# Asymptotics for the critical level
and a strong invariance principle
for high intensity shot noise fields

Raphael Lachieze-Rey1 1Université Paris Descartes / Université de Paris
<EMAIL_ADDRESS>and  Stephen Muirhead2
<EMAIL_ADDRESS>2School of Mathematics and Statistics, University of
Melbourne

###### Abstract.

We study fine properties of the convergence of a high intensity shot noise
field towards the Gaussian field with the same covariance structure. In
particular we (i) establish a strong invariance principle, i.e. a quantitative
coupling between a high intensity shot noise field and the Gaussian limit such
that they are uniformly close on large domains with high probability, and (ii)
use this to derive an asymptotic expansion for the critical level above which
the excursion sets of the shot noise field percolate.

Keywords: Shot noise fields, Gaussian fields, percolation, strong invariance
principle

MSC subject classifications: 60G60, 60K35

##  1\. Introduction

Let $g(x)\in L^{1}(\mathbb{R}^{d})$ be a continuous kernel and let
$\mathcal{P}_{\lambda}$ denote a Poisson point process on $\mathbb{R}^{d}$
with intensity $\lambda$ with respect to the Lebesgue measure. The shot noise
field with kernel $g$ is the stationary random field

$F_{\lambda}(x)=\sum_{i\in\mathcal{P}_{\lambda}}g(x-i),$

well-defined almost surely by the integrability of $g$.

We are interested in the global connectivity of the excursion sets

$\\{F_{\lambda}\leq\ell\\}:=\\{x\in\mathbb{R}^{d}:F_{\lambda}(x)\leq\ell\\}$

and in particular the range of levels for which the excursion sets have
unbounded connected components. By monotonicity there exists a critical level
$\ell_{c}=\ell_{c}(F_{\lambda})\in[-\infty,\infty]$ such that

(1.1) $\mathbb{P}[\\{F_{\lambda}\leq\ell\\}\text{ contains an unbounded
component}]=\begin{cases}0&\text{if }\ell<\ell_{c},\\\ >0&\text{if
}\ell>\ell_{c},\end{cases}$

and under mild conditions on $g$ the critical level $\ell_{c}$ is known to be
finite [MS83, BM17]. However very little is known about the value of
$\ell_{c}$ or how it depends on $\lambda$.

In this paper we study the asymptotic expansion of $\ell_{c}(F_{\lambda})$ in
the ‘high intensity’ limit, i.e. as $\lambda\to\infty$. In this limit
$F_{\lambda}$ approximates, after suitable rescaling and centering, the
stationary Gaussian field $f$ with the same covariance structure (see Section
1.2). Restricting our attention to the planar case $d=2$, since it is known
that $\ell_{c}(f)=0$ for reasons of symmetry and planar duality, one naturally
expects that the critical level of the rescaled shot noise field is
$\ell_{c}\approx\ell_{c}(f)=0$, and our main result gives a quantitative
version of this approximation (see Theorem 1.3).

Towards this, our second main result is a quantitative coupling between high
intensity shot noise fields and the limiting Gaussian field which is strong
enough to control the error between their critical levels (see Theorem 1.11).
The coupling is valid in all dimensions $d\geq 1$, and the sup norm of the
error is presumably optimal (up to logarithmic factors) in dimensions $d=1,2$,
in that it corresponds to the rate of convergence of marginals (up to
logarithmic factors), with exponential tail decay.

This result can be considered a strong invariance principle for shot noise
fields, which extends existing convergence results available in the literature
(see the seminal papers by [Pap71, HS85, Lan87] about convergence of marginals
and functional central limit theorems, and [BD12a, KM95] for some
generalisations), and we believe it to be of independent interest. First, the
proximity between Gaussian fields and high intensity shot noise fields is
important in many applied fields (see, e.g., [KM95] in actuarial sciences,
[BD20, GGM11] in image analysis and texture synthesis, [Hei07] in stereology,
[AY10] in the study of wireless networks, and further references therein) and
the strong invariance principle gives a means to control this approximation.
Second, this result could, in principle, be used to estimate other geometric
quantities of shot noise field excursions; see, e.g., the many recent works in
probability theory and image analysis on this topic [BD12a, BD12b, BD20].
Finally, since high intensity shot noise fields are also used as a way to
simulate Gaussian fields with the same mean and covariance, this result gives
uniform control over the error in the simulation.

###  1.1. Asymptotic expansion of the critical level

As mentioned above, in this paper we consider the asymptotic expansion of
$\ell_{c}(F_{\lambda})$ as $\lambda\to\infty$. Most of our attention will be
on the planar case $d=2$, although some of the intermediate results hold in
all dimensions.

To state our main result we introduce some assumptions on the kernel:

######  Assumption 1.1.

Suppose $g$ satisfies the following:

  * •

(Symmetry) $g$ is isotropic;

  * •

(Smoothness) $g\in C^{4}(\mathbb{R}^{d})$;

  * •

(Decay) There exist $\beta>d$ such that, for every multi-index $\alpha$ such
that $|\alpha|\leq 4$,

${|\partial^{\alpha}g(x)|\leq(1+|x|)^{-\beta}\ ,\quad x\in\mathbb{R}^{d}.}$

Among other things, the latter two conditions ensure that $F_{\lambda}$ is
almost surely $C^{3}$-smooth. We use the assumption of isotropy mainly for
simplicity, and because it gives us access to results in the literature,
although we believe isotropy could be relaxed to symmetry under negation and
permutation of coordinate axes with minor adjustments to the arguments.

For technical reasons we shall further assume:

######  Assumption 1.2.

The random vector $(F_{\lambda}(0),\nabla F_{\lambda}(0))$ has a bounded
density.

We give examples of kernels satisfying Assumption 1.2 in Remark 1.5 below. A
necessary condition for Assumption 1.2 is that $g$ has unbounded support.

We can now state our main result, which concerns the rescaled and centred shot
noise field

$\displaystyle
f_{\lambda}(x)=\frac{F_{\lambda}(x)-\mathbb{E}[F_{\lambda}(0)]}{\lambda^{1/2}}.$

As discussed in detail in the next section, the rescaled field $f_{\lambda}$
converges as $\lambda\to\infty$ to a limiting stationary Gaussian field $f$.
By the Campbell-Mecke formula
$\mathbb{E}[F_{\lambda}(0)]=\lambda\int_{\mathbb{R}^{d}}g(x)dx$.

For functions $j,k:\mathbb{R}^{+}\to\mathbb{R}$ we write $j(x)=o(k(x))$ if
$|j(x)|/|k(x)|\to 0$ as $x\to\infty$, and $j(x)=O(k(x))$ if there exists a
$c>0$ such that $\limsup_{x\to\infty}|j(x)|/|k(x)|\leq c$. For $h\in
L^{1}(\mathbb{R}^{d})$ we write $\int h$ to abbreviate
$\int_{\mathbb{R}^{d}}h(x)dx$.

######  Theorem 1.3 (Asymptotic expansion of the critical level).

Let $d=2$ and assume $g$ satisfies Assumption 1.1 and 1.2. Then, as
$\lambda\to\infty$,

(1.2) $\ell_{c}(f_{\lambda})\to 0.$

If moreover $g\geq 0$ then, as $\lambda\to\infty$,

(1.3)
$\ell_{c}(f_{\lambda})=O\big{(}\lambda^{-1/2}(\log\lambda)^{3/2}\big{)}.$

######  Remark 1.4.

By rescaling, (1.2) and (1.3) are equivalent to the asymptotic expansions
$\ell_{c}(F_{\lambda})=\lambda\int g+o(\lambda^{1/2})$ and
$\ell_{c}(F_{\lambda})=\lambda\int g+O((\log\lambda)^{3/2})$ respectively.

######  Remark 1.5 (Examples).

For examples of shot noise fields $F_{\lambda}$ to which Theorem 1.3 applies,
one can take the kernels

$g(x)=(1+|x|^{2})^{-\beta/2}\,,\ \beta>2,\quad\text{or}\quad
g(x)=\exp(-(1+|x|^{2})^{-\gamma/2})\,,\ \gamma\in(0,1).$

It is clear that these kernels satisfy Assumption 1.1 (perhaps after
multiplication by a constant), and see [LM22, Appendix A.2] for a proof that
the corresponding shot noise fields satisfy Assumption 1.2.

Since $f_{\lambda}$ converges to the limiting Gaussian field $f$, and
$\ell_{c}(f)=0$, it is natural to expect that $\ell_{c}(f_{\lambda})\to 0$.
The main content of Theorem 1.3 is to control the rate of decay of
$\ell_{c}(f_{\lambda})$. Nevertheless, we do not expect Theorem 1.3 to be
optimal. In fact we believe that $\ell_{c}(f_{\lambda})$ is at most
${O(\lambda^{-1/2}\log\lambda)}$, and perhaps even smaller, even if we do not
assume $g\geq 0$ (see Remark 1.12).

######  Question 1.6.

What is the order of $\ell_{c}(f_{\lambda})$ as $\lambda\to\infty$? How does
it depend on the kernel $g$?

######  Remark 1.7 (Possible extensions).

A more general model of shot noise is the random field

$F_{\lambda}(x)=\sum_{i\in\mathcal{P}_{\lambda}}Y_{i}g(x-i)$

where $\\{Y_{i}\\}_{i\in\mathcal{P}_{\lambda}}$ are independent copies of a
random variable $Y$ with finite mean; this reduces to the model we consider if
$Y_{i}\equiv 1$. Under sufficient moment conditions on $Y$ we expect that our
proof can be adapted to show that (1.2)–(1.3) hold. In the case that $Y$ is
symmetric then it is expected, and in some cases known (see [LM22]), that
$\ell_{c}(f_{\lambda})=0$ for all $\lambda>0$ for reasons of planar duality.

###  1.2. Strong invariance principle for shot noise fields

As briefly described above, our proof of Theorem 1.3 relies on the fact that
$F_{\lambda}$ converges, after suitable centring and rescaling, to a
stationary Gaussian field $f$. Indeed a key intermediate step in the proof is
to establish a ‘strong invariance principle’ for this convergence, which
improves on the qualitative convergence available in the literature.

To state the result we return to the setting of shot noise fields in arbitrary
dimension $d\geq 1$ and assume that $g\in L^{1}(\mathbb{R}^{d})\cap
L^{2}(\mathbb{R}^{d})$. As above we consider the centred and rescaled shot
noise field

$f_{\lambda}(x)=\frac{F_{\lambda}(x)-\mathbb{E}[F_{\lambda}(0)]}{\lambda^{1/2}}.$

An application of the Campbell-Mecke formula shows that
$\mathbb{E}[F_{\lambda}(0)]=\lambda\int g$, and also that

$\textrm{Cov}[f_{\lambda}(0),f_{\lambda}(x)]=\int g(-y)g(x-y)\,dy,$

which is independent of $\lambda$ and finite by our assumption that
$\textrm{Var}[f_{\lambda}(0)]=\|g\|_{L^{2}}<\infty$. Hence it is natural to
compare $f_{\lambda}$ to the stationary centred Gaussian field $f$ with
covariance kernel

$K(x)=\mathbb{E}[f(0)f(x)]=\int g(-y)g(x-y)\,dy.$

Indeed, under a certain subset of the conditions in Assumption 1.1, it is
known that $f_{\lambda}$ satisfies a functional central limit theorem
(‘invariance principle’) in the high intensity limit.

######  Assumption 1.8.

Suppose $g$ satisfies the following:

  * •

(Smoothness) $g\in C^{2}(\mathbb{R}^{d})$

  * •

(Decay) There exist $\beta>d$ such that, for every multi-index $\alpha$ such
that $|\alpha|\leq 2$,

$|\partial^{\alpha}g(x)|\leq(1+|x|)^{-\beta}\ ,\quad x\in\mathbb{R}^{d}.$

In particular these assumptions imply (by dominated convergence) that $K\in
C^{4}(\mathbb{R}^{d})$, and that both the shot noise field $f_{\lambda}$ and
the Gaussian field $f$ are almost surely $C^{1}$-smooth.

######  Theorem 1.9 (Invariance principle for shot noise fields; [HS85,
Theorem 8]).

Assume $g$ satisfies Assumption 1.8. Then

$f_{\lambda}\Rightarrow f$

in law in the topology of uniform convergence on compact sets.

######  Remark 1.10.

The invariance principle in Theorem 1.9 actually holds under slightly weaker
conditions (see [HS85, Theorem 8]) but for simplicity we do not state the most
general version.

To prove our main result (Theorem 1.3) we need to quantify the invariance
principle in terms of a ‘coupling distance’, that is, we want that
$f_{\lambda}$ and $f$ may be coupled on some sufficiently rich probability
space so that they are close with high probability. For $R\geq 1$, let $B(R)$
denote the Euclidean ball of radius $R$ centred at the origin. For
$D\subset\mathbb{R}^{d}$, let $\|\cdot\|_{\infty,D}$ denote the sup-norm on
$D$, abbreviating $\|\cdot\|_{\infty}=\|\cdot\|_{\infty,\mathbb{R}^{d}}$.

######  Theorem 1.11 (Strong invariance principle for shot noise fields).

  1. (1)

Assume $g$ satisfies Assumption 1.8 for $\beta>d$. Then there exist
$c_{1},c_{2}>0$, depending only on $\beta$ and the dimension $d$, such that
for every $\lambda\geq 1$ there is a coupling of $f_{\lambda}$ and $f$
satisfying, for every $R,t\geq 1$,

$\mathbb{P}\Big{[}\|f_{\lambda}-f\|_{\infty,B(R)}\geq\sigma_{d}(\lambda)t\Big{]}\leq
c_{1}R^{d}\lambda^{c_{d,\beta}}e^{-c_{2}t},$

where

(1.4) $\sigma_{d}(\lambda)=\begin{cases}\lambda^{-1/2}&d=1,\\\
\lambda^{-1/2}\sqrt{\log\lambda}&d=2,\\\ \lambda^{-1/d}&d\geq 3,\\\
\end{cases}\quad\text{and}\quad c_{d,\beta}=1+\frac{d}{2}+\frac{d}{\beta-d}.$

  2. (2)

Moreover, suppose $I$ is a finite set of multi-indices, and there is a
$\beta>d$ such that each of the partial derivatives
$(\partial^{\alpha}g)_{\alpha\in I}$ satisfy Assumption 1.8 for $\beta$. Then
for every $\lambda\geq 1$ there is a coupling of $f_{\lambda}$ and $f$
satisfying, for every $R,t\geq 1$,

$\mathbb{P}\Big{[}\exists\alpha\in
I:\|\partial^{\alpha}f_{\lambda}-\partial^{\alpha}f\|_{\infty,B(R)}\geq\sigma_{d}(\lambda)t\Big{]}\leq
c_{1}|I|R^{d}\lambda^{c_{d,\beta}}e^{-c_{2}t},$

where $c_{1},c_{2}>0$ depend only on $\beta$ and $d$.

######  Remark 1.12.

The upshot of Theorem 1.11 is that the convergence of $f_{\lambda}$ to $f$
takes place on the scale $\sigma_{d}(\lambda)\log\lambda$, in the sense that
one can couple $f_{\lambda}$ and $f$ so that their difference on a compact set
$D\subseteq\mathbb{R}^{d}$ (or even a polynomially-growing ball
$B(\lambda^{c})$) is $O(\sigma_{d}(\lambda)\log\lambda)$ with high
probability. We do not expect this scale to be optimal for all $d\geq 1$,
although we believe it to be optimal up to a logarithmic factor if
$d\in\\{1,2\\}$.

To justify the optimality, at least heuristically, recall the celebrated
result of Komlós-Major-Tusnády (KMT) [KMT75] on coupling the empirical process
$\widetilde{\mathcal{E}_{n}}(s)=n^{-1/2}(\sum_{i=1}^{n}\mathds{1}_{X_{i}\leq
s}-ns)$ on $[0,1]$, where $X_{i}$ denote i.i.d. random variables uniformly
distributed on $[0,1]$, with a Brownian bridge
$\widetilde{\mathcal{W}}_{0}(s)$. One version of the result states that, for
each $n\geq 1$ the coupling can be done so that, for all $t\geq 0$,

$\mathbb{P}\Big{[}\max_{s\in[0,1]}|\widetilde{\mathcal{E}_{n}}(s)-\widetilde{\mathcal{W}}_{0}(s)|\geq
n^{-1/2}(c_{1}\log n+t)\Big{]}\leq c_{2}e^{-c_{3}t},$

where $c_{1},c_{2},c_{3}>0$ are constants independent of $n$. This implies
that the coupling error is at most $O(n^{-1/2}\log n)$ with high probability,
and this is thought to be best possible. Further, Beck [Bec85] has shown that,
in general dimension $d\geq 1$, the analogous strong invariance principle has
error that is at least of order $n^{-1/2}(\log n)^{(d-1)/2}$. Hence it is
natural to expect that the error in the strong invariance principle for shot
noise fields is at least of order $\lambda^{-1/2}(\log\lambda)^{c_{d}}$ where
$c_{d}\geq 1$ may depend on the dimension. Indeed one cannot hope to couple
$f_{\lambda}$ and $f$ on any scale smaller than $\lambda^{-1/2}$, since that
is the scale on which the marginals converge (by the Berry-Esseen theorem for
instance).

######  Remark 1.13.

Although we are unaware of any result in the literature directly comparable to
Theorem 1.11, there is a line of related work that has established strong
invariance principles for other functionals of the Poisson point process
$\mathcal{P}_{\lambda}$, in particular of the form
$(\sum_{i\in\mathcal{P}_{\lambda}}\mathds{1}_{i\in S})_{S\in\mathcal{S}}$,
where $\mathcal{S}$ is a class of subsets of $[0,1]^{d}$ [CR75, Rév76, Mas89].
For classes $\mathcal{S}$ that (i) are not too big, and (ii) contain sets with
smooth enough boundaries, a strong invariance principle analogous to Theorem
1.11 has been shown to hold with $\sigma_{d}(\lambda)$ replaced by
$\lambda^{-1/(2d)}$ up to logarithmic factors [Mas89]; this result is based on
the dyadic coupling scheme of [KMT75]. The fact that the error bound in
Theorem 1.11 is smaller (at least in $d\geq 2$) is due to the smoothness of
the kernel $g$ compared to the test functions $\mathds{1}_{i\in S}$. One can
also obtain faster rates (although not as fast as in Theorem 1.11 in general)
by restricting $\mathcal{S}$ to the class of rectangles, which partially
compensates for the lack of smoothness (see, e.g., [CR75, Tus77, Rio96] in the
setting of empirical processes).

Interestingly, the works [CR75, Rév76, Mas89] use the strong invariance
principle for the Poisson measure as a means to prove a strong invariance
principle for the empirical process via ‘Poissonisation’, whereas our approach
follows the reverse pathway, deducing Theorem 1.11 from known results on the
empirical process due to Koltchinskii [Kol94] (and see also [Rio94] for
related results), which are ultimately based on the dyadic scheme of [KMT75].

######  Remark 1.14.

In dimensions $d\geq 3$ it might be possible to improve Theorem 1.11 by
following the alternative approach in [Rio96] based on the Strassen-Dudley
representation theorem and the multivariate central limit theorems of Zaitsev
[Zai87a, Zai87b] rather than the dyadic coupling scheme of [KMT75]; see also
[BM06] for related results. In fact, assuming sufficient smoothness of $g$, we
believe that one could achieve the bound

$\mathbb{P}\Big{[}\|f_{\lambda}-f\|_{\infty,B(R)}\geq\lambda^{-1/2+\delta}t\Big{]}\leq
c_{1}R^{d}\lambda^{c_{2}}e^{-c_{3}t},$

for arbitrary small $\delta>0$. Since we are mainly interested in the case
$d=2$, and since in $d\in\\{1,2\\}$ the KMT-based strategy leads to a sharper
bound, we do not pursue this here.

###  1.3. Remarks on higher dimensions

Although our strategy to prove Theorem 1.3 is essentially planar, since the
convergence to a Gaussian limit occurs in all dimensions we believe that if
$d\geq 2$

(1.5)
$\ell_{c}(f_{\lambda})=\ell_{c}(f)+O\big{(}\sigma_{d}(\lambda)\lambda^{1/2}\log\lambda\big{)},$

where $\ell_{c}(f)\in\mathbb{R}$ is the critical level of the limiting
Gaussian field $f$ (defined by analogy with (1.1), and recall that
$\ell_{c}(f)$ depends on the kernel $g$). Only in dimension $d=2$ do we have
$\ell_{c}(f)=0$ for reasons of planar duality (see Theorem 2.1 below), whereas
in general we expect $\ell_{c}(f)<0$ (see [DRRV21] for a recent proof for a
class of Gaussian fields, and c.f. Bernoulli percolation on $\mathbb{Z}^{d}$,
for which the critical parameter satisfies
$p_{c}(\mathbb{Z}^{d})<p_{c}(\mathbb{Z}^{2})=1/2$ for $d\geq 3$ [Gri99]). We
also do not rule out that $\sigma_{d}(\lambda)\lambda^{1/2}\log\lambda$ in
(1.5) could be replaced by $(\log\lambda)^{c_{d}}$ or even something smaller.

######  Question 1.15.

What is the order of $\ell_{c}(f_{\lambda})-\ell_{c}$? How does it depend on
the kernel $g$ and on the dimension $d$?

Although even the qualitative convergence
$\ell_{c}(f_{\lambda})\to\ell_{c}(f)$ remains out of reach if $d\geq 3$, using
the recent results of Severo [Sev21] in place of Theorem 2.1 below, our
strategy could be adapted to prove the rigorous lower bound

(1.6) $\ell_{c}(f_{\lambda})\geq\ell_{c}(f)+o(1),$

under the conditions in Theorem 1.3 and assuming $g\geq 0$ (the latter
condition is needed to apply the results in [Sev21]). However the matching
upper bound does not follow from our strategy, essentially because one lacks a
finite-size criterion for percolation in $d\geq 3$ (see Section 5.1).

###  1.4. Overview of the rest of the paper

In Section 2 we prove Theorem 1.3 subject to two intermediate results, namely
the strong invariance principle stated in Theorem 1.11 above, and a continuity
criterion for the critical level stated in Proposition 2.8 below. These
intermediate results are the focus of the subsequent sections: in Section 3 we
collect preliminary results on shot noise and Gaussian fields, in Section 4 we
prove the strong invariance principle in Theorem 1.11, and finally in Section
5 we prove the continuity criteria in Proposition 2.8.

###  1.5. Acknowledgements

The second author was supported by the Australian Research Council (ARC)
Discovery Early Career Researcher Award DE200101467. The authors thank Chinmoy
Bhattacharjee, Michael Goldman, Martin Huesmann, Manjunath Krishnapur and
Felix Otto for helpful discussions on strong invariance principles for Poisson
random measures, and Giovanni Peccati for suggesting to extend the strong
invariance principle to derivatives of the field.

##  2\. The asymptotic expansion of the critical level

In this section we prove our main result (Theorem 1.3) subject to intermediate
statements which are established in the subsequent sections. Throughout this
section we restrict our attention to the planar case $d=2$.

The proof consists of using the convergence of shot noise fields to a limiting
Gaussian field, described in Section 1.2 above, to ‘pull back’ known results
on level set percolation of the Gaussian limit (see Theorems 2.1 and 2.3
below). In order to achieve this we use a quantitative continuity result (see
Proposition 2.8 below) that relates the critical level of shot noise fields
with that of the Gaussian limit.

###  2.1. Level set percolation of Gaussian fields

In this section we recall the relevant results on level set percolation of
planar Gaussian fields $f$. By planar duality, and the fact that $f$ and $-f$
are equal in law, it is natural to expect that $\ell_{c}(f)=0$ for centred
stationary planar Gaussian fields. Indeed this, and more precise descriptions
of the phase transition at $\ell=0$, have recently been established for a wide
class of Gaussian fields (see [BG17, RV20, MV20, MRV20] and references
therein).

Let us introduce first the concept of a sharp threshold for rectangle crossing
events. For a level $\ell\in\mathbb{R}$ and $a,b>0$, let
$\text{{Cross}}_{\ell}[a,b]$ denote the event that
$\\{f\leq\ell\\}\cap([0,a]\times[0,b])$ contains a path that crosses the
rectangle $[0,a]\times[0,b]$ from left to right (i.e. intersects both
$\\{0\\}\times[0,b]$ and $\\{a\\}\times[0,b]$). We say that $f$ has a sharp
threshold (at $\ell=0$) if, for every $w>0$ and $\rho>0$,

(2.1) $\lim_{R\to\infty}\mathbb{P}[\textrm{Cross}_{-w}[\rho
R,R]]=0\quad\text{and}\quad\lim_{R\to\infty}\mathbb{P}[\textrm{Cross}_{w}[\rho
R,R]]=1.$

If $f$ has symmetry under rotation by $\pi/2$, then since $f$ and $-f$ are
equal in law each of the statements in (2.1) implies the other (see the
discussion at the beginning of Section 2.2).

######  Theorem 2.1 ([MRV20, Theorem 1.3]).

Let $f$ be a $C^{3}$-smooth centred stationary isotropic planar Gaussian field
with covariance $K$ satisfying:

  * •

$K(x)(\log\log|x|)^{3}\to 0$ as $|x|\to\infty$;

  * •

The vector $(f(0),f(x),\nabla f(0),\nabla f(x))$ is non-degenerate for each
$x\in\mathbb{R}^{d}\setminus\\{0\\}$.

Then $\ell_{c}(f)=0$ and $f$ has a sharp threshold in the sense of (2.1).

######  Remark 2.2.

The fact that $\ell_{c}(f)=0$ and the existence of a sharp threshold in the
sense of (2.1) are closely related, but it is not known that they are
equivalent in general. In [MRV20] a quantitative version of (2.1) was used to
deduce that $\ell_{c}(f)=0$, and for our purposes it turns out that
$\ell_{c}(f)=0$ alone is insufficient (see however Remark 2.10).

While the conclusions of Theorem 2.1 will be sufficient to prove (1.2) in
Theorem 1.3, for the stronger result (1.3) we will need a quantitative
improvement on (2.1) proven in [MV20] under stronger conditions (although this
improvement is believed to be true more generally). We say that $f$ has a
near-critical window of size $w$ (at $\ell=0$) if there exists a positive
function $w=w(R)>0$ such that, for every $\rho>0$,

(2.2) $\lim_{R\to\infty}\mathbb{P}[\textrm{Cross}_{-w(R)}[\rho
R,R]]=0\quad\text{and}\quad\lim_{R\to\infty}\mathbb{P}[\textrm{Cross}_{w(R)}[\rho
R,R]]=1.$

This property is monotonic in $w$ in the sense that if it holds for some $w$
it holds for any $w^{\prime}\geq w$. As for (2.1), if $f$ is Gaussian and has
symmetry under rotation by $\pi/2$ then if one of the statement holds for all
$\rho>0$, so does the other. Note that if $f$ has a sharp threshold then $f$
has a near-critical window of size $w$ for arbitrary small constant
$w(R)=w^{\prime}>0$. We say that $f$ has a polynomial near-critical window if
it has a near-critical window of size $w(R)=R^{-c}$ for some $c>0$.

######  Theorem 2.3 ([MV20, Theorem 1.15]).

Let $g:\mathbb{R}^{2}\to\mathbb{R}$ be a $C^{3}$-smooth positive isotropic
function, and assume there exists a $\beta>2$ such that, for every multi-index
$\alpha$ such that $|\alpha|\leq 3$,

$|\partial^{\alpha}g(x)|\leq(1+|x|)^{-\beta},\quad x\in\mathbb{R}^{2}.$

Let $f$ be the $C^{2}$-smooth centred stationary isotropic planar Gaussian
field whose covariance is $K(x)=\int g(-y)g(x-y)\,dy$. Then $\ell_{c}(f)=0$
and $f$ has a polynomial near-critical window.

######  Remark 2.4.

By analogy with planar Bernoulli percolation, it is natural to conjecture that
the true near-critical window is of polynomial order with exponent $1/\nu=3/4$
(where $\nu=4/3$ is the correlation length exponent for Bernoulli percolation,
see [Gri99, Chapters 9 & 10]), in the sense that (2.2) holds for
$w(R)R^{3/4}\to\infty$ but does not hold if $w(R)R^{3/4}\to 0$. In [MV20] it
was shown that the exponent satisfies $1/\nu\leq 1$ if it exists. On the other
hand, if one relaxes the decay assumption on $g$ but instead requires that $g$
be regularly varying at infinity with index $-\beta\in(-\infty,-1)$ then one
might expect that in general the critical window is of polynomial order with
exponent $1/\nu=\min\\{\beta-1,3/4\\}$.

###  2.2. Continuity of the critical level

In order to take advantage of the convergence of the shot noise field to a
Gaussian limit (Theorem 1.11), we need a quantitative continuity result that
relates critical levels through the limit. We state this continuity result in
a general form that applies to arbitrary sequences of stationary planar random
fields that converge to a limit.

Let us first introduce a mild regularity assumption on the fields. Recall the
crossing event $\textrm{Cross}_{\ell}[a,b]$, and let
$\textrm{Cross}^{tb}_{\ell}[a,b]$ denote the ‘top-bottom’ version of this
event, that is, the event that $\\{f\leq\ell\\}\cap([0,a]\times[0,b])$
contains a path that intersects both $[0,a]\times\\{b\\}$ and
$[0,a]\times\\{0\\}$. We say that a planar random field $f$ is regular if, for
each $\ell\in\mathbb{R}$ and $a,b\geq 0$, all translations and rotations of
the events $\\{\pm f\in\textrm{Cross}_{\ell}[a,b]\\}$ are measurable, and the
events

$\\{f\in\textrm{Cross}_{\ell}[a,b]\\}\quad\text{and}\quad\\{-f\in\textrm{Cross}^{tb}_{-\ell}[a,b]\\}$

partition the probability space up to a null set. In other words, for any
rectangle $R$ and $\ell\in\mathbb{R}$ almost surely exactly one of the
following holds: (i) $R\cap\\{f\leq\ell\\}$ contains a path from left to
right, or (ii) $R\cap\\{f\geq\ell\\}$ contains a path from top to bottom.
Under Assumptions 1.2 and 1.8 the shot noise fields $F_{\lambda}$ and the
Gaussian field $f$ are regular. This is a consequence of their level lines
being $C^{1}$-smooth curves which almost surely do not have tangential
intersections with fixed line-segments (see [LM22, Lemma 3.1] and [RV19, Lemma
A.9] respectively for details).

Let us also introduce the concept of sprinkled decoupling, which is the main
property we use to guarantee continuity of the critical level (see also [PR15,
MV20] for other appearances of this property). Recall the crossing events
$\textrm{Cross}_{\ell}[a,b]$, which are increasing with respect to the field.
We say that a planar regular random field $f$ satisfies the sprinkled
decoupling property with sprinkling function $h(R)\geq 0$ and error function
$e(R)\geq 0$ if, for any $R\geq 1$ and $\ell\in\mathbb{R}$, and any events
$A,B$ which are translations and rotations of
$\\{f\in\textrm{Cross}_{\ell}[R,3R]\\}$ such that the distance between the
associated rectangles on which $A$ and $B$ are defined is at least $R$, it
holds that

(2.3) $\mathbb{P}[f\in A\cap B]\leq\mathbb{P}[f+h(R)\in A]\mathbb{P}[f+h(R)\in
B]+e(R).$

This property is monotonic in $(h,e)$, in the sense that if it holds for
$(h,e)$ it also holds for any $(h^{\prime},e^{\prime})$ such that
$h^{\prime}\geq h$ and $e^{\prime}\geq e$.

If the sprinkled decoupling property holds for $h=0$ and $e(R)\to 0$, and if
the field $f$ is positively-associated so that $\mathbb{P}[f\in A\cap
B]\geq\mathbb{P}[f\in A]\mathbb{P}[f\in B]$, then we recover the usual notion
of asymptotic independence

$|\mathbb{P}[f\in A\cap B]-\mathbb{P}[f\in A]\mathbb{P}[f\in B]|\leq e(R)\to
0.$

We shall rather be interested in the case that $(h,e)$ can be chosen to decay
polynomially, and we say that $f$ satisfies the polynomial sprinkled
decoupling property if there exist constants $c_{h},c_{e,1},c_{e,2}>0$ such
that $f$ satisfies the sprinkled decoupling property with $h(R)=R^{-c_{h}}$
and $e(R)=c_{e,1}R^{-c_{e,2}}$. We say that a collection $(f_{n})$ of random
fields satisfies the uniform polynomial sprinkled decoupling property if each
$f_{n}$ satisfies the polynomial sprinkled decoupling property for constants
$(c_{h},c_{e,1},c_{e,2})$ chosen uniformly over the collection.

The next proposition verifies that, under the conditions in Assumption 1.8,
for any $\eta>1/2$ the rescaled shot noise fields
$(\lambda^{-\eta}f_{\lambda})_{\lambda\geq 1}$ satisfy this property (see
Section 3.4 for the proof):

######  Proposition 2.5.

Assume that the kernel $g$ satisfies Assumption 1.8 for some $\beta>2$ and let
$\eta>1/2$. Then the collection of random fields
$(\lambda^{-\eta}f_{\lambda})_{\lambda\geq 1}$ satisfies the uniform
polynomial sprinkled decoupling property.

######  Remark 2.6.

As shown in the proof, the implicit constant $c_{h}$ in Proposition 2.5 can be
chosen to be any $c_{h}<\beta-2$, although we do not use this fact.

######  Remark 2.7.

While it may be the case that the fields $(f_{\lambda})_{\lambda\geq 1}$ also
satisfy the uniform polynomial sprinkled decoupling property, our proof only
establishes it for the rescaled fields
$(\lambda^{-\eta}f_{\lambda})_{\lambda\geq 1}$ if $\eta>1/2$; see Remark 3.5.

We can now state our quantitative continuity result for sequences of
stationary planar random fields which converge to a limit (see Section 5 for
the proof):

######  Proposition 2.8.

Let $(f_{n})_{n\geq 1}$ and $f$ be stationary planar regular random fields.
Suppose there exists positive functions $s(n),u(n),R(n)$ and $w(R)$ such that:

  * •

The collection of rescaled fields $(s(n)f_{n})_{n\geq 1}$ satisfies the
uniform polynomial sprinkled decoupling property with positive constants
$(c_{h},c_{e,1},c_{e,2})$.

  * •

The random field $f$ has a near-critical window (at $\ell=0$) of size $w(R)$.

  * •

$R(n)\to\infty$ as $n\to\infty$, and for every $n\geq 1$ there exists a
coupling of $f_{n}$ and $f$ satisfying, as $n\to\infty$,

(2.4) $\mathbb{P}[\|f_{n}-f\|_{\infty,B(2R(n))}\geq u(n)]\to 0.$

Then there exists a $c>0$, depending only on $c_{h}$, such that

(2.5) $|\ell_{c}(f_{n})|\leq
3\max\\{u(n),w(R(n)),c\,s(n)^{-1}R(n)^{-{c_{h}}}\\}$

for sufficiently large $n$.

######  Remark 2.9.

It is not essential that the uniform sprinkled decoupling satisfied by the
fields $f_{n}$ is ‘polynomial’ for a continuity result such as Proposition 2.8
(with suitable adjustments) to hold; what is required is that $h$ and $e$ are
summable over a dyadic sequence, i.e.

$\sum_{n\geq 1}h(2^{n})<\infty\qquad\text{and}\qquad\sum_{n\geq
1}e(2^{n})<\infty,$

so for instance it would be enough if $h$ and $e$ decay at order $(\log
R)^{-c}$ for some $c>1$; see the proof of Proposition 5.1 for details. We use
polynomial sprinkled decoupling for convenience and since it already holds
under our assumptions.

###  2.3. Proof of Theorem 1.3

Fix $\eta>1/2$, and recall that by Proposition 2.5 the collection of random
fields $(\lambda^{-\eta}f_{\lambda})_{\lambda\geq 1}$ satisfies the uniform
polynomial sprinkled decoupling property with some constant $c_{h}>0$.
Moreover, one can check that Assumption 1.1 guarantees that the Gaussian field
$f$ satisfies the conditions of Theorem 2.1, and hence $f$ has a near-critical
window of size $w$ for arbitrary small constant $w(R)=w^{\prime}>0$. Define
$R(\lambda)=\lambda^{c_{3}}$ and
$u(\lambda)=c_{4}\lambda^{-1/2}(\log\lambda)^{3/2}$ for constants
$c_{3}>\eta/c_{h}$ and $c_{4}>(c_{2,\beta}+2c_{3})/c_{2}$, where
$c_{2,\beta}>0$ is defined in (1.4) and $c_{2}>0$ is the constant appearing in
Theorem 1.11. Then by the strong invariance principle in Theorem 1.11 (applied
to $t=c_{4}\log\lambda)$ there exists a $c_{1}>0$ (independent of the choice
of $c_{3},c_{4}$) such that, for every $\lambda\geq 1$ sufficiently large
there exists a coupling of $f_{\lambda}$ and $f$ satisfying

(2.6) $\mathbb{P}\Big{[}\|f_{\lambda}-f\|_{\infty,2B(R(\lambda))}\geq
u(\lambda)\Big{]}\leq
c_{1}\lambda^{c_{2,\beta}}R(\lambda)^{2}e^{-c_{2}t}=c_{1}\lambda^{c_{2,\beta}+2c_{3}}e^{-c_{2}c_{4}\log\lambda}.$

By the choice of $c_{4}>0$, the right-hand side of (2.6) tends to zero as
$\lambda\to\infty$. Hence the assumptions in Proposition 2.8 are satisfied,
and we deduce that

$|\ell_{c}(f_{\lambda})|\leq
c_{5}\max\\{\lambda^{-1/2}(\log\lambda)^{3/2},w^{\prime},\lambda^{\eta-
c_{h}c_{3}}\\}$

for some $c_{5}>0$. Recalling that $c_{3}>\eta/c_{h}$, by taking
$\lambda\to\infty$ and then $w^{\prime}\to 0$ we see that
$\ell_{c}(f_{\lambda})\to 0$, which is (1.2).

For the second statement (1.3), observe that Assumption 1.1 and the extra
condition that $g\geq 0$ implies that $f$ satisfies the assumptions of Theorem
2.3, and hence $f$ has a near-critical window of size $w(R)=R^{-c_{6}}$ for
some $c_{6}>0$. Redefine $R=R(\lambda)=\lambda^{c_{7}}$ and
$u(\lambda)=c_{8}\lambda^{-1/2}(\log\lambda)^{3/2}$ for constants
$c_{7}>\max\\{(1/2+\eta)/c_{h},1/(2c_{6})\\}$ and
$c_{8}>(c_{2,\beta}+2c_{7})/c_{2}$. Then by the strong invariance principle in
Theorem 1.11, for every $\lambda\geq 1$ sufficiently large there exists a
coupling of $f_{\lambda}$ and $f$ satisfying

(2.7) $\mathbb{P}\Big{[}\|f_{\lambda}-f\|_{\infty,2B(R(\lambda))}\geq
u(\lambda)\Big{]}\leq
c_{1}\lambda^{c_{2,\beta}}R(\lambda)^{2}e^{-c_{2}t}=c_{1}\lambda^{c_{2,\beta}+2c_{7}}e^{-c_{2}c_{8}\log\lambda}.$

Again by the choice of $c_{4}>0$ the right-hand side of (2.7) tends to zero as
$\lambda\to\infty$. Hence the assumptions in Proposition 2.8 are satisfied,
and we deduce that

$|\ell_{c}(f_{\lambda})|\leq
c_{9}\max\\{\lambda^{-1/2}(\log\lambda)^{3/2},w(\lambda^{c_{7}}),\lambda^{1/2-c_{h}c_{7}}\\}$

for some $c_{9}>0$. Since
$w(\lambda^{c_{7}})=\lambda^{-c_{6}c_{7}}=o(\lambda^{-1/2})$ and
$\lambda^{\eta-c_{h}c_{7}}=o(\lambda^{-1/2})$ (recall that
$c_{7}>\max\\{(1/2+\eta)/c_{h},1/(2c_{6})\\}$), this implies that

$|\ell_{c}(f_{\lambda})|=O(\lambda^{-1/2}(\log\lambda)^{3/2}).$

######  Remark 2.10.

To deduce (1.2) from Proposition 2.8 we used the existence of a sharp
threshold in (2.1). On the other hand, it is perhaps more natural to work with
the assumption $\ell_{c}(f)=0$ rather than (2.1). As we now explain, if we
additionally assume that $f$ is positively-associated (equivalent to assuming
$K(x)\geq 0$ [Pit82], and hence implied by $g(x)\geq 0$) then in fact
$\ell_{c}(f)=0$ implies (2.1). Indeed let us consider the contrapositive
statement, i.e., we choose an $\ell>0$ such that

(2.8) $\liminf_{R\to\infty}\mathbb{P}[\textrm{Cross}_{-\ell}[R,2R]]>0,$

and aim to show that this implies $\ell_{c}(f)\geq\ell/2>0$. By applying
Russo-Seymour-Welsh theory [ATT18, KST20], available since we assume
stationarity, isotropy, and positive associations, we can deduce from (2.8)
that

$\liminf_{R\to\infty}\mathbb{P}(\textrm{Ann}_{-\ell}[R,2R])>0,$

where $\textrm{Ann}_{-\ell}[R,2R]$ is the event that
$\\{f\leq-\ell\\}\cap([-2R,2R]^{2}\setminus[-R,R]^{2})$ contains a circuit.
Then an ergodic argument (see the ‘box lemma’ in [GKR88], and recall that we
assume $K(x)\to 0$ as $|x|\to\infty$ which implies that $f$ is ergodic) shows
that

$\mathbb{P}[\text{every bounded set $A\subseteq\mathbb{R}^{2}$ is surrounded
by a circuit in }\\{f\leq-\ell\\}]=1.$

This implies that $\\{f>-\ell\\}$ almost surely does not have an unbounded
component, and by symmetry in law of $f$ and $-f$, also that $\\{f<\ell\\}$
does not have an unbounded component. Hence $\ell_{c}(f)\geq\ell>0$, as we
aimed to show.

##  3\. Properties of shot noise and Gaussian fields

In this section we collect some preliminary estimates on shot noise and
Gaussian fields on $\mathbb{R}^{d}$, in particular (i) we show how to couple
these fields to approximations that have finite-range dependence, and (ii) we
give bounds on their $C^{1}$-norm. We then use the finite-range approximation
to establish the polynomial sprinkled decoupling property in Proposition 2.5.

Throughout this section we assume for simplicity that the kernel $g$ satisfies
Assumption 1.8 for some $\beta>d$, although some of the results hold under
weaker assumptions. All of the constants defined in this section depend only
on $\beta$ and $d$.

###  3.1. Convolution representations of shot noise and Gaussian fields

We begin by recalling a convenient representation of shot noise and Gaussian
fields. Recall that $f_{\lambda}$ denotes the rescaled and centred shot noise
field

$f_{\lambda}(x)=\frac{F_{\lambda}(x)-\mathbb{E}[F_{\lambda}(0)]}{\lambda^{1/2}}=\frac{\sum_{i\in\mathcal{P}_{\lambda}}g(x-i)-\lambda\smallint
g}{\lambda^{1/2}},$

and that $f$ is the stationary centred Gaussian field $f$ with covariance

$K(x)=\mathbb{E}[f(0)f(x)]=\int g(-y)g(x-y)\,dy.$

Clearly we may represent $f_{\lambda}$ as

$f_{\lambda}(x)=(g\star\widetilde{\mathcal{P}_{\lambda}})(x)=\widetilde{\mathcal{P}_{\lambda}}(g(x-\cdot)),$

where $\star$ denotes convolution, $\widetilde{\mathcal{P}_{\lambda}}$ is the
normalised compensated Poisson measure

$\widetilde{\mathcal{P}_{\lambda}}=\frac{\sum_{i\in\mathcal{P}_{\lambda}}\delta_{i}-\lambda
dx}{\lambda^{1/2}}\ ,\qquad(\widetilde{\mathcal{P}_{\lambda}}(g))_{g\in
L^{1}(\mathbb{R}^{d})}=\Big{(}\frac{\sum_{i\in\mathcal{P}_{\lambda}}g(i)-\lambda\int
g}{\lambda^{1/2}}\Big{)}_{g\in L^{1}(\mathbb{R}^{d})},$

$\delta_{i}$ denotes a Dirac mass at $i\in\mathbb{R}^{d}$, and $dx$ is the
Lebesgue measure. Similarly, recall the ‘stationary moving average’
representation of the Gaussian field $f$

(3.1) $\displaystyle
f(x)=(g\star\widetilde{\mathcal{W}})(x)=\widetilde{\mathcal{W}}(g(x-\cdot)),$

where $\widetilde{\mathcal{W}}$ denotes the white noise on $\mathbb{R}^{d}$,
i.e. $(\widetilde{\mathcal{W}}(g))_{g\in L^{2}(\mathbb{R}^{d})}$ is the
centred Gaussian process such that
$\mathbb{E}[\widetilde{\mathcal{W}}(g_{1})\widetilde{\mathcal{W}}(g_{2})]=\int
g_{1}g_{2}$. To justify this representation, observe that
$g\star\widetilde{\mathcal{W}}$ is a stationary centred Gaussian field with
covariance kernel

$K(x)=\mathbb{E}[f(0)f(x)]=\mathbb{E}\big{[}\widetilde{\mathcal{W}}(g(0-\cdot))\widetilde{\mathcal{W}}(g(x-\cdot))\big{]}=\int
g(-y)g(x-y)dy,$

and so has the same law as the Gaussian field $f$ defined in Section 1.2.

###  3.2. Finite-range approximation

The representations $f_{\lambda}=g\star\widetilde{\mathcal{P}_{\lambda}}$ and
$f=g\star\widetilde{\mathcal{W}}$ allow us to construct approximations
$f^{r}_{\lambda}$ and $f^{r}$ of the fields $f_{\lambda}$ and $f$
respectively, that are (i) coupled on the same probability space, and (ii)
$r$-range dependent, meaning that events supported on sets separated by a
distance $r$ are independent. In fact we may construct these approximations
simultaneously for all $r>0$.

For this we introduce a smooth cut-off function $\chi:\mathbb{R}^{d}\to[0,1]$
such that $\chi(x)=1$ for $x\in B(1/4)$ and $\chi(x)=0$ for $x\notin B(1/2)$;
the choice of $\chi$ is arbitrary but the unspecified constants in our results
depend on it (including Theorem 1.3). Then for $r>0$, define the truncated
kernel $g^{r}(x)=g(x)\chi(x/r)$, and the fields

$f_{\lambda}^{r}(x)=(g^{r}\star\widetilde{\mathcal{P}_{\lambda}})(x)=\widetilde{\mathcal{P}_{\lambda}}(g^{r}(x-\cdot))=\frac{\sum_{i\in\mathcal{P}_{\lambda}}g^{r}(x-i)-\lambda\smallint
g^{r}}{\lambda^{1/2}}$

and

$f^{r}(x)=(g^{r}\star\widetilde{\mathcal{W}})(x)=\widetilde{\mathcal{W}}(g^{r}(x-\cdot)).$

Since $g^{r}$ is supported on $B(r/2)$, the fields $f_{\lambda}^{r}$ and
$f^{r}$ are $r$-range dependent, and they are naturally coupled to
$f_{\lambda}$ and $f$, respectively, via $\widetilde{\mathcal{P}_{\lambda}}$
and $\widetilde{\mathcal{W}}$.

In the following two propositions we control the quality of these
approximations. Recall that $\beta>d$ is the constant in Assumption 1.8.

######  Proposition 3.1.

There exist $c_{1},c_{2}>0$ such that, for every $\lambda,r,u\geq 1$

$\mathbb{P}\big{[}\|f_{\lambda}-f^{r}_{\lambda}\|_{[0,1]^{d},\infty}\geq
u\lambda^{1/2}r^{d-\beta}\big{]}\leq c_{1}\lambda e^{-c_{2}\sqrt{u}}.$

###### Proof.

By the definition of $f_{\lambda}^{r}$ and $f_{\lambda}$ we have

$(f_{\lambda}^{r}-f_{\lambda})(x)=((g-g^{r})\star\widetilde{\mathcal{P}_{\lambda}})(x)=\widetilde{\mathcal{P}_{\lambda}}((g-g^{r})(x-\cdot))=\frac{\hat{F}^{r}_{\lambda}(x)-\lambda\smallint(g^{r}-g)}{\lambda^{1/2}},$

where
$\hat{F}^{r}_{\lambda}(x)=\sum_{i\in\mathcal{P}_{\lambda}}(g-g^{r})(x-i)$ is a
shot noise field with kernel $g-g^{r}$. By the decay of $g$ in Assumption 1.8,
there is a $c_{3}>0$ such that, for $r\geq 1$,

$|\smallint(g^{r}-g)|\leq c_{3}r^{d-\beta},$

and hence

(3.2) $\mathbb{P}\big{[}\|f_{\lambda}-f^{r}_{\lambda}\|_{[0,1]^{d},\infty}\geq
u\lambda^{1/2}r^{d-\beta}\big{]}\leq\mathbb{P}\big{[}\|\hat{F}^{r}_{\lambda}\|_{[0,1]^{d},\infty}\geq(u-c_{3})\lambda
r^{d-\beta}\big{]}.$

To control this latter quantity we apply the concentration bounds for shot
noise fields in [LM22, Proposition A.2]. Consider the rescaled field
$\hat{G}(\cdot)=\hat{F}^{r}_{\lambda}(\cdot/\lambda^{1/d})$ which is a unit-
intensity shot noise field with kernel
$h(\cdot)=(g-g^{r})(\cdot/\lambda^{1/d})$, and define the auxiliary function
on $\mathbb{R}^{d}$

$j(x)=\sup_{y\in[-1/2,1/2]^{d}}|h(x+y)|.$

Applying [LM22, Proposition A.2, (A.1)] to $\hat{G}$ (where $j$ corresponds to
$\hat{h}$ using the notation in this proposition, and we may take
$\beta=\infty$ in this proposition since we work with degenerate marks), there
exist $c_{4},c_{5}>0$ depending only on $d$ such that, for all $v\geq 1$,

$\displaystyle\mathbb{P}\Big{[}\|\hat{F}^{r}_{\lambda}\|_{[0,1]^{d},\infty}\geq
2v\Big{(}\|j\|_{L^{1}}+\sqrt{2v}\|j\|_{L^{2}}+\frac{v}{3}\|j\|_{\infty}\Big{)}\Big{]}$
$\displaystyle\qquad=\mathbb{P}\Big{[}\|\hat{G}\|_{[0,\lambda^{1/d}]^{d},\infty}\geq
2v\Big{(}\|j\|_{L^{1}}+\sqrt{2v}\|j\|_{L^{2}}+\frac{v}{3}\|j\|_{\infty}\Big{)}\Big{]}$
$\displaystyle\qquad\leq c_{4}\lambda e^{-c_{5}v}.$

By the decay of $g$ in Assumption 1.8, there is a $c_{6}>0$ such that, for
$r\geq 1$,

$\max\big{\\{}\|j\|_{L^{1}},\|j\|_{L^{2}},\|j\|_{\infty}\big{\\}}\leq
c_{6}\lambda r^{d-\beta},$

so the bound reduces to

$\mathbb{P}\Big{[}\|\hat{F}^{r}_{\lambda}\|_{[0,1]^{d},\infty}\geq
c_{7}v^{2}\lambda r^{d-\beta}\Big{]}\leq c_{4}\lambda e^{-c_{5}v}$

for some $c_{7}>0$. Assuming $u\geq c_{3}+c_{7}$ we may apply this bound to
$v^{2}=(u-c_{3})/c_{7}\geq 1$, and combining with (3.2) we deduce that

$\mathbb{P}\big{[}\|f_{\lambda}-f^{r}_{\lambda}\|_{[0,1]^{d},\infty}\geq
u\lambda^{1/2}r^{d-\beta}\big{]}\leq c_{4}\lambda e^{-c_{8}\sqrt{u-c_{3}}}$

for some $c_{8}>0$. This proves the result for sufficiently large $u$, and
hence for all $u\geq 1$ after adjusting constants. ∎

######  Proposition 3.2.

There exist $c_{1},c_{2}>0$ such that, for every $r,u\geq 1$

$\mathbb{P}\big{[}\|f-f^{r}\|_{[0,1]^{d},\infty}\geq ur^{d/2-\beta}\big{]}\leq
c_{1}e^{-c_{2}u^{2}}.$

###### Proof.

The case $d=2$ is proven in [MV20, Proposition 3.11], and the proof extends
immediately to arbitrary dimension $d\geq 1$ (with obvious adjustments). ∎

###  3.3. Derivative bounds

For a domain $D\subset\mathbb{R}^{d}$, let
$\|h\|_{C^{1}(D)}=\sup_{|\alpha|\leq 1}\|\partial^{\alpha}h\|_{\infty,D}$. The
following results bound the $C^{1}$-norm of the fields $f_{\lambda}$ and $f$:

######  Proposition 3.3.

There exist $c_{1},c_{2}>0$ such that, for $\lambda,u\geq 1$,

$\mathbb{P}\big{[}\|f_{\lambda}\|_{C^{1}([0,1]^{d})}\geq
u\lambda^{1/2}\big{]}\leq c_{1}\lambda e^{-c_{2}\sqrt{u}}.$

###### Proof.

For each multi-index $\alpha$ such that $|\alpha|\leq 1$,
$\lambda^{1/2}(\partial^{\alpha}f_{\lambda})$ is distributed as a shot noise
field with kernel $\partial^{\alpha}g$ (since we may interchange derivative
and summation by dominated convergence). Hence we can control the $C^{1}$-norm
of $f_{\lambda}$ using the concentration bounds for shot noise fields in
[LM22, Proposition A.2] as in the proof of Proposition 3.1.

For $|\alpha|\leq 1$, consider the unit-intensity shot noise field
$\hat{G}_{\alpha}(\cdot)=\lambda^{1/2}(\partial^{\alpha}f_{\lambda})(\cdot/\lambda^{1/d})$
and define the auxiliary function on $\mathbb{R}^{d}$

$j_{\alpha}(x)=\sup_{y\in[-1/2,1/2]^{d}}|\partial^{\alpha}g((x+y)/\lambda^{1/d})|.$

According to [LM22, Proposition A.2, (A.1)] (again we may take $\beta=\infty$
in this proposition since we work with degenerate marks), there exist
$c_{3},c_{4}>0$ depending only on $d$ such that, for all $u\geq 1$,

$\displaystyle\mathbb{P}\Big{[}\lambda^{1/2}\|\partial^{\alpha}f_{\lambda}\|_{[0,1]^{d},\infty}\geq
2u\Big{(}\|j_{\alpha}\|_{L^{1}}+\sqrt{2u}\|j_{\alpha}\|_{L^{2}}+\frac{u}{3}\|j_{\alpha}\|_{\infty}\Big{)}\Big{]}$
$\displaystyle\qquad=\mathbb{P}\Big{[}\|\hat{G}_{\alpha}\|_{[0,\lambda^{1/d}]^{d},\infty}\geq
2u\Big{(}\|j_{\alpha}\|_{L^{1}}+\sqrt{2u}\|j_{\alpha}\|_{L^{2}}+\frac{u}{3}\|j_{\alpha}\|_{\infty}\Big{)}\Big{]}$
$\displaystyle\qquad\leq c_{3}\lambda e^{-c_{4}u}.$

By the decay of $g$ and its derivatives in Assumption 1.8,

$\max\big{\\{}\|j_{\alpha}\|_{L^{1}},\|j_{\alpha}\|_{L^{2}},\|j_{\alpha}\|_{\infty}\big{\\}}\leq
c_{5}\lambda,$

for some $c_{5}>0$, so the bound reduces to

$\mathbb{P}\Big{[}\lambda^{1/2}\|\partial^{\alpha}f_{\lambda}\|_{[0,1]^{d},\infty}\geq
c_{6}\lambda u^{2}\Big{]}\leq c_{3}\lambda e^{-c_{4}u}$

for some $c_{6}>0$. We obtain the desired result by summing up the bound for
all $|\alpha|\leq 1$ and adjusting constants. ∎

######  Proposition 3.4.

There exist $c_{1},c_{2}>0$ such that, for $u\geq 1$,

$\mathbb{P}\big{[}\|f\|_{C^{1}([0,1]^{d})}\geq u\big{]}\leq
c_{1}e^{-c_{2}u^{2}}.$

###### Proof.

By Kolmogorov’s theorem (see [NS16, Appendix A]),
$\mathbb{E}[\|f\|_{C^{1}([0,1]^{d})}]$ is bounded by a constant depending only
on $\beta$ and $d$, and the result then follows from the BTIS theorem [AW09,
Theorem 2.9]. ∎

###  3.4. Sprinkled decoupling for shot noise fields

In this section we prove the sprinkled decoupling property for shot noise
fields stated in Proposition 2.5. We shall make use of the following
observation: if $A$ is an increasing event supported on
$D\subset\mathbb{R}^{d}$, and $f,g$ are fields that are coupled on the same
probability space, then for any $\varepsilon>0$,

$\displaystyle\mathbb{P}[f\in A]$ $\displaystyle=\mathbb{P}[f\in
A,\|f-g\|_{D,\infty}\leq\varepsilon]+\mathbb{P}[f\in
A,\|f-g\|_{D,\infty}>\varepsilon]$ (3.3)
$\displaystyle\leq\mathbb{P}[g+\varepsilon\in
A]+\mathbb{P}[\|f-g\|_{D,\infty}\geq\varepsilon].$

###### Proof of Proposition 2.5.

Fix $\eta>1/2$ and $\delta\in(0,\beta-d)$, and let $\lambda\geq 1$, $R\geq 1$,
$\ell\in\mathbb{R}$, and $A,B$ be events which are translations or rotations
of $\\{f\in\textrm{Cross}_{\ell}[R,3R]\\}$ such that the distance between
their associated rectangles, which we denote by $R_{A}$ and $R_{B}$
respectively, is at least $R$. Since $A$ and $B$ are increasing, by (3.3) we
have

$\displaystyle\mathbb{P}[\lambda^{-\eta}f_{\lambda}\in A\cap B]$
$\displaystyle\leq\mathbb{P}\big{[}\lambda^{-\eta}f^{R}_{\lambda}+R^{d-\beta+\delta}\in
A\cap
B\big{]}+\mathbb{P}\big{[}\|f_{\lambda}-f^{R}_{\lambda}\|_{\infty,R_{A}\cup
R_{B}}\geq\lambda^{\eta}R^{d-\beta+\delta}\big{]}$
$\displaystyle\leq\mathbb{P}\big{[}\lambda^{-\eta}f^{R}_{\lambda}+R^{d-\beta+\delta}\in
A\big{]}\mathbb{P}\big{[}\lambda^{-\eta}f^{R}_{\lambda}+R^{d-\beta+\delta}\in
B\big{]}+\mathbb{P}[\|f_{\lambda}-f^{R}_{\lambda}\|_{\infty,R_{A}\cup
R_{B}}\geq\lambda^{\eta}R^{d-\beta+\delta}],$

where in the last step we used that $R_{A}$ and $R_{B}$ are separated by
distance $\geq R$ and that the field $f_{\lambda}^{R}$ is $R$-range dependent.
By Proposition 3.1 (applied to $u=R^{\delta}\lambda^{\eta-1/2}$) and the union
bound we have

$\mathbb{P}[\|f_{\lambda}-f^{R}_{\lambda}\|_{\infty,R_{A}\cup
R_{B}}\geq\lambda^{\eta}R^{d-\beta+\delta}]\leq c_{1}R^{d}\lambda
e^{-c_{2}\sqrt{R^{\delta}\lambda^{\eta-1/2}}}$

for some $c_{1},c_{2}>0$. Hence the field $\lambda^{-\eta}f_{\lambda}$
satisfies sprinkled decoupling with polynomial sprinkling function
$h(R)=R^{d-\beta+\delta}$ and error function $e(R)=c_{1}R^{d}\lambda
e^{-c_{2}\sqrt{R^{\delta}\lambda^{\eta-1/2}}}\leq c_{3}R^{-c_{4}}$ for some
$c_{3},c_{4}>0$ which are uniform in $\lambda\geq 1$. ∎

######  Remark 3.5.

The reason that we restrict $\eta>1/2$ in Proposition 2.5 is to ensure that
the error function $e(R)=c_{1}R^{d}\lambda
e^{-c_{2}\sqrt{R^{\delta}\lambda^{\eta-1/2}}}$ is uniformly decaying in $R$;
indeed if $\eta\leq 1/2$ then $e(R)$ is only small if

$R\gg\lambda^{(1/2-\eta)/\delta}(\log\lambda)^{2/\delta}\to\infty.$

##  4\. A strong invariance principle for shot noise fields

In this section we prove the strong invariance principle for shot noise fields
in arbitrary dimension $d\geq 1$, stated in Theorem 1.11. Recall the
representations $f_{\lambda}(x)=\widetilde{\mathcal{P}_{\lambda}}(g(x-\cdot))$
and $f(x)=\widetilde{\mathcal{W}}(g(x-\cdot))$ from Section 3.1. Our strategy
is to first establish a coupling of $\widetilde{\mathcal{P}_{\lambda}}$ and
$\widetilde{\mathcal{W}}$ such that $\widetilde{\mathcal{P}_{\lambda}}(g)$ and
$\widetilde{\mathcal{W}}(g)$ are close with high probability for a single
compactly-supported test function $g$ (see Proposition 4.1 below). This is
achieved by applying the results of Koltchinskii [Kol94] on strong invariance
principles for the empirical process, based on the dyadic coupling scheme of
[KMT75]. We then apply this to translations of the (truncated) kernel
$g^{r}(x-\cdot)$ on a sufficiently fine mesh $x\in\varepsilon\mathbb{Z}^{d}$,
and use continuity properties of the fields $f_{\lambda}$ and $f$ to deduce
the result.

###  4.1. Strong invariance principle for a single test function

Let us introduce the quantities which control the quality of the coupling. Let
$I=[0,1)^{d}$ denote the unit cube. A binary expansion of $I$ is a sequence
$\Delta=(\Delta_{j})_{j\geq 0}$ of partitions of $I$ satisfying:

  * •

For $j\geq 0$, $\Delta_{j}=(\Delta_{j,k})_{k=0,\ldots,2^{j}-1}$ with each
$\Delta_{j,k}$ a rectangle $[a_{i},b_{i})\times\ldots\times[a_{d},b_{d})$;

  * •

$\Delta_{0,0}=I$, and for all $j\geq 0$ and $k=0,\ldots,2^{j}-1$,

$\Delta_{j,k}=\Delta_{j+1,2k}\cup\Delta_{j+1,2k+1}\quad\text{and}\quad\Delta_{j+1,2k}\cap\Delta_{j+1,2k+1}=\emptyset;$

  * •

For $j\geq 0$ and $k=0,\ldots,2^{j}-1$, $\textrm{Vol}(\Delta_{j,k})=2^{-j}$;

  * •

The Borel $\sigma$-algebra on $I$ is the completion with respect to the
Lebesgue measure on $I$ of the $\sigma$-algebra generated by $\cup_{i\geq
0}\Delta_{i}$.

In other words, $\Delta$ is an iterative division of $I$ into rectangles of
equal volume that generate the Borel $\sigma$-algebra. By sequential division
along the $d$ coordinate directions, one can check that there exists a binary
expansion of $I$ such that (i)
$\textrm{diam}(\Delta_{j,k})\leq\sqrt{d}2^{-\lfloor
j/d\rfloor}\leq\sqrt{d}2^{1-j/d}$ for all $j\geq 0$, and (ii) $\Delta_{j,k}$
are homothetic to a finite class of rectangles, and we henceforth fix $\Delta$
to be one such expansion.

For a measurable function $h:I\to\mathbb{R}$ and a Borel subset $D\subset I$,
define the $L^{2}$-modulus

$\omega^{2}(h;D)=\frac{1}{\textrm{Vol}(D)}\int_{D}(h-\bar{h})^{2},$

where $\bar{h}=\textrm{Vol}(D)^{-1}\int_{D}h.$ For $m\geq 0$, define the
‘level-$m$’ $L^{2}$-modulus

(4.1)
$q_{m}(h)=\Big{(}\sum_{j=0}^{m}\sum_{k=0}^{2^{j}-1}\omega^{2}(h;\Delta_{j,k})\Big{)}^{1/2}.$

This quantity is homogeneous in the sense that if
$h^{\prime}(\cdot)=ah(\cdot)$ for a constant $a>0$ then
$q_{m}(h^{\prime})=aq_{m}(h)$.

Although not necessary to state our result, for later use we observe that the
asymptotic behaviour of $q_{m}(h)$ as $m\to\infty$ depends on the regularity
of $h$ as well as the dimension $d\geq 1$. In particular, for $h$
differentiable almost everywhere, the Poincaré-Wirtinger inequality yields

$\omega^{2}(h;\Delta_{j,k})\leq\frac{c_{d}\text{\rm{diam}}(\Delta_{j,k})^{2}\|\nabla
h\|_{L^{2}(\Delta_{j,k})}^{2}}{\textrm{Vol}(\Delta_{j,k})},$

where $c_{d}>0$ is a constant depending only on the dimension, and so

$\displaystyle q_{m}(h)$ $\displaystyle\leq
c^{\prime}_{d}\Big{(}\sum_{j=0}^{m}2^{(1-2/d)j}\sum_{k}\|\nabla
h\|_{L^{2}(\Delta_{j,k})}^{2}\Big{)}^{1/2}$ (4.2) $\displaystyle\leq
c^{\prime}_{d}\|\nabla h\|_{L^{2}(I)}\times\begin{cases}\sqrt{2}&d=1,\\\
\sqrt{m+1}&d=2,\\\ \sqrt{2}\times 2^{(1/2-1/d)m}&d\geq 3,\end{cases}$

where $c^{\prime}_{d}=2\sqrt{dc_{d}}$.

Finally, for $i\in\mathbb{Z}^{d}$ define the translated cubes $C_{i}=i+I$,
which collectively tile $\mathbb{R}^{d}$, and for a function
$g:\mathbb{R}^{d}\to\mathbb{R}$ we define $g_{i}=g|_{C_{i}}$ to be the
restriction of $g$ to the cube $C_{i}$, so that
$g=\sum_{i\in\mathbb{Z}^{d}}g_{i}$. We further define $q_{m}(g_{i})$ in the
natural way (i.e. by identifying $C_{i}$ with $I$).

######  Proposition 4.1.

There exists $c_{1},c_{2}>0$ depending only on the dimension such that, for
every $\lambda\geq 1$ there exists a coupling of
$\widetilde{\mathcal{P}_{\lambda}}$ and $\widetilde{\mathcal{W}}$ satisfying,
for every $r\geq 1$, measurable function $g$ that is supported on $B(r)$, and
every $t\geq 1$,

$\mathbb{P}\Big{[}|\widetilde{\mathcal{P}_{\lambda}}(g)-\widetilde{\mathcal{W}}(g)|\geq
t\lambda^{-1/2}\sum_{i\in\mathbb{Z}^{d}}\big{(}q_{[\log_{2}(2\lambda/t)]^{+}}(g_{i})+\|g_{i}\|_{\infty}\big{)}\Big{]}\leq
c_{1}r^{d}\lambda e^{-c_{2}t}$

where $q_{m}(h)$ is defined in (4.1) and $[x]^{+}=\max\\{0,\lfloor
x\rfloor\\}$.

To prove Proposition 4.1 we rely on the results of Koltchinskii [Kol94] on
strong invariance principles for the empirical process (see also [Rio94] for
related results), which we now recall. For $n\geq 1$, the empirical process on
$I$ (equipped with the Lebesgue measure) is the sequence of random measures

(4.3)
$\widetilde{\mathcal{E}_{n}}=\frac{\sum_{i=1}^{n}\delta_{X_{i}}-ndx}{\sqrt{n}}\
,\qquad(\widetilde{\mathcal{E}_{n}}(h))_{h\in
L^{1}(I)}=\Big{(}\frac{\sum_{i=1}^{n}h(X_{i})-n\int h}{\sqrt{n}}\Big{)}_{h\in
L^{1}(I)},$

where $(X_{i})_{i\geq 1}$ is an i.i.d. sequence of random variables
distributed uniformly on $I$. Further let $\widetilde{\mathcal{W}}_{0}$ denote
the ‘Brownian bridge process’ on $I$, i.e.
$(\widetilde{\mathcal{W}}_{0})_{h\in L^{2}(I)}$ is the centred Gaussian
process such that
$\mathbb{E}[\widetilde{\mathcal{W}}_{0}(h_{1})\widetilde{\mathcal{W}}_{0}(h_{2})]=\int
h_{1}h_{2}-\int h_{1}\int h_{2}$.

######  Proposition 4.2 (Special case of [Kol94, Theorem 3.5]).

There exists $c_{1},c_{2}>0$ depending only on the dimension such that, for
every $n\geq 1$ there exists a coupling of $\widetilde{\mathcal{E}_{n}}$ and
$\widetilde{\mathcal{W}}_{0}$ satisfying, for every measurable
$h:I\to\mathbb{R}$ such that $\|h\|_{\infty}\leq 1$, and every $t\geq 1$,

$\mathbb{P}\Big{[}|\widetilde{\mathcal{E}_{n}}(h)-\widetilde{\mathcal{W}}_{0}(h)|\geq
tn^{-1/2}(q_{[\log_{2}(n/t)]^{+}}(h)+1)\Big{]}\leq c_{1}ne^{-c_{2}t}.$

######  Remark 4.3.

[Kol94, Theorem 3.5] is stated in the general setting of a probability space
$(X,\mathcal{A},P)$ that admits a binary expansion, i.e. a sequence
$\Delta=(\Delta_{j})_{j\geq 0}$ of partitions of $X$ satisfying

  * •

For $i\geq 0$, $\Delta_{j}=(\Delta_{j,k})_{k=0,\ldots,2^{j}-1}$ with
$\Delta_{j,k}\in\mathcal{A}$ for all $k$;

  * •

$\Delta_{0,0}=X$, and for all $j\geq 0$ and $k=0,\ldots,2^{j}-1$,

$\Delta_{j,k}=\Delta_{j+1,2k}\cup\Delta_{j+1,2k+1}\quad\text{and}\quad\Delta_{j+1,2k}\cap\Delta_{j+1,2k+1}=\emptyset;$

  * •

For $j\geq 0$ and $k=0,\ldots,2^{j}-1$, $P(\Delta_{j,k})=2^{-j}$;

  * •

$\mathcal{A}$ is the completion with respect to $P$ of the $\sigma$-algebra
generated by $\cup_{i\geq 0}\Delta_{i}$.

Denote by $\widetilde{\mathcal{E}_{n}}$ the empirical process on $X$, defined
by analogy with (4.3), and by $\widetilde{\mathcal{W}}_{0}$ the Brownian
bridge process on $X$. Then [Kol94, Theorem 3.5] states that there exists
$c_{1},c_{2}>0$ such that, for every $n\geq 1$ there exists a coupling of
$\widetilde{\mathcal{E}_{n}}$ and $\widetilde{\mathcal{W}}_{0}$ satisfying,
for every measurable $h:X\to\mathbb{R}$ such that $\|h\|_{\infty}\leq 1$, and
every $x>0$ and $y\geq 1$,

(4.4)
$\mathbb{P}\Big{[}n^{1/2}|\widetilde{\mathcal{E}_{n}}(h)-\widetilde{\mathcal{W}}_{0}(h)|\geq
x+x^{1/2}y^{1/2}\big{(}\hat{q}_{[\log_{2}(n/y)]^{+}}(h)+1\big{)}\Big{]}\leq
c_{1}(e^{-c_{2}x}+ne^{-c_{2}y}),$

where

$\hat{q}_{m}(g)=\Big{(}\sum_{j=0}^{m}2^{j}\sum_{k=0}^{2^{j}-1}\int_{\Delta_{j,k}}\Big{|}h(u)-2^{j}\int_{\Delta_{j,k}}h(v)P(dv)\Big{|}^{2}P(du)\Big{)}^{1/2}.$

In the Euclidean setting $(X,\mathcal{A},P)=(I,\mathcal{B},dx)$, we have
$\hat{q}_{m}(h)=q_{m}(h)$, hence we obtain Proposition 4.2 by setting $x=y=t$
in (4.4) and adjusting the constants.

######  Remark 4.4.

The Euclidean setting was also considered in [Kol94], however in deriving the
bounds in (4.1), [Kol94] used the trivial inequality
$|h-\bar{h}|\leqslant\text{\rm{diam}}(\Delta)\|\nabla h\|_{\infty,I}$ rather
than the Poincaré-Wirtinger inequality, which led to a term $\|\nabla
h\|_{I,\infty}$ in place of our $\|\nabla h\|_{L^{2}(I)}$.

Let us make the connection between shot noise fields and empirical processes.
Fix $\lambda\geq 1$, and let $N=\mathcal{P}_{\lambda}(I)$, so that
$N\sim\textrm{Pois}(\lambda)$. Then, for $h\in L^{1}(I)$,

$\sum_{i\in\mathcal{P}_{\lambda}}h(i)\stackrel{{\scriptstyle
d}}{{=}}\sum_{i=1}^{N}h(X_{i})$

where $(X_{i})_{i\geq 1}$ is an i.i.d. sequence of random variables
independent of $\mathcal{P}_{\lambda}$ distributed uniformly on $I$. Hence if
we let $\widetilde{\mathcal{E}_{N}}$ be, conditionally on $N$, an independent
copy of the $N^{\text{th}}$ coordinate of the empirical process on $I$, then

$(\widetilde{\mathcal{P}_{\lambda}}(h))_{h\in L^{1}(I)}\stackrel{{\scriptstyle
d}}{{=}}\Big{(}\frac{\sum_{i=1}^{N}h(X_{i})-\lambda\int
h}{\sqrt{\lambda}}\Big{)}_{h\in
L^{1}(I)}=\Big{(}\frac{\sqrt{N}\widetilde{\mathcal{E}_{N}}(h)+(N-\lambda)\int
h}{\sqrt{\lambda}}\Big{)}_{h\in L^{1}(I)}.$

Similarly, if we let $Z$ denote a standard Gaussian random variable
independent of $\widetilde{\mathcal{W}}_{0}$ (the Brownian bridge process on
$I$), then

$(\widetilde{\mathcal{W}}(h))_{h\in L^{2}(I)}\stackrel{{\scriptstyle
d}}{{=}}\Big{(}\widetilde{\mathcal{W}}_{0}(h)+Z\int h\Big{)}_{h\in L^{2}(I)}.$

Hence every coupling of
$(N,\widetilde{\mathcal{E}_{N}},Z,\widetilde{\mathcal{W}}_{0})$ induces a
coupling of $(\widetilde{\mathcal{P}}(h),\widetilde{\mathcal{W}}(h))_{h\in
L^{1}(I)\cap L^{2}(I)}$ such that

(4.5)
$\displaystyle\sqrt{\lambda}\big{(}\widetilde{\mathcal{P}}(h)-\widetilde{\mathcal{W}}(h)\big{)}$
$\displaystyle\qquad\qquad=\sqrt{N}\big{(}\widetilde{\mathcal{E}_{N}}(h)-\widetilde{\mathcal{W}}_{0}(h)\big{)}+(\sqrt{N}-\sqrt{\lambda})\widetilde{\mathcal{W}}_{0}(h)+(N-\lambda-\sqrt{\lambda}Z)\int
h.$

To help analyse the expression in (4.5) we use the following univariate
coupling lemma for Poisson random variables, whose proof is deferred to the
end of the section:

######  Lemma 4.5.

There exists universal constants $c_{1},c_{2}>0$ such that, for every
$\lambda\geq 1$ there is a coupling of $N\sim\textrm{Pois}(\lambda)$ and a
standard Gaussian $Z$ such that, for $t\geq 1$,

$\mathbb{P}[|N-\lambda-\sqrt{\lambda}Z|\geq t]\leq c_{1}e^{-c_{2}t}.$

We shall also use the following standard Poisson and Gaussian concentration
bounds

(4.6) $\mathbb{P}[|N-\lambda|\geq t]\leq
2e^{-\frac{t^{2}}{2(\lambda+t)}}\quad\text{and}\quad\mathbb{P}[|Z|\geq t]\leq
2e^{-t^{2}/2}$

valid for all $\lambda,t>0$, which can be derived using a standard Chernoff
bound

$\mathbb{P}[Y\geq t]\leq e^{-at}\mathbb{E}[e^{aY}]\,,\quad\text{for all
}a,t>0\text{ and random variables }Y,$

either setting $Y=\pm Z$ and $a=\pm t$ in the Gaussian case, or
$Y=\pm(N-\lambda)$ and $a=\pm\log(1\pm t/\lambda)$ in the Poisson case, using
the fact that $(1+x)\log(1+x)-x\geq x^{2}/(2+2x/3))$ for all $x>-1$.

We are now ready to give the proof of Proposition 4.1:

###### Proof of Proposition 4.1.

In the proof $c_{1},c_{2}>0$ will denote positive constants depending only on
the dimension which may change from line to line.

We begin by constructing the necessary coupling. Fix $\lambda\geq 1$, let
$(Z^{i})_{i\in\mathbb{Z}^{d}}$ be an i.i.d. sequence of standard Gaussians,
and let $(\widetilde{\mathcal{W}}^{i}_{0})_{i\in\mathbb{Z}^{d}}$ be a sequence
of independent Brownian bridge processes on $i+I$. For each
$i\in\mathbb{Z}^{d}$, let $N^{i}$ be a $\textrm{Pois}(\lambda)$-distributed
random variable coupled to $Z^{i}$ so as to satisfy the conclusion of Lemma
4.5, and conditionally on $N^{i}$, let $\widetilde{\mathcal{E}^{i}_{N^{i}}}$
denote the empirical process on $i+I$ coupled to
$\widetilde{\mathcal{W}}^{i}_{0}$ so as to satisfy the conclusion of
Proposition 4.2. This defines a coupling of
$(N^{i},\widetilde{\mathcal{E}^{i}_{N^{i}}},Z^{i},\widetilde{\mathcal{W}}_{0}^{i})_{i\in\mathbb{Z}^{d}}$,
which, by (4.5) and the decomposition $g=\sum_{i\in\mathbb{Z}^{d}}g_{i}$,
induces for every $g\in L^{1}(\mathbb{R}^{d})\cap L^{2}(\mathbb{R}^{d})$ a
coupling of
$\widetilde{\mathcal{P}_{\lambda}}(g)=\sum_{i\in\mathbb{Z}^{d}}\widetilde{\mathcal{P}_{\lambda}}(g_{i})$
and
$\widetilde{\mathcal{W}}(g)=\sum_{i\in\mathbb{Z}^{d}}\widetilde{\mathcal{W}}(g_{i})$
such that, for every $i\in\mathbb{Z}^{d}$,

$\displaystyle\sqrt{\lambda}(\widetilde{\mathcal{P}_{\lambda}}(g_{i})-\widetilde{\mathcal{W}}(g_{i}))$
(4.7)
$\displaystyle\qquad=\Big{(}\sqrt{N^{i}}(\widetilde{\mathcal{E}^{i}_{N^{i}}}(g_{i})-\widetilde{\mathcal{W}}^{i}_{0}(g_{i}))+(\sqrt{N^{i}}-\sqrt{\lambda})\widetilde{\mathcal{W}}^{i}_{0}(g)+(N^{i}-\lambda-\sqrt{\lambda}Z^{i})\int
g_{i}\Big{)}.$

Now let $t\geq 1$ be given and abbreviate
$m^{\prime}=[\log_{2}(2\lambda/t)]^{+}$. Then by the union bound and (4.1)

$\displaystyle\mathbb{P}\Big{[}\sqrt{\lambda}|\widetilde{\mathcal{P}_{\lambda}}(g)-\widetilde{\mathcal{W}}(g)|\geq
3t\sum_{i\in\mathbb{Z}^{d}}(q_{m^{\prime}}(g_{i})+\|g_{i}\|_{\infty})\Big{]}$
$\displaystyle\quad\leq\sum_{i\in\mathbb{Z}^{d}:\|g_{i}\|_{\infty}\neq
0}\mathbb{P}\Big{[}\sqrt{\lambda}|\widetilde{\mathcal{P}_{\lambda}}(g_{i})-\widetilde{\mathcal{W}}(g_{i})|]\geq
3t(q_{m^{\prime}}(g_{i})+\|g_{i}\|_{\infty})\Big{]}$
$\displaystyle\quad\leq|\\{i\in\mathbb{Z}^{d}:(i+I)\cap B(r)\neq\emptyset\\}|$
$\displaystyle\qquad\times\max_{i\in\mathbb{Z}^{d}:\|g_{i}\|_{\infty}\neq
0}\Big{\\{}\mathbb{P}\Big{[}\big{|}\sqrt{N^{i}}(\widetilde{\mathcal{E}^{i}_{N^{i}}}(g_{i})-\widetilde{\mathcal{W}}^{i}_{0}(g_{i}))\big{|}\geq
t(q_{m^{\prime}}(g_{i})+\|g_{i}\|_{\infty})\Big{]}$
$\displaystyle\qquad\qquad\qquad+\mathbb{P}\Big{[}\big{|}(\sqrt{N^{i}}-\sqrt{\lambda})\widetilde{\mathcal{W}}^{i}_{0}(g)\big{|}\geq
t\|g_{i}\|_{\infty}]\Big{]}+\mathbb{P}\Big{[}\big{|}(N^{i}-\lambda-\sqrt{\lambda}Z^{i})\smallint
g_{i}\big{|}\geq t\|g_{i}\|_{\infty}\Big{]}\Big{\\}}$ $\displaystyle\quad\leq
c_{1}r^{d}\times\max_{i\in\mathbb{Z}^{d}:\|g_{i}\|_{\infty}\neq
0}\Big{\\{}\sup_{n\leq\max\\{2\lambda,t\\}}\mathbb{P}\Big{[}\sqrt{n}|\widetilde{\mathcal{E}^{i}_{n}}(g_{i})-\widetilde{\mathcal{W}}_{0}^{i}(g_{i})|\geq
t(q_{m^{\prime}}(g_{i})+\|g_{i}\|_{\infty})\,\big{|}\,N^{i}=n\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad+\mathbb{P}\big{[}N^{i}>\max\\{2\lambda,t\\}\big{]}+\mathbb{P}\Big{[}\big{|}\big{(}\sqrt{N^{i}}-\sqrt{\lambda}\big{)}\widetilde{\mathcal{W}}^{i}_{0}(g_{i})\big{|}\geq
t\|g_{i}\|_{\infty}]\Big{]}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\mathbb{P}\Big{[}\big{|}N^{i}-\lambda-\sqrt{\lambda}Z^{i}\big{|}\|g_{i}\|_{L^{1}}\geq
t\|g_{i}\|_{\infty}\Big{]}\Big{\\}}.$

We analyse the four summands in the above bound separately. First observe
that, since $m\mapsto q_{m}(h)$ is non-decreasing,

$\sup_{n\leq\max\\{2\lambda,t\\}}q_{[\log_{2}(n/t)]^{+}}(h)\leq
q_{[\log_{2}(2\lambda/t)]^{+}}(h)=q_{m^{\prime}}(h).$

Hence applying Proposition 4.2 to the function
$h_{i}=g_{i}/\|g_{i}\|_{\infty}$, for which

$q_{m^{\prime}}(h_{i})=q_{m^{\prime}}(g_{i})/\|g_{i}\|_{\infty},$

we have

$\displaystyle\sup_{n\leq\max\\{2\lambda,t\\}}\mathbb{P}\big{[}\sqrt{n}|\widetilde{\mathcal{E}^{i}_{n}}(g_{i})-\widetilde{\mathcal{W}}_{0}^{i}(g_{i})|\geq
t(q_{m^{\prime}}(g_{i})+\|g_{i}\|_{\infty})\,\big{|}\,N^{i}=n\big{]}$
$\displaystyle\qquad\leq\sup_{n\leq\max\\{2\lambda,t\\}}\mathbb{P}\big{[}\sqrt{n}|\widetilde{\mathcal{E}^{i}_{n}}(h_{i})-\widetilde{\mathcal{W}}_{0}^{i}(h_{i})|\geq
t(q_{[\log_{2}(n/t)]^{+}}(h_{i})+1)\,\big{|}\,N^{i}=n\big{]}$
$\displaystyle\qquad\leq c_{1}\max\\{2\lambda,t\\}e^{-c_{2}t}\leq c_{1}\lambda
e^{-c_{2}t}.$

Next we have

$\displaystyle\mathbb{P}\Big{[}\big{|}\big{(}\sqrt{N^{i}}-\sqrt{\lambda}\big{)}\widetilde{\mathcal{W}}^{i}_{0}(g_{i})\big{|}\geq
t\|g_{i}\|_{\infty}\Big{]}$
$\displaystyle\leq\mathbb{P}\Big{[}|\sqrt{N^{i}}-\sqrt{\lambda}|\geq\sqrt{t}\Big{]}+\mathbb{P}\big{[}|\widetilde{\mathcal{W}}^{i}_{0}(g_{i})|\geq\sqrt{t}\|g_{i}\|_{\infty}\big{]}$
$\displaystyle\leq\mathbb{P}[|\sqrt{N}-\sqrt{\lambda}|\geq\sqrt{t}]+\mathbb{P}[|Z|\geq\sqrt{t}],$

where in the last inequality we used that
$\mathbb{E}[\widetilde{\mathcal{W}}_{0}^{i}(g_{i})^{2}]=\int g_{i}^{2}-(\int
g_{i})^{2}\leq\|g_{i}\|^{2}_{\infty}$. Using the fact that, for $a,b,c>0$,

$|\sqrt{a}-\sqrt{b}|\geq\sqrt{c}\quad\Longrightarrow\quad|a-b|\geq\max\\{\sqrt{bc},c\\},$

and applying the concentration bounds in (4.6) we have

$\mathbb{P}[|\sqrt{N}-\sqrt{\lambda}|\geq\sqrt{t}]+\mathbb{P}[|Z|\geq\sqrt{t}]\leq\mathbb{P}[|N-\lambda|\geq\max\\{\sqrt{\lambda
t},t\\}]+\mathbb{P}[|Z|\geq\sqrt{t}]\leq c_{1}e^{-c_{2}t}.$

Finally

$\mathbb{P}[N^{i}\geq\max\\{2\lambda,t\\}]\leq\mathbb{P}[|N-\lambda|\geq\max\\{\lambda,t-\lambda\\}]\leq
c_{1}e^{-c_{2}\max\\{\lambda,t\\}},$

and

$\mathbb{P}\Big{[}\big{|}N^{i}-\lambda-\sqrt{\lambda}Z^{i}\big{|}\|g_{i}\|_{L^{1}}\geq\|g_{i}\|_{\infty}t\Big{]}\leq\mathbb{P}\Big{[}\big{|}N^{i}-\lambda-\sqrt{\lambda}Z^{i}\big{|}\geq
t\Big{]}\leq c_{1}e^{-c_{2}t},$

where we used the concentration bounds in (4.6) and Lemma 4.5. Combining the
above we deduce that

$\mathbb{P}\Big{[}\lambda^{1/2}|\widetilde{\mathcal{P}_{\lambda}}(g)-\widetilde{\mathcal{W}}(g)|\geq
3t\sum_{i\in\mathbb{Z}^{d}}(q_{m^{\prime}}(g_{i})+\|g_{i}\|_{\infty})\Big{]}\leq
c_{1}r^{d}\lambda e^{-c_{2}t},$

and we obtain the result by adjusting constants.

∎

###  4.2. Strong invariance principle for shot noise fields; proof of Theorem
1.11

We are now ready to prove Theorem 1.11. Let us first prove (1). In the proof
$c_{1},c_{2}>0$ will denote positive constants depending only on $\beta$ and
$d$ which may change from line to line. Let $\lambda\geq 1$ be given and
recall the coupling between $\widetilde{\mathcal{P}_{\lambda}}$ and
$\widetilde{\mathcal{W}}$ in Proposition 4.1; we shall use the same coupling
(which is in fact independent of the kernel $g$). Recall also the definition
of $\sigma_{d}(\lambda)$ in (1.4). Fix $R,t\geq 1$ and parameters
$r,\varepsilon>0$ to be chosen later. Then by first using that

$\|f_{\lambda}-f\|_{\infty,B(R)}\leq\|f_{\lambda}-f\|_{\infty,\varepsilon\mathbb{Z}^{d}\cap
B(R)}+\sqrt{d}\varepsilon\|f_{\lambda}\|_{C^{1}(B(R))}+\sqrt{d}\varepsilon\|f\|_{C^{1}(B(R))},$

and then approximating $f_{\lambda}$ and $f$ by the $r$-dependent fields
$f^{r}_{\lambda}$ and $f^{r}$ respectively, we have by the union bound,

$\displaystyle\mathbb{P}\Big{[}\|f_{\lambda}-f\|_{\infty,B(R)}\geq(3+2\sqrt{d})t\sigma_{d}(\lambda)\Big{]}$
$\displaystyle\qquad\leq\mathbb{P}\Big{[}\|f^{r}_{\lambda}-f^{r}\|_{\infty,\varepsilon\mathbb{Z}^{d}\cap
B(R)}\geq t\sigma_{d}(\lambda)\Big{]}$
$\displaystyle\qquad\quad+\mathbb{P}\Big{[}\|f_{\lambda}-f^{r}_{\lambda}\|_{\infty,B(R)}\geq
t\sigma_{d}(\lambda)\Big{]}+\mathbb{P}\Big{[}\|f-f^{r}\|_{\infty,B(R)}\geq
t\sigma_{d}(\lambda)\Big{]}$
$\displaystyle\qquad\quad\quad+\mathbb{P}\Big{[}\|f_{\lambda}\|_{C^{1}(B(R))}\geq
t\varepsilon^{-1}\sigma_{d}(\lambda)\Big{]}+\mathbb{P}\Big{[}\|f\|_{C^{1}(B(R))}\geq
t\varepsilon^{-1}\sigma_{d}(\lambda)\Big{]}$ (4.8)
$\displaystyle\qquad\quad=:E_{1}+E_{2}+E_{3}+E_{4}+E_{5}.$

We next substitute parameters

(4.9) $\varepsilon=\min\\{t^{-1},\lambda^{-1/2}\\}\qquad\text{and}\qquad
r=(t\lambda)^{1/(\beta-d)}$

and control each of the summands in the right-hand side of (4.8) in turn (see
Remark 4.6 for some comments on this choice).

For $i\in\mathbb{Z}^{d}$, recall the translated cube $C_{i}=i+I$, and define
$g_{i}=g^{r}|_{C_{i}}$. Recall the definition of $q_{m}(h)$ in (4.1). Since
$g\in C^{1}(\mathbb{R}^{d})$, using the bounds in (4.1) we have, for any
$i\in\mathbb{Z}^{d}$,

$q_{m}(g_{i})\leq c_{1}\|\nabla
g_{i}\|_{L^{2}(i+I)}\times\begin{cases}1&d=1,\\\ \sqrt{m+1}&d=2,\\\
2^{(1/2-1/d)m}&d\geq 3.\end{cases}$

By the decay of $g$ in Assumption 1.8, we have the following bounds (uniformly
over $r\geq 1$),

$\sum_{i\in\mathbb{Z}^{d}}\|\nabla g_{i}\|_{L^{2}(i+I)}=\|\nabla
g\|_{L^{2}(\mathbb{R}^{d})}\leq
c_{1}\quad\text{and}\quad\sum_{i\in\mathbb{Z}^{d}}\|g_{i}\|_{\infty}\leq
c_{1}$

and so, considering the definition of $\sigma_{d}(\lambda)$ in (1.4), we
obtain

$\displaystyle\lambda^{-1/2}\sum_{i\in\mathbb{Z}^{d}}\big{(}q_{[\log_{2}(2\lambda/t)]^{+}}(g_{i})+\|g_{i}\|_{\infty}\big{)}$
$\displaystyle\leq c_{1}\lambda^{-1/2}\begin{cases}1&d=1,\\\
\sqrt{\log\lambda}&d=2,\\\ \lambda^{1/2-1/d}&d\geq 3,\end{cases}$
$\displaystyle=c_{1}\sigma_{d}(\lambda).$

Applying Proposition 4.1 to the functions $g^{r}(\cdot-i)$ for
$i\in\varepsilon\mathbb{Z}^{d}\cap B(R)$, by the union bound we obtain

(4.10)
$E_{1}=\mathbb{P}\Big{[}\|f^{r}_{\lambda}-f^{r}\|_{\infty,\varepsilon\mathbb{Z}^{d}\cap
B(R)}\geq t\sigma_{d}(\lambda)\Big{]}\leq
c_{1}\varepsilon^{-d}R^{d}r^{d}\lambda e^{-c_{2}t}.$

We move on to estimating the remaining terms $E_{i}$, $i=2,3,4,5$. Recalling
the definition of $r$ in (4.9) we observe that

$t\sigma_{d}(\lambda)\geq t\lambda^{-1/2}=t^{2}\lambda^{1/2}r^{d-\beta}.$

Hence applying Proposition 3.1 (to $u=t^{2}$) and the union bound

$E_{2}=\mathbb{P}\Big{[}\|f_{\lambda}-f^{r}_{\lambda}\|_{\infty,B(R)}\geq
t\sigma_{d}(\lambda)\Big{]}\leq\mathbb{P}\Big{[}\|f_{\lambda}-f^{r}_{\lambda}\|_{\infty,B(R)}\geq
t^{2}\lambda^{1/2}r^{d-\beta}\Big{]}\leq c_{1}R^{d}\lambda e^{-c_{2}t}.$

Similarly

$t\sigma_{d}(\lambda)\geq
t\lambda^{-1/2}=t^{2}\lambda^{1/2}r^{d-\beta}\geq\sqrt{t}r^{d/2-\beta},$

and hence applying Proposition 3.2 (to $u=\sqrt{t}$) and the union bound

(4.11) $E_{3}=\mathbb{P}\Big{[}\|f-f^{r}\|_{\infty,B(R)}\geq
t\sigma_{d}(\lambda)\Big{]}\leq\mathbb{P}\Big{[}\|f-f^{r}\|_{\infty,B(R)}\geq\sqrt{t}r^{d/2-\beta}\Big{]}\leq
c_{1}R^{d}e^{-c_{2}t}.$

Finally recalling the definition of $\varepsilon$ in (4.9) we have

$t\varepsilon^{-1}\sigma_{d}(\lambda)\geq
t\varepsilon^{-1}\lambda^{-1/2}\geq\max\\{t^{2}\lambda^{-1/2},\sqrt{t}\\}$

and so by Propositions 3.3 and 3.4 and the union bound

(4.12) $E_{4}=\mathbb{P}\Big{[}\|f_{\lambda}\|_{C^{1}(B(R))}\geq
t\varepsilon^{-1}\sigma_{d}(\lambda)\Big{]}\leq\mathbb{P}\Big{[}\|f_{\lambda}\|_{C^{1}(B(R))}\geq
t^{2}\lambda^{-1/2}\Big{]}\leq c_{1}R^{d}\lambda e^{-c_{2}t}$

and

(4.13) $E_{5}=\mathbb{P}\Big{[}\|f\|_{C^{1}(B(R))}\geq
t\varepsilon^{-1}\sigma_{d}(\lambda)\Big{]}\leq\mathbb{P}\Big{[}\|f\|_{C^{1}(B(R))}\geq\sqrt{t}\Big{]}\leq
c_{1}R^{d}e^{-c_{2}t}.$

Putting (4.10)–(4.13) into (4.8) we have, for sufficiently large $t$,

$\mathbb{P}\Big{[}\|f_{\lambda}-f\|_{\infty,B(R)}\geq(3+2\sqrt{d})t\sigma_{d}(\lambda)\Big{]}\leq\sum_{i=1}^{5}E_{i}\leq
c_{1}\varepsilon^{-d}R^{d}r^{d}\lambda e^{-c_{2}t}.$

Recalling the definition of the parameters $\varepsilon$ and $r$ in (4.9), and
also the definition of $c_{d,\beta}$ in (1.4), the right-hand side of the
above display equals

$c_{1}R^{d}t^{d}\max\\{t^{d},\lambda^{d/2}\\}(\lambda t)^{d/(\beta-d)}\lambda
e^{-c_{2}t}\leq c_{1}R^{d}\lambda^{c_{d,\beta}}t^{d+d/(\beta-d)}e^{-c_{2}t},$

which implies the result by adjusting constants.

The proof of (2) is a straightforward extension. The key observations are that
(i) by the convolution representations in Section 3.1 (the exchange of
integration and differentiation is justified by the dominated convergence
theorem, in light of the decay in Assumption 1.8)

$\partial^{\alpha}f_{\lambda}=(\partial^{\alpha}g)\star\widetilde{P_{\lambda}}\quad\text{and}\quad\partial^{\alpha}f=(\partial^{\alpha}g)\star\widetilde{W},$

and (ii) the coupling in the proof of (1) is chosen independently of the
kernel $g$. Hence by applying (1) to $\partial^{\alpha}g$ instead of $g$ for
each $\alpha\in I$, the result follows by the union bound.

######  Remark 4.6.

The main contribution to the right-hand side of (4.8) is from $E_{1}$, and we
make the choice of parameters in (4.9) to balance $E_{1}$ with the
contribution from the other summands $E_{i}$, $i\geq 2$, controlling the
truncation and discretisation errors, uniformly over $\lambda$ and $t$. Note
however that this choice is not optimal for all $\lambda$ and $t$
simultaneously, and the bounds on $E_{i}$, $i\geq 2$, could be sharpened in
some cases.

###  4.3. Proof of the univariate coupling lemma for Poisson random variables

In this section we obtain Lemma 4.5 by applying the following ‘Tusnády-type’
lemma on coupling a simple random walk to Brownian motion:

######  Lemma 4.7 ([KMT75, Lemma 1 and Section 6]).

Let $(X_{i})_{i\geq 1}$ be a sequence of i.i.d. random variables with zero
mean, unit variance, and a finite moment generating function in a
neighbourhood of the origin. Then there are constants
$\varepsilon,c_{1},c_{2}>0$ such that, for every $n\geq 1$, there is a
coupling of $S_{n}=\sum_{i=1}^{n}X_{i}$ and a standard Gaussian $Z$ satisfying

$|S_{n}-\sqrt{n}Z|\leq\frac{c_{1}S_{n}^{2}}{n}+c_{2}\quad\text{if}\quad|S_{n}|<\varepsilon
n.$

###### Proof of Lemma 4.5.

Let $(X_{i})_{i\geq 1}$ be a sequence of i.i.d. random variables distributed
as $X_{i}\stackrel{{\scriptstyle d}}{{=}}\textrm{Pois}(1)-1$, and let
$\varepsilon,c_{1},c_{2}>0$ be the constants appearing in Lemma 4.7 for this
$X_{i}$. By adjusting constants it suffices to prove the result for $t$
sufficiently large, and we split the proof into two cases: (i) $4\leq
t\leq\varepsilon\lfloor\lambda\rfloor$, and (ii)
$t\geq\varepsilon\lfloor\lambda\rfloor$.

For the case $4\leq t\leq\varepsilon\lfloor\lambda\rfloor$, note that
$N\sim\textrm{Pois}(\lambda)\stackrel{{\scriptstyle
d}}{{=}}\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}+\lfloor\lambda\rfloor+Y$,
where $Y$ is an independent random variable distributed as
$\textrm{Pois}(\lambda-\lfloor\lambda\rfloor)$. Consider the coupling of
$\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}$ and $Z$ guaranteed by Lemma 4.7 and
define $N=\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}+\lfloor\lambda\rfloor+Y$.
Then since

$N-\lambda-\sqrt{\lambda}Z=\Big{(}\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}-\sqrt{\lfloor\lambda\rfloor}Z\Big{)}+Y+(\lfloor\lambda\rfloor-\lambda)+(\sqrt{\lfloor\lambda\rfloor}-\sqrt{\lambda})Z,$

we have

$\displaystyle\mathbb{P}[|N-\lambda-\sqrt{\lambda}Z|\geq
t]\leq\mathbb{P}\Big{[}\Big{|}\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}-\sqrt{\lfloor\lambda\rfloor}Z\Big{|}\geq
t/4\Big{]}+\mathbb{P}[Y\geq t/4]+\mathbb{P}[|Z|\geq t/4]$
$\displaystyle\leq\mathbb{P}\Big{[}\Big{|}\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}-\sqrt{\lfloor\lambda\rfloor}Z\Big{|}\geq
t/4,\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}<\varepsilon\lfloor\lambda\rfloor\Big{]}+\mathbb{P}\Big{[}\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}\geq\varepsilon\lfloor\lambda\rfloor\Big{]}$
$\displaystyle\qquad\qquad\qquad+\mathbb{P}[Y\geq t/4]+\mathbb{P}[|Z|\geq
t/4]$
$\displaystyle\leq\mathbb{P}\Big{[}\Big{|}\frac{c_{1}(\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i})^{2}}{\lfloor\lambda\rfloor}+c_{2}\Big{|}\geq
t/4\Big{]}+\mathbb{P}\Big{[}\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}\geq\varepsilon\lfloor\lambda\rfloor\Big{]}+\mathbb{P}[Y\geq
t/4]+\mathbb{P}[|Z|\geq t/4]$ $\displaystyle\leq
2\mathbb{P}\Big{[}\Big{|}\textrm{Pois}(\lfloor\lambda\rfloor)-\lfloor\lambda\rfloor\Big{|}\geq\min\Big{\\{}\sqrt{\lfloor\lambda\rfloor/c_{1}}\sqrt{t/4-c_{2}},\varepsilon\lfloor\lambda\rfloor\Big{\\}}\Big{]}\\!+\mathbb{P}[\textrm{Pois}(1)\geq
t/4]+\mathbb{P}[|Z|\geq t/4]$

where in the first inequality we used $|\lambda-\lfloor\lambda\rfloor|\leq
1\leq t/4$ and $|\sqrt{\lambda}-\sqrt{\lfloor\lambda\rfloor}|\leq 1$, in the
third inequality we used Lemma 4.7, and in the last inequality we used that
$\sum_{i=1}^{\lfloor\lambda\rfloor}X_{i}\sim\textrm{Pois}(\lfloor\lambda\rfloor)-\lfloor\lambda\rfloor$
and that $Y$ is stochastically dominated by $\textrm{Pois}(1)$. The
concentration bounds in (4.6) imply that the above is at most
$c_{3}e^{-c_{4}\min\\{t,\lambda\\}}$ for some $c_{3},c_{4}>0$ and $t$
sufficiently large, which gives the result in this case by taking $t$
sufficient large and adjusting constants.

In the case $t\geq\varepsilon\lfloor\lambda\rfloor$, we instead bound

$\mathbb{P}[|N-\lambda-\sqrt{\lambda}Z|\geq t]\leq\mathbb{P}[|N-\lambda|\geq
t/2]+\mathbb{P}[|Z|\geq t/(2\sqrt{\lambda})].$

Again the concentration bounds in (4.6) imply that the above is at most
$c_{5}e^{-c_{6}\min\\{t,t^{2}/\lambda\\}}$, which gives the result also in
this case after adjusting constants. ∎

##  5\. Continuity of the critical level

In this section we prove the quantitative continuity result for sequences of
stationary planar random fields stated in Proposition 2.8 above.

###  5.1. Finite-size criterion

We first establish a finite-size criterion that applies to a single planar
random field. This states roughly that, as soon as crossing events occur with
sufficiently small (resp. large) probability at a given level, one can deduce
that the critical level is not much smaller (resp. bigger) than this level.
The proof is similar to classical bootstrapping arguments for Bernoulli
percolation [Kes82, Section 5.1], although in our setting the sprinkled
decoupling property replaces independence (see also [PR15, MV20] for similar
arguments).

Recall that $\textrm{Cross}_{\ell}[a,b]$ (resp.
$\textrm{Cross}^{tb}_{\ell}[a,b]$) denotes the event that $\\{f\leq\ell\\}$
crosses the rectangle $[0,a]\times[0,b]$ from left to right (resp. top to
bottom).

######  Proposition 5.1 (Finite-size criterion).

Let $f$ be a stationary planar regular random field that satisfies the
polynomial sprinkled decoupling property with constants
$(c_{h},c_{e,1},c_{e,2})$. Then there exists $c>0$ depending only on $c_{h}$,
and $\delta,R_{0}>0$ depending only on $(c_{e,1},c_{e,2})$, such that, for all
$\ell\in\mathbb{R}$ and $R\geq R_{0}$,

$\max\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell}[R,3R]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell}[3R,R]]\big{\\}}\leq\delta\
\ \quad\quad\Longrightarrow\quad\ell_{c}(f)\geq\ell-cR^{-c_{h}}\ \ $

and

$\min\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell}[3R,R]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell}[R,3R]]\big{\\}}\geq
1-\delta\quad\Longrightarrow\quad\ell_{c}(f)\leq\ell+cR^{-c_{h}}.$

###### Proof.

Fix $\delta>0$ and $R_{0}\geq 1$ such that
$49(\delta+\sqrt{c_{e,1}}R_{0}^{-c_{e,2}/2})<1$. By the classical construction
of Kesten [Kes82, Section 5.1] (see Figure 1), for every $R\geq 1$ one can
find collections $(A_{i})_{i=1,2,\ldots,7}$ and $(B_{j})_{j=1,2,\ldots,7}$ of
translated copies of the events $\textrm{Cross}_{\ell}[R,3R]$ or
$\textrm{Cross}^{tb}_{\ell}[3R,R]$ such that the following holds for every
$\ell\in\mathbb{R}$:

  * •

$\textrm{Cross}_{\ell}[3R,9R]$ implies $\cup_{i,j=1,\ldots,7}(A_{i}\cap
B_{j})$; and

  * •

For each $i,j=1,\ldots,7$, the distance between the rectangles associated with
$A_{i}$ and $B_{j}$ is at least $R$.

Then the union bound, sprinkled decoupling, and stationarity imply that, for
every $R\geq 1$ and $\ell\in\mathbb{R}$,

$\displaystyle\mathbb{P}[\textrm{Cross}_{\ell}[3R,9R]]$ $\displaystyle\leq
49\max_{i,j=1,2,\ldots,7}\mathbb{P}[f\in A_{i}\cap B_{j}]$ $\displaystyle\leq
49\max_{i,j=1,2,\ldots,7}\mathbb{P}[f+h(R)\in A_{i}]\mathbb{P}[f+h(R)\in
B_{j}]+e(R)$ (5.1) $\displaystyle\leq
49\max\\{\mathbb{P}[\textrm{Cross}_{\ell+R^{-c_{h}}}[R,3R]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell+R^{-c_{h}}}[3R,R]]\big{\\}}^{2}+c_{e,1}R^{-c_{e,2}}.$

Figure 1. On the left, a realisation of $\text{{Cross}}_{\ell}[3R,9R]$. On the
right, the $14$ translated copies of $[R,3R]$ and $[3R,R]$ on which the events
$(A_{i})_{i=1,\ldots,7}$ and $(B_{j})_{j=1,\ldots,7}$ are supported, and in
bold a pair $(A_{i},B_{j})$ such that $A_{i}\cap B_{j}$ occurs.

Now suppose $R\geq R_{0}$ and $\ell\in\mathbb{R}$ are such that

$\max\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell}[R,3R]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell}[3R,R]]\big{\\}}\leq\delta.$

For $n\geq 0$, define the sequences $r_{n}=R3^{n}$ and
$\ell_{n}=\ell-\sum_{i=0}^{n-1}r_{i}^{-c_{h}}$ with $\ell_{0}=\ell$.
Abbreviating

$a_{n}=\max\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell_{n}}[r_{n},3r_{n}]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell_{n}}[3r_{n},r_{n}]]\big{\\}},$

we deduce from (5.1) that, for every $n\geq 0$,

(5.2) $a_{n+1}\leq 49a_{n}^{2}+c_{e,1}r_{n}^{-c_{e,2}}.$

Defining $a_{n}^{\prime}=49(a_{n}+\sqrt{c_{e,1}}r_{n}^{-c_{e,2}/2})\geq
a_{n}$, one checks from (5.2) that

$\displaystyle a^{\prime}_{n+1}$
$\displaystyle=49(a_{n+1}+\sqrt{c_{e,1}}r_{n}^{-c_{e,2}/2})\leq
49(49a_{n}^{2}+c_{e,1}r_{n}^{-c_{e,2}}+\sqrt{c_{e,1}}r_{n}^{-c_{e,2}/2})$
$\displaystyle\leq
49^{2}(a_{n}+\sqrt{c_{e,1}}r_{n}^{-c_{e,2}/2})^{2}=(a_{n}^{\prime})^{2}.$

Since also

$a_{0}^{\prime}=49(a_{0}+\sqrt{c_{e,1}}R^{-c_{e,2}/2})\leq
49(\delta+\sqrt{c_{e,1}}R_{0}^{-c_{e,2}/2})<1$

by the choice of $\delta$ and $R_{0}$, we deduce that $a_{n}\leq
a_{n}^{\prime}$ decays exponentially in $n$. Defining $\ell^{\prime}=\ell-
cR^{-c_{h}}$, where $c=\sum_{i=0}^{\infty}3^{-c_{h}i}\in(0,\infty)$, so that
$\ell_{n}\downarrow\ell^{\prime}$, by monotonicity we conclude that

(5.3)
$\max\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell^{\prime}}[r_{n},3r_{n}]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell^{\prime}}[3r_{n},r_{n}]]\big{\\}}\leq
c_{1}e^{-c_{2}n}$

for all $n\geq 0$ and some $c_{1},c_{2}>0$.

To finish the proof of the first statement, define the event $A_{n}$ that
$\\{f\leq\ell^{\prime}\\}$ contains a path that intersects both
$[-r_{n}/2,r_{n}/2]^{2}$ and $\partial[-3r_{n}/2,3r_{n}/2]^{2}$. By the union
bound and stationarity

$\mathbb{P}[A_{n}]\leq
4\max\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell^{\prime}}[r_{n},3r_{n}]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell^{\prime}}[3r_{n},r_{n}]]\big{\\}}\to
0$

as $n\to\infty$. Since $\ell^{\prime}>\ell_{c}(f)$ would imply that
$\liminf_{n\to\infty}\mathbb{P}[A_{n}]>0$, we deduce that
$\ell_{c}(f)\geq\ell^{\prime}=\ell-cR^{-c_{h}}$ as required.

For the second statement, by replacing $f\mapsto-f$ and using the regularity
of the field, the same reasoning that led to (5.3) shows that if $R\geq R_{0}$
and $\ell\in\mathbb{R}$ are such that

$\min\big{\\{}\mathbb{P}[\textrm{Cross}_{\ell}[3R,R]],\mathbb{P}[\textrm{Cross}^{tb}_{\ell}[R,3R]]\big{\\}}\geq
1-\delta$

then, defining $r_{n}=R3^{n}$ and $\ell^{\prime}=\ell+cR^{-c_{h}}$,

$\min\big{\\{}\mathbb{P}[f\in\textrm{Cross}_{\ell^{\prime}}[3r_{n},r_{n}]],\mathbb{P}[f\in\textrm{Cross}^{tb}_{\ell^{\prime}}[r_{n},3r_{n}]]\big{\\}}\geq
1-c_{1}e^{-c_{2}n}$

for all $n\geq 0$. By a classical construction (see Figure 2) one can define
events $(A_{n})_{n\geq 0}$ satisfying:

  * •

If $n$ is even (resp. odd) then $A_{n}$ is a translation of
$\text{{Cross}}_{\ell^{\prime}}[3r_{n},r_{n}]$ (resp.
$\text{{Cross}}^{tb}_{\ell^{\prime}}[r_{n},3r_{n}]$);

  * •

For any $n_{0}\geq 0$, $\cap_{n\geq n_{0}}A_{n}$ implies the existence of an
infinite path in $\\{f\leq\ell^{\prime}\\}$.

Then by the Borel-Cantelli lemma $\\{f\leq\ell^{\prime}\\}$ contains an
infinite component almost surely, and thus
$\ell_{c}(f)\leq\ell^{\prime}=\ell+cR^{-c_{h}}$ as required.

Figure 2. A depiction of the events $A_{n}$ for $n=1,2,3$.

∎

###  5.2. Proof of Proposition 2.8

Let $s(n),u(n),R(n)$ and $w(R)$ be as in the statement of the proposition, and
let $R_{0},\delta>0$ be the constants defined by Proposition 5.1 applied to
the fields $(s(n)f_{n})_{n\geq 1}$; by the uniform sprinkled decoupling
property these can be chosen uniformly over $n$. By (3.3), (2.4), and the
definition of the near-critical window, we have

$\displaystyle\min\\{\mathbb{P}[f_{n}\in\textrm{Cross}_{w(R(n))+u(n)}[3R(n),R(n)]],\mathbb{P}[f_{n}\in\textrm{Cross}^{tb}_{w(R(n))+u(n)}[R(n),3R(n)]]\\}$
$\displaystyle\qquad\geq\min\\{\mathbb{P}[f\in\textrm{Cross}_{w(R(n))}[3R(n),R(n)]],\mathbb{P}[f\in\textrm{Cross}^{tb}_{w(R(n))}[R(n),3R(n)]]\\}$
$\displaystyle\qquad\qquad\qquad-\mathbb{P}[\|f_{n}-f\|_{\infty,B(2R(n))}\geq
u(n)]$ $\displaystyle\qquad\to 1$

as $n\to\infty$. Via a rescaling, this implies that

$\displaystyle\min\\{\mathbb{P}[s(n)f_{n}\in\textrm{Cross}_{s(n)(w(R(n))+u(n))}[3R(n),R(n)]],\mathbb{P}[s(n)f_{n}\in\textrm{Cross}^{tb}_{s(n)(w(R(n))+u(n))}[R(n),3R(n)]]\\}$
$\displaystyle\qquad\geq 1-\delta$

for sufficiently large $n$. Hence taking $n$ sufficiently large so that also
$R(n)\geq R_{0}$, by the finite-size criterion in Proposition 5.1 applied to
$s(n)f_{n}$ we deduce that

$\ell_{c}(s(n)f_{n})\leq s(n)(w(R(n))+u(n))+cR(n)^{-c_{h}}$

where $c>0$ depends only on $c_{h}$. Via rescaling, this implies that

$\ell_{c}(f_{n})\leq w(R(n))+u(n)+cs(n)^{-1}R(n)^{-c_{h}}$

for sufficiently large $n$. An analogous argument for the events

$\textrm{Cross}_{-w(R(n))-u(n)}[R(n),3R(n)]\quad\text{and}\quad\textrm{Cross}^{tb}_{-w(R(n))-u(n)}[3R(n),R(n)]$

shows that

$\ell_{c}(f_{n})\geq-w(R(n))-u(n)-cs(n)^{-1}R(n)^{-c_{h}}$

for sufficiently large $n$, which completes the proof.

## References

  * [ATT18] D. Ahlberg, V. Tassion, and A. Teixeira.  Sharpness of the phase transition for continuum percolation.  Probab. Theory Related Fields, 172(1–2):525–281, 2018.
  * [AW09] J. Azaïs and M. Wschebor.  Level sets and extrema of random processes and fields.  John Wiley & Sons, Inc., Hoboken, NJ, 2009.
  * [AY10] M. Aljuaid and H. Yanikomeroglu.  Investigating the Gaussian convergence of the distribution of the aggregate interference power in large wireless networks.  IEEE Trans. Vehicular Tech., 59(9):4418–4424, 2010.
  * [BD12a] H. Biermé and A. Desolneux.  Crossings of smooth shot noise processes.  Ann. Appl. Probab., 22(6):2240–2281, 2012.
  * [BD12b] H. Biermé and A. Desolneux.  A Fourier approach for the level crossings of shot noise processes with jumps.  J. Appl. Prob., 49:100–113, 2012.
  * [BD20] H. Biermé and A. Desolneux.  Mean geometry for 2D random fields: level perimeter and level total curvature integrals.  Ann. Appl. Probab., 30(2):561–607, 2020.
  * [Bec85] J. Beck.  Lower bounds on the approximation of the multivariate empirical process.  Z. Wahrsch. verw. Gebiete, 70:289–306, 1985.
  * [BG17] V. Beffara and D. Gayet.  Percolation of random nodal lines.  Publ. Math. IHES, 126:131–176, 2017.
  * [BM06] P. Berthet and D. M. Mason.  Revisiting two strong approximation results of Dudley and Philipp.  In IMS Lecture Notes–Monograph Series Vol 51: High Dimensional Probability, pages 155–172. Institute of Mathematical Sciences, 2006.
  * [BM17] E. Broman and R. Meester.  Phase transition and uniqueness of levelset percolation.  J. Stat. Phys., 167(6):1376–1400, 2017.
  * [CR75] M. Csörgő and P. Révész.  A new method to prove Strassen-type laws of invariance principle. II.  Z. Wahrsch. verw. Gebiete, 31:261–269, 1975.
  * [DRRV21] H. Duminil-Copin, A. Rivera, P.-F. Rodriguez, and H. Vanneuville.  Existence of unbounded nodal hypersurface for smooth Gaussian fields in dimension $d\geq 3$.  arXiv preprint arXiv:2108.08008, 2021.
  * [GGM11] B. Galerne, Y. Gousseau, and J. Morel.  Random phase textures: Theory and synthesis.  IEEE Trans. Image Proc., 20(1):257–267, 2011.
  * [GKR88] A. Gandolfi, M. Keane, and L. Russo.  On the uniqueness of the infinite occupied cluster in dependent two-dimensional site percolation.  Ann. Probab., 16(3):1147–1157, 1988.
  * [Gri99] G. R. Grimmett.  Percolation.  Springer: Berlin, Germany, 1999.
  * [Hei07] L. Heinrich.  Limit distributions of some stereological estimators in Wicksell’s corpuscle problem.  Image Anal. Stereol., 26:63–71, 2007.
  * [HS85] L. Heinrich and V. Schmidt.  Normal convergence of multidimension shot noise and rates of this convergence.  Adv. Appl. Prob., 17:709–730, 1985.
  * [Kes82] H. Kesten.  Percolation theory for mathematicians.  Progress in Probability and Statistics Vol. 2. Springer, 1982.
  * [KM95] C. Kluppelberg and T. Mikosch.  Explosive Poisson shot noise processes with applications to risk reserves.  Bernoulli, 1(1):125–147, 1995.
  * [KMT75] J. Komlós, P. Major, and G. Tusnády.  An approximation of partial sums of independent RV’s and the sample DF.  Z. Wahrsch. verw. Gebiete, 32:111–131, 1975.
  * [Kol94] V. I. Koltchinskii.  Komlos-Major-Tusnady approximation for the general empirical process and Haar expansions of classes of functions.  J. Theor. Probab., 7(1):73–118, 1994.
  * [KST20] L. Köhler-Schindler and V. Tassion.  Crossing probabilities for planar percolation.  arXiv preprint arXiv:2011.04618, 2020.
  * [Lan87] J. A. Lane.  The Berry-Esseen bound for Poisson shot noise.  Adv. Appl. Prob., 19:512–514, 1987.
  * [LM22] R. Lachièze-Rey and S. Muirhead.  Percolation of the excursion sets of planar symmetric shot noise fields.  Stoch. Proc. Appl., 147:175–209, 2022.
  * [Mas89] P. Massart.  Strong approximation for multivariate empirical and related processes, via KMT constructions.  Ann. Probab., 17:266–291, 1989.
  * [MRV20] S. Muirhead, A. Rivera, and H. Vanneuville.  The phase transition for planar Gaussian percolation models without FKG.  arXiv preprint arXiv:2010.11770, 2020.  With an appendix by L. Köhler-Schindler.
  * [MS83] S. A. Molchanov and A. K. Stepanov.  Percolation in random fields. II.  Theor. Math. Phys., 55(3):592–599, 1983.
  * [MV20] S. Muirhead and H. Vanneuville.  The sharp phase transition for level set percolation of smooth planar Gaussian fields.  Ann. I. Henri Poincaré Probab. Stat., 56(2):1358–1390, 2020.
  * [NS16] F. Nazarov and M. Sodin.  Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions.  J. Math. Phys. Anal. Geo., 12(3):205–278, 2016.
  * [Pap71] A. Papoulis.  High density shot noise and Gaussianity.  J. Appl. Prob., 8:118–127, 1971.
  * [Pit82] L. D. Pitt.  Positively correlated normal variables are associated.  Ann. Probab., 10(2):496–499, 1982.
  * [PR15] S. Popov and B. Ráth.  On decoupling inequalities and percolation of the excursion sets of the Gaussian free field.  J. Stat. Phys., 159:312–320, 2015.
  * [Rév76] P. Révész.  On strong approximation of the multidimensional empirical process.  Ann. Probab., 4:729–743, 1976.
  * [Rio94] E. Rio.  Local invariance principles and their application to density estimation.  Probab. Theory Related Fields, 98:21–45, 1994.
  * [Rio96] E. Rio.  Vitesse de convergence dans le principe d’invariance faible pour la fonction de répartition empirique multivariée.  C. R. Acad. Sci. Paris Sér. I. Math., 322(2):169–172, 1996\.
  * [RV19] A. Rivera and H. Vanneuville.  Quasi-independence for nodal lines.  Ann. Henri Poincaré, 55(3):1679–1711, 2019.
  * [RV20] A. Rivera and H. Vanneuville.  The critical threshold for Bargmann-Fock percolation.  Ann. Henri Lebesgue, 3:169–215, 2020.
  * [Sev21] F. Severo.  Sharp phase transition for Gaussian percolation in all dimensions.  arXiv preprint arXiv:2105.05219, 2021.
  * [Tus77] G. Tusnády.  A remark on approximation of the sample DF in the multidimensional case.  Period. Math. Hungar., 8:53–55, 1977.
  * [Zai87a] A. Y. Zaitsev.  Estimates of the Lévy-Prokhorov distance in the multivariate central limit theorem for random variables with finite exponential moments.  Theory Probab. Appl., 31:203–220, 1987.
  * [Zai87b] A. Y. Zaitsev.  On the Gaussian approximation of convolutions under multidimensional analogues of S.N. Bernstein’s inequality conditions.  Z. Wahrsch. verw. Gebiete, 74:535–566, 1987.