n}\frac{\mathrm{d}\theta^{h}_{n}(p)}{\mathrm{d}p}\leq\left(2(b^{h}_{r}(\epsilon))^{-1}n^{-1}\int_{0}^{n}\theta^{h}_{s}(p)\mathrm{d}s\right)\frac{\mathrm{d}\theta^{h}_{n}(p)}{\mathrm{d}p}.$
(9.7)

Now, we will complete the proof by showing that the terms in (9.7) approximate
those of (9.5). Define

$b_{r}(p):=\mathbb{P}(\\{0\longleftrightarrow\partial
B_{2r}(0)\\}\cap\\{\partial B_{2r}(0)\subset\mathcal{O}_{p}\\}).$

By the same reasoning as in the forthcoming arguments in (9.8), we have that
for every $s>0$ and $p\in[\epsilon,1-\epsilon]$,

$\theta^{h}_{s}(p)\uparrow\theta_{s}(p),b^{h}_{r}(p)\uparrow b_{r}(p).$

Thus, to show that the differential inequality (9.7) converges to the
differential inequality (9.5), it remains to show that $\theta_{s}(p)$ is
differentiable in $p$ and that the derivatives
$\frac{\mathrm{d}\theta^{h}_{n}(p)}{\mathrm{d}p}$ converge as $h\to\infty$. To
prove this, we will now show that derivatives
$\frac{\mathrm{d}\theta^{h}_{n}(p)}{\mathrm{d}p}$ converge uniformly as
$h\to\infty$ and thus proving both differentiability and convergence of the
derivatives.

Set
${\mathbb{X}}_{h}:={\mathbb{R}}^{d}\times[0,h]\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r},{\mathbb{X}}^{\prime}:={\mathbb{R}}^{d}\times(h,\infty)\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}$.
Define,

$\displaystyle g_{n}^{h}(p)$
$\displaystyle:=\int_{{\mathbb{X}}_{h}}\mathbb{E}[f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x$
$\displaystyle g_{n}(p)$
$\displaystyle:=\int_{{\mathbb{X}}}\mathbb{E}[f_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x,$

By the Russo-Margulis formula [49, Exercise 19.8],
$\frac{\mathrm{d}\theta^{h}_{n}(p)}{\mathrm{d}p}=g^{h}_{n}(p)$. Hence to show
uniform convergence of the derivatives, it is enough to show uniform
convergence of $g^{h}_{n}(p)$ to $g_{n}(p)$ for $p\in[\epsilon,1-\epsilon]$.
Now, using the fact that $f_{n}^{h}$ depends only on $\tilde{\eta}^{h}$, we
derive that

$\displaystyle|g_{n}(p)-g_{n}^{h}(p)|$
$\displaystyle\leq\int_{{\mathbb{X}}^{\prime}}\mathbb{E}[f_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x$
$\displaystyle\quad+\int_{{\mathbb{X}}_{h}}\mathbb{E}[|f_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})|]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x$
$\displaystyle\quad+\int_{{\mathbb{X}}_{h}}\mathbb{E}[|f^{h}_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})-f_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})|]\mathbb{Q}_{1}(\mathrm{d}K_{1})\mathbb{Q}_{2}(\mathrm{d}K_{2})\mathrm{d}t\mathrm{d}x.$

We sub-divide $B_{n+2r}(0)$ into small cubes $R_{n1},\ldots,R_{nk_{a}}$ of
side-length $a$ and $a$ small enough such that the diameter of each cube is at
most $r_{0}/4$ for $r_{0}$ as in assumption (8.0). Set
${\mathcal{K}}^{\prime}_{r_{0}}:=\\{K\in{\mathcal{K}}_{r}:B_{r_{0}}(0)\subset
K\\}$ and $\beta_{i}=\mathbb{Q}_{i}({\mathcal{K}}^{\prime}_{r_{0}}),i=1,2$.
Note that $\beta_{i}>0$ for $i=1,2$ because of assumption (8.0).

Our key observation is that if for some $t$ and all $i$,
$\tilde{\eta}_{h}(R_{ni}\times[0,t)\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})\neq
0$, then
$\mathcal{O}_{p}(\tilde{\eta}_{h}+\delta_{(x,t^{\prime},K_{1},K_{2},0)})\cap
B_{n}(0)=\mathcal{O}_{p}(\tilde{\eta}+\delta_{(x,t^{\prime},K_{1},K_{2},1)})\cap
B_{n}(0)$ for $t^{\prime}>t,x\in B_{n+2r}(0)$ and for all $p$. Now from this
observation, union bound over $R_{ni}$’s and the fact that
$\tilde{\eta}_{h}(R_{ni}\times[0,t)\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})$
is a Poisson random variable with mean $a^{d}\beta_{1}\beta_{2}t$, we derive
that for $x^{\prime}=(x,t,K_{1},K_{2})$,

$\mathbb{E}[f_{n}(\tilde{\eta}+\delta_{(x^{\prime},0)})-f_{n}(\tilde{\eta}+\delta_{(x^{\prime},1)})]\leq
k_{a}\exp\\{-a^{d}\beta_{1}\beta_{2}t\\}.$

Similarly, if for some $h$ and all $i$,
$\tilde{\eta}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})\neq
0$, then $\mathcal{O}_{p}(\tilde{\eta}_{h})\cap
B_{n}(0)=\mathcal{O}_{p}(\tilde{\eta})\cap B_{n}(0)$ and
$\mathcal{O}_{p}(\tilde{\eta}_{h}+\delta_{(x,t,K_{1},K_{2},a)})\cap
B_{n}(0)=\mathcal{O}_{p}(\tilde{\eta}+\delta_{(x,t,K_{1},K_{2},a)})\cap
B_{n}(0)$ for $t\leq h,a\in\\{0,1\\},x\in B_{n+2r}(0)$ and for all $p$.
Arguing as above, the latter two integrands are bounded by
$k_{a}\exp\\{-a^{d}\beta_{1}\beta_{2}h\\}$. Thus substituting these bounds in
the above inequality and observing that by the definition of
$f_{n},f_{n}^{h}$, one can restrict to $x\in B_{n+2r}(0)$, we obtain that

$|g_{n}(p)-g_{n}^{h}(p)|\leq
k_{a}|B_{n+2r}(0)|\left(\int_{h}^{\infty}\exp\\{-a^{d}\beta_{1}\beta_{2}t\\}\mathrm{d}t+2\exp\\{-a^{d}\beta_{1}\beta_{2}h\\}\right).$
(9.8)

From the above bounds, we obtain uniform convergence of $g^{h}_{n}(p)$ to
$g_{n}(p)$ for $p\in[\epsilon,1-\epsilon]$ and as argued below (9.7), this
completes the proof of the (9.5) and hence that of the theorem as well. ∎

A powerful consequence of the above sharp phase transition result is the exact
determination of the critical probabilities in the planar case when
$\mathbb{Q}_{1}=\mathbb{Q}_{2}=\mathbb{Q}$. By considerations of self-duality,
it was conjectured that $p_{c}=p^{*}_{c}=\frac{1}{2}$ in the case of fixed
balls by Benjamini and Schramm [10, Problem 5]. It was proven in the case of
fixed boxes by Hirsch [40] and then in the case of fixed balls by Müller [56].
Recently, it was proven in the case of random boxes by Ghosh and Roy [31]. Our
next result will include all of the above as special cases.

######  Theorem 9.3 (Critical probability for planar confetti percolation).

Consider the black occupied region $\mathcal{O}_{p}$ in the planar (i.e.,
$d=2$) confetti percolation model as defined in (9.1) with grain distribution
$\mathbb{Q}_{1}=\mathbb{Q}_{2}=\mathbb{Q}$ satisfying assumptions in (8.0).
Further, we assume that $\mathbb{Q}$ is invariant under $\pi/2$-rotations and
reflections at the coordinate axes. Then we have that
$p_{c}(\mathbb{Q},\mathbb{Q})=p_{c}^{*}(\mathbb{Q},\mathbb{Q})=\frac{1}{2}$.

Further, under a certain transitivity condition, it was shown in [31, Theorem
1.2] for the planar case that
$p_{c}(\mathbb{Q}_{1},\mathbb{Q}_{2})=\frac{|B_{1}|}{|B_{1}|+|B_{2}|}$ when
$\mathbb{Q}_{1}=\delta_{B_{1}}$ and $\mathbb{Q}_{2}=\delta_{B_{2}}$ for
$B_{1},B_{2}\in{\mathcal{K}}_{r}$ and $|.|$ denotes the Lebesgue volume. It is
not yet known if the transitivity condition holds always.

###### Proof.

Using the observation
$\mathcal{V}_{p}(\mathbb{Q}_{1},\mathbb{Q}_{2})\overset{d}{=}\mathcal{O}_{1-p}(\mathbb{Q}_{2},\mathbb{Q}_{1})$,
we obtain that
$p^{*}_{c}(\mathbb{Q}_{1},\mathbb{Q}_{2})=1-p_{c}(\mathbb{Q}_{2},\mathbb{Q}_{1})$
and thus $p^{*}_{c}(\mathbb{Q},\mathbb{Q})+p_{c}(\mathbb{Q},\mathbb{Q})=1$
when $\mathbb{Q}_{1}=\mathbb{Q}_{2}=\mathbb{Q}$. Hence, to prove the theorem
it suffices to show that
$p_{c}(\mathbb{Q},\mathbb{Q})=p_{c}^{*}(\mathbb{Q},\mathbb{Q})$.

Now from the sharp phase transition result (Theorem 9.1), we can obtain that

$\mathbb{P}(Cross_{2n,2n}(p))\overset{n\to\infty}{\to}0\,\,\mbox{for}\,\,p<p_{c}\,\,\mbox{and}\,\,\mathbb{P}(Cross_{2n,2n}(p))\overset{n\to\infty}{\to}1\,\,\mbox{for}\,\,p>p_{c},\,\,$
(9.9)

and similarly for $\mathbb{P}(Cross^{*}_{2n,2n}(p))$ with respect to
$p_{c}^{*}$. We will now conclude the proof of the theorem assuming (9.9) and
later prove the same. By planarity and the definition of the model, we have
that there exists a left-right crossing of $\mathcal{O}_{p}$ iff there exists
no top-down crossing of $\mathcal{V}_{p}$. From this observation and
$\pi/2$-rotation invariance of $\mathcal{O}_{p}$ and $\mathcal{V}_{p}$, we
obtain that $\mathbb{P}(Cross_{2n,2n}(p))=1-\mathbb{P}(Cross^{*}_{2n,2n}(p))$.
Thus, we also have that

$\mathbb{P}(Cross_{2n,2n}(p))\overset{n\to\infty}{\to}0\,\,\mbox{for}\,\,p<p^{*}_{c}\,\,\mbox{and}\,\,\mathbb{P}(Cross_{2n2,n}(p))\overset{n\to\infty}{\to}1\,\,\mbox{for}\,\,p>p^{*}_{c},\,\,$

and so $p_{c}=p_{c}^{*}$ as required to complete the proof of the theorem.

We now prove (9.9) for $\mathbb{P}(Cross_{2n,2n}(p))$ and that for
$\mathbb{P}(Cross^{*}_{2n,2n}(p))$ follows the same arguments. For the first
statement, observe that a left-right crossing in $[0,2n]^{d}$ has to pass
through one of the $(n-1)$ balls $B_{(n,2i+1)}(1),i=0,2,\ldots,n-1$ and hence
there is a path from one of these balls to $\\{2n\\}\times[0,2n]$. As in the
proof of $\tilde{\gamma}_{c}=\gamma_{c}$ in Theorem 8.1, by union bound,
stationarity of the Poisson process and reasoning as in the derivation of
(9.6), we obtain that

$\displaystyle\mathbb{P}(Cross_{2n,2n}(p))$
$\displaystyle\leq\mathbb{P}(\cup_{i=0}^{n-1}\\{B_{1}(n,2i+1)\longleftrightarrow\\{\\{2n\\}\times[0,2n]\\})$
$\displaystyle\leq\sum_{i=0}^{n-1}\mathbb{P}(B_{1}(n,2i+1)\longleftrightarrow\partial
B_{n}(n,2i+1))$ $\displaystyle\leq
n\mathbb{P}(B_{0}(1)\longleftrightarrow\partial B_{n}(0))\leq
b_{1/2}(p)^{-1}n\theta_{n}(p),$

where $b_{r}(p)$ is defined similarly as below (9.6). Now, by exponential
decay in Theorem 9.1(1), the RHS above converges to $0$ as $n\to\infty$ for
$p<p_{c}$. For the second statement, we shall follow Zhang’s argument (see
[22, Proposition 4.1] and [40, Proposition 3]). For Zhang’s argument, it
suffices that $\mathcal{O}_{p}$ is invariant under translations of
${\mathbb{R}}^{d}$ and rotation by $\pi/2$, $\mathcal{O}_{p}$ is ergodic with
respect to translations of ${\mathbb{R}}^{d}$ and black-increasing events are
positively correlated. We shall now sketch Zhang’s argument. Let $n\geq 2k\geq
1$. Any $\mathcal{O}_{p}$-path from $B^{\infty}_{k}$ to $\partial
B^{\infty}_{n}$ ends in one of the four sides of $\partial B^{\infty}_{n}$.
Now, by $\pi/2$-rotation invariance of $\mathcal{O}_{p}$ and square-root trick
[74, Proposition 4.1], we have that

$\displaystyle\mathbb{P}(\mbox{there is a $\mathcal{O}_{p}$-path from
$B^{\infty}_{k}$ to the left of $\partial B^{\infty}_{n}$})$
$\displaystyle\geq 1-\mathbb{P}(B^{\infty}_{k}\nleftrightarrow\partial
B_{n}^{\infty})^{1/4}$ $\displaystyle\geq
1-\mathbb{P}(B^{\infty}_{k}\nleftrightarrow\infty)^{1/4}.$

Let $A_{n,k}$ denote the event that $B^{\prime}_{n,k}:=(n,n)+B^{\infty}_{k}$
is connected to the left of $[0,2n]^{2}$ and the right of $[0,2n]^{2}$ via
paths in $[0,2n]^{2}$. Then from the above bound and positive correlation of
black-increasing events, we derive that

$\mathbb{P}(A_{n,k})\geq
1-2\mathbb{P}(B^{\infty}_{k}\nleftrightarrow\infty)^{1/4}.$

Observe that $A_{n,k}\setminus Cross_{2n,2n}(p)\subset A^{\prime}_{n,k},$
where the latter is defined as

$A^{\prime}_{n,k}:=\\{\mbox{there are two distinct components in
$\mathcal{O}_{p}$ that intersect $B^{\prime}_{n,k}$ and
$\partial[0,2n]^{2}$}\\}.$

Further, $\cap_{n\geq 1}A^{\prime}_{n,k}\subset\\{\mbox{there are two distinct
infinite components in $\mathcal{O}_{p}$}\\}$ and the latter event has zero
probability by uniqueness of the infinite component of $\mathcal{O}_{p}$ (see
Remark 9.4). Thus, we obtain that

$\lim_{n\to\infty}\mathbb{P}(Cross_{2n,2n}(p))=\lim_{n\to\infty}\mathbb{P}(A_{n,k})\geq
1-2\mathbb{P}(B^{\infty}_{k}\nleftrightarrow\infty)^{1/4}.$

By ergodicity and that $\theta(p)>0$ as $p>p_{c}$, we have that
$\mathbb{P}(\mbox{$\mathcal{O}_{p}$ has an infinite component})=1$ and so
$\mathbb{P}(B^{\infty}_{k}\longleftrightarrow\infty)\to 1$ as $k\to\infty$.
Substituting this in the above bound, we obtain the second statement in (9.9)
and thus completing the proof of (9.9). ∎

######  Remark 9.4.

We do not provide a detailed proof for the uniqueness of the infinite occupied
component in the confetti percolation model. To prove this claim, one can
follow the arguments in [27]. The uniqueness of the infinite component of
$\mathcal{O}_{p}$ follows by ergodicity (with respect to translations of
${\mathbb{R}}^{d}$), invariant under $\pi/2$-rotations and reflections at the
coordinate axes, positive association of black increasing events and
asymptotic independence property of $\tilde{\eta}$ (see [40, Section 3.2]).

We now give a simple criterion for verifying noise sensitivity and exceptional
times in the confetti percolation model analogous to the one for
$k$-percolation in the Poisson Boolean model.

######  Theorem 9.5 (Noise sensitivity criteria).

Consider the occupied region $\mathcal{O}_{p}$ in the confetti percolation
model defined in (9.1) with grain distributions
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ satisfying the assumptions in (8.0). Let
$f_{n}:=f_{n}(p)=\mathbf{1}\\{Cross_{\kappa n,n}(p)\\}$ for $\kappa>0$ where
$Cross_{\kappa n,n}(p)$ is the crossing event defined in (9.2). Assume that
$\\{f_{n}(p_{c})\\}_{n\geq 1}$ is non-degenerate as in (7.1) and
$\mathbb{P}(Arm_{r,s}(p_{c}))\to 0$ as $s\to\infty$. Then we have the
following.

  1. (i)

$\\{f_{n}(p_{c})\\}_{n\geq 1}$ is noise sensitive.

  2. (ii)

$\\{f_{n}(p_{c})\\}_{n\geq 1}$ has an infinite set of exceptional times in
$[0,1]$ i.e., $|S_{n}\cap[0,1]|\overset{\mathbb{P}}{\longrightarrow}\infty$
where the set of exceptional times $S_{n}$ is defined below (7.1).

  3. (iii)

Assume that there exists $C<\infty$ and $c_{arm}>0$ such that for all $s>2r$,
the following holds :

$\mathbb{P}(Arm_{r,s}(p_{c}))\leq Cs^{-c_{arm}}.$

Then, $\mathbb{P}(Cross_{\kappa n,n}((1+c_{n})p_{c}))\to 1$ and
$\mathbb{P}(Cross_{\kappa n,n}((1-c_{n})p_{c}))\to 0$ where
$c_{n}=n^{-\frac{c_{arm}}{2}+\epsilon}$ for some $\epsilon>0$.

######  Remark 9.6.

  1. 1.

Similarly to what was observed in Remark 8.4-1, one can use Theorem 7.4 to
show that the conclusion of Point (iii) in the previous statement continues to
hold for sequences of the type $c_{n}=n^{-c_{arm}+\epsilon}$.

  2. 2.

Reasoning as in Remark 8.4(i), using the Harris-FKG inequality and
monotonicity of the events, we can derive that
$\mathbb{P}(Arm_{r,s}(p_{c}))\leq b_{r}^{-1}\theta_{s-r}(p_{c})$ where $b_{r}$
is as defined below (9.6). This will be useful in Corollary 9.8.

###### Proof.

As with the proof of Theorem 8.3, we construct a suitable randomized stopping
set and then appeal to our general results in Proposition 6.3 and Corollary
7.2 to prove the first two statements. The third statement does not directly
follow from Theorem 7.3 but needs adaptation of the techniques therein.

We fix a $\kappa>0$. Let $R_{n}=[0,\kappa n]\times[0,n]^{d-1}$ be the
rectangle. Given $s\in(0,\kappa n)$, let $L=\\{s\\}\times[0,n]^{d-1}$ and let
$S$ be union of the connected components of $\mathcal{O}_{p}\cap R_{n}$ that
intersect $L$. Again if no such component exists, $S=\emptyset$. Define
$Z_{n}^{s}:=(S\cup L)\oplus B_{r}(0)$. By arguments as in the proof of Theorem
8.3 and Remark A.8, we have that
$\tilde{Z}_{n}^{s}=Z_{n}^{s}\times{\mathbb{R}}_{+}\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0,1\\}$
is a stopping set and determines $f_{n}$. As usual, we choose our randomized
stopping set $Z_{n}$ by first choosing $Y\in(0,n)$ uniformly at random and
then setting $\tilde{Z}_{n}^{s}=\tilde{Z}_{n}^{Y}$. As in Theorem 8.3, we
derive the following bound on the revealment probability :

$\delta_{n}:=\sup_{\tilde{x}=(x,t,K_{1},K_{2},a)\in\tilde{\mathbb{X}}}\mathbb{P}(\tilde{x}\in\tilde{Z}_{n})\leq
2(\kappa n)^{-1}\int_{0}^{\kappa n}\mathbb{P}(Arm_{r,s}(p_{c}))\mathrm{d}s.$

Thus the proof of the first statement of our theorem is complete using our
assumption on decay of arm probability and Proposition 6.3. Now the second
statement will follow from Corollary 7.2, provided we can show jump regularity
of $f_{n}$. We will show jump-regularity at the end of the proof.

Now, we will prove the third statement. Set $c_{n}=n^{-c_{arm}/2+\epsilon}$
for some $\epsilon>0$. Observe that by our assumption on decay of arm-
probabilities in the third statement, Corollary 7.2 and the jump-regularity of
$f_{n}$ (to be proved below), we have that

$|S_{n}\cap[0,c_{n}]|\overset{\mathbb{P}}{\longrightarrow}\infty.$ (9.10)

Abusing notation, we will let $\tilde{\eta}$ be the Poisson point process on
$R^{+}_{n}\times{\mathbb{R}}_{+}\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0,1\\}$
with $p=p_{c}$ and $R_{n}^{+}=R_{n}\oplus B_{0}(2r)$. Let $\tilde{\eta}^{t}$
be the Markov process driven by Ornstein-Uhlenbeck semigroup as in Section 6.1
with stationary distribution same as that of $\tilde{\eta}$ and also
$\tilde{\eta}^{0}=\tilde{\eta}$. To make explicit the dependence on the point
process, we set $f^{t}_{n}(p_{c}):=f_{n}(\tilde{\eta}^{t})$ to denote the
left-right crossing event of $R_{n}$ in
$\mathcal{O}_{p_{c}}(\tilde{\eta}^{t})$ i.e., $Cross_{kn,n}(p_{c})$ for
$\mathcal{O}_{p_{c}}(\tilde{\eta}^{t})$. Now, using (9.10), we have that

$\displaystyle\mathbb{P}(f^{t}_{n}(p_{c})=1\,\mbox{for some
$t\in[0,c_{n}]$})\to 1\,\mbox{as $n\to\infty$}.$ (9.11)

For $t\geq 0$, let $\tilde{\eta}^{t}_{+}$ be the Poisson process of points
born before time $t$ and $\tilde{\eta}^{t}_{-}\subset\tilde{\eta}$ be the
Poisson process of points that have been removed by time $t$. Let
$\tilde{\eta}^{t}_{b+}\subset\tilde{\eta}^{t}_{+}$ be the subset of ‘black
grains’ i.e.,
$\tilde{\eta}^{t}_{b+}=\tilde{\eta}^{t}_{+}\cap(R^{+}_{n}\times{\mathbb{R}}_{+}\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0\\})$.
Similarly, we denote $\tilde{\eta}^{t}_{w-}\subset\tilde{\eta}^{t}_{-}$ be the
subset of ‘white grains’ that were removed by time $t$. As the process
$\tilde{\eta}^{t}_{+}$ is independent of $\tilde{\eta}$, so is the process
$\tilde{\eta}^{t}_{b+}$ and the latter process has intensity measure
$p_{c}t\tilde{\gamma}$, where recall that $\tilde{\gamma}$ is the intensity
measure of the Poisson point process $\tilde{\eta}$ in the above paragraph.
Then $\zeta^{t}:=\tilde{\eta}-\tilde{\eta}^{t}_{w-}+\tilde{\eta}^{t}_{b+}$ is
a Poisson process with intensity measure
$(p_{c}+e^{-t}(1-p_{c})+tp_{c})\tilde{\gamma}$ and the intensity measure of
the ‘black grains’ is $(1+t)p_{c}\tilde{\gamma}$. So, the probability of a
grain being black is $\frac{(1+t)p_{c}}{p_{c}+e^{-t}(1-p_{c})+tp_{c}}$. We let
$\tilde{\eta}^{t}_{b},\tilde{\eta}^{t}_{w}$ denote respectively the black and
white grains in $\tilde{\eta}^{t}$. Similarly, we use the notation
$\zeta^{t}_{b},\zeta^{t}_{w}$. By construction of $\tilde{\eta}^{t}$ and
$\zeta^{t}$, the ‘black grains’ are never removed in $\zeta^{t}$ while they
may be removed in $\tilde{\eta}^{t}$ depending on their lifetimes. Also,
‘white grains’ are removed in $\zeta^{t}$ and not added while newer white
grains may be added in $\tilde{\eta}^{t}$. Thus,
$\tilde{\eta}^{t}_{b}\subset\zeta^{t}_{b}\subset\zeta^{c_{n}}_{b}$ and
$\tilde{\eta}^{t}_{w}\supset\zeta^{t}_{w}\supset\zeta^{c_{n}}_{w}$ for $t\leq
c_{n}$. Since $f_{n}$ is monotonic (9.3), $f_{n}(\eta^{t})\leq
f_{n}(\zeta^{t})\leq f_{n}(\zeta^{c_{n}}),t\in[0,c_{n}]$. Now, by (9.11), we
have that $\mathbb{E}[f_{n}(\zeta^{c_{n}})]\to 1$ as $n\to\infty$. Since the
probability of a grain being black in $\zeta^{c_{n}}$ is at most
$(1+c_{n})p_{c}$, again by monotonicity, we have that
$\mathbb{P}(Cross_{kn,n}((1+c_{n})p_{c}))\to 1$ as $n\to\infty$. Adapting the
arguments in Theorem 7.3 as we did above, we can obtain the second part of the
statement also.

Only jump-regularity remains to be proven now to complete the proof of the
theorem. Unlike in the $k$-percolation case, it is not true that
$\tilde{\eta}\cap(R_{n}^{+}\times{\mathbb{R}}_{+}\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0,1\\})$
has a finite intensity measure for any $n$. Nevertheless, we can still show
that $f_{n}$ is well-approximate by functionals that depend on $\tilde{\eta}$
restricted to a compact set and are hence jump-regular.

Let $\tilde{\eta}_{h}:=\tilde{\eta}\cap{\mathbb{X}}_{h}$ where
${\mathbb{X}}_{h}:=R^{+}_{n}\times[0,h]\times{\mathcal{K}}_{r}\times{\mathcal{K}}_{r}\times\\{0,1\\}$
i.e., $\tilde{\eta}_{h}$ is the Poisson process restricted to height $h$ for
some $h>0$. Note that
$\tilde{\eta}^{t}_{h}:=\tilde{\eta}^{t}\cap{\mathbb{X}}_{h}$ is the Markov
process driven by Ornstein-Uhlenbeck semigroup with stationary distribution
same as that of $\tilde{\eta}_{h}$. Analogous to
$f^{t}_{n}(p_{c}):=f_{n}(\tilde{\eta}^{t})$, define
$f^{t}_{n,h}(p_{c}):=f_{n}(\tilde{\eta}^{t}_{h})$ to be the indicator of the
crossing event $Cross_{\kappa n,n}(p_{c})$ in
$\mathcal{O}_{p}(\tilde{\eta}_{h}^{t})$ for $h,t>0$. Since
$\tilde{\gamma}({\mathbb{X}}_{h})<\infty$, $f^{t}_{n,h}(p_{c})$ is jump-
regular as it depends only on the Poisson process inside the compact set
${\mathbb{X}}_{h}$. Suppose we show that

$\lim_{h\to\infty}\sup_{t\in[0,1]}|f^{t}_{n}(p_{c})-f^{t}_{n,h}(p_{c})|\overset{p}{=}0.$
(9.12)

Then, we claim that $f_{n}^{t}(p_{c})$ is also jump-regular. We argue this as
follows : Let $S_{n}$ be the set of exceptional times of $f_{n}^{t}$
restricted to $[0,1]$ (see the definition of exceptional times in Section 7).
Similarly, set $S_{n,h}$ to be the set of exceptional times of $f_{n,h}^{t}$
restricted to $[0,1]$. To show jump-regularity of $f_{n}^{t}(p_{c})$, it
suffices to show that $\mathbb{P}(S_{n}<\infty)=1$ as $f_{n}^{t}(p_{c})$ is
$\\{0,1\\}$-valued. Fix $M,h>0$. Using that $S_{n}=S_{n,h}$ if
$f^{t}_{n}\equiv f^{t}_{n,h}$ for $t\in[0,1]$ and that $f^{t}_{n},f^{t}_{n,h}$
are $\\{0,1\\}$-valued, we derive that

$\displaystyle\mathbb{P}(S_{n}\geq M)$
$\displaystyle\leq\mathbb{P}(S_{n,h}\geq
M)+\mathbb{P}(\sup_{t\in[0,1]}|f^{t}_{n}(p_{c})-f^{t}_{n,h}(p_{c})|\neq 0)$
$\displaystyle=\mathbb{P}(S_{n,h}\geq
M)+\mathbb{P}(\sup_{t\in[0,1]}|f^{t}_{n}(p_{c})-f^{t}_{n,h}(p_{c})|\geq 1)$

Now, by letting $M\to\infty$ first and then letting $h\to\infty$ and using
jump-regularity of $f^{t}_{n,h}(p_{c})$ and (9.12), we obtain that
$\lim_{M\to\infty}\mathbb{P}(S_{n}\geq M)=0$. Thus, we have that
$\mathbb{P}(S_{n}<\infty)=1$ and hence that $f_{n}^{t}$ is jump-regular.

We are now left to prove (9.12). We sub-divide $R^{+}_{n}$ into small cubes
$R_{n1},\ldots,R_{nk_{a}}$ of side-length $a$ and $a$ small enough such that
the diameter of each cube is at most $r_{0}/4$ for $r_{0}$ as in assumption
(8.0). Set
${\mathcal{K}}^{\prime}_{r_{0}}:=\\{K\in{\mathcal{K}}_{r}:B_{r_{0}}(0)\subset
K\\}$. Our key observation is that if for some $t$ and all $i$,
$\tilde{\eta}^{t}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})\neq
0$, then
$\mathcal{O}_{p}(\tilde{\eta}^{t}_{h})=\mathcal{O}_{p}(\tilde{\eta}^{t})$ and
so $f_{n}^{t}(p_{c})=f_{n,h}^{t}(p_{c})$. Thus, we have that

$\\{\sup_{t\in[0,1]}|f^{t}_{n}(p_{c})-f^{t}_{n,h}(p_{c})|\neq
0\\}\subset\bigcup_{i=1}^{k_{a}}\bigcup_{t\in[0,1]}\\{\tilde{\eta}^{t}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})=0\\}.$
(9.13)

We now show that the probability of the latter event can be made small by
choosing a large $h$. By stationarity of the Poisson point process
$\tilde{\eta}_{h}$, it suffices to show that for each $i\leq k_{a}$,

$\lim_{h\to\infty}\mathbb{P}(\bigcup_{t\in[0,1]}\\{\tilde{\eta}^{t}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})=0\\})=0.$
(9.14)

Let us fix a $i\leq k_{a}$ to prove (9.14). Set
$N_{h}:=\tilde{\eta}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})$.
Note that $N_{h}$ is Poisson distributed with mean $a^{d}\beta_{1}\beta_{2}h$
where $\beta_{i}=\mathbb{Q}_{i}({\mathcal{K}}^{\prime}_{r_{0}}),i=1,2$. By
assumption (8.0), $\beta_{i}>0,i=1,2$. By the definition of the Markov process
$\tilde{\eta}^{t}_{h}$, if
$\tilde{\eta}^{t}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})=0$
for some $t\in[0,1]$, then either
$\tilde{\eta}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})=0$
or the life-time of every particle in $\tilde{\eta}_{h}\cap
R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\}$
is at most $1$. Since the lifetimes are all i.i.d. exponential($1$) random
variables and $N_{h}$ is a Poisson random variable, we can now compute the
required probability.

$\displaystyle\mathbb{P}(\bigcup_{t\in[0,1]}\\{\tilde{\eta}^{t}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})=0\\})$
$\displaystyle\leq\mathbb{P}(\tilde{\eta}_{h}(R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\})=0)$
$\displaystyle\quad+\mathbb{P}(\mbox{life-time of every particle in
$\tilde{\eta}_{h}\cap
R_{ni}\times[0,h]\times{\mathcal{K}}^{\prime}_{r_{0}}\times{\mathcal{K}}^{\prime}_{r_{0}}\times\\{0,1\\}$
is at most $1$})$
$\displaystyle=\exp\\{-a^{d}\beta_{1}\beta_{2}h\\}+\mathbb{E}[(1-e^{-1})^{N_{h}}]$
$\displaystyle=\exp\\{-a^{d}\beta_{1}\beta_{2}h\\}+\exp\\{-a^{d}\beta_{1}\beta_{2}he^{-1}\\}.$
(9.15)

Thus letting $h\to\infty$, we obtain (9.14) which gives (9.12) via (9.13).
This completes the proof of jump-regularity of $f_{n}^{t}(p_{c})$ and hence
the proof of the theorem. ∎

Combining the above theorem with [4, Theorem 8.3], we get the following
corollary in the case of planar confetti percolation model with grains
supported on balls centred at origin. Recall that ${\mathcal{K}}_{r}^{b}$ is
the space of all balls of radius at most $r$ centred at origin (see above
(8.0)).

######  Corollary 9.7.

Consider the occupied region $\mathcal{O}_{p}$ in the planar confetti
percolation model (i.e., $d=2$) as defined in (9.1) with grain distributions
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ satisfying assumption (8.0) and further
$\mathbb{Q}_{1},\mathbb{Q}_{2}$ are supported on ${\mathcal{K}}_{r}^{b}$. Then
for $f_{n}:=f_{n}(p_{c})$ as defined in Theorem 9.5, the conclusions of
Theorem 9.5 hold.

We now give a second application to noise sensitivity of confetti percolation
using the arguments in Theorem 9.3. This applies to more general grains
distribution than balls but is less quantitative.

######  Corollary 9.8.

Consider the occupied region $\mathcal{O}_{p}$ in the planar confetti
percolation model (i.e., $d=2$) as defined in (9.1) with grain distributions
$\mathbb{Q}_{1}=\mathbb{Q}_{2}=\mathbb{Q}$ satisfying assumption (8.0).
Further, assume that $\mathbb{Q}$ is invariant under $\pi/2$-rotations and
reflections at the coordinate axes. Then for
$f_{n}:=f_{n}(1/2)=\mathbf{1}\\{Cross_{n,n}(1/2)\\}$, the first two statements
in the conclusions of the Theorem 9.5 hold.

###### Proof.

Observe that by planar duality, a left-right crossing of $[0,n]^{2}$ by
$\mathcal{O}_{p}$ exists iff there is no top-down crossing of $[0,n]^{2}$ by
$\mathcal{V}_{p}$. Further using $\pi/2$-rotation invariance of $\mathbb{Q}$,
we have that $\mathbb{P}(Cross_{n,n}(p))=1-\mathbb{P}(Cross^{*}_{n,n}(p))$ for
all $p\in[0,1]$. By self-duality at $p=1/2$, we obtain that
$\mathbb{P}(Cross_{n,n}(1/2))=\mathbb{P}(Cross^{*}_{n,n}(1/2))=1/2$ for all
$n\geq 1$. Thus, $f_{n}$ is non-degenerate as in (7.1).

As remarked below Theorem 9.5, we have that

$\mathbb{P}(Arm_{r,s}(1/2))\leq b_{r}^{-1}\theta_{s-r}(1/2),$

where $b_{r}$ is as defined below (9.6) and $s>2r$. From Zhang’s argument as
elaborated in the proof of Theorem 9.3, we obtain that if $\theta(1/2)>0$ then
$\mathbb{P}(Cross_{n,n}(1/2))\to 1$ as $n\to\infty$. Since
$\mathbb{P}(Cross_{n,n}(1/2))=1/2$ for all $n\geq 1$, we have that
$\theta(1/2)=0$. Thus, we also obtain that
$\lim_{s\to\infty}\theta_{s-r}(1/2)=0$ and thereby verifying the assumptions
of Theorem 9.5. ∎

###  10 Noise sensitivity for the planar Boolean model with unbounded balls

While the previous two sections studied sharp phase transition and noise
sensitivity for some percolation models with bounded range of dependence, we
will now show that our noise sensitivity results can be extended to a model
with a unbounded range of dependence. We will focus on a very specific model -
the planar Poisson Boolean model - as many explicit estimates are known here
and this will best illustrate the usefulness of our results and also the proof
ideas.

We shall adopt the framework of [4]. Let
$\tilde{\eta}:=\\{(X_{i},R_{i})\\}_{i\geq
1}\subset{\mathbb{R}}^{2}\times{\mathbb{R}}_{+}$ be a Poisson point process of
intensity $\gamma\,\mathrm{d}x\,\mathbb{Q}(dr)$ where $\mathbb{Q}$ is a
probability distribution on ${\mathbb{R}}_{+}$ called the radii distribution
such that for some $\alpha>0$,

$\int_{0}^{\infty}r^{2+\alpha}\mathbb{Q}(dr)\in(0,\infty).$ (10.1)

In other words, $\\{X_{i}\\}_{i\geq 1}$ is a stationary Poisson point process
of intensity $\gamma$ and the points are equipped with i.i.d. marks
$R_{i},i\geq 1$. The occupied region of the planar Poisson Boolean model is
defined as

$\mathcal{O}(\gamma):=\mathcal{O}(\tilde{\eta})=\bigcup_{i\geq
1}B_{R_{i}}(X_{i}).$ (10.2)

Similarly, we will use other notations from Section 8. In particular, one can
define a critical intensity $\gamma_{c}$ as in (8.3). It was shown in [37]
that (10.1) with $\alpha=0$ is necessary for $\gamma_{c}>0$ and much later
[33] showed that (10.1) with $\alpha=0$ is sufficient for $\gamma_{c}>0$.
Thus, using a standard coupling with the bounded radius case, one can obtain
that $\gamma_{c}\in(0,\infty)$ under condition (10.1) with $\alpha=0$; see for
example, [4, Theorem 1.1]. We have assumed $\alpha>0$ as the more quantitative
estimates on arm probabilities in [4] require this assumption.

Our main theorem in this section yields quantitative noise sensitivity,
existence of exceptional times and bounds on critical window for crossing
probabilities in the planar Poisson Boolean model as defined above. This was
suggested as an open problem in [4].

######  Theorem 10.1 (Noise sensitivity for planar Poisson Boolean model).

Let $\tilde{\eta}$ be the Poisson point process as defined above with
$\gamma=\gamma_{c}$ and $\mathcal{O}(\gamma)$ be the occupied region of the
Poisson Boolean model as defined in (10.2) with the radii distribution
$\mathbb{Q}$ satisfying (10.1). Let $\kappa>0$ be given. Let
$f_{n}(\gamma_{c})=f_{n}(\tilde{\eta}):=\mathbf{1}\\{Cross_{\kappa
n,n}(\tilde{\eta})\\}$ where $Cross_{\kappa n,n}(\tilde{\eta})$ is the
crossing event defined in (8.2). Then the following statements hold.

  1. (i)

$\\{f_{n}(\gamma_{c})\\}_{n\geq 1}$ is noise sensitive.

  2. (ii)

$\\{f_{n}(\gamma_{c})\\}_{n\geq 1}$ has an infinite set of exceptional times
in $[0,1]$ i.e., $|S_{n}\cap[0,1]|\overset{\mathbb{P}}{\longrightarrow}\infty$
where the set of exceptional times $S_{n}$ is as defined below (7.1).

  3. (iii)

There exists $b>0$ such that $\mathbb{E}[f_{n}((1+n^{-b})\gamma_{c})]\to 1$
and $\mathbb{E}[f_{n}((1-n^{-b})\gamma_{c})]\to 0$ as $n\to\infty$.

The last statement can be used to provide an alternative proof of [4, Theorem
5.1] which shows sharpness of phase transition of crossing probabilities via
discretization, an inequality of Talagrand [73] and the Russo-Margulis
formula.

###### Proof.

The proof of the theorem relies on our exact estimates for the case of bounded
balls in Theorem 8.3 and the following observation relating noise sensitivity
of a sequence of functions to its approximation. Suppose that
$f_{n},f^{\prime}_{n}:{\mathbf{N}}\to[0,1],n\geq 1$ are two sequences of
functions. Then, using the triangle inequality, we have that

$|\operatorname{{\mathbb{C}ov}}[f_{n}(\tilde{\eta}^{t}),f_{n}(\tilde{\eta})]-\operatorname{{\mathbb{C}ov}}[f^{\prime}_{n}(\tilde{\eta}^{t}),f^{\prime}_{n}(\tilde{\eta})]|\leq
4\mathbb{P}(f_{n}(\tilde{\eta})\neq f^{\prime}_{n}(\tilde{\eta})),$ (10.3)

where $\tilde{\eta}^{t},t\geq 0$ is the Markov process defined in Section 6.1.
This is the Markov process driven by Ornstein-Uhlenbeck semigroup with
stationary distribution same as that of $\tilde{\eta}$ and also
$\tilde{\eta}^{0}=\tilde{\eta}$.

Choose a sequence $r_{n}=n^{1-\epsilon},n\geq 0$ for an
$\epsilon<\frac{\alpha}{2+\alpha}$. Define
$\tilde{\eta}^{\prime}_{n}:=\\{(X_{i},R_{i})\in\tilde{\eta}:R_{i}\leq
r_{n}\\}$ be the restricted Poisson point process and
$f^{\prime}_{n}(\tilde{\eta}):=\mathbf{1}\\{Cross_{\kappa
n,n}(\tilde{\eta}^{\prime}_{n})\\}.$ Because of (10.3), the proof of the
theorem is complete if we show that $f^{\prime}_{n}$ is noise sensitive and
that

$\lim_{n\to\infty}\mathbb{P}(f_{n}(\tilde{\eta})\neq
f^{\prime}_{n}(\tilde{\eta}))=0$ (10.4)

Firstly, we show that $f^{\prime}_{n}(\tilde{\eta})$ is noise sensitive.
Following the proof of Theorem 8.3, we have that there exists a randomized
stopping set $Z^{\prime}_{n}$ determining $f^{\prime}_{n}(\tilde{\eta})$ and
satisfying the assumptions of Proposition 5.3 such that

$\delta^{\prime}_{n}:=\delta(Z^{\prime}_{n})\leq\frac{2}{\kappa
n}\int_{0}^{\kappa
n}\mathbb{P}(Arm_{r_{n},s}(\tilde{\eta}^{\prime}_{n}))\mathrm{d}s,$

where $Arm_{r,s}(\cdot)$ is the one-arm crossing event as defined in (8.2).
Since the one-arm crossing event is increasing and further using [4, Corollary
4.4], we obtain that for some $c_{arm}>0$,

$\delta^{\prime}_{n}\leq\frac{2}{\kappa n}\int_{0}^{\kappa
n}\mathbb{P}(Arm_{r_{n},s}(\tilde{\eta}))\mathrm{d}s\leq\frac{C}{\kappa
n}\left(2r_{n}+\int_{2r_{n}}^{\kappa
n}\Big{(}\frac{r_{n}}{s}\Big{)}^{c_{arm}}\mathrm{d}s\right)\to 0,$

where the convergence is due to the choice of $r_{n}$. Thus $f^{\prime}_{n}$
is noise sensitive by Proposition 5.3. Now we will show (10.4).

Observe that by definition,

$\mathbb{P}(f_{n}(\tilde{\eta})\neq
f^{\prime}_{n}(\tilde{\eta}))\leq\mathbb{P}(\exists\,i\in{\mathbb{N}},R_{i}\geq
r_{n}\,\,\mbox{and}\,\,B_{R_{i}}(X_{i})\cap[0,\kappa n]^{2}\neq\emptyset).$
(10.5)

From arguments similar to that in [4, (2.13)], we have that for some constant
$C$,

$\displaystyle\mathbb{P}(\exists\,i\in{\mathbb{N}},R_{i}\geq
r_{n}\,\,\mbox{and}\,\,B_{R_{i}}(X_{i})\cap[0,\kappa n]^{2}\neq\emptyset)$
$\displaystyle\leq\mathbb{P}(\tilde{\eta}([0,\kappa
n+r_{n}]^{2}\times[r_{n},\infty))\neq 0)$
$\displaystyle\quad+\mathbb{P}(\exists\,i\in{\mathbb{N}},X_{i}\notin[0,\kappa
n+r_{n}]^{2}\,\,\mbox{and}\,\,B_{R_{i}}(X_{i})\cap[0,\kappa
n]^{2}\neq\emptyset)$ $\displaystyle\leq C\gamma_{c}\Big{(}1+\frac{\kappa
n+1}{r_{n}}\Big{)}^{2}\left(r_{n}^{2}\mathbb{Q}((r_{n},\infty))+\int_{r_{n}}^{\infty}r^{2}\mathbb{Q}(\mathrm{d}r)\right)$
$\displaystyle\leq 2C\gamma_{c}\Big{(}1+\frac{\kappa
n+1}{r_{n}}\Big{)}^{2}r_{n}^{-\alpha}\int_{r_{n}}^{\infty}r^{2+\alpha}\mathbb{Q}(\mathrm{d}r)\leq
8C\gamma_{c}n^{-\alpha+\epsilon(2+\alpha)}\int_{r_{n}}^{\infty}r^{2+\alpha}\mathbb{Q}(\mathrm{d}r)\to
0,$ (10.6)

where in the last inequality we have again used the choice of $r_{n}$ and in
the convergence, we have used the choice of $\epsilon$. Thus, using (10.5), we
have proven (10.4) and hence the proof of the first statement is complete via
(10.3). Further, by using the above bounds and (6.8) for covariance of
$f^{\prime}_{n}$, we obtain that

$\operatorname{{\mathbb{C}ov}}[f_{n}(\tilde{\eta}^{t}),f_{n}(\tilde{\eta})]\leq
8C\gamma_{c}n^{-\alpha+\epsilon(2+\alpha)}\int_{r_{n}}^{\infty}r^{2+\alpha}\mathbb{Q}(\mathrm{d}r)+C\delta^{\prime}_{n}\frac{e^{-t}}{(1-e^{-t})^{2}}.$
(10.7)

The second and third statement follow from (10.7), the explicit bounds for
$\delta^{\prime}_{n}$ and Theorems 7.1 and 7.3 (see remark below the theorem)
if we show jump-regularity of $f_{n}$. We will do so now by using a strategy
similar to that in the proof of Theorem 9.5.

Fix $h>0$. Let $\tilde{\eta}_{h}:=\tilde{\eta}\cap([0,\kappa
n+h]^{2}\times[0,h])$ be the truncated Poisson process and
$g_{n,h}(\tilde{\eta})=\mathbf{1}\\{Cross_{\kappa n,n}(\tilde{\eta}_{h})\\}$
denote the indicator of the crossing event of the window $[0,\kappa
n]\times[0,n]$ by $\mathcal{O}(\tilde{\eta}_{h})$. Since $\tilde{\eta}_{h}$
has finite intensity measure, $g_{n,h}$ is jump-regular. As argued in Theorem
9.5 (see (9.12) and below), the proof of jump-regularity of $f_{n}$ is
complete if we show that

$\lim_{h\to\infty}\sup_{t\in[0,1]}|f_{n}(\tilde{\eta}^{t})-g_{n,h}(\tilde{\eta}^{t})|\overset{p}{=}0.$
(10.8)

Let $\tilde{\eta}_{+}$ denote the new points born in time interval $[0,1]$ in
the Markov process $\tilde{\eta}^{t}$. By the definition of the Markov
process, $\tilde{\eta}_{+}$ is independent of $\tilde{\eta}$ and has intensity
measure $\gamma_{c}\,\mathrm{d}x\,\mathbb{Q}(\mathrm{d}r)$ i.e.,
$\tilde{\eta}_{+}\overset{d}{=}\tilde{\eta}$. Using this and the definitions
of $f_{n},g_{n,h}$, we derive that

$\displaystyle\mathbb{P}(\sup_{t\in[0,1]}|f_{n}(\tilde{\eta}^{t})-g_{n,h}(\tilde{\eta}^{t})|\geq
1)$ $\displaystyle\leq\mathbb{P}(\exists
t\in[0,1],\tilde{\eta}^{t}(\\{(x,r):r>h,B_{r}(x)\cap[0,\kappa
n]^{2}\neq\emptyset\\})\neq 0)$
$\displaystyle\leq\mathbb{P}(\tilde{\eta}(\\{(x,r):r>h,B_{r}(x)\cap[0,\kappa
n]^{2}\neq\emptyset\\})\neq 0)$
$\displaystyle\quad+\mathbb{P}(\tilde{\eta}_{+}(\\{(x,r):r>h,B_{r}(x)\cap[0,\kappa
n]^{2}\neq\emptyset\\})\neq 0)$ $\displaystyle\leq
2\mathbb{P}(\tilde{\eta}(\\{(x,r):r>h,B_{r}(x)\cap[0,\kappa
n]^{2}\neq\emptyset\\})\neq 0)$ $\displaystyle\leq
4C\gamma_{c}\Big{(}1+\frac{\kappa
n+1}{h}\Big{)}^{2}\int_{h}^{\infty}r^{2}\mathbb{Q}(\mathrm{d}r)$

where in the last inequality we have used the bounds as derived in (10.6) but
for $h$ instead of $r_{n}$. From our last bound the required convergence in
(10.8) follows and hence the proof of jump-regularity of $f_{n}$ and also the
proof of the Theorem is complete. ∎

The $2+\alpha$ moment condition (10.1) can be replaced by $d+\alpha$ moment
condition in $d$-dimensions and while most of the above proof would go
through, the arm probability estimates of [4, Corollary 4.4] are not known and
this is the main obstacle in extending the theorem to higher-dimensions.

###  11 Further applications

We now briefly indicate some further possible applications of our main
theorems without going into details.

Vacant-set percolation: Our proof methods can also be adapted to show sharp
phase transition for $\mathcal{O}(\gamma)^{c}$ as well i.e., the region
covered by at most $(k-1)$-balls. If $k=0$, this is the vacancy region in the
usual Boolean model and one can find sharp phase transition results for vacant
region of the Boolean model with random radii (including the case of unbounded
support in the planar case) in [4, 5, 63, 25]. Further, phase transition
results for the Boolean model with random radii with unbounded support have
been proven in [33, 35, 25]. A challenging question would be to extend the
results of [25] to Poisson Boolean model with general grains and possibly to
$k$-percolation model.

Voronoi Percolation: The results of the previous section on confetti
percolation can also be shown to hold for the Voronoi percolation model on the
Poisson process. These results were proven in [23, 3] by suitable
discretization and application of OSSS and Schramm-Steif inequalities
respectively. Our construction of stopping sets and proof methodology in the
previous section can be extended to the case of Voronoi percolation as well.
This will yield sharp phase transition as well as noise sensitivity and
exceptional times at criticality more directly from the results of [74].
Further, it should be possible to obtain analogues of Corollary 8.5 for more
general Voronoi percolation models as defined in [4, Section 8] and [31,
Section 1.2]. The necessary estimates for non-degeneracy of crossing events
and decay of arm-probabilities were established in [4, Theorem 8.1].

Level set percolation: Percolation of level sets of Poissonian shot-noise
random fields has been another model that has received some attention (see
[55, 8, 16]). More recently, in [46] RSW-type estimates, non-degeneracy of
crossing events, decay of arm-probabilities and sharp phase transition were
proven. In particular, [46, Proposition 4.2] is proved using the discrete OSSS
inequality and our continuum analogue of OSSS inequality can again be used to
avoid discretization. Further, one would expect to deduce noise sensitivity
and exceptional times for crossings in this model at criticality analogous to
Corollary 8.5.

Other continuum models: One can apply our methods also to prove a sharp phase
transition in random connection model with bounded edges and Miller-Abrahams
random resistor network with lower-bounded conductances more straightforwardly
(see [26]). Another model where it would be interesting to apply our results
is the Poisson stick model [67]. We had mentioned that $k$-percolation in the
Boolean model corresponds to face percolation in the Cěch complex. One can
also analogously define percolation of faces in the Vietoris-Rips complex
built on the Poisson point process. This is same as clique percolation on the
random graph induced by the Boolean model. Clique percolation was introduced
in [19]. See [43] for the above two models as well as two other models
describing connectivity of faces. Percolation in all these models can be
studied using our results.

### Appendices

In the forthcoming Appendices A & B, we consider the general setup of Section
1.5. In particular, $({\mathbb{X}},{\mathcal{X}})$ denotes a Borel space
endowed with a localizing ring ${\mathcal{X}}_{0}$ defined in terms of an
increasing sequence $\\{B_{n}:n\geq 1\\}$; see Subsection 1.5. Given a locally
finite measure $\lambda$, we write $\Pi_{\lambda}$ to indicate the law of a
Poisson process $\eta$ on ${\mathbb{X}}$ such that
$\lambda=\mathbb{E}[\eta(\cdot)]$.

####  A Graph-measurable mappings and stopping sets

A mapping $Z\colon{\mathbf{N}}\to{\mathcal{X}}$ is called graph-measurable
(see [54]) if $(x,\mu)\mapsto{\mathbf{1}}\\{x\in Z(\mu)\\}$ is a measurable
mapping on ${\mathbb{X}}\times{\mathbf{N}}$. Note that we do not equip
${\mathcal{X}}$ with a $\sigma$-field and that the mapping $Z$ is not assumed
to be a measurable mapping in any sense. All that is required in our paper is
the following measurability property.

######  Lemma A.1.

Suppose that $Z\colon{\mathbf{N}}\to{\mathcal{X}}$ is graph-measurable. Then,
writing $Z(\mu)^{c}={\mathbb{X}}\backslash Z(\mu)$, the mapping

${\mathbf{N}}\times{\mathbf{N}}\rightarrow{\mathbf{N}}\times{\mathbf{N}}\times{\mathbf{N}}\times{\mathbf{N}}:(\nu,\mu)\mapsto(\nu_{Z(\mu)},\mu_{Z(\mu)},\nu_{Z(\mu)^{c}},\mu_{Z(\mu)^{c}})$

is measurable.

###### Proof.

It suffices to prove that the first and third components of the above mapping
are measurable. The proof proceeds as that of Lemma 115-(i) in [45]. Let
$B\in{\mathcal{X}}$. We need to show that $(\nu,\mu)\mapsto\nu_{Z(\mu)}(B)$ is
measurable. By definition of a localizing ring and monotone convergence we may
assume that $B\in{\mathcal{X}}_{0}$. By the monotone class theorem (using
Dynkin systems) it can be shown that the mapping

$\displaystyle(\nu,\mu)\mapsto\int{\mathbf{1}}\\{(x,\mu)\in A,\,x\in
B\\}\,\nu(dx)$

is measurable for each $A\in{\mathcal{X}}\otimes{\mathcal{N}}$. In particular,
writing $H(\mu)$ for either $Z(\mu)$ or $Z(\mu)^{c}$ and applying the graph-
measurability of $Z$ (and therefore of $Z^{c}$), one deduces that
$\int{\mathbf{1}}\\{x\in H(\mu),\,x\in B\\}\,\nu(dx)$ is a measurable function
of $(\nu,\mu)$, as asserted. ∎

We say that a graph-measurable mapping $Z\colon{\mathbf{N}}\to{\mathcal{X}}$
is a stopping set, if

$\displaystyle
Z(\mu)=Z(\mu_{Z(\mu)}+\psi_{Z(\mu)^{c}}),\quad\mu,\psi\in{\mathbf{N}}.$ (A.1)

In particular a stopping set satisfies

$\displaystyle Z(\mu)=Z(\mu_{Z(\mu)}),\quad\mu\in{\mathbf{N}},$ (A.2)

so that $(x,\mu)\mapsto{\mathbf{1}}\\{x\in Z(\mu)\\}$ is
${\mathcal{X}}\otimes{\mathcal{N}}_{Z}$-measurable, where ${\mathcal{N}}_{Z}$
is the $\sigma$-field on ${\mathbf{N}}$ generated by $\mu\mapsto\mu_{Z(\mu)}$.
Observe that, if $Z^{\prime}$ is another stopping set such that $Z(\mu)\subset
Z^{\prime}(\mu)$ for every $\mu\in{\mathbf{N}}$, then (A.2) implies that
$\mathcal{N}_{Z}\subset\mathcal{N}_{Z^{\prime}}$.

We say that a measurable function $f\colon{\mathbf{N}}\to{\mathbb{R}}$ is
determined by a stopping set $Z$ if $f(\mu)=f(\mu_{Z(\mu)})$ for each
$\mu\in{\mathbf{N}}$. This means that $f$ is ${\mathcal{N}}_{Z}$-measurable.
In this case we even have that

$\displaystyle
f(\mu)=f(\mu_{Z(\mu)}+\psi_{Z(\mu)^{c}}),\quad\mu,\psi\in{\mathbf{N}}.$ (A.3)

This follows by applying the determination property of $f$ with $\mu$ replaced
by $\mu_{Z(\mu)}+\psi_{Z(\mu)^{c}}$ and then (A.1).

In contrast to the classical theory of stopping sets (see e.g. [81]) we do not
assume that $Z$ is taking its values in the space of closed (or compact)
subsets of a locally compact Polish space. But if this is the case and if $Z$
is measurable with respect to the Borel $\sigma$-field generated by the Fell
topology, then [9, Proposition A.1] shows that (A.1) is equivalent to the
standard definition of a stopping set, at least in the case of simple point
measures. Moreover, in this case the $\sigma$-field ${\mathcal{N}}_{Z}$
coincides with the classical stopped $\sigma$-field, given as the system of
all sets $A\in{\mathcal{N}}$ such that $A\cap\\{Z\subset
K\\}\in{\mathcal{N}}_{K}$ for each compact $K\subset{\mathbb{X}}$. Our
definition of a stopping set seems to be new in this generality. As already
observed, it is very convenient for the applications developed in our work,
since it allows one to avoid several restrictive (and technical) measurability
and topological assumptions that typically emerge when using the classical
theory.

But in the case $Z$ is measurable with respect to the Borel $\sigma$-field
generated by the Fell topology, then [9, Proposition A.1] shows that (A.1) is
equivalent to the standard definition of a stopping set.

We will need the following simple lemma.

######  Lemma A.2.

Suppose that $Z$ is a stopping set and let $\varphi,\psi\in{\mathbf{N}}$. Then
$\mu_{Z(\mu)}=\psi$ if and only if $\mu_{Z(\psi)}=\psi$. In this case
$Z(\mu)=Z(\psi)$.

###### Proof.

If $\mu_{Z(\mu)}=\psi$, then (A.1) yields $Z(\mu)=Z(\psi)$ and in particular
$\mu_{Z(\psi)}=\psi$. If, conversely, the latter holds, we have that $\psi$ is
supported by $Z(\psi)$ and $\mu_{Z(\psi)}=\psi_{Z(\psi)}$. Applying (A.1)
(with $\psi$ in place of $\mu$) we obtain that

$\displaystyle
Z(\psi)=Z(\psi_{Z(\psi)}+\mu_{Z(\psi)^{c}})=Z(\mu_{Z(\psi)}+\mu_{Z(\psi)^{c}})=Z(\mu).$

This finishes the proof.∎

We now fix a Poisson process $\eta$ on ${\mathbb{X}}$ with locally finite
intensity $\lambda$. The following result is well-known for classical stopping
sets and can be attributed to [68, Theorem 4]; see also [81, 82]. In our
Poisson setting (and still adopting the classical definition of stopping sets
from [81]) it can be derived from Lemma A.3 in [9] and the multivariate Mecke
equation (1.9). Our proof shows that such an approach applies to the more
general definiton stopping set adopted in this paper.

######  Theorem A.3.

Suppose that $Z$ is a stopping set and let $A\in{\mathcal{N}}$. Then

$\displaystyle\mathbb{P}(\eta_{Z(\eta)^{c}}\in
A\mid\eta_{Z(\eta)})=\Pi_{\lambda_{Z(\eta)^{c}}}(A),\quad\mathbb{P}\text{-a.s.\
on $\\{\eta(Z(\eta))<\infty\\}$}.$ (A.4)

###### Proof.

Let $g\colon{\mathbf{N}}\times{\mathbf{N}}\to{\mathbb{R}}_{+}$ be measurable
and $k\in{\mathbb{N}}$. Given $x_{1},\dots,x_{k}\in{\mathbb{X}}$ we write
$\mathbf{x}=(x_{1},\dots,x_{k})$ and
$\delta_{\mathbf{x}}:=\delta_{x_{1}}+\cdots+\delta_{x_{k}}$. We have that (see
e.g. equation (4.19) in [49])

$\displaystyle
I:=\mathbb{E}[{\mathbf{1}}\\{\eta(Z(\eta))=k\\}g(\eta_{Z(\eta)},\eta_{Z(\eta)^{c}})]=\frac{1}{k!}\mathbb{E}\bigg{[}\int
g(\delta_{\mathbf{x}},\eta_{Z(\eta)^{c}}){\mathbf{1}}\\{\eta_{Z(\eta)}=\delta_{\mathbf{x}}\\}\,\eta^{(k)}(\mathrm{d}\mathbf{x})\bigg{]}.$

By Lemma A.2 we have that $\eta_{Z(\eta)}=\delta_{\mathbf{x}}$ iff
$\eta_{Z(\delta_{\mathbf{x}})}=\delta_{\mathbf{x}}$ in which case
$Z(\eta)=Z(\delta_{\mathbf{x}})$. Therefore,

$\displaystyle I$ $\displaystyle=\frac{1}{k!}\mathbb{E}\bigg{[}\int
g(\delta_{\mathbf{x}},\eta_{Z(\delta_{\mathbf{x}})^{c}}){\mathbf{1}}\\{\eta_{Z(\delta_{\mathbf{x}})}=\delta_{\mathbf{x}}\\}\,\eta^{(k)}(\mathrm{d}\mathbf{x})\bigg{]}$
$\displaystyle=\frac{1}{k!}\mathbb{E}\bigg{[}\int
g(\delta_{\mathbf{x}},(\eta+\delta_{\mathbf{x}})_{Z(\delta_{\mathbf{x}})^{c}}){\mathbf{1}}\\{(\eta+\delta_{\mathbf{x}})_{Z(\delta_{\mathbf{x}})}=\delta_{\mathbf{x}}\\}\,\lambda^{k}(\mathrm{d}\mathbf{x})\bigg{]},$

where we have used the the multivariate Mecke equation (1.9) and the required
measurability of the mappings
$(\mu,\mathbf{x})\mapsto{\mathbf{1}}\\{\mu=\delta_{\mathbf{x}}\\}$,
$\mathbf{x}\mapsto\delta_{\mathbf{x}}$ and
$(\mu,\mathbf{x})\mapsto\mu_{Z(\delta_{\mathbf{x}})^{c}}$ follows from the
Borel property of ${\mathbb{X}}$ and Lemma A.1. Since
$(\eta+\delta_{\mathbf{x}})_{Z(\delta_{\mathbf{x}})}=\delta_{\mathbf{x}}$ iff
$\mathbf{x}\in Z(\delta_{\mathbf{x}})$ and $\eta(Z(\delta_{\mathbf{x}}))=0$ we
can use the complete independence property of $\eta$ (and Fubini’s theorem) to
obtain that

$\displaystyle I=\frac{1}{k!}\mathbb{E}\bigg{[}\iint
g(\delta_{\mathbf{x}},\psi){\mathbf{1}}\\{\mathbf{x}\in
Z(\delta_{\mathbf{x}}),\eta(Z(\delta_{\mathbf{x}}))=0\\}\,\Pi_{\lambda_{Z(\delta_{\mathbf{x}})^{c}}}(\mathrm{d}\psi)\,\lambda^{k}(\mathrm{d}\mathbf{x})\bigg{]}.$
(A.5)

Reversing the preceding arguments yields

$\displaystyle
I=\mathbb{E}\bigg{[}\int{\mathbf{1}}\\{\eta(Z(\eta))=k\\}g(\eta_{Z(\eta)},\psi)\,\Pi_{\lambda_{Z(\eta)^{c}}}(\mathrm{d}\psi)\bigg{]}.$

A simplified version of the preceding proof shows that this remains true for
$k=0$. Therefore

$\displaystyle\mathbb{E}[{\mathbf{1}}\\{\eta(Z(\eta))<\infty\\}g(\eta_{Z(\eta)},\eta_{Z(\eta)^{c}})]=\mathbb{E}\bigg{[}\int{\mathbf{1}}\\{\eta(Z(\eta))<\infty\\}g(\eta_{Z(\eta)},\psi)\,\Pi_{\lambda_{Z(\eta)^{c}}}(\mathrm{d}\psi)\bigg{]}.$

Taking $g$ of product form, this yields (A.4), provided that
$\Pi_{\lambda_{Z(\eta)^{c}}}(A)$ depends measurably on $\eta_{Z(\eta)}$. Since
$Z(\eta)=Z(\eta_{Z(\eta)})$ this follows from the measurability of
$\mu\mapsto\Pi_{\lambda_{Z(\mu)^{c}}}(A)$, a fact that can be verified as in
the the proof of [49, Lemma 13.4]. Indeed, it suffices to take

$A:=\\{\psi\in{\mathbf{N}}:\psi(C_{1})=m_{1},\dots,\psi(C_{n})=m_{n}\\},$

where $C_{1},\ldots,C_{n}\in{\mathcal{X}}$ are pairwise disjoint and
$m_{1},\ldots,m_{n}\in{\mathbb{N}}_{0}$. Then

$\Pi_{\lambda_{Z(\mu)^{c}}}(A)=\prod^{n}_{i=1}\frac{g_{i}(\mu)^{m_{i}}}{m_{i}!}\exp[-g_{i}(\mu)],$

where $g_{i}(\mu):=\int{\mathbf{1}}\\{x\in C_{i},\,x\in
Z(\mu)^{c}\\}\,\lambda(dx)$. Since $(x,\mu)\mapsto{\mathbf{1}}\\{x\notin
Z(\mu)\\}$ is measurable, the functions $g_{i}$ are measurable. ∎

######  Remark A.4.

The proof of Theorem A.3 shows that the full stopping set property (A.2) is
not required to conclude (A.4). We only need that
$Z\colon{\mathbf{N}}\to{\mathcal{X}}$ is graph-measurable and that the
following holds for all $\mu,\psi\in{\mathbf{N}}$ with
$\psi({\mathbb{X}})<\infty$. We have $\mu_{Z(\mu)}=\psi$ if and only if
$\mu_{Z(\psi)}=\psi$. In this case $Z(\mu)=Z(\psi)$.

It is worth mentioning that (A.5) yields a formula for the distribution of
$\eta_{Z(\eta)}$. For a constant mapping $Z$ this reduces to an elementary
property of a Poisson process.

######  Proposition A.5.

Suppose that $Z$ is a stopping set. Then

$\displaystyle\mathbb{P}(\eta(Z(\eta))<\infty,\eta_{Z(\eta)}\in\cdot)=\sum^{\infty}_{k=0}\frac{1}{k!}\int{\mathbf{1}}\\{\delta_{\mathbf{x}}\in\cdot\\}{\mathbf{1}}\\{\mathbf{x}\in
Z(\delta_{\mathbf{x}})\\}\exp[-\lambda(Z(\delta_{\mathbf{x}}))]\,\lambda^{k}(\mathrm{d}\mathbf{x}),$

where the summand for $k=0$ is interpreted as ${\mathbf{1}}\\{{\bf
0}\in\cdot\\}\exp[-\lambda(Z({\bf 0}))]$, where $\bf{0}$ stands for the zero
measure.

Next we extend Theorem A.3 so as to allow for $\eta(Z(\eta))=\infty$.

######  Theorem A.6.

Suppose that $Z_{1},Z_{2},\ldots$ are stopping sets such that $Z_{n}\uparrow
Z$ for some $Z\colon{\mathbf{N}}\to{\mathcal{X}}$. Then $Z$ is a stopping set.
If, moreover, $\mathbb{P}(\eta(Z_{n}(\eta))<\infty)=1$ for each
$n\in{\mathbb{N}}$ then we have for each $A\in\mathcal{N}$ that

$\displaystyle\mathbb{P}(\eta_{Z(\eta)^{c}}\in
A\mid\eta_{Z(\eta)})=\Pi_{\lambda_{Z(\eta)^{c}}}(A),\quad\mathbb{P}\text{-a.s.}$
(A.6)

###### Proof.

Graph-measurability of $Z$ is obvious. To check (A.1) we take
$\mu,\psi\in{\mathbf{N}}$ such that $\psi(Z(\mu))=0$. Then

$\displaystyle Z(\mu_{Z(\mu)}+\psi)$
$\displaystyle=\bigcup^{\infty}_{n=1}Z_{n}(\mu_{Z(\mu)}+\psi)=\bigcup^{\infty}_{n=1}Z_{n}(\mu_{Z_{n}(\mu)}+\mu_{Z(\mu)\setminus
Z_{n}(\mu)}+\psi)$
$\displaystyle=\bigcup^{\infty}_{n=1}Z_{n}(\mu_{Z(\mu)})=Z(\mu_{Z(\mu)}),$

concluding the proof of (A.1).

To prove (A.6) we need to show for each bounded and measurable
$f\colon{\mathbf{N}}\times{\mathbf{N}}\to{\mathbb{R}}_{+}$ that

$\displaystyle\mathbb{E}[f(\eta_{Z(\eta)},\eta_{{\mathbb{X}}\setminus
Z(\eta)})]=\mathbb{E}\bigg{[}\int f(\eta_{Z(\eta)},\mu_{{\mathbb{X}}\setminus
Z(\eta)})\,\Pi_{\lambda}(d\mu)\bigg{]}.$ (A.7)

By a monotone class argument we can assume that there exists
$B\in{\mathcal{X}}_{0}$ such that $f(\mu,\nu)=f(\mu_{B},\nu_{B})$ for all
$\mu,\nu\in{\mathbf{N}}$. It follows from (A.4) (and a monotone class
argument) that

$\displaystyle\mathbb{E}[f(\eta_{Z_{n}(\eta)},\eta_{{\mathbb{X}}\setminus
Z_{n}(\eta)})]=\mathbb{E}\bigg{[}\int
f(\eta_{Z_{n}(\eta)},\mu_{{\mathbb{X}}\setminus
Z_{n}(\eta)})\,\Pi_{\lambda}(d\mu)\bigg{]}.$ (A.8)

Since $\eta(B)<\infty$ we have that
$f(\eta_{Z_{n}(\eta)},\eta_{{\mathbb{X}}\setminus Z_{n}(\eta)})\to
f(\eta_{Z(\eta)},\eta_{{\mathbb{X}}\setminus Z(\eta)})$ with respect to the
discrete topology. By bounded convergence the left-hand side of (A.8) tends to
the left-hand side of (A.7). For the right-hand sides we note that bounded
convergence implies for each $\nu\in{\mathbf{N}}$ that

$\displaystyle\lim_{n\to\infty}\int
f(\nu_{Z_{n}(\nu)},\mu_{{\mathbb{X}}\setminus
Z_{n}(\nu)})\,\Pi_{\lambda}(d\mu)=\int
f(\nu_{Z(\nu)},\mu_{{\mathbb{X}}\setminus Z(\nu)})\,\Pi_{\lambda}(d\mu).$

Again by bounded convergence the right-hand side of (A.8) tends to the right-
hand side of (A.7). ∎

Property (A.6) is sometimes referred to as the Markov property of $\eta$ (see
again [68, Theorem 4] and [82]). To express it in a different way, we let
$\eta^{\prime}$ be an independent copy of $\eta$. Then (A.6) is equivalent to

$\displaystyle\eta\overset{d}{=}\eta_{Z(\eta)}+\eta^{\prime}_{{\mathbb{X}}\backslash
Z(\eta)},$ (A.9)

where the equality in distribution is in the sense of point processes.

######  Remark A.7.

The preceding results can be generalized as follows. Suppose that
$({\mathbb{M}},\mathcal{M})$ is another Borel space. Consider the product
${\mathbb{X}}\times{\mathbb{M}}$ equipped with the product $\sigma$-field and
some localizing ring. For each measure $\nu$ on
${\mathbb{X}}\times{\mathbb{M}}$ and each $B\in\mathcal{X}$ we denote by
$\nu_{B}:=\nu_{B\times{\mathbb{M}}}$ the restriction of $\nu$ to
$B\times{\mathbb{M}}$. We say that a graph-measurable mapping
$\tilde{Z}\colon{\mathbf{N}}({\mathbb{X}}\times{\mathbb{M}})\to{\mathcal{X}}$
is a stopping set on ${\mathbb{X}}$ if (A.1) holds. Then Theorems A.3 and A.6
remain true in an obviously modified form and essentially unchanged proofs.

######  Remark A.8.

Let $\tilde{Z}$ be a stopping set as in Remark A.7. Then
$Z:=\tilde{Z}\times{\mathbb{M}}$ is graph-measurable and a stopping set on
$({\mathbb{X}}\times{\mathbb{M}},{\mathcal{X}}\otimes\mathcal{M})$, in the
sense of (A.1).

A weaker version of the next result (corresponding to equation (A.12) below)
was proved in [81, Theorem 3]; see also [65, formula (3.3)].

######  Proposition A.9.

Suppose that $Z,Z_{1},Z_{2},\ldots$ are stopping sets such that
$\mathbb{P}(\eta(Z_{n}(\eta))<\infty)=1$ for each $n\in{\mathbb{N}}$ and
$Z_{n}\uparrow Z$. Then

$\displaystyle{\mathbf{1}}\\{x\in Z(\eta)\\}={\mathbf{1}}\\{x\in
Z(\eta+\delta_{x})\\},\quad\mathbb{P}\text{-a.s.},\,\lambda\text{-a.e.\ $x$}.$
(A.10)

###### Proof.

Let $B\in{\mathcal{X}}_{0}$. Then

$\displaystyle\mathbb{E}[\eta(B\cap Z(\eta))]$
$\displaystyle=\lambda(B)-\mathbb{E}[\mathbb{E}[\eta(B\cap
Z(\eta)^{c})\mid\eta_{Z(\eta)}]]$
$\displaystyle=\lambda(B)-\mathbb{E}\bigg{[}\int\mu(B)\,\Pi_{\lambda_{Z(\eta)^{c}}}(d\mu)\bigg{]}$
$\displaystyle=\lambda(B)-\mathbb{E}[\lambda_{Z(\eta)^{c}}(B)]=\mathbb{E}[\lambda(B\cap
Z(\eta))],$ (A.11)

where we have used (A.6) for the second identity. On the other hand we obtain
from the Mecke equation (1.9) that

$\displaystyle\mathbb{E}[\eta(B\cap
Z(\eta))]=\mathbb{E}\bigg{[}\int{\mathbf{1}}\\{x\in B\\}{\mathbf{1}}\\{x\in
Z(\eta+\delta_{x})\\}\,\lambda(dx)\bigg{]}.$

Therefore,

$\displaystyle\int{\mathbf{1}}\\{x\in B\\}\mathbb{P}(x\in
Z(\eta))\,\lambda(dx)=\int{\mathbf{1}}\\{x\in B\\}\mathbb{P}(x\in
Z(\eta+\delta_{x}))\,\lambda(dx),$

and since $B$ is arbitrary this shows that

$\displaystyle\mathbb{P}(x\in Z(\eta))=\mathbb{P}(x\in
Z(\eta+\delta_{x})),\quad\lambda\text{-a.e.\ $x$}.$ (A.12)

Take $\mu\in{\mathbf{N}}$ and $x\in{\mathbb{X}}$ such that $x\notin
Z(\mu+\delta_{x})$. By (A.1),

$\displaystyle
Z(\mu+\delta_{x})=Z((\mu+\delta_{x})_{Z(\mu+\delta_{x})}+\mu_{Z(\mu+\delta_{x})^{c}})=Z(\mu_{Z(\mu+\delta_{x})}+\mu_{Z(\mu+\delta_{x})^{c}})=Z(\mu),$

so that $x\notin Z(\mu)$. Therefore $\\{x\in Z(\eta)\\}\subset\\{x\in
Z(\eta+\delta_{x})\\}$ and (A.12) implies the asserted result (A.10). ∎

####  B Non-attainable stopping sets

For the rest of the section, we fix a locally finite measure $\lambda$ on
$({\mathbb{X}},{\mathcal{X}})$ and denote by $\eta$ a Poisson process on
${\mathbb{X}}$ with intensity $\lambda$. Recall from Remark 3.4 that a
stopping set $Z$ is said to be $\lambda$-attainable if $Z=\cup_{t}Z_{t}$ for
some $\lambda$-continuous (that is, verifying (3.4)) CTDT $\\{Z_{t}\\}$. The
fact that the OSSS inequality (3.6) has only been proved for
$\lambda$-attainable stopping sets begs the question of whether there exist
stopping sets $Z$ that are not $\lambda$-attainable (and therefore for which
the validity of the OSSS inequality is not established). The principal aim of
this section is to answer positively to this question by proving the next
statement.

######  Proposition B.1 (Existence of non-attainable stopping sets).

Let $Z$ be a stopping set such that $\lambda(Z(\mu))>0$ for every
$\mu\in{\mathbf{N}}$. Assume that there exists a measurable partition
$\\{C_{i}:i=1,...,m\\}$ of ${\mathbb{X}}$ such that, for every $i$,
$\mathbb{P}(\lambda(C_{i}\cap Z(\eta))=0)>0$. Then, there is no
$\lambda$-continuous CTDT $\\{Z_{t}\\}$ such that $Z=\cup_{t}Z_{t}$.

######  Example 2.

The following example is inspired by the discussion contained in [77, Example
1.28], that was brought to our attention by Laurin Koehler-Schindler. Let
$\lambda({\mathbb{X}})\in(0,\infty)$ and assume that ${\mathbb{X}}$ equals the
disjoint union of three measurable sets $C_{1},C_{2},C_{3}$ such that
$\lambda(C_{i})>0$, $i=1,2,3$. For $\mu\in{\mathbf{N}}$, set
$X_{i}(\mu):={\mathbf{1}}\\{\mu(C_{i})>0\\}$ for $i=1,2,3$. We define

$\displaystyle Z(\mu):=\begin{cases}C_{1}\cup C_{2}\cup
C_{3}={\mathbb{X}},&\text{if $X_{1}(\mu)=X_{2}(\mu)=X_{3}(\mu)$},\\\ C_{1}\cup
C_{2},&\text{if $X_{1}(\mu)=1,X_{2}(\mu)=0$},\\\ C_{1}\cup C_{3},&\text{if
$X_{1}(\mu)=0,X_{3}(\mu)=1$},\\\ C_{2}\cup C_{3},&\text{if
$X_{2}(\mu)=1,X_{3}(\mu)=0$}.\end{cases}$

Then, it is easily seen that $Z$ is a stopping set such that $\lambda(Z)>0$,
and $\mathbb{P}(\lambda(C_{i}\cap Z(\eta))=0)>0$ for every $i=1,2,3$.

The proof of Proposition B.1 is based on a non-trivial zero-one law for CTDTs,
stated in the forthcoming Theorem B.3. We first prove an ancillary result.

######  Lemma B.2.

Suppose that $\\{Z_{t}:t\in{\mathbb{R}}_{+}\\}$ is a $\lambda$-continuous
CTDT. Define the mapping $\tau\colon{\mathbf{N}}\to[0,\infty]$ by

$\displaystyle\tau(\mu):=\inf\\{t\geq
0:\lambda(Z_{t}(\mu))>0\\},\quad\mu\in{\mathbf{N}},$ (B.13)

with the usual convention $\inf\emptyset:=+\infty$. Then, writing
$Z_{\infty}(\mu):=\cup_{t}Z_{t}(\mu)$, one has that $\mu\mapsto
Z^{u}(\mu):=Z_{\tau(\mu)+u}(\mu)$ is a stopping set for every $u\geq 0$.
Moreover, one has necessarily that $\lambda(Z^{0}(\mu))=0$, for every
$\mu\in{\mathbf{N}}$.

###### Proof.

Given $\mu\in{\mathbf{N}}$ and $t\geq 0$ we have $\tau(\mu)<t$ if and only if
$\lambda(Z_{t}(\mu))>0$. Hence, $\tau$ is measurable and $Z^{u}$ is graph-
measurable since, for every $(\mu,x)\in{\mathbf{N}}\times{\mathbb{X}}$, the
quantity ${\bf 1}\\{x\in Z^{u}(\mu)\\}$ is the limit of

$\displaystyle{\bf 1}\\{\tau(\mu)\\!=\\!+\infty,\,x\in
Z_{\infty}\\}\\!+\\!{\bf 1}\\{\tau(\mu)\\!=\\!0,\,x\in
Z_{u}\\}\\!+\\!\sum_{k=0}^{\infty}{\bf
1}\left\\{\tau(\mu)\\!\in\\!\left(\frac{k}{n},\frac{k+1}{n}\right],\,x\in
Z_{\frac{k+1}{n}+u}\right\\},$

as $n\to\infty$, where we have used (2.2). It is also immediately checked that
$\lambda(Z^{0}(\mu))=0$, because of the $\lambda$-continuity property (3.4).
Now take $u\geq 0$ and $\mu,\psi\in{\mathbf{N}}$ such that
$\psi(Z^{u}(\mu))=0$; we claim that

$\displaystyle\tau(\mu)=\tau(\mu_{Z^{u}(\mu)}+\psi).$ (B.14)

Reasoning as in the first part of the proof of Theorem A.6, one sees that
(B.14) is true whenever $\tau(\mu)=+\infty$. Now assume $\tau(\mu)<\infty$; we
will prove (B.14) for $u=0$, from which the general case follows at once.
Abbreviate $s:=\tau(\mu)$, in such a way that $Z_{s}(\mu)=Z^{0}(\mu)$. Since
$\lambda(Z_{s}(\mu))=0$ and $Z_{s}(\mu)=Z(\mu_{Z_{s}(\mu)}+\psi)$ we obtain
that $\tau(\mu_{Z^{0}(\mu)}+\psi)\geq s$. On the other hand, we have for each
$t>s$ that $\lambda(Z_{t}(\mu))>0$. Since the $Z_{t}$ are
${\mathcal{X}}_{0}$-valued and $\mu,\psi$ are locally finite, we can exploit
the right-continuity (2.2) of a CTDT to find a $t>s$ satisfying

$\displaystyle\mu(Z_{t}(\mu)\setminus Z_{s}(\mu))=\psi(Z_{t}(\mu)\setminus
Z_{s}(\mu))=0.$

But then, $\psi(Z_{t}(\mu))=0$ and

$\displaystyle\lambda(Z_{t}(\mu_{Z_{s}(\mu)}))=\lambda(Z_{t}(\mu_{Z_{t}(\mu)}+\psi))=\lambda(Z_{t}(\mu))>0,$

which shows that $\tau(\mu_{Z^{0}(\mu)}+\psi)\leq t$ for every $t>s$ such that
$t-s$ is sufficiently small, yielding (B.14). To conclude, set
$\mu^{\prime}:=\mu_{Z^{u}(\mu)}+\psi$, with $\psi(Z^{u}(\mu))=0$. By (B.14) we
have $\tau(\mu^{\prime})=\tau(\mu)$. Therefore,

$\displaystyle
Z^{u}(\mu_{Z^{u}(\mu)}+\psi)=Z_{\tau(\mu^{\prime})+u}(\mu^{\prime})=Z_{\tau(\mu)+u}(\mu^{\prime})=Z_{\tau(\mu)+u}(\mu_{Z^{u}(\mu)}+\psi)=Z_{\tau(\mu)+u}(\mu)=Z^{u}(\mu).$

Hence, $Z^{u}$ is a stopping set. ∎

The next statement is the most important result of the section.

######  Theorem B.3 (Zero-one laws for CTDT).

Let $\\{Z_{t}:t\in{\mathbb{R}}_{+}\\}$ be a $\lambda$-continuous CTDT, and
define the mapping $\tau(\cdot)$ as in (B.13). Assume that $\tau(\mu)<\infty$
for every $\mu\in{\mathbf{N}}$ and write

$\mathcal{G}_{u}:=\sigma(\eta_{Z^{u}(\eta)}),\quad u\geq 0,$ (B.15)

where the stopping sets $Z^{u}$ are defined in the statement of Lemma B.2.
Then $\mathcal{G}_{u}\subset\mathcal{G}_{v}$ for every $u<v$. Moreover,
$\eta_{Z^{0}(\eta)}$ is equal to the zero measure with probability one, and
the $\sigma$-field

$\mathcal{G}_{0+}:=\bigcap_{u>0}\mathcal{G}_{u},$

is $\mathbb{P}$-trivial, that is: for all $B\in\mathcal{G}_{0+}$,
$\mathbb{P}(B)\in\\{0,1\\}$.

###### Proof.

The first part of the statement follows immediately from Lemma A.1 and the
fact that, for every $u<v$, $Z^{u}$ and $Z^{v}$ are stopping sets such that
$Z^{u}(\mu)\subset Z^{v}(\mu)$ for all $\mu\in{\mathbf{N}}$. The fact that
$\eta_{Z^{0}(\eta)}$ equals the zero measure 0, $\mathbb{P}$-a.s., is a direct
consequence of Proposition A.5 and of the fact that $\lambda(Z^{0}({\bf
0}))=0$, by virtue of Lemma B.2. To show the triviality of $\mathcal{G}_{0+}$,
it is enough to show that, for every bounded $\sigma(\eta)$-measurable random
variable $Y$,

$\mathbb{E}[Y\,|\,\mathcal{G}_{0+}]=\mathbb{E}[Y],\quad\mathbb{P}\mbox{\rm-a.s.}\;$
(B.16)

indeed, if (B.16) is in order then, for every $B\in\mathcal{G}_{0+}$, one has
that ${\bf 1}_{B}=\mathbb{P}(B)$, $\mathbb{P}$-a.s., from which the triviality
follows. By a monotone class argument, it is sufficient to prove (B.16) for
every $Y=\exp\\{i\int fd\eta\\}$, where $f$ is a bounded deterministic kernel,
whose support is contained in some $A\in{\mathcal{X}}_{0}$. Now select a
sequence $\epsilon_{n}\downarrow 0$. By the backwards martingale convergence
theorem, one has that

$\mathbb{E}[Y\,|\,\mathcal{G}_{0+}]=\lim_{n\to\infty}\mathbb{E}[Y\,|\,\mathcal{G}_{\epsilon_{n}}],\quad\mathbb{P}\mbox{\rm-a.s.}.$

Exploiting the Markov property (A.6) one has also that, for $\epsilon>0$,

$\mathbb{E}[Y\,|\,\mathcal{G}_{\epsilon}]=\exp\left\\{i\int_{Z^{\epsilon}(\eta)}fd\eta\right\\}\times\int\exp\left\\{i\int_{{\mathbb{X}}\backslash
Z^{\epsilon}(\eta)}fd\mu\right\\}\Pi_{\lambda}(d\mu),$

and the right-hand side of the previous equality converges $\mathbb{P}$-a.s.
to

$\int\exp\left\\{i\int_{{\mathbb{X}}}fd\mu\right\\}\Pi_{\lambda}(d\mu)=\mathbb{E}[Y],$

as $\epsilon\downarrow 0$, since $Z^{\epsilon}(\mu)\downarrow Z^{0}(\mu)$, for
all $\mu\in{\mathbf{N}}$, and
$\int_{Z^{0}(\eta)}fd\eta=0=\lambda(Z^{0}(\eta))$, $\mathbb{P}$-a.s.. ∎

###### Proof of Proposition B.1.

The proof is by contradiction. Assume that $\\{Z_{t}:t\in{\mathbb{R}}_{+}\\}$
is a $\lambda$-continuous CTDT such that $Z=\cup_{t}Z_{t}$, and observe that
the assumptions on $Z$ imply that $\tau(\mu)<\infty$ for every
$\mu\in{\mathbf{N}}$. For every $\mu\in{\mathbf{N}}$, we define

$\mathcal{U}(\mu)=\\{A\in\mathcal{X}:\lambda(A\cap
Z^{\epsilon}(\mu))>0,\,\,\forall\epsilon>0\\}.$

It is readily checked that, for every $D\in\mathcal{X}$, the event
$\\{D\in\mathcal{U}(\eta)\\}$ is in $\mathcal{G}_{0+}$, and therefore
$\mathbb{P}[D\in\mathcal{U}(\eta)]\in\\{0,1\\}$, by virtue of Theorem B.3. By
the definition of $\tau(\eta)$, there exists at least one index
$i\in\\{1,...,m\\}$ such that $\mathbb{P}[C_{i}\in\mathcal{U}(\eta)]>0$, and
therefore $\mathbb{P}[C_{i}\in\mathcal{U}(\eta)]=1$. But this is absurd, since
then

$0=\mathbb{P}[C_{i}\notin\mathcal{U}(\eta)]\geq\mathbb{P}[\lambda(C_{i}\cap
Z(\eta))=0]>0,$

where we have used the assumptions on $C_{i}$ in the statement. ∎

Example 2 gives explicitly a stopping set that is not $\lambda$-attainable by
any CTDT. However, one may wonder whether there are stopping sets that are not
$\lambda$-attainable by any randomized CTDT as defined in Section 4. We will
now show that this is true for the stopping set in Example 2.

######  Example 3.

Let $Z$ be the stopping set defined in Example 2. Suppose that
$\\{Z_{t}=Z_{t}^{Y}:t\in{\mathbb{R}}_{+}\\}$ is a $\lambda$-continuous
randomized CTDT for an independent random variable $Y$ such that
$Z=\cup_{t}Z_{t}$. Define $\mathcal{U}(\eta)=\mathcal{U}^{Y}(\eta)$ as in the
proof of Proposition B.1. Without loss of generality assume that
$\mathbb{P}(C_{1}\in\mathcal{U}(\eta))>0$.

Observe that $\lambda(Z^{0}(\eta))=0$ for $Z^{0}$ as defined in Lemma B.2 and
so as in the proof of Theorem B.3, we have that $\eta(Z^{0}(\eta))={\bf 0}$,
$\mathbb{P}$-a.s.. Thus, we derive that, $\mathbb{P}$-a.s.,

$\displaystyle\mathbb{P}(\eta(C_{2})>0,\eta(C_{3})=0\,|\,\eta_{Z^{0}(\eta)})$
$\displaystyle=\mathbb{P}(\eta(C_{2}\setminus
Z^{0}(\eta))>0,\eta(C_{3}\setminus Z^{0}(\eta))=0\,|\,\eta_{Z^{0}(\eta)})$
$\displaystyle=(1-e^{-\lambda(C_{2})})e^{-\lambda(C_{3})}=:\kappa$ (B.17)

where in deriving the second equality, we have used the strong Markov property
(A.6) and the complete independence property of the Poisson process as well as
that $Z^{0}(\eta)$ is a stopping set with $\lambda(Z^{0}(\eta))=0$. Now, for
any $\epsilon>0$, using (B.17), we derive that

$\displaystyle 0$
$\displaystyle<\kappa\mathbb{P}(C_{1}\in\mathcal{U}(\eta))\leq\kappa\mathbb{P}(C_{1}\cap
Z^{\epsilon}(\eta)\neq\emptyset)$
$\displaystyle\leq\mathbb{E}[\mathbb{P}(C_{1}\cap
Z^{\epsilon}(\eta)\neq\emptyset\,|\,\eta_{Z^{0}(\eta)})\mathbb{P}(\eta(C_{2}\setminus
Z^{0}(\eta))>0,\eta(C_{3}\setminus Z^{0}(\eta))=0\,|\,\eta_{Z^{0}(\eta)})]$
$\displaystyle=\mathbb{E}[\mathbb{P}(C_{1}\cap
Z^{\epsilon}(\eta)\neq\emptyset,\eta(C_{2})>0,\eta(C_{3})=0\,|\,\eta_{Z^{0}(\eta)})]$
$\displaystyle=\mathbb{P}(C_{1}\cap
Z^{\epsilon}(\eta)\neq\emptyset,\eta(C_{2})>0,\eta(C_{3})=0)$
$\displaystyle\leq\mathbb{P}(C_{1}\cap Z\neq\emptyset,Z=C_{2}\cup C_{3})=0,$

where in the first equality we have again used the strong Markov property
(A.6) and the complete independence property of the Poisson process, as well
as the fact that $Z^{0}(\eta)$ is a stopping set. Thus, we cannot realize $Z$
by a randomized CTDT as well.

[Acknowledgments] The authors thank Sergei Zuyev for sharing an earlier
version of his continuum noise sensitivity notes. We are thankful to Raphael
Lachiéze-Rey for discussions regarding CTDT and suggesting some simplified
CTDT constructions. We are thankful to Laurin Koehler-Schindler, Stephen
Muirhead, Vincent Tassion and Hugo Vanneuville for some comments regarding the
discrete OSSS and Schramm-Steif inequalities. We thank Srikanth K. Iyer for
discussing and pointing out percolation theory literature related to RSW-type
estimates and comments on a preliminary draft.

DY’s research was supported by DST-INSPIRE faculty award and CPDA from the
Indian Statistical Institute. GP’s research was supported by the FNR grant
FoRGES (R-AGR3376-10) at Luxembourg University. The work significantly
benefitted from reciprocal visits to our respective institutions and we wish
to thank all the three institutions for hosting us and supporting our visits.

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