# Three central limit theorems for the unbounded excursion component of a
Gaussian field

Michael McAuley1 1School of Mathematics and Statistics, Technological
University Dublin<EMAIL_ADDRESS>
###### Abstract.

For a smooth, stationary Gaussian field $f$ on Euclidean space with fast
correlation decay, there is a critical level $\ell_{c}$ such that the
excursion set $\\{f\geq\ell\\}$ contains a (unique) unbounded component if and
only if $\ell<\ell_{c}$. We prove central limit theorems for the volume,
surface area and Euler characteristic of this unbounded component restricted
to a growing box. For planar fields, the results hold at all supercritical
levels (i.e. all $\ell<\ell_{c}$). In higher dimensions the results hold at
all sufficiently low levels (all $\ell<-\ell_{c}<\ell_{c}$) but could be
extended to all supercritical levels by proving the decay of truncated
connection probabilities. Our proof is based on the martingale central limit
theorem.

###### Key words and phrases:

Gaussian fields, percolation, excursion sets, central limit theorem

###### 2010 Mathematics Subject Classification:

60G60, 60G15, 58K05

##  1\. Introduction and main results

###  1.1. Background and motivation

Percolation theory studies the long-range connectivity properties of random
processes. Aside from being an elegant mathematical theory, this topic is
motivated by a desire to understand phase transitions in different areas of
science [52], particularly statistical physics [23]. Classical percolation
models typically consist of random subsets of a lattice or a large graph. (See
[24] for a thorough background on classical percolation or [19] for a shorter
overview.)

More recently, progress has been made in understanding the percolation
properties of continuous models, in particular the excursion sets of Gaussian
fields (see [7] for a recent survey). Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a
$C^{2}$-smooth stationary Gaussian field. The excursion sets and level sets of
$f$ are defined respectively as

$\\{f\geq\ell\\}:=\left\\{x\in\mathbb{R}^{d}\;\middle|\;f(x)\geq\ell\right\\}\quad\text{and}\quad\\{f=\ell\\}:=\left\\{x\in\mathbb{R}^{d}\;\middle|\;f(x)=\ell\right\\}\quad\text{for
}\ell\in\mathbb{R}.$

From the perspective of percolation, one may then ask about the topological
behaviour of these sets on large scales and how this depends on $\ell$ or the
distribution of $f$.

Let us mention a few sources of motivation for studying percolation of
Gaussian fields:

  1. (1)

Applications
Smooth Gaussian fields are used for modelling phenomena across science (e.g.
in medical imaging [55] and quantum chaos [27]) and so understanding their
topological/geometric properties frequently has applications in such areas.
For example, the ‘component count’ of a field (defined as the number of
connected components of $\\{f\geq\ell\\}$ in some large domain) is a natural
object of study in percolation theory and has been used in statistical testing
of cosmological data [48].

  2. (2)

Stimulating mathematical theory
Gaussian fields offer new and interesting challenges in percolation theory
because they often lack some of the important properties of classical discrete
models (such as independence on disjoint domains, positive association, finite
energy etc - see [24] for definitions). Overcoming these obstacles requires
new ideas, leading to a richer mathematical theory. As an example, [39]
developed new general methods to prove the existence of a phase transition for
Gaussian fields without positive association.

  3. (3)

Exploring universality
Certain properties of percolation models on discrete lattices have been
observed numerically to be ‘universal’ in the sense that they depend on the
dimension of the model but not on the individual lattice [33, 35]. Some of
these properties also seem to be invariant under certain changes to the
dependence structure of the model [17, 12]. There is great interest in
understanding this universality; in particular, can we classify the models
which have the same behaviour? Gaussian fields provide a new family of
percolation models with which to probe these concepts, and numerical evidence
[5] suggests their behaviour may match that of some simple lattice models (at
least for one important example of a Gaussian field). It would be of great
interest to prove rigorous results in this direction for Gaussian fields.

The most fundamental result in percolation theory is the existence of a (non-
trivial) phase transition for long range connections. For a given Gaussian
field $f$, this can be stated as follows: there exists $\ell_{c}\in\mathbb{R}$
such that

(1.1) $\mathbb{P}\big{(}\\{f\geq\ell\\}\text{ has an unbounded
component}\big{)}=\begin{cases}1&\text{if }\ell<\ell_{c},\\\ 0&\text{if
}\ell>\ell_{c}.\end{cases}$

In recent years, the existence of such a phase transition for Gaussian fields
has been proven under increasingly general assumptions. We will describe
precise conditions on $f$ which ensure this existence below. For now we just
mention that if the covariance function of $f$ decays sufficiently quickly at
infinity (and some other regularity conditions hold) then the phase transition
occurs and the unbounded component of $\\{f\geq\ell\\}$ is unique almost
surely for each $\ell<\ell_{c}$ (see Figure 1 for an illustration).

(a) $\ell=0.05$

(b) $\ell=\ell_{c}=0$

(c) $\ell=-0.05$

Figure 1. The excursion sets $\\{f\geq\ell\\}$ are shown in white for the
Matérn Gaussian field $f:\mathbb{R}^{2}\to\mathbb{R}$ with parameter $\nu=10$
(see Example 1.4 for a description of this field). The largest connected
component of the excursion set is highlighted in green. The critical level
$\ell_{c}$ for this model is known to be zero.

Once this phase transition has been established, it is then a very natural
progression to consider the geometry of the unbounded component in the
supercritical regime (i.e. when $\ell<\ell_{c}$). In particular one might ask;

######  Question 1.1.

How do basic geometric statistics for the unbounded component behave?

######  Question 1.2.

How does this behaviour depend on the distribution of $f$?

Perhaps the most elementary geometric property of a subset of Euclidean space
is its size or volume and this turns out to be an interesting quantity to
study in the context of percolation theory. Let ${\\{f\geq\ell\\}_{\infty}}$
denote the unbounded component of $\\{f\geq\ell\\}$ (when it exists) and let
$\Lambda_{n}=[-n,n]^{d}$. An application of the ergodic theorem (after
imposing suitable conditions on $f$) shows that

$(2n)^{-d}\mathrm{Vol}\big{[}\\{f\geq\ell\\}_{\infty}\cap\Lambda_{n}\big{]}\to\mathbb{P}\big{(}0\in\\{f\geq\ell\\}_{\infty}\big{)}\qquad\text{as
}n\to\infty,$

where $\mathrm{Vol}[\cdot]$ denotes the $d$-dimensional volume (Hausdorff
measure) of a set and convergence occurs almost surely and in $L^{1}$. The
latter term is the _percolation probability_ for the field $f$. We therefore
see that, aside from being a natural quantity of interest in its own right,
the normalised volume of the infinite component provides a consistent
estimator for an important theoretical expression. Could one find the
asymptotic variance and distribution of this estimator?

Another very natural geometric quantity for a set is the size of its boundary
(according to some appropriate Hausdorff measure). The boundary of our
excursion sets are $C^{2}$-smooth and so we consider their $(d-1)$-dimensional
Hausdorff measure which we refer to as surface area. Since the boundary of
excursion sets (of a continuous function) are level sets, they have a clear
interpretation. This also leads to the question of whether an unbounded
component of the level set $\\{f=\ell\\}$ exists and, if so, what its
geometric properties are. We discuss this briefly in Section 1.4.

(a) $\ell=-0.01$

(b) $\ell=-0.1$

Figure 2. The excursion sets $\\{f\geq\ell\\}$ are shown in white for the
Bargmann-Fock field $f:\mathbb{R}^{2}\to\mathbb{R}$ (this model is described
in Example 1.4). The boundary of the largest connected component of the
excursion set is highlighted in orange.

One might also ask; what is the topological shape of the unbounded excursion
component? The Euler characteristic is an integer-valued quantity which
provides topological information about a set. For a ‘nice’ set
$A\subset\mathbb{R}^{2}$, the Euler characteristic is the number of components
of $A$ minus the number of ‘holes’ in $A$ (see Figure 3). Analogous
definitions hold in higher dimensions. For a thorough background on the Euler
characteristic and its study in the context of excursion sets of Gaussian
fields we recommend [1].

Figure 3. The largest component (highlighted in green) of the excursion set
$\\{f\geq\ell\\}\cap\Lambda_{n}$ surrounds three ‘holes’ and hence has an
Euler characteristic of minus two. The Euler characteristic is well defined
for the class of ‘basic complexes’ which includes excursion sets of smooth
Gaussian fields (see [1, Chapter 6] for details).

In this paper we study the previous three geometric quantities (volume,
surface area and Euler characteristic) for the unbounded excursion component
restricted to a large box. We show that if the covariance function of $f$
decays rapidly at infinity then each of these quantities satisfies a central
limit theorem (CLT) as the size of the box increases. We discuss the
significance of these results in Section 1.4, after stating them precisely.

###  1.2. Statement of main results

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a stationary Gaussian field with mean
zero, variance one and continuous covariance function
$K(x-y):=\mathbb{E}[f(x)f(y)]$. We assume that $f$ can be represented as

$f=q\ast W$

for some $q\in L^{2}(\mathbb{R}^{d})$ which is Hermitian (i.e. $q(x)=q(-x)$)
where $W$ denotes a Gaussian white noise on $\mathbb{R}^{d}$ and $\ast$
denotes convolution. We note that such a representation always exists if $K\in
L^{1}(\mathbb{R}^{d})$; by Bochner’s theorem $K$ is the Fourier transform of a
probability measure and if $K$ is integrable this measure has a density $\rho$
with respect to Lebesgue measure, one can then set
$q=\mathcal{F}[\sqrt{\rho}]$ (where
$\mathcal{F}[g](x)=\int_{\mathbb{R}^{d}}e^{it\cdot x}g(t)\;dt$ denotes the
Fourier transform) and observe that $q\ast W$ will have covariance $q\ast q=K$
as required.

Throughout our work we make the following assumptions:

######  Assumption 1.3.

  1. (1)

(Smoothness 1) $K\in C^{2k+}$ for some $k\geq 4$,

  2. (2)

(Smoothness 2) $q\in C^{3}(\mathbb{R}^{d})$ and $\partial^{\alpha}q\in
L^{2}(\mathbb{R}^{d})$ for every $|\alpha|\leq 3$,

  3. (3)

(Decay) There exist $\beta>d$ and $c\geq 1$ such that, for all $|x|\geq 1$,

$\max_{|\alpha|\leq 2}|\partial^{\alpha}q(x)|<c|x|^{-\beta},$

  4. (4)

(Symmetry) $q(x)$ is invariant under sign changes and permuting the
coordinates of $x$,

  5. (5)

(Positivity) If $d=2$ then $K=q\ast q\geq 0$, if $d\geq 3$ then $q\geq 0$.

We note that the first smoothness condition above implies that $f$ is $C^{k}$
almost surely and the decay condition implies that

$\max_{\lvert\alpha\rvert\leq 2}\lvert\partial^{\alpha}K(x)\rvert<C\lvert
x\rvert^{-\beta}$

for all $\lvert x\rvert\geq 1$. Conditions (2), (4) and (5) here will not play
any role in our arguments, but we include them as they have been used to
establish the phase transition for $f$.

We consider some examples of Gaussian fields which satisfy the above
assumption:

######  Example 1.4.

  1. (1)

The _Bargmann-Fock field_ is the centred, real-analytic Gaussian field with
covariance function

$K(x)=e^{-\frac{1}{2}\lvert x\rvert^{2}}.$

This field is of interest in algebraic geometry, where it arises as a local
scaling limit of random homogeneous polynomials of high degree (see the
introduction to [4] for further details and motivation). The percolation
properties of the Bargmann-Fock field are well studied, in part because its
super-exponential decay of correlations is helpful in extending arguments from
discrete percolation which rely on independence on disjoint domains. Using
basic properties of the Fourier transform, one can verify that the Bargmann-
Fock field has a white noise decomposition with $q(x):=(2/\pi)^{d/4}e^{-\lvert
x\rvert^{2}}$ which evidently satisfies Assumption 1.3 for any $\beta>d$.

  2. (2)

The class of _Matérn fields_ are defined by the covariance functions

$K(x)=\frac{1}{2^{\nu-1}\Gamma(\nu)}\lvert x\rvert^{\nu}K_{\nu}(\lvert
x\rvert)$

where $\Gamma$ denotes the Gamma function, $K_{\nu}$ is the modified Bessel
function of the second kind and $\nu>0$ is a parameter. These fields are
widely used in machine learning [49, Chapter 4]. The spectral density of such
a field is given by

$\rho(t)\propto(1+\lvert t\rvert^{2})^{-\nu-d/2}.$

Hence the covariance function is $k$ times differentiable if and only if
$\nu>k$, so that the parameter $\nu$ governs the smoothness of the field.
Since $q=\mathcal{F}[\sqrt{\rho}]$ it follows that

$q(x)\propto\lvert x\rvert^{\nu^{\prime}}K_{\nu^{\prime}}(\lvert x\rvert)$

where $\nu^{\prime}:=\frac{\nu}{2}-\frac{d}{4}$. Using properties of the
modified Bessel functions, one can then verify that all the conditions of
Assumption 1.3 are satisfied for $\nu>6+d/2$ and any $\beta>d$.

  3. (3)

As an example of a field satisfying Assumption 1.3 for a fixed value of
$\beta$, one can set

$q(x)=(1+\lvert x\rvert^{2})^{-\beta/2}.$

These fields have been less used in applications, but are of interest
theoretically for understanding how the rate of decay of correlations (i.e.
the value of $\beta$) affects the geometric properties of the field.

Assumption 1.3 ensures that the field undergoes a sharp version of the phase
transition described in (1.1), which we now describe. Recall that
$\Lambda_{n}=[-n,n]^{d}$ is the centred cube of side length $2n$ and for three
sets $A,B,C\subseteq\mathbb{R}^{d}$ let us write
$A\overset{B}{\longleftrightarrow}C$ for the event that there exists a
continuous path in $B$ which starts in $A$ and ends in $C$.

######  Theorem 1.5.

Let $f$ satisfy Assumption 1.3, then there exists $\ell_{c}\in\mathbb{R}$ such
that the following holds:

  * •

for each $\ell>\ell_{c}$ there exists $C,c>0$ such that for all $n>2$

$\mathbb{P}\Big{(}\Lambda_{1}\overset{\\{f\geq\ell\\}}{\longleftrightarrow}\partial\Lambda_{n}\Big{)}<Ce^{-cn},$

  * •

for each $\ell<\ell_{c}$ there exists $C,c>0$ such that for all $n>2$

$\mathbb{P}\Big{(}\Lambda_{n}\cap\\{f\geq\ell\\}_{\infty}=\emptyset\Big{)}<Ce^{-cn^{d-1}}$

where $\\{f\geq\ell\\}_{\infty}$ denotes the union of all unbounded components
of $\\{f\geq\ell\\}$.

Moreover for $\ell<\ell_{c}$, with probability one the excursion set
$\\{f\geq\ell\\}$ has precisely one unbounded component (i.e.
$\\{f\geq\ell\\}_{\infty}$ is connected).

###### Proof.

These claims were established for $d=2$ in [42] and [50] (the former used
slightly stronger assumptions on $q$, the latter relaxed these). The results
were extended to all higher values of $d$ in [53] and [54]. ∎

We remark that in the planar case (i.e. when $d=2$) it was proven that
$\ell_{c}=0$. In higher dimensions ($d\geq 3$) it is expected that
$\ell_{c}>0$ and this has been proven [20] when Assumption 1.3 is satisfied
for a sufficiently large $\beta$ .

One more key ingredient is required for our results.

######  Assumption 1.6 (Truncated connection decay).

For a given $\ell<\ell_{c}$, the probability that $\Lambda_{1}$ intersects a
bounded component of $\\{f\geq\ell\\}$ of diameter greater than $n$ decays
super-polynomially in $n$.

Letting
$\\{f\geq\ell\\}_{<\infty}:=\\{f\geq\ell\\}\setminus\\{f\geq\ell\\}_{\infty}$
be the union of all bounded excursion components, Assumption 1.6 is equivalent
to saying that for any $k\in\mathbb{N}$

$n^{k}\cdot\mathbb{P}\Big{(}\Lambda_{1}\overset{\\{f\geq\ell\\}_{<\infty}}{\longleftrightarrow}\partial\Lambda_{n}\Big{)}\to
0\qquad\text{as }n\to\infty.$

Truncated connection decay is closely related to a stronger local form of
uniqueness for the unbounded component which is conjectured to hold for all
supercritical levels. We discuss this more in Section 1.4. When
$\ell<-\ell_{c}$, truncated connection decay is an easy consequence of the
subcritical connection decay of Theorem 1.5 and symmetry of the Gaussian
distribution.

######  Proposition 1.7.

Let $f$ satisfy Assumption 1.3, then Assumption 1.6 holds for all
$\ell<-\ell_{c}$. In particular if $d=2$ then Assumption 1.6 holds for the
entire supercritical regime $\ell<\ell_{c}=0$.

This result essentially follows because any bounded component of
$\\{f\geq\ell\\}$ must be ‘surrounded’ by a component of $\\{f\leq\ell\\}$. If
the former has large diameter then so must the latter, but for
$\ell<-\ell_{c}$ this probability decays exponentially in the diameter by
Theorem 1.5 (and the fact that $\\{f\leq\ell\\}$ has the same distribution as
$\\{f\geq-\ell\\}$). The complete argument is given in Section 3.

We can now state our main results. Let $f$ satisfy Assumption 1.3 for some
given $\ell<\ell_{c}$. (This choice of $f$ and $\ell$ is fixed throughout most
of the paper.) For any compact Borel set $A$, we define our three quantities
of interest as

$\displaystyle\mu_{\mathrm{Vol}}(A)$
$\displaystyle:=\mathrm{Vol}[A\cap\\{f\geq\ell\\}_{\infty}]$
$\displaystyle\mu_{\mathrm{EC}}(A)$
$\displaystyle:=\mathrm{EC}[A\cap\\{f\geq\ell\\}_{\infty}]$
$\displaystyle\mu_{\mathrm{SA}}(A)$
$\displaystyle:=\mathcal{H}^{d-1}[A\cap\partial(\\{f\geq\ell\\}_{\infty})]$

where $\mathrm{Vol}[\cdot]$ denotes the volume, $\mathrm{EC}[\cdot]$ denotes
the Euler characteristic and $\mathcal{H}^{k}(\cdot)$ denotes the
$k$-dimensional Hausdorff measure. We begin by describing the first order
behaviour of these functionals on large scales:

######  Proposition 1.8.

Let $f$ satisfy Assumption 1.3 and $\ell<\ell_{c}$. Given
$\star\in\\{\mathrm{Vol},\mathrm{EC},\mathrm{SA}\\}$

$\frac{\mu_{\star}(\Lambda_{n})}{(2n)^{d}}\to\mathbb{E}[\mu_{\star}([0,1]^{d})]=:c_{\star}(\ell)\qquad\text{as
}n\to\infty,$

where convergence occurs almost surely and in $L^{1}$. Furthermore

$c_{\mathrm{Vol}}(\ell)=\mathbb{P}(0\in\\{f\geq\ell\\}_{\infty})\in(0,1),\qquad
c_{\mathrm{SA}}(\ell)>0,$

and as $\ell\to-\infty$,

$c_{\mathrm{Vol}}(\ell)\to 1\qquad\text{and}\qquad c_{\mathrm{SA}}(\ell)\to
0.$

Next we describe the variance and limiting distribution of our functionals.
Given $k\geq 4$, we define

$\beta_{\mathrm{Vol}}(k)=3d,\qquad\beta_{\mathrm{EC}}(k)=\frac{k-1}{k-3}3d,\qquad\beta_{\mathrm{SA}}(k)=\frac{18k(k+1)-84}{k(k+1)-8}d.$

######  Theorem 1.9.

Let $\star\in\\{\mathrm{Vol},\mathrm{EC},\mathrm{SA}\\}$ and suppose
Assumption 1.3 holds for some $k\geq 4$ and $\beta>\beta_{\star}(k)$. Suppose
also that Assumption 1.6 holds, then there exists
$\sigma=\sigma_{\star}(\ell)\geq 0$ such that as $n\to\infty$

$\frac{\mathrm{Var}[\mu_{\star}(\Lambda_{n})]}{(2n)^{d}}\to\sigma^{2}\qquad\text{and}\qquad\frac{\mu_{\star}(\Lambda_{n})-\mathbb{E}[\mu_{\star}(\Lambda_{n})]}{(2n)^{d/2}}\xrightarrow{d}\sigma
Z$

where $Z$ is a standard normal random variable.

A semi-explicit expression for $\sigma_{\star}(\ell)$ is given in Theorem 2.2.
By imposing slightly stronger conditions, we can ensure that the limiting
variance in Theorem 1.9 is strictly positive and hence the CLT is non-
degenerate:

######  Theorem 1.10.

Let $\star\in\\{\mathrm{Vol},\mathrm{SA}\\}$ and let $\ell<-\ell_{c}$. Suppose
the conditions for Theorem 1.9 are satisfied and in addition that $q\geq 0$,
then

$\sigma_{\star}(\ell)>0.$

The assumption that $\ell<-\ell_{c}$ is of course vacuous for planar fields
since $\ell_{c}=0$. We discuss how this condition could be weakened and
explain the omission of the case $\star=\mathrm{EC}$ at the beginning of
Section 5.

Whilst the geometric quantities $\mu_{\star}$ for
$\star\in\\{\mathrm{Vol},\mathrm{SA},\mathrm{EC}\\}$ are very natural from a
theoretical perspective, they are less useful in applications as they cannot
be observed from a bounded domain. That is, if we look at a realisation of
$\\{f\geq\ell\\}$ on a large domain $\Lambda_{n}$ for some $\ell<\ell_{c}$, we
expect the set to be dominated by one large component which is part of
$\\{f\geq\ell\\}_{\infty}$. However we cannot tell from within $\Lambda_{n}$
which parts of the excursion set that intersect the boundary are contained in
the unbounded component. Thus it would be useful to have a ‘finitary’ version
of our CLT. Fortunately such a statement follows without much difficulty from
our main result.

$\Lambda_{(1+\epsilon)n}$$\Lambda_{n}$$\\{f\geq\ell\\}$ Figure 4. The dashed
area shows the excursion set $\\{f\geq\ell\\}$ restricted to
$\Lambda_{(1+\epsilon)n}$. The dark grey area corresponds to
$\\{f\geq\ell\\}_{n,\epsilon}$.

Let $\epsilon>0$ be fixed and for $n\in\mathbb{N}$ let
$\\{f\geq\ell\\}_{n,\epsilon}$ be the union of all components of
$\\{f\geq\ell\\}\cap\Lambda_{n}$ which are connected to
$\partial\Lambda_{(1+\epsilon)n}$ in $\\{f\geq\ell\\}$ (see Figure 4). We then
define

$\displaystyle\overline{\mu}_{\mathrm{Vol}}(\Lambda_{n},\epsilon)$
$\displaystyle:=\mathrm{Vol}[\\{f\geq\ell\\}_{n,\epsilon}]$
$\displaystyle\overline{\mu}_{\mathrm{EC}}(\Lambda_{n},\epsilon)$
$\displaystyle:=\mathrm{EC}[\\{f\geq\ell\\}_{n,\epsilon}]$
$\displaystyle\overline{\mu}_{\mathrm{SA}}(\Lambda_{n},\epsilon)$
$\displaystyle:=\mathcal{H}^{d-1}[\\{f\geq\ell\\}_{n,\epsilon}\cap\\{f=\ell\\}].$

As a consequence of Assumption 1.6, components of
$\\{f\geq\ell\\}\cap\Lambda_{n}$ have very small probability of connecting to
$\partial\Lambda_{(1+\epsilon)n}$ unless they are part of the unbounded
component. One can therefore show that $\\{f\geq\ell\\}_{n,\epsilon}$ and
$\\{f\geq\ell\\}_{\infty}\cap\Lambda_{n}$ have essentially the same geometric
statistics, meaning that the former also satisfies a CLT:

######  Corollary 1.11.

Let $\epsilon>0$ be fixed, then the statement of Theorem 1.9 holds verbatim if
$\mu_{\star}(\Lambda_{n})$ is replaced by
$\overline{\mu}_{\star}(\Lambda_{n},\epsilon)$. Moreover the limiting variance
$\sigma^{2}_{\star}(\ell)$ takes the same value after this replacement.

###  1.3. Outline of the proof

We now describe our method of proof for Theorems 1.9 and 1.10:

White noise CLT: Let $W$ be a Gaussian white noise on $\mathbb{R}^{d}$ and for
$v\in\mathbb{Z}^{d}$ let $W_{v}$ be the restriction of $W$ to the cube
$B_{v}:=v+[0,1)^{d}$. For each $n\in\mathbb{N}$ and $v\in\mathbb{Z}^{d}$ let
$\mu(v+\Lambda_{n}):=F(v+\Lambda_{n},W)$ where $F$ is some deterministic
functional. We call $\mu$ a _stationary white-noise functional_ if it is
invariant under a common translation of both arguments; in other words, for
all $u,v\in\mathbb{Z}^{d}$ and $n\in\mathbb{N}$;

$F(u+v+\Lambda_{n},W(\cdot-u))=F(v+\Lambda_{n},W),$

where $W(\cdot-u)$ denotes the functional defined by $A\mapsto W(A-u)$ for any
Borel set $A$. We note that $\mu_{\star}$ satisfies this condition for each
$\star\in\\{\mathrm{Vol},\mathrm{SA},\mathrm{EC}\\}$ (courtesy of the white
noise representation for $f$).

Our first step is to apply (a slight generalisation of) the classical
martingale-array CLT to the stationary white-noise functional $\mu$. Let
$\preceq$ denote the (weak) lexicographic ordering on $\mathbb{Z}^{d}$. We
define a family of $\sigma$-algebras $(\mathcal{F}_{v})_{v\in\mathbb{Z}^{d}}$
by $\mathcal{F}_{v}:=\sigma(W_{u}\;|\;u\preceq v)$ and a collection of random
variables

$S_{n,v}:=\frac{\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}_{v}]-\mathbb{E}[\mu(\Lambda_{n})]}{(2n)^{d/2}}\qquad
n\in\mathbb{N},v\in\mathbb{Z}^{d}.$

This collection forms a martingale array with respect to the lexicographic
ordering (we define this notion precisely in Section 2). Moreover if we take
each coordinate of $v$ to infinity then one can show that $S_{n,v}$ converges
to
$S_{n,\infty^{\ast}}:=(2n)^{-d/2}(\mu(\Lambda_{n})-\mathbb{E}[\mu(\Lambda_{n})])$.
For $v\in\mathbb{Z}^{d}$ let $v^{-}$ denote the element of $\mathbb{Z}^{d}$
immediately preceding $v$ in the lexicographic ordering. We write
$U_{n,v}:=S_{n,v}-S_{n,v^{-}}$ for the increment of our lexicographic
martingale array. The martingale CLT states that $S_{n,\infty}$ is
asymptotically Gaussian, provided that the increments $U_{n,v}$ satisfy
certain probabilistic bounds. We show that these follow from corresponding
bounds on the change in $\mu$ when locally resampling the white noise, which
we now describe.

For $v\in\mathbb{Z}^{d}$ let $\widetilde{W}^{(v)}$ denote $W$ after resampling
$W_{v}$ independently. That is, let $W^{\prime}$ be an independent copy of $W$
and for any Borel set $A$ we define

$\widetilde{W}^{(v)}(A)=W(A\setminus B_{v})+W^{\prime}(A\cap B_{v}).$

We then define

$\Delta_{v}(u+\Lambda_{n})=F(u+\Lambda_{n},W)-F(u+\Lambda_{n},\widetilde{W}^{(v)})$

to be the change in our functional when resampling the white noise on the cube
$B_{v}$. Then by definition of $\mathcal{F}_{v}$

$U_{n,v}=(2n)^{-d/2}\mathbb{E}[\Delta_{v}(\Lambda_{n})\;|\;\mathcal{F}_{v}]\quad\text{a.s.},$

which allows us to related the conditions of the martingale CLT to
$\Delta_{v}(\Lambda_{n})$. By assuming also that $\mu$ is additive over
distinct unit cubes, we finally show that a CLT holds for $S_{n,\infty}$
provided that for all $v,w\in\mathbb{Z}^{d}$

(1.2)
$\mathbb{E}\left[\lvert\Delta_{v}(B_{w})\rvert+\lvert\Delta_{v}(B_{w})\rvert^{2+\epsilon}\right]\leq
C(1+\lvert v-w\rvert)^{-3d-\delta}$

where $\epsilon,\delta>0$ are independent of $v$ and $w$.

Application to unbounded component: Theorem 1.9 is proven by verifying
condition (1.2) for $\mu_{\star}$ when
$\star\in\\{\mathrm{Vol},\mathrm{SA},\mathrm{EC}\\}$. In this setting,
resampling the white noise on $B_{v}$ is equivalent to replacing $f$ by
$\widetilde{f}_{v}:=q\ast\widetilde{W}^{(v)}$. We can control the difference
between the excursion sets of these two functions using a basic Morse
theoretic argument: consider the family of excursion sets
$\\{(1-t)f+t\widetilde{f}_{v}\geq\ell\\}$ for $t\in[0,1]$, as $t$ increases,
the level/excursion sets will deform continuously unless they pass through a
critical point.

We now focus on the case $\star=\mathrm{Vol}$ and ask, how can the volume of
the unbounded component contained in some fixed cube $B_{w}$ change as we vary
$t$? If the level sets $\\{(1-t)f+t\widetilde{f}_{v}=\ell\\}$ pass through any
critical points inside $B_{w}$ (including ‘boundary critical points’ which
will be defined later), then their topology may change (see case (i) in Figure
5). We call such a cube _locally unstable_. In this case we do not know which
components of $\\{f\geq\ell\\}\cap B_{w}$ and
$\\{\widetilde{f}_{v}\geq\ell\\}\cap B_{w}$ are contained in the corresponding
unbounded components. We say that there is a _local topological contribution_
to the change in volume of the unbounded component and we bound this
contribution trivially:

(1.3) $\lvert\Delta_{v}(B_{w})\rvert\leq 1.$

If the level sets do not pass through any critical points, then their topology
inside $B_{w}$ is preserved and in this case the volume of the unbounded
component (restricted to $B_{w}$) can change due to excursion components
changing in volume (see case (ii) in Figure 5). We describe this as a
_geometric contribution_ to the change in volume. It is not difficult to
control this contribution in terms of the regions where $f-\ell$ is small
(relative to $f-\widetilde{f}_{v}$).

However there is one more way in which the volume of the unbounded component
may change. Even if the topology of level sets within $B_{w}$ is unchanged, it
is possible that topological changes elsewhere will disconnect some of the
excursion components in $B_{w}$ from the unbounded component (see case (iii)
in Figure 5). We describe this as a _non-local topological contribution_. In
this case one of the components of $\\{f\geq\ell\\}\cap B_{w}$ must be
contained in $\\{f\geq\ell\\}_{\infty}$ and the corresponding component of
$\\{\widetilde{f}_{v}\geq\ell\\}\cap B_{w}$ must not be contained in
$\\{\widetilde{f}_{v}\geq\ell\\}_{\infty}$ (or vice versa). This implies that
$B_{w}$ must be connected by a bounded component of $\\{f\geq\ell\\}$ (or
$\\{\widetilde{f}_{v}\geq\ell\\}$) to some locally unstable cube $B_{u}$. We
note that $B_{u}$ can, in principle, be arbitrarily far from $B_{w}$. Once
again, we bound such a contribution trivially by one.

$B_{w}$$\infty$$\\{f\geq\ell\\}_{\infty}$$\\{f\geq\ell\\}_{<\infty}$(i)$B_{w}$$\infty$$\\{\widetilde{f}_{v}\geq\ell\\}_{\infty}$$\\{\widetilde{f}_{v}\geq\ell\\}_{<\infty}$$B_{w}$(ii)$\infty$$\\{\widetilde{f}_{v}\geq\ell\\}_{\infty}$$\\{\widetilde{f}_{v}\geq\ell\\}_{<\infty}$$B_{w}$(iii)$\infty$$\\{\widetilde{f}_{v}\geq\ell\\}_{\infty}$
Figure 5. Three ways in which the volume of the unbounded component restricted
to a cube can change: in case (i) the topology within the cube changes and we
bound $\lvert\Delta_{v}(B_{w})\rvert$ by $1$. In case (ii) there is no
topological change in $B_{w}$ or in any cube which is connected to $B_{w}$ by
a finite excursion component so we bound $\lvert\Delta_{v}(B_{w})\rvert$ by
the volume of the solid grey region in the figure. In case (iii) there is a
topological change outside $B_{w}$ affecting which components inside $B_{w}$
are part of the unbounded component, so we bound
$\lvert\Delta_{v}(B_{w})\rvert$ by $1$.

To prove condition (1.2) for $\mu_{\mathrm{Vol}}$, we simply sum each of the
three contributions above. This requires us to have quantitative decay bounds
on (i) the probability of $B_{w}$ being locally unstable when we resample
$W_{v}$ as $\lvert w-v\rvert\to\infty$ and (ii) the probability of $B_{w}$
being connected to $B_{u}$ by a bounded component of $\\{f\geq\ell\\}$ as
$\lvert u-w\rvert\to\infty$. The latter is precisely Assumption 1.6. The
former follows from the fact that

(1.4)
$p_{v}(x):=f(x)-\widetilde{f}_{v}(x)=\int_{B_{v}}q(x-u)\;d(W-\widetilde{W}^{(v)})(u)$

and its derivatives are small when $x\in B_{w}$, which in turn follows from
our decay assumptions on $q$.

We prove condition (1.2) for $\star=\mathrm{SA}$ and $\star=\mathrm{EC}$ in a
very similar way by decomposing the change into local/non-local topological
and geometric contributions. In place of the trivial upper bound (1.3) we use
the total surface area for $f$ and $\widetilde{f}_{v}$ in $B_{w}$ or their
total number of critical points when $\star=\mathrm{SA},\mathrm{EC}$
respectively. This is the underlying reason why Theorem 1.9 requires stronger
covariance decay (i.e. larger values of $\beta$) when
$\star=\mathrm{SA},\mathrm{EC}$ as compared to $\star=\mathrm{Vol}$: our
argument interpolates between moment bounds for these upper bounds and decay
of the probability that a cube is unstable. The finiteness of moments of
surface area and critical points follows from smoothness of the underlying
field (i.e. larger $k$ in Assumption 1.3) whereas the trivial constant bound
on the volume has finite moments of all order without any smoothness
assumptions.

Positivity of variance: The proof of Theorem 1.9 gives the following semi-
explicit expression for the limiting variance:

(1.5)
$\sigma_{\star}^{2}(\ell)=\mathbb{E}\Big{[}\mathbb{E}\big{[}\lim_{n\to\infty}\Delta_{0}(\Lambda_{n})\;|\;\mathcal{F}_{0}\big{]}^{2}\Big{]}=\mathbb{E}\Big{[}\mathbb{E}\big{[}\lim_{n\to\infty}\mu_{\star}(\Lambda_{n},f)-\mu_{\star}(\Lambda_{n},\widetilde{f}_{0})\;|\;\mathcal{F}_{0}\big{]}^{2}\Big{]}.$

Applying the result to a rescaled version of the field $f$, we show that (1.5)
still holds if we instead resample the white noise on $\Lambda_{m}$, for some
large but fixed $m$, rather than $B_{0}$. We then find a lower bound for
$\sigma^{2}$ by resampling only the _average value_ of the white noise on
$\Lambda_{m}$ (i.e. the value of
$W(\Lambda_{m})=\int\mathds{1}_{\Lambda_{m}}dW(u)$). Since $f=q\ast W$, this
resampling is equivalent to adding $Z_{0}q\ast\mathds{1}_{\Lambda_{m}}$ to $f$
where $Z_{0}$ is a standard Gaussian variable. When $\int q(x)\;dx\neq 0$ and
$m$ is large, the latter is roughly equivalent to perturbing $f$ by a constant
on $\Lambda_{m}$ and by some decaying term on
$\mathbb{R}^{d}\setminus\Lambda_{m}$.

Assuming that there is some $\ell^{\prime}$ for which
$c_{\star}(\ell^{\prime})\neq c_{\star}(\ell)$, where $c_{\star}$ is defined
in Proposition 1.8, the perturbation on $\Lambda_{m}$ can cause a change in
$\mu_{\star}(\Lambda_{n})$ (where $n>>m$) of order $m^{d}$. This assumption is
verified in Proposition 1.8 for $\star=\mathrm{Vol}$ and $\star=\mathrm{SA}$
but not for $\star=\mathrm{EC}$, explaining why the latter case is omitted
from Theorem 1.10. Finally we show that the effect on
$\mu_{\star}(\Lambda_{n})$ of the perturbation outside $\Lambda_{m}$ is of
order $o(m^{d})$, so that by taking $m$ sufficiently large the overall effect
of the perturbation is non-zero, yielding a positive variance. The latter
argument relies on having a statement about truncated connection decay for a
perturbed version of $f$. Such a statement follows from subcritical decay
(i.e. the first part of Theorem 1.5) when $\ell<-\ell_{c}$, which explains why
we impose this assumption in Theorem 1.10. However we note that the statement
could alternatively be proven using a ‘sprinkled’ version of truncated arm
decay which would conjecturally hold for all $\ell<\ell_{c}$.

###  1.4. Discussion

We now discuss some possible extensions of our work and related open
questions.

Related work: Our proof of Theorem 1.9 builds upon a versatile CLT for
stationary functionals of spatial white noise due to Penrose [47]. This result
unified the approach of several other works which applied the classical
martingale CLT to spatial probabilistic processes [30, 31, 34, 56]. In
particular we note that Zhang [56] proved a CLT for the size of the unbounded
component of Bernoulli percolation on $\mathbb{Z}^{d}$ (a discrete analogue of
our volume CLT) and Penrose [47] proved a corresponding result for the largest
component in a finite box.

Additional complications arise in our setting: the functionals $\mu_{\star}$
have infinite range of dependence on the underlying white noise process and
controlling the behaviour of the unbounded component in this continuous
setting presents some technical difficulties. Moreover, verifying positivity
of the limiting variance requires new techniques. A central limit theorem for
the number of connected components of the excursion/level set of a Gaussian
field was proven in [9] using methods closely related to those in this work.
Although controlling changes in the latter functional under perturbation is
simpler as there are no geometric or non-local topological contributions.

Truncated arm decay: Whilst our results are valid for all supercritical levels
in the planar case, the assumption of truncated connection decay has not been
verified (for all levels) in higher dimensions. Based on the behaviour of
Bernoulli percolation [18] and similarities between this model and fast-decay
Gaussian fields (see [7] and references therein) it is natural to conjecture
that truncated connection probabilities should decay exponentially at any
supercritical level for fields satisfying Assumption 1.3. That is, we would
expect for each $\ell<\ell_{c}$ there exists $C,c>0$ such that

$\mathbb{P}\Big{(}\Lambda_{1}\overset{\\{f\geq\ell\\}_{<\infty}}{\longleftrightarrow}\partial\Lambda_{n}\Big{)}<Ce^{-cn}\qquad\text{for
all }n\in\mathbb{N}.$

See [53, Section 1.3] for further details and references. A proof of the
latter statement, which seems to require new ideas when compared to the case
of Bernoulli percolation, would immediately extend our CLTs to all
supercritical levels in $d\geq 3$.

Martingale methods for non-local functionals: A geometric functional of a
field is said to be _local_ if its value on a domain is determined by summing
the value of the functional over a fine partition of the domain. For example,
the volume of the excursion set of a field is local but the volume of the
unbounded component of the excursion set is _non-local_ since one needs to
know which parts of the excursion set belong to this component to compute the
volume. The statistics of many local functionals are well understood: under
conditions roughly equivalent to Assumption 1.3, CLTs have been proven for
local functionals including the Lipschitz-Killing curvatures of excursion sets
[21, 43, 32] and the number of critical points [45]. These results are proven
using the Wiener chaos expansion (see [28, Chapter 2] for background). The
Wiener chaos expansion of non-local functionals is generally intractable, and
so these quantities have proven more challenging to analyse.

One of the key takeaways we would like to impress upon the reader, is the
potential value of martingale methods in studying non-local functionals. Our
results show that such methods offer significant insight into
$\mu_{\mathrm{Vol}}$, $\mu_{\mathrm{EC}}$ and $\mu_{\mathrm{SA}}$ (all of
which are non-local). Moreover the same general approach was applied to the
component count (which is also non-local) in [9]. Minor extensions of the
methods in this work would likely be sufficient to prove CLTs for the
following quantities (restricted to $\Lambda_{n}$ as $n\to\infty$):

  * •

the number of critical points of $f$ in $\\{f\geq\ell\\}_{\infty}$,

  * •

the volume of $\\{f\geq\ell\\}_{\infty}$ restricted to some fixed lower
dimensional hyperplane,

  * •

the volume of $\\{f_{1}\geq\ell\\}_{\infty}\cap\\{f_{2}\geq\ell\\}_{\infty}$
for two independent fields $f_{1}$ and $f_{2}$,

  * •

the (higher order) Betti numbers of $\\{f\geq\ell\\}$.

One could also study ‘supercritical geometry’ of Gaussian fields via other
unbounded sets such as:

  * •

the unbounded component of $\\{f=\ell\\}$, or

  * •

the unbounded component of $\\{\ell_{1}\leq f\leq\ell_{2}\\}$.

These components have been proven to exist for certain fields and values of
$\ell$ [20, 39] although more work is needed for a full characterisation.

The use of martingale methods in our work relies on the existence of a
convolution-white noise representation (i.e. $f=q\ast W$ for suitable $q$).
This imposes certain restrictions on $f$, including the fact that it must have
integrable covariance function. It is an open question as to whether there are
alternative representations of more general Gaussian fields which would also
be amenable to martingale arguments.

Geometry and dependence structure: Recall that the motivation for our work
(Questions 1.1 and 1.2) was to understand the geometry of the unbounded
excursion component and its dependence on the distribution of $f$. Our results
do not provide any evidence for major differences in supercritical geometric
behaviour amongst fields, but known results for other functionals suggest that
we should expect this. Let us consider three broad classes of dependence
structure and their resulting geometric properties:

  1. (1)

Short-range correlations
We call a (smooth, stationary) Gaussian field _short-range correlated_ if
$K\in L^{1}(\mathbb{R}^{d})$ (where $K$ is the covariance function). Morally
speaking, this is roughly equivalent to Assumption 1.3 for some $\beta>d$. The
quintessential example is the Bargmann-Fock field although we also mention the
family of Cauchy fields with covariance function $K_{\beta}(x):=(1+\lvert
x\rvert^{2})^{-\beta/2}$ for $\beta>d$.

A variety of geometric functionals restricted to growing boxes $\Lambda_{n}$
are known to have similar behaviour for short-range correlated fields; namely,
the mean and variance of these quantities scale like the volume $n^{d}$ and an
asymptotic CLT holds with Gaussian limit. Specifically under Assumption 1.3,
this behaviour has been verified for the Lipschitz-Killing curvatures [21, 43,
32], the number of critical points [45] and the component count [9] (although
the last result was proven only in the case that $\beta>3d$).

  2. (2)

Regularly varying long-range correlations
We say that a field $f$ is _regularly varying with index $\beta\in(0,d)$ and
remainder $L$_ if $K(x)=\lvert x\rvert^{-\beta}L(\lvert x\rvert)$ where $L$ is
slowly varying at infinity. (The reader unfamiliar with this terminology can
consult [11] for a definition or just think of $K$ as decaying like $\lvert
x\rvert^{-\beta}$ at infinity.). Cauchy fields with covariance kernel
$K_{\beta}$ for $\beta\in(0,d)$ fall within this class.

Geometric properties of such fields are less well characterised compared to
the short-range correlated case but may be expected to exhibit super-volume
order variance scaling. That is, the variance of the geometric functionals
considered above (Lipschitz-Killing curvatures, component count etc)
restricted to the box $\Lambda_{n}$ could have variance of order
$n^{2d-\beta}L(n)$ for some slowly varying function $L$. It has been proven
that the variance of the component count has an upper bound of this order [6]
and the corresponding lower bound has been proven in the planar case $d=2$ for
certain levels [8]. No CLTs have been proven for such functionals.

  3. (3)

Oscillating long-range correlations
We say that a field has _oscillating long-range correlations_ if $K$ is not
integrable and can take negative values. The most important representative of
this class is the _monochromatic random wave_ : the field with spectral
measure supported uniformly on $\mathbb{S}^{d-1}$.

Fascinating geometric results have been proven for monochromatic random waves,
mainly in the two-dimensional case. The broad picture which emerges for a
number of functionals related to excursion sets is the following: at most
levels $\ell$ the functionals of $\\{f\geq\ell\\}$ have super-volume order
scaling for the variance and a CLT is known. However for a small number of
_anomalous levels_ the functionals have variance of lower order (typically
proportional to the volume, up to some logarithmic factors) and CLTs are
sometimes known in these cases. This pattern of behaviour has been observed
for the area [36, 38], boundary length [46, 51], Euler characteristic [13, 14]
and number of critical points [16, 15] of two-dimensional monochromatic random
waves. (To be more precise, most of these results hold for families of
Gaussian fields on the sphere which converge locally to the monochromatic
random wave but we expect the behaviour to be similar.)

The phenomenon of anomalous levels was first observed in [10] for the length
of the nodal set (i.e. the zero level set) and is known, at least in this
case, as _Berry cancellation_. The previous results have been proven using
either the Kac-Rice formula (see [3, Chapter 6] for background) or the Wiener
chaos expansion. Both of these methods rely on locality of the functionals.

So should we expect the previous relationship between geometric functionals
and dependence structure to manifest for the unbounded excursion component?

Our results verify analogous behaviour for a fairly general subset of the
short-range correlated fields, so we expect this holds for the entire short-
range correlated class. For a large class of fields with regularly varying
long-range correlations, a sharp phase transition is known to occur [41, 40]
whilst for the planar monochromatic random wave (also known as the random
plane wave) a phase transition at $\ell=0$ has been proven [39]. Hence one can
legitimately ask about the geometry of the unbounded excursion component, but
our results provide no insight as such fields fail to have a convolution-white
noise decomposition. We can only pose the following questions:

######  Question 1.12.

If a field is regularly varying with index $\beta\in(0,d)$, does the volume of
its unbounded excursion component restricted to $\Lambda_{n}$ have variance of
order $n^{2d-\beta}$? Are there anomalous levels for which the variance has
lower order? Does the volume satisfy a CLT?

######  Question 1.13.

For monochromatic random waves in dimension $d$, does the volume of the
unbounded excursion component restricted to $\Lambda_{n}$ have variance of
order $n^{d+1}$? Are there anomalous levels with lower order variance? Does
the volume satisfy a CLT?

Of course a prerequisite to answering the last question would be extending the
phase transition for monochromatic random waves to $d\geq 3$. The suggested
variance order $n^{d+1}$ is based on results for the volume of excursion sets
of a model related to monochromatic random waves in dimensions $d\geq 2$ [37].

Proving the existence of anomalous levels for non-local variables, such as
those related to the unbounded component, could be particularly challenging as
the Wiener chaos expansion becomes intractable. We note that upper and lower
bounds on the variance of the component count (a non-local variable) have been
proven for fields with long-range correlations [8, 6] and the former gives a
necessary condition for anomalous levels. However it seems that the methods
used in these works cannot be extended to prove the existence of anomalous
levels. Whilst our methods cannot be applied directly to answer any of the
above questions, the general approach of using an abstract martingale CLT
along with some decomposition of the field might hold promise.

###  1.5. Acknowledgements

Part of this work was completed at the Department of Mathematics and
Statistics at the University of Helsinki and was supported by the European
Research Council Advanced Grant QFPROBA (grant number 741487). I would like to
thank Stephen Muirhead for helpful conversations and suggesting Proposition
1.8 and Dmitry Beliaev for previous collaboration which inspired this work.

##  2\. A CLT for white noise functionals

In this section we prove a CLT for stationary white noise functionals which
generalises the result of Penrose [47] to the case of infinite range of
dependence. We then derive a simpler sufficient condition for this CLT to hold
when the white noise functional is (approximately) additive.

We begin by stating an abstract CLT for martingale arrays. Recall that
$\preceq$ denotes the standard lexicographic ordering on $\mathbb{Z}^{d}$. Let
$\\{\mathcal{F}_{n,v}\;|\;v\in\mathbb{Z}^{d},n\in\mathbb{N}\\}$ be a
collection of $\sigma$-algebras such that for each $n$, $v\preceq w$ implies
$\mathcal{F}_{n,v}\subseteq\mathcal{F}_{n,w}$. We say that a collection of
random variables $\\{S_{n,v}\;|\;v\in\mathbb{Z}^{d},n\in\mathbb{N}\\}$ is a
_lexicographic martingale array_ if the following holds:

  1. (1)

$S_{n,v}$ is $\mathcal{F}_{n,v}$-measurable for each $n$ and $v$,

  2. (2)

if $v\preceq w$ then $\mathbb{E}[S_{n,w}\;|\;\mathcal{F}_{n,v}]=S_{n,v}$ for
each $n$.

We say that the array is mean-zero if $\mathbb{E}[S_{n,v}]=0$ for all $n,v$
and square integrable if
$\sup_{v\in\mathbb{Z}^{d}}\mathbb{E}[S_{n,v}^{2}]<\infty$ for each
$n\in\mathbb{N}$. We say that a sequence of points in $\mathbb{Z}^{d}$
converges to $\pm\infty^{*}$ if all coordinates of the points converge to
$\pm\infty$. For a square-integrable martingale array, the $L^{p}$-convergence
theorems for forward/reverse martingales imply that the following limits exist

$S_{n,\infty^{*}}:=\lim_{v\to\infty^{*}}S_{n,v}\qquad\text{and}\qquad
S_{n,-\infty^{*}}:=\lim_{v\to-\infty^{*}}S_{n,v}$

where convergence occurs almost surely and in $L^{2}$.

Finally we wish to impose a condition on our array which ensures that the
martingale (or equivalently the family of $\sigma$-algebras
$\mathcal{F}_{n,v}$) does not make any ‘jumps’ when some coordinates tend to
infinity. (To illustrate; for $d=2$ we would like to know that
$\lim_{k\to\infty}S_{n,(0,k)}=\lim_{j\to-\infty}S_{n,(1,j)}$ almost surely, so
that the important behaviour of the martingale can be captured on bounded
domains.) We therefore say that $S_{n,v}$ is _regular at infinity_ if for all
$n\in\mathbb{N}$, $i\in\\{1,\dots,d-1\\}$ and
$(v_{1},\dots,v_{i})\in\mathbb{Z}^{i}$

$\lim_{v_{i+1},\dots,v_{d}\to\infty}S_{n,v}=\lim_{v_{i+1},\dots,v_{d}\to-\infty}S_{n,v^{\prime}}$

almost surely where $v=(v_{1},\dots,v_{d})$ and
$v^{\prime}=(v_{1},\dots,v_{i-1},v_{i}+1,v_{i+1},\dots,v_{d})$. For
$v\in\mathbb{Z}^{d}$ we let $v^{-}$ denote the element of $\mathbb{Z}^{d}$
immediately preceding $v$ in the lexicographic order and we define the
differences of the martingale array as $U_{n,v}:=S_{n,v}-S_{n,v^{-}}$. A
straightforward consequence of being regular at infinity is that

$S_{n,\infty^{*}}-S_{n,-\infty^{*}}=\sum_{v\in\mathbb{Z}^{d}}U_{n,v}$

whenever $\sum_{v\in\mathbb{Z}^{d}}\lvert U_{n,v}\rvert<\infty$.

The following result was proven in [9]. It is a straightforward generalisation
of a classical CLT for finite martingale arrays which can be found, for
example, in [25, Chapter 3].

######  Theorem 2.1.

Let $\\{S_{n,v},\mathcal{F}_{n,v}:v\in\mathbb{Z}^{d},n\in\mathbb{N}\\}$ be a
mean-zero square-integrable lexicographic martingale array which is regular at
infinity such that $S_{n,-\infty^{*}}=0$ for each $n$. Suppose that

(2.1) $\displaystyle\sup_{v\in\mathbb{Z}^{d}}\lvert
U_{n,v}\rvert\xrightarrow{p}0\quad\text{as }n\to\infty$ (2.2)
$\displaystyle\sup_{n}\mathbb{E}\Big{[}\sup_{v\in\mathbb{Z}^{d}}U_{n,v}^{2}\Big{]}<\infty$
(2.3)
$\displaystyle\sum_{v\in\mathbb{Z}^{d}}U_{n,v}^{2}\xrightarrow{L^{1}}\eta^{2}\in[0,\infty)\quad\text{as
}n\to\infty$ (2.4)
$\displaystyle\mathbb{E}\Big{[}\sum_{v\in\mathbb{Z}^{d}}\lvert
U_{n,v}\rvert\Big{]}<\infty\quad\text{for all }n\in\mathbb{N}.$

Then $\mathrm{Var}[S_{n,\infty^{*}}]\to\eta^{2}$ and
$S_{n,\infty^{*}}\xrightarrow{d}Z$ as $n\to\infty$ where
$Z\sim\mathcal{N}(0,\eta^{2})$.

We now state our generalised CLT. Recall that $W$ is a Gaussian white noise on
$\mathbb{R}^{d}$ and $W_{v}$ denotes its restriction to the unit cube $B_{v}$
for $v\in\mathbb{Z}^{d}$. For each $n\in\mathbb{N}$ and $v\in\mathbb{Z}^{d}$
we let $\mu(v+\Lambda_{n}):=F(v+\Lambda_{n},W)$ where $F$ is some
deterministic functional. We say that $\mu$ is a _stationary white-noise
functional_ if $\mu(v+\Lambda_{n})$ is unchanged when we translate $v$ and $W$
by the same vector in $\mathbb{Z}^{d}$ (i.e.
$F(u+v+\Lambda_{n},W(\cdot-u))=F(v+\Lambda_{n},W)$ for any
$u\in\mathbb{Z}^{d}$). We define the differences of $\mu$ as

$\Delta_{u}(v+\Lambda_{n})=F(v+\Lambda_{n},W)-F(v+\Lambda_{n},\widetilde{W}^{(u)})$

where we recall that $\widetilde{W}^{(u)}$ denotes the white noise $W$ with
$W_{u}$ resampled independently.

######  Theorem 2.2.

Let $\mu$ be a stationary white-noise functional satisfying the following:

  1. (1)

(Finite second moments) $\mathbb{E}[\mu(v+\Lambda_{n})^{2}]<\infty$ for all
$v\in\mathbb{Z}^{d}$ and $n\in\mathbb{N}$,

  2. (2)

(Stabilisation) there exists a random variable $\Delta_{0}$ such that for any
sequence of cubes $D_{n}:=v_{n}+\Lambda_{n}$ satisfying
$D_{n}\to\mathbb{R}^{d}$ (i.e. $\cup_{N}\cap_{n>N}D_{n}=\mathbb{R}^{d}$) we
have

$\Delta_{0}(D_{n})\to\Delta_{0}$

in probability as $n\to\infty$,

  3. (3)

(Bounded moments) There exists $\epsilon>0$ such that

$\sup_{v\in\mathbb{Z}^{d},n\in\mathbb{N}}\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\right]<\infty,$

  4. (4)

(Moment decay) There exists $c,\epsilon,\zeta>0$ and
$\gamma>\max\\{d,\zeta\\}$ such that for all
$v\in\mathbb{Z}^{d}\setminus\Lambda_{n}$

$\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert+\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\right]\leq
c\;n^{\zeta}\;\mathrm{dist}(v,\Lambda_{n})^{-\gamma}.$

Let $\mathcal{F}_{v}$ be the $\sigma$-algebra generated by
$\\{W_{u}\;|\;u\preceq v\\}$. Then as $n\to\infty$,

$\frac{\mathrm{Var}[\mu(\Lambda_{n})]}{(2n)^{d}}\to\sigma^{2}\quad\text{and}\quad\frac{\mu(\Lambda_{n})-\mathbb{E}[\mu(\Lambda_{n})]}{(2n)^{d/2}}\xrightarrow{d}\sigma
Z$

where $Z\sim\mathcal{N}(0,1)$ and
$\sigma^{2}=\mathbb{E}\big{[}\mathbb{E}[\Delta_{0}|\mathcal{F}_{0}]^{2}\big{]}$.

Our proof follows that of Penrose [47, Theorem 2.1] with some additional
arguments to deal with functionals having an infinite range of dependence.
More or less the same proof was given for one particular such functional (the
component count) in [9].

First we specify the martingale to which we will apply Theorem 2.1:

######  Lemma 2.3.

Let $\mu$ satisfy the conditions of Theorem 2.2, then

$S_{n,v}:=(2n)^{-d/2}(\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}_{v}]-\mathbb{E}[\mu(\Lambda_{n})])\quad\text{and}\quad\mathcal{F}_{n,v}:=\mathcal{F}_{v}$

define a mean-zero, square-integrable lexicographic martingale array which is
regular at infinity. Moreover $S_{n,-\infty^{*}}=0$ and
$S_{n,\infty^{*}}=(2n)^{-d/2}(\mu(\Lambda_{n})-\mathbb{E}[\mu(\Lambda_{n})])$.

###### Proof.

It is clear from the definition that $S_{n,v}$ is a mean-zero lexicographic
martingale array. Since $\mu$ has finite second moments, the array is square
integrable. By Lévy’s downward and upward convergence theorems respectively

$\displaystyle\lim_{v\to-\infty^{*}}\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}_{v}]$
$\displaystyle=\mathbb{E}[\mu(\Lambda_{n})\;|\;\cap_{v\in\mathbb{Z}^{d}}\mathcal{F}_{v}]=\mathbb{E}[\mu(\Lambda_{n})]$
$\displaystyle\lim_{v\to\infty^{*}}\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}_{v}]$
$\displaystyle=\mathbb{E}[\mu(\Lambda_{n})\;|\;\sigma(\cup_{v\in\mathbb{Z}^{d}}\mathcal{F}_{v})]=\mu(\Lambda_{n})$

where the right-most equalities hold because the tail $\sigma$-algebra
$\cap_{v\in\mathbb{Z}^{d}}\mathcal{F}_{v}$ is trivial and $W$ is measurable
with respect to $\sigma(\cup_{v\in\mathbb{Z}^{d}}\mathcal{F}_{v})$. Hence
$S_{n,-\infty^{*}}=0$ and $S_{n,\infty^{*}}=\mu(\Lambda_{n})$, as required.

It remains to show that $S_{n,v}$ is regular at infinity. We fix two sequences
$j^{(m)}$ and $k^{(m)}$ in $\mathbb{Z}^{d}$ such that

$\displaystyle j^{(m)}$
$\displaystyle=(u_{1},\dots,u_{i},a^{(m)}_{1},\dots,a^{(m)}_{d-i})$
$\displaystyle k^{(m)}$
$\displaystyle=(u_{1},\dots,u_{i}+1,b^{(m)}_{1},\dots,b^{(m)}_{d-i})$

where $a^{(m)}_{1},\dots,a^{(m)}_{d-i}\to\infty$ and
$b^{(m)}_{1},\dots,b^{(m)}_{d-i}\to-\infty$ with $m$. By Lévy’s upward and
downward theorems, as $m\to\infty$ we have

$\displaystyle\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}_{j^{(m)}}]$
$\displaystyle\to\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}^{-}]$
$\displaystyle\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}_{k^{(m)}}]$
$\displaystyle\to\mathbb{E}[\mu(\Lambda_{n})\;|\;\mathcal{F}^{+}]$

where

$\mathcal{F}^{-}=\sigma\Big{(}\bigcup_{m\in\mathbb{N}}\mathcal{F}_{j^{(m)}}\Big{)}\quad\text{and}\quad\mathcal{F}^{+}=\bigcap_{m\in\mathbb{N}}\mathcal{F}_{k^{(m)}}.$

Regularity at infinity then follows if we can show that the completions of
$\mathcal{F}^{+}$ and $\mathcal{F}^{-}$ coincide. We note that
$\mathcal{F}^{+}$ is generated by events in $\mathcal{F}^{-}$ together with
those measurable with respect to a ‘tail’ of independent variables. We can
therefore argue by generalising the proof of Kolmogorov’s zero-one law.
Specifically for $A\in\mathcal{F}^{+}$, defining

$\displaystyle\mathcal{G}_{m}:=\sigma(W_{v}\;|\;k^{(m)}\preceq v\preceq
k^{(1)})\quad\text{and}\quad\mathcal{G}_{\infty}=\sigma\Big{(}\bigcup_{m}\mathcal{G}_{m}\Big{)}$

we may apply Lévy’s upward theorem once more to see that

(2.5)
$\mathbb{E}[\mathds{1}_{A}\;|\;\sigma(\mathcal{F}^{-},\mathcal{G}_{m})]\to\mathbb{E}[\mathds{1}_{A}\;|\;\sigma(\mathcal{F}^{-},\mathcal{G}_{\infty})]=\mathds{1}_{A}$

where the final equality follows since
$\sigma(\mathcal{F}^{-},\mathcal{G}_{\infty})\supseteq\mathcal{F}_{k^{(1)}}\supseteq\mathcal{F}^{+}$.
However since $\mathcal{F}^{+}$ is independent of $\mathcal{G}_{m}$ we have

$\mathbb{E}[\mathds{1}_{A}\;|\;\sigma(\mathcal{F}^{-},\mathcal{G}_{m})]=\mathbb{E}[\mathds{1}_{A}\;|\;\mathcal{F}^{-}].$

Combined with (2.5) this implies that $A$ is measurable with respect to the
completion of $\mathcal{F}^{-}$, as required. ∎

Next we record an elementary lemma which we will use repeatedly in
calculations. Recall that $\mathrm{dist}(\cdot,\cdot)$ denotes the Euclidean
distance between two sets (or a point and a set).

######  Lemma 2.4.

Let $\gamma>d$, then there exists a constant $c>0$ depending only on $d$ and
$\gamma$ such that for $n\in\mathbb{N}$ and $r\geq 1$

$\sum_{v\in\mathbb{Z}^{d}\;:\;\mathrm{dist}(v,\Lambda_{n})>r}\mathrm{dist}(v,\Lambda_{n})^{-\gamma}\leq
cr^{-\gamma+1}\max\\{r,n\\}^{d-1}.$

###### Proof.

For $v\in\mathbb{Z}^{d}$ we write $C(v)$ for the closest point to $v$ in
$\Lambda_{n}$. The idea of the proof is to partition
$\\{v\;|\;\mathrm{dist}(v,\Lambda_{n})>r)\\}$ according to the point in
$\Lambda_{n}$ which $v$ is closest to.

For $i=0,1,\dots,d$ let $\mathrm{Face}_{i}$ denote the points
$x\in\Lambda_{n}$ such that $d+i$ of their nearest neighbours (in
$\mathbb{Z}^{d}$) are also contained in $\Lambda_{n}$. So for example
$\mathrm{Face}_{0}$ denotes the corners of the cube $\Lambda_{n}$ and
$\mathrm{Face}_{d}$ denotes the points of $\mathbb{Z}^{d}$ in the interior of
$\Lambda_{n}$. We note that the number of points in $\mathrm{Face}_{i}$ is at
most $c_{d}n^{i}$ by elementary geometric considerations.

We define $S_{x}=\\{v\;|\;\mathrm{dist}(v,\Lambda_{n})>r,C(v)=x\\}$. Observe
that $S_{x}=\emptyset$ whenever $x\in\mathrm{Face}_{d}$. Moreover when
$x\in\mathrm{Face}_{i}$ and $v\in S_{x}$, $x-v$ must be orthogonal to each of
the $i$ directions in which both neighbours of $x$ are contained in
$\Lambda_{n}$. In other words $S_{x}-x$ is contained in a subspace of
dimension $d-i$ and hence

$\sum_{v\in S_{x}}\lvert
v-x\rvert^{-\gamma}\leq\sum_{y\in\mathbb{Z}^{d-i}\;:\;\lvert y\rvert>r}\lvert
y\rvert^{-\gamma}\leq c_{\gamma}r^{-\gamma+d-i}.$

We then conclude that

$\displaystyle\sum_{v\in\mathbb{Z}^{d}\;:\;\mathrm{dist}(v,\Lambda_{n})>r}\mathrm{dist}(v,\Lambda_{n})^{-\gamma}=\sum_{i=0}^{d-1}\sum_{x\in\mathrm{Face}_{i}}\sum_{v\in
S_{x}}\lvert v-x\rvert^{-\gamma}$
$\displaystyle\leq\sum_{i=0}^{d-1}c_{d}n^{i}c_{\gamma}r^{-\gamma+d-i}$
$\displaystyle\leq cr^{-\gamma+1}\max\\{r,n\\}^{d-1}$

as required. ∎

Finally we state a version of the ergodic theorem (given in [29, Theorem
25.12]) which we will make use of in proving Theorem 2.2:

######  Theorem 2.5 (Multi-variate ergodic theorem).

Let $\xi$ be a random element in some set $S$ with distribution $\nu$. Let
$T_{1},\dots,T_{d}$ be $\nu$-preserving transformations of $S$. Assume that
the invariant $\sigma$-algebra of each $T_{i}$ is trivial and let $G\in
L^{p}(\nu)$ for some $p>1$, then as $n\to\infty$

$\frac{1}{n^{d}}\sum_{i=1}^{d}\sum_{k_{i}\leq n}G(T_{1}^{k_{1}}\dots
T_{d}^{k_{d}}\xi)\to\mathbb{E}[G(\xi)]$

where convergence occurs almost surely and in $L^{p}$.

###### Proof of Theorem 2.2.

By Lemma 2.3, the variance convergence and CLT that we wish to prove will
follow if we can verify the numbered conditions in Theorem 2.1 (along the way
we will also derive the claimed expression for the limiting variance
$\sigma^{2}$).

First we observe, since $W$ is independent on disjoint domains, that

$U_{n,v}=(2n)^{-d/2}\mathbb{E}[\Delta_{v}(\Lambda_{n})\;|\;\mathcal{F}_{v}]$

almost surely. Next we note that combining the bounded moments and moment
decay assumptions with Lemma 2.4 shows that

(2.6)
$\sum_{v\in\mathbb{Z}^{d}}\mathbb{E}\big{[}\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\big{]}\leq
cn^{d}+\sum_{v\in\mathbb{Z}^{d}\setminus\Lambda_{2n}}cn^{\zeta}\;\mathrm{dist}(v,\Lambda_{n})^{-\gamma}\leq
c^{\prime}n^{d}$

for some $\epsilon,c,c^{\prime}>0$ (since $\gamma>\max\\{d,\zeta\\}$). By
interpolating the moment decay assumption (i.e. using Littlewood’s $L^{p}$
inequality) we have

(2.7) $\mathbb{E}\big{[}\lvert\Delta_{v}(\Lambda_{n})\rvert^{2}\big{]}\leq
cn^{\zeta}\mathrm{dist}(v,\Lambda_{n})^{-\gamma}.$

Hence by Jensen’s inequality and the bounded moments condition, we see that
(2.6) also holds for $\epsilon=0$ (with a different constant $c^{\prime}$).
Applying Markov’s inequality and the conditional form of Jensen’s inequality,
for any $\delta>0$

$\displaystyle\mathbb{P}\Big{[}\sup_{v\in\mathbb{Z}^{d}}\lvert
U_{n,v}\rvert>\delta\Big{]}\;\delta^{2+\epsilon}\leq\sum_{v\in\mathbb{Z}^{d}}\mathbb{E}\big{[}\lvert
U_{n,v}\rvert^{2+\epsilon}\big{]}\leq(2n)^{-d\frac{2+\epsilon}{2}}\sum_{v\in\mathbb{Z}^{d}}\mathbb{E}\big{[}\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\big{]}\leq
cn^{-d\frac{\epsilon}{2}}$

which tends to zero, verifying (2.1). By identical reasoning in the case
$\epsilon=0$, we have

$\displaystyle\mathbb{E}\Big{[}\sup_{v\in\mathbb{Z}^{d}}U_{n,v}^{2}\Big{]}\leq\mathbb{E}\Big{[}\sum_{v\in\mathbb{Z}^{d}}U_{n,v}^{2}\Big{]}\leq(2n)^{-d}\mathbb{E}\Big{[}\sum_{v\in\mathbb{Z}^{d}}\lvert\Delta_{v}(\Lambda_{n})\rvert^{2}\Big{]}\leq
c$

where $c$ is independent of $n$, verifying (2.2). Applying the conditional and
unconditional forms of Jensen’s inequality, the bounded moment/moment decay
assumptions and Lemma 2.4 yields that for each $n\in\mathbb{N}$

$\displaystyle(2n)^{d/2}\mathbb{E}\Big{[}\sum_{v\in\mathbb{Z}^{d}}\lvert
U_{n,v}\rvert\Big{]}$
$\displaystyle\leq\sum_{v\in\mathbb{Z}^{d}}\mathbb{E}\Big{[}\lvert\Delta_{v}(\Lambda_{n})\rvert\Big{]}$
$\displaystyle\leq\sum_{\mathrm{dist}(v,\Lambda_{n})\leq
n}\mathbb{E}\Big{[}\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\Big{]}^{\frac{1}{2+\epsilon}}+\sum_{\mathrm{dist}(v,\Lambda_{n})>n}cn^{\zeta}\;\mathrm{dist}(v,\Lambda_{n})^{-\gamma}<\infty$

verifying (2.4).

It remains to verify (2.3). For $v\in\mathbb{Z}^{d}$ let $\tau_{v}$ denote
translation by $v$. Let $D_{n}=w_{n}+\Lambda_{n}$ where
$w_{1},w_{2},\dots\in\mathbb{Z}^{d}$ then by the definitions of $\mu$ and
$\Delta_{v}$, for any $v\in\mathbb{Z}^{d}$ the sequences of random variables
$\Delta_{v}(D_{n})$ and $\Delta_{0}(\tau_{-v}D_{n})$ have the same
distribution. Hence if $D_{n}\to\mathbb{R}^{d}$, then by the stabilisation
assumption there exists a random variable $\Delta_{v}$ such that
$\Delta_{v}(D_{n})\to\Delta_{v}$ in probability. We now simplify notation by
defining

$X_{v}(\Lambda_{n}):=\mathbb{E}[\Delta_{v}(\Lambda_{n})\;|\;\mathcal{F}_{v}]\qquad\text{and}\qquad
X_{v}:=\mathbb{E}[\Delta_{v}\;|\;\mathcal{F}_{v}]$

so that the statement we need to prove is

(2.8)
$(2n)^{-d}\sum_{v\in\mathbb{Z}^{d}}X_{v}^{2}(\Lambda_{n})\xrightarrow{L^{1}}\sigma^{2},\qquad\text{as
}n\to\infty.$

We do this in three steps: (i) we show that the contribution to this sum from
terms $X_{v}^{2}(\Lambda_{n})$ where $v$ is outside $\Lambda_{n}$ or near the
boundary of $\Lambda_{n}$ is small, (ii) we show that the remaining terms are
well approximated in $L^{1}$ by their limits $X_{v}^{2}$ and (iii) we apply
the ergodic theorem to the sum of these limit terms.

If we choose $\lambda\in(0,1)$ sufficiently small, then by (2.7) and the
conditional Jensen inequality

(2.9)
$(2n)^{-d}\sum_{v\;:\;\mathrm{dist}(v,\Lambda_{n})>n^{1-\lambda}}\mathbb{E}[X_{v}^{2}(\Lambda_{n})]\leq(2n)^{-d}\sum_{v:\mathrm{dist}(v,\Lambda_{n})>n^{1-\lambda}}cn^{\zeta}\;\mathrm{dist}(v,\Lambda_{n})^{-\gamma}\to
0$

as $n\to\infty$ (using Lemma 2.4). By the bounded moments condition, for the
same value of $\lambda>0$ we have

(2.10)
$\displaystyle(2n)^{-d}\sum_{v\;:\;\mathrm{dist}(v,\partial\Lambda_{n})\leq
n^{1-\lambda}}\mathbb{E}[X_{v}^{2}(\Lambda_{n})]$
$\displaystyle\leq(2n)^{-d}\sum_{v\;:\;\mathrm{dist}(v,\partial\Lambda_{n})\leq
n^{1-\lambda}}\mathbb{E}[X_{v}^{2+\epsilon}(\Lambda_{n})]^{\frac{1}{2+\epsilon}}$
$\displaystyle\leq cn^{-d}n^{d-\lambda}\to 0$

as $n\to\infty$.

Now let $n^{-}=n-n^{1-\lambda}$. We claim that

(2.11)
$\max_{v\in\mathbb{Z}^{d}\cap\Lambda_{n^{-}}}\mathbb{E}\Big{[}\big{\lvert}X_{v}^{2}(\Lambda_{n})-X_{v}^{2}\big{\rvert}\Big{]}\to
0\quad\text{as }n\to\infty.$

Let $v_{n}\in\mathbb{Z}^{d}$ be the vertex which maximises the moment above
for a given $n$ and let $D_{n}=-v_{n}+\Lambda_{n}$. Observe that
$D_{n}\to\mathbb{R}^{d}$ since
$\mathrm{dist}(v_{n},\partial\Lambda_{n})\to\infty$. Therefore
$\Delta_{v}(D_{n})\to\Delta_{v}$ in probability. Since the
$(2+\epsilon)$-moments of $\Delta_{v}(D_{n})$ are bounded uniformly over $n$,
Vitali’s convergence theorem implies that $\Delta_{v}(D_{n})\to\Delta_{v}$ in
$L^{2}$. Then using the fact that conditional expectation is a contraction on
$L^{p}$ we have

$X_{v}(D_{n})=\mathbb{E}[\Delta_{v}(D_{n})\;|\;\mathcal{F}_{v}]\xrightarrow{L^{2}}\mathbb{E}[\Delta_{v}\;|\;\mathcal{F}_{v}]=X_{v}$

and in particular (2.11) holds.

Combining (2.9), (2.10) and (2.11) we see that

(2.12)
$(2n)^{-d}\sum_{v\in\mathbb{Z}^{d}}X_{v}^{2}(\Lambda_{n})-(2n)^{d}\sum_{v\in\mathbb{Z}^{d}\cap\Lambda_{n^{-}}}X_{v}^{2}\xrightarrow{L^{1}}0.$

It remains for us to apply the ergodic theorem to the finite sum above. Let
$\xi=(W_{v},W_{v}^{\prime})_{v\in\mathbb{Z}^{d}}$ denote the pair of white
noise processes used to define $\mu$ and $\Delta_{v}(\cdot)$. Let
$T_{1},\dots,T_{d}$ respectively denote translation by distance one in the
positive direction of the $d$ axes of $\mathbb{R}^{d}$. Since $W$ and
$W^{\prime}$ are stationary, each $T_{i}$ is a measure-preserving
transformation of $\xi$. Since $(W_{v},W^{\prime}_{v})$ are independent for
different $v\in\mathbb{Z}^{d}$, the $\sigma$-algebra of invariant events
associated with each $T_{i}$ is trivial (by Kolmogorov’s 01-law). We now
augment our previous notation to include the dependency on $\xi$; so we write
$X_{v}(D_{n},\xi)=X_{v}(D_{n})$ and $X_{v}(\xi)=X_{v}$ etc. We then claim that
for any $v\in\mathbb{Z}^{d}$

(2.13) $X_{v}(\xi)=X_{0}(\tau_{-v}\xi)$

where
$\tau_{-v}\xi=(\tau_{-v}W_{u},\tau_{-v}W^{\prime}_{u})_{u\in\mathbb{Z}^{d}}=(W_{u+v},W^{\prime}_{u+v})_{u\in\mathbb{Z}^{d}}$.
To prove the claim, we first note that for any $n$

$\Delta_{v}(\Lambda_{n},\xi)=\Delta_{0}(-v+\Lambda_{n},\tau_{-v}\xi)$

using the fact that $\mu$ is stationary (recall the definition before Theorem
2.2). Then taking $n\to\infty$ and using the stabilisation assumption, we see
that the term on the left converges in probability to $\Delta_{v}(\xi)$ while
the term on the right converges to $\Delta_{0}(\tau_{-v}\xi)$ and so these
limits must coincide almost surely.

Next we observe that by definition of the lexicographic ordering

(2.14) $\mathcal{F}_{0}(\tau_{-v}W):=\sigma(\tau_{-v}W_{u}\;|\;u\preceq
0)=\sigma(W_{u+v}\;|\;u\preceq 0)=\sigma(W_{u}\;|\;u\preceq
v)=:\mathcal{F}_{v}(W).$

Therefore

$X_{v}(\xi)=\mathbb{E}[\Delta_{v}(\xi)\;|\;\mathcal{F}_{v}(W)]=\mathbb{E}[\Delta_{0}(\tau_{-v}\xi)\;|\;\mathcal{F}_{0}(\tau_{-v}W)]=X_{0}(\tau_{-v}\xi)$

completing the proof of (2.13). Finally we note that by Fatou’s lemma and the
bounded moments assumption

$\mathbb{E}\left[\lvert
X_{0}(\xi)\rvert^{2+\epsilon}\right]\leq\liminf_{n\to\infty}\mathbb{E}\left[\lvert
X_{0}(\Lambda_{n},\xi)\rvert^{2+\epsilon}\right]<\infty$

so $X_{0}^{2}\in L^{1+\epsilon/2}(\nu)$ where $\nu$ denotes the distribution
of $\xi$. Then since $\tau_{v}=T_{1}^{v_{1}}\dots T_{d}^{v_{d}}$ for
$v\in\mathbb{Z}^{d}$, we may apply Theorem 2.5 to conclude that

$(2n)^{-d}\sum_{v\in\mathbb{Z}^{d}\cap\Lambda_{n}^{-}}X_{v}^{2}=(2n)^{-d}\sum_{v\in\mathbb{Z}^{d}\cap\Lambda_{n}^{-}}X_{0}^{2}(\tau_{-v}\xi)\xrightarrow{L^{1}}\mathbb{E}[X_{0}^{2}]=\mathbb{E}\big{[}\mathbb{E}[\Delta_{0}\;|\;\mathcal{F}_{0}]^{2}\big{]}=:\sigma^{2}.$

Combining this with (2.12) proves (2.8) and hence completes the proof of the
theorem. ∎

######  Remark 2.6.

When applying the martingale CLT, the most difficult condition to verify is
often the convergence of the sum of squared increments (i.e. the analogue of
(2.3)). Penrose [47] recognised that for stationary white-noise functionals,
this convergence could be elegantly dealt with using the ergodic theorem and
this insight is crucial for the proof we have just given. It is also
interesting to note the importance of the lexicographic filtration in this
context: (2.14) was essential in our application of the ergodic theorem and
this property holds only for the standard lexicographic ordering of
$\mathbb{Z}^{d}$ (up to reflections and reorderings of axes).

If the white-noise functional in the statement of Theorem 2.2 is additive,
which is the case for our three functionals of interest, it is natural to
expect that one could find somewhat simpler versions of conditions (2)-(4) in
terms of how the functional changes on _unit cubes_ when resampling the white-
noise (i.e. conditions on $\Delta_{v}(B_{w})$ rather than
$\Delta_{v}(\Lambda_{n})$). The following result shows that this is indeed
true and will simplify the proof of the CLTs for $\mu_{\mathrm{Vol}}$,
$\mu_{\mathrm{SA}}$ and $\mu_{\mathrm{EC}}$.

######  Proposition 2.7.

Let $\mu$ be a stationary white-noise functional such that with probability
one the following holds: for any distinct $w_{1},\dots,w_{n}\in\mathbb{Z}^{d}$

(2.15)
$\mu\left(\bigcup_{i=1}^{n}B_{w_{i}}\right)=\sum_{i=1}^{n}\mu(B_{w_{i}}).$

Suppose that there exists a family of random variables $Y_{v}(w)\geq 0$ for
$v,w\in\mathbb{Z}^{d}$ such that

(2.16) $\left\lvert\Delta_{v}\left(B_{w}\right)\right\rvert\leq Y_{v}(w)$

and

(2.17) $\mathbb{E}[Y_{v}(w)]+\mathbb{E}\big{[}Y_{v}(w)^{2+\epsilon}\big{]}\leq
c(1+\lvert v-w\rvert)^{-(3d+\delta)}$

for some $\epsilon,\delta,c>0$ which depend only on the distribution of $f$.
Then $\mu$ satisfies conditions (2)-(4) of Theorem 2.2.

If we replace (2.15) and (2.16) with the weaker assumption that for any
$D:=u+\Lambda_{n}$ (where $u\in\mathbb{Z}^{d}$)

(2.18)
$\left\lvert\Delta_{v}\left(D\right)\right\rvert\leq\sum_{w\in\mathbb{Z}^{d}\cap
D}Y_{v}(w),$

where $Y_{v}(w)$ again satisfies (2.17), then $\mu$ satisfies conditions
(3)-(4) of Theorem 2.2.

###### Proof.

Suppose that (2.18) and (2.17) hold. Taking $v\in\mathbb{Z}^{d}$ and
$n\in\mathbb{N}$ as given, we write $Y(w)=Y_{v}(w)$. By (2.18) and the
inequality $\left(\sum_{i=1}^{m}y_{i}\right)^{p}\leq\sum_{i=1}^{m}y_{i}^{p}$,
which holds for any $p\in(0,1)$ and $y_{1},\dots,y_{m}\geq 0$, we have

$\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\right]\leq\mathbb{E}\left[\left(\sum_{w,x,y\in\mathbb{Z}^{d}\cap\Lambda_{n}}Y(w)Y(x)Y(y)\right)^{\frac{2+\epsilon}{3}}\right]\\\
\leq\sum_{w,x,y\in\mathbb{Z}^{d}\cap\Lambda_{n}}\mathbb{E}\left[\big{(}Y(w)Y(x)Y(y)\big{)}^{\frac{2+\epsilon}{3}}\right].$

Applying Hölder’s inequality to the right hand side along with (2.17)

(2.19)
$\displaystyle\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\right]\leq\left(\sum_{w\in\Lambda_{n}}\mathbb{E}\left[Y(w)^{2+\epsilon}\right]^{\frac{1}{3}}\right)^{3}\leq\left(\sum_{w\in\Lambda_{n}}c(1+\lvert
v-w\rvert)^{\frac{-3d-\delta}{3}}\right)^{3}.$

Hence

$\sup_{v\in\mathbb{Z}^{d},n\in\mathbb{N}}\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\right]\leq\left(\sum_{w\in\mathbb{Z}^{d}}c(1+\lvert
w\rvert)^{-d-\frac{\delta}{3}}\right)^{3}<\infty$

verifying the bounded moments condition.

Now suppose that $v\notin\Lambda_{n}$. Then by (2.19)

$\displaystyle\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert^{2+\epsilon}\right]$
$\displaystyle\leq\left(cn^{d}\;\mathrm{dist}(v,\Lambda_{n})^{-d-\delta/3}\right)^{3}$

which verifies the moment decay condition for the $(2+\epsilon)$-moments. By
(2.16) and (2.17)

$\displaystyle\mathbb{E}\left[\lvert\Delta_{v}(\Lambda_{n})\rvert\right]\leq\sum_{w\in\Lambda_{n}}\mathbb{E}[Y_{v}(w)]$
$\displaystyle\leq cn^{d}\;\mathrm{dist}(v,\Lambda_{n})^{-3d-\delta}$

which verifies the other part of the moment decay condition.

Now assume that (2.15) holds and fix a sequence $D_{n}:=v_{n}+\Lambda_{n}$
such that $D_{n}\to\mathbb{R}^{d}$. By (2.17) and the Borel-Cantelli lemma

$\sum_{w\in\mathbb{Z}^{d}}Y_{0}(w)<\infty$

almost surely. Given $m\in\mathbb{N}$, we may choose $N\in\mathbb{N}$ large
enough that $\cap_{n\geq N}D_{n}\supset\Lambda_{m}$. Then for any
$n_{1},n_{2}>N$ by (2.15) and (2.16)

$\displaystyle\lvert\Delta_{0}(D_{n_{1}})-\Delta_{0}(D_{n_{2}})\rvert\leq\lvert\Delta_{0}(D_{n_{1}}\setminus
D_{n_{2}})\rvert+\lvert\Delta_{0}(D_{n_{2}}\setminus
D_{n_{1}})\rvert\leq\sum_{v\in\mathbb{Z}^{d}\setminus\Lambda_{m}}Y_{0}(w).$

Since the right-hand side converges to zero as $m\to\infty$, we see that
$\Delta_{0}(D_{n})$ is almost surely Cauchy and hence convergent to some limit
$\Delta_{0}$.

To see that the limit does not depend on the choice of domains $D_{n}$, we may
take any other suitable sequence $D_{n}^{\prime}$ and consider
$(D_{1},D_{1}^{\prime},D_{2},D_{2}^{\prime},\dots)$. By our above argument we
see that $\Delta_{0}(D_{n}^{\prime})$ and $\Delta_{0}(D_{n})$ converge almost
surely to some $\Delta_{0}^{*}$ but the latter convergence implies that
$\Delta_{0}^{*}=\Delta_{0}$ almost surely. ∎

##  3\. Topological stability

As described in Section 1.3 there are three different potential contributions
to the change in each of our functionals (volume, surface area and Euler
characteristic) under perturbation: local/non-local topological contributions
and geometric contributions. In this subsection, we control the probability of
topological contributions using concepts from (stratified) Morse theory.

Dealing first with the local contributions; if we consider the level sets
$\\{f+tp=\ell\\}\cap B_{v}$ as $t$ varies in $[0,1]$, it is intuitively clear
that the level sets should deform continuously, preserving their topology,
unless they pass through a critical point. Thinking more carefully, one might
realise that we also need to control how the topology changes near the
boundary of $B_{v}$, which can be done by considering ‘boundary critical
points’. This is essentially the logic of the first fundamental result of
stratified Morse theory which we now make rigorous.

Each unit cube $B_{v}$ for $v\in\mathbb{Z}^{d}$ can be viewed as a _stratified
set_ by partitioning it into the finite union of each of its open faces of
dimension $0,1,\dots,d$ which we refer to as _strata_. For example, if $d=2$
then the stratification of $B_{v}$ consists of the (two-dimensional) interior
of the square, four one-dimensional edges and four zero-dimensional corners.
Let $A$ be a stratum of $B_{v}$ and $x\in A$. We say that $x$ is a _stratified
critical point_ of a function $g\in C^{1}(B_{v})$ if $\nabla_{A}g(x)=0$ where
$\nabla_{A}$ denotes the gradient restricted to the affine space $A$. If $A$
is a zero-dimensional stratum then $\nabla_{A}g\equiv 0$ by convention. We say
that the _level_ of the critical point is $g(x)$.

Given $g,p\in C^{1}(B_{v})$, we say that $(g,p)$ is _locally topologically
stable_ (for domain $B_{v}$ at level $\ell$) if for all $t\in[0,1]$, $g+tp$
has no stratified critical points in $D$ at level $\ell$. Note that local
topological stability is equivalent for $(g,p)$ and $(g+p,-p)$.

We define a _stratified isotopy_ of a stratified set $D$ to be a continuous
map $H:D\times[0,1]\to D$ such that

  1. (1)

$H(\cdot,t):D\to D$ is a homeomorphism for each $t\in[0,1]$, and

  2. (2)

for any stratum $A$ of $D$, $H(A)=A$.

With these definitions we may state our first stability result:

######  Lemma 3.1.

For $v_{1},\dots,v_{n}\in\mathbb{Z}^{d}$, let $D:=\cup_{i=1}^{n}B_{v_{i}}$ be
equipped with the stratification consisting of all open faces of
$B_{v_{1}},\dots,B_{v_{n}}$ and let $g,p\in C^{2}(D)$. If $(g,p)$ is locally
topologically stable on each $B_{v_{i}}$ at level $\ell$ then there exists a
stratified isotopy $H$ of $D$ such that

$H(\\{g\geq\ell\\},t)=\\{g+tp\geq\ell\\}\quad\text{for all }t\in[0,1].$

###### Proof.

This was proven for boxes in [6, Lemma 4.1] using Thom’s isotopy lemma. The
proof remains valid for $D$ since the latter is a Whitney stratified space. ∎

This result provides us with a sufficient condition for topological stability
in terms of pointwise behaviour of $g$, $p$ and their derivatives. We next
give a probabilistic statement for stability of $f$ under the perturbation
$p_{v}$ (recall that the latter is defined in (1.4)). Actually we will
consider slightly more general perturbations which will be needed later in
proving positivity of $\sigma^{2}$ (i.e. Theorem 1.10). Given $g,p\in
C^{2}(\mathbb{R}^{d})$ we define the locally unstable set as

$\mathcal{U}_{\mathrm{Loc}}(g,p)=\big{\\{}w\in\mathbb{Z}^{d}\;\big{|}\;(g,p)\text{
is locally unstable for }B_{w}\text{ at level }\ell\big{\\}}.$

######  Lemma 3.2.

Let $f$ satisfy Assumption 1.3 and let $\rho:\mathbb{R}^{d}\to\mathbb{R}$ be a
$C^{1}$ deterministic function. Then for each $\delta>0$ there exists $c>0$
(which is independent of $\rho$ but may depend on the distribution of $f$),
such that

$\mathbb{P}(w\in\mathcal{U}_{\mathrm{Loc}}(f,\rho))\leq
c\|\rho\|^{1-\delta}_{C^{1}(w+\Lambda_{2})}\qquad\text{for all
}w\in\mathbb{Z}^{d}$

and

$\mathbb{P}\left(w\in\mathcal{U}_{\mathrm{Loc}}(f,p_{v}+\rho)\right)\leq
c\|\rho\|_{C^{1}(w+\Lambda_{2})}^{1-\delta}+c(1+\lvert
v-w\rvert)^{-\beta+\delta}\qquad\text{for all }v,w\in\mathbb{Z}^{d}.$

###### Proof.

The first statement follows from [9, Lemma 2.4]. The same lemma also states
that for any $\epsilon>0$, the probability that
$w\in\mathcal{U}_{\mathrm{Loc}}(f,p_{v}+\rho)$ is at most

(3.1)
$c_{\epsilon}\inf_{\tau>M_{1}+c\sqrt{M_{2}}}\left(\tau^{1-\epsilon}+e^{-\frac{(\tau-
M_{1}-c\sqrt{M_{2}})^{2}}{2M_{2}}}\right)$

where

$M_{1}:=\|\rho\|_{C^{1}(w+\Lambda_{2})},\qquad M_{2}:=\sup_{x,y\in
w+\Lambda_{2}}\sup_{\lvert\alpha\rvert,\lvert\gamma\rvert\leq
2}\left\lvert\partial^{\alpha}_{x}\partial^{\gamma}_{y}\mathrm{Cov}[p_{v}(x),p_{v}(y)]\right\rvert$

and $c>0$ is an absolute constant. (The proof of the cited lemma uses standard
estimates for the supremum of a Gaussian field, given in [9, Section 3], and a
bound on the probability of $f$ and $\nabla f$ being simultaneously small,
from [44].) From the white noise representation for $p_{v}$ (in (1.4))

$\lvert\partial_{x}^{\alpha}\partial_{y}^{\gamma}\mathrm{Cov}[p_{v}(x),p_{v}(y)]\rvert=2\left\lvert\int_{B_{v}}\partial_{x}^{\alpha}q(x-u)\partial_{y}^{\gamma}q(y-u)\;du\right\rvert$

(where taking the derivative inside the integral follows from dominated
convergence since $q\in C^{3}(\mathbb{R}^{d})$ and $B_{v}$ is compact).
Therefore by Assumption 1.3, $M_{2}\leq C(1+\lvert v-w\rvert)^{-2\beta}$ for
some $C>0$ independent of $v$ and $w$. Using this observation and setting
$\tau=2(M_{1}+c\sqrt{M_{2}})^{1-\epsilon}$ in (3.1) we have

$\mathbb{P}\left(w\in\mathcal{U}_{\mathrm{Loc}}(f,p_{v}+\rho)\right)\leq
c_{1}\Big{(}\|\rho\|_{C^{1}(w+\Lambda_{2})}^{1-2\epsilon}+(1+\lvert
v-w\rvert)^{-\beta(1-2\epsilon)}+e^{-c_{2}(1+\lvert
v-w\rvert)^{2\epsilon\beta}}\Big{)}$

for some $c_{1},c_{2}>0$ which depend on $\epsilon$. Choosing $\epsilon>0$
sufficiently small (depending on $\beta$ and $\delta$) then proves the bound
in the statement of the lemma. ∎

Next we turn to non-local topological contributions. Recall that such
contributions occur whenever the topology of the level sets within a unit cube
$B_{v}$ do not change but topological changes in another cube
connect/disconnect some components in $B_{v}$ to/from infinity. Our next
result shows that, for such a change to occur, $B_{v}$ must be connected to a
locally unstable cube via a bounded component.

Recall that for three sets $A,B,C\subset\mathbb{R}^{d}$ we write
$A\overset{B}{\longleftrightarrow}C$ if there exists a continuous curve in $B$
joining a point in $A$ to a point in $C$. For a $C^{2}$ function
$g:\mathbb{R}^{d}\to\mathbb{R}$, let $\\{g\geq\ell\\}_{\infty}$ denote the
union of any unbounded components of $\\{g\geq\ell\\}$ and let
$\\{g\geq\ell\\}_{<\infty}:=\\{g\geq\ell\\}\setminus\\{g\geq\ell\\}_{\infty}$
be the union of the bounded components. Then for a pair of $C^{2}$ functions
$(g,p)$ we define the _finite-range unstable set_ as

$\mathcal{U}_{\mathrm{FR}}(g,p)=\mathcal{U}_{\mathrm{Loc}}(g,p)\cup\bigcup_{v\in\mathcal{U}_{\mathrm{Loc}}(g,p)}\Big{\\{}w\in\mathbb{Z}^{d}\;\Big{|}\;B_{w}\overset{\\{g\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{v}\text{
or
}B_{w}\overset{\\{g+p\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{v}\Big{\\}}.$

We say that $B_{w}$ is finite-range stable if
$w\notin\mathcal{U}_{\mathrm{FR}}(g,p)$.

######  Lemma 3.3.

Let $D:=\cup_{i=1}^{n}B_{v_{i}}$ be equipped with the stratification
consisting of all open faces of $B_{v_{1}},\dots,B_{v_{n}}$ and let $g,p\in
C^{2}(D)$. Assume that $\\{g\geq\ell\\}\cap D$ and $\\{g+p\geq\ell\\}\cap D$
have at most a finite number of components. If $B_{v_{i}}$ is finite-range
stable for each $i$, then there exists a stratified isotopy $H$ of $D$ such
that

$H(\\{g\geq\ell\\},t)=\\{g+tp\geq\ell\\}\quad\text{for all }t\in[0,1]$

and

$H(\\{g\geq\ell\\}_{\infty},1)=\\{g+p\geq\ell\\}_{\infty}.$

###### Proof.

Let $D^{+}$ be the union of all unit cubes which are connected to $D$ by
bounded excursion components of $g$ or $g+p$, that is

$D^{+}:=D\cup\bigcup_{i=1}^{n}\bigcup\Big{\\{}B_{v}\;\Big{|}\;B_{v}\overset{\\{g\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{v_{i}}\;\text{or}\;B_{v}\overset{\\{g+p\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{v_{i}}\Big{\\}}.$

Since $\\{g\geq\ell\\}\cap D$ and $\\{g+p\geq\ell\\}\cap D$ have finitely many
components, we see that $D^{+}$ is the union of finitely many unit cubes. By
definition all of these cubes are locally topologically stable, so we may
apply Lemma 3.1 to obtain a stratified isotopy $H:D^{+}\times[0,1]\to D^{+}$
such that $H(\\{g\geq\ell\\},t)=\\{g+tp\geq\ell\\}\cap D^{+}$.

Let $A$ be a component of $\\{g\geq\ell\\}_{<\infty}$ which intersects $D$.
Observe that $A$ cannot intersect the boundary of $D^{+}$ (if it did, then
some cube not contained in $D^{+}$ would be connected to $D$ by a bounded
excursion component). Hence, since $H$ preserves strata, $H(A,1)$ cannot
intersect the boundary of $D^{+}$. In particular $H(A,1)$ is bounded and so is
a subset of $\\{g+p\geq\ell\\}_{<\infty}$. This is true for any such component
$A$, and so

$H(\\{g\geq\ell\\}_{<\infty}\cap D,1)\subseteq\\{g+p\geq\ell\\}_{<\infty}.$

By the same reasoning, any component of $\\{g+p\geq\ell\\}_{<\infty}$ which
intersects $D$ must be the image under $H(\cdot,1)$ of a component of
$\\{g\geq\ell\\}_{<\infty}$. Therefore, recalling that $H(D,1)=D$ since $H$
preserves strata, we have

$H(\\{g\geq\ell\\}_{<\infty}\cap D,1)=\\{g+p\geq\ell\\}_{<\infty}\cap D.$

The restriction of $H$ to $D$ then satisfies the conditions in the statement
of the lemma. ∎

Our next objective is to obtain probabilistic bounds for finite-range
stability. As an input to such a bound, we will require the following
generalisation of Proposition 1.7 which allows the field $f$ to be perturbed
by a deterministic, non-negative function.

######  Proposition 3.4.

Let $f$ satisfy Assumption 1.3 and let $\rho:\mathbb{R}^{d}\to\mathbb{R}$ be
non-negative, continuously differentiable and deterministic. For each
$\ell<-\ell_{c}$ there exists $C,c>0$ such that for all $n\in\mathbb{N}$ and
$v\in\mathbb{Z}^{d}$

$\mathbb{P}\left(B_{v}\overset{\\{f+\rho\geq\ell\\}_{<\infty}}{\longleftrightarrow}v+\partial\Lambda_{n}\right)\leq
Ce^{-cn}.$

###### Proof of Proposition 1.7.

Set $\rho\equiv 0$ and apply Proposition 3.4. ∎

###### Proof of Proposition 3.4.

Since the distribution of $f$ is translation invariant (as are our assumptions
on the function $\rho$) it is enough to prove the bound only for $v=0$.
Suppose that $B_{0}$ is connected to $\partial\Lambda_{n}$ by some bounded
component of $\\{f+\rho\geq\ell\\}$. Let $A$ denote this component and let $N$
be the largest integer such that $A$ intersects $\partial\Lambda_{N+1}$. So
the diameter of $A$ is at least $N$ and $N\geq n-1$. By Bulinskaya’s lemma [1,
Lemma 11.2.10] $f+\rho$ almost surely has no critical points at level $\ell$.
Hence the boundary of $A$ is $C^{1}$-smooth and so there is a component $E$ of
$\\{f+\rho\leq\ell\\}$ which ‘surrounds’ $A$. More precisely, $E$ is the
component of $\\{f+\rho\leq\ell\\}$ which intersects the outer boundary of
$A$. Then by definition of $N$, there exists some
$x\in\mathbb{Z}^{d}\cap\Lambda_{N+2}\setminus\Lambda_{N}$ such that $E$
intersects $B_{x}$. Moreover since the diameter of $E$ is at least as large as
the diameter of $A$ we see that $E$ connects $B_{x}$ to
$x+\partial\Lambda_{N/\sqrt{d}-1}$. By non-negativity of $\rho$ and symmetry
of the normal distribution

$\mathbb{P}\left(B_{x}\overset{\\{f+\rho\leq\ell\\}}{\longleftrightarrow}x+\partial\Lambda_{\frac{N}{\sqrt{d}}-1}\right)\leq\mathbb{P}\left(B_{x}\overset{\\{f\leq\ell\\}}{\longleftrightarrow}x+\partial\Lambda_{\frac{N}{\sqrt{d}}-1}\right)=\mathbb{P}\left(B_{x}\overset{\\{f\geq-\ell\\}}{\longleftrightarrow}x+\partial\Lambda_{\frac{N}{\sqrt{d}}-1}\right).$

Since $-\ell>\ell_{c}$, using sharpness of the phase transition (i.e. Theorem
1.5) the latter probability is bounded above by $Ce^{-cN}$ where $C,c>0$
depend only on $\ell$ and the distribution of $f$. Therefore by the union
bound

$\mathbb{P}\Big{(}\Lambda_{1}\overset{\\{f\geq\ell\\}_{<\infty}}{\longleftrightarrow}\partial\Lambda_{n}\Big{)}\leq\sum_{N\geq
n-1}\sum_{x\in\Lambda_{N+2}\setminus\Lambda_{N}}Ce^{-cN}\leq\sum_{N\geq
n-1}C^{\prime}N^{d-1}e^{-cN}\leq C^{\prime\prime}e^{-cn}$

for constants $C,C^{\prime},C^{\prime\prime},c>0$, as required. ∎

We now use this result to get a probabilistic bound for finite-range
stability.

######  Lemma 3.5.

Let $f$ satisfy Assumptions 1.3 and 1.6, then for all $\delta>0$ there exists
$c>0$ such that

$\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{v}))\leq c(1+\lvert
v-w\rvert)^{-\beta+\delta}\qquad\text{for }v,w\in\mathbb{Z}^{d}.$

In particular, by the Borel-Cantelli lemma,
$\mathcal{U}_{\mathrm{FR}}(f,p_{0})$ is almost surely finite.

Let $\rho:\mathbb{R}^{d}\to\mathbb{R}$ be a deterministic function such that

$\max_{\lvert\alpha\rvert\leq 2}\lvert\partial^{\alpha}\rho(x)\rvert\leq
C(1+\lvert x\rvert)^{-\beta}$

for some $C>0$ and all $x\in\mathbb{R}^{d}$. Suppose in addition that
$\ell<-\ell_{c}$, then for all $\delta>0$ there exists $c>0$ such that

$\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{0}+\rho))\leq c(1+\lvert
w\rvert)^{-\beta+\delta}\qquad\text{for }w\in\mathbb{Z}^{d}.$

###### Proof.

We first argue for $p_{0}+\rho$. By definition of the finite-range unstable
set and the union bound

(3.2)
$\displaystyle\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{0}+\rho))\leq$
$\displaystyle\mathbb{P}(w\in\mathcal{U}_{\mathrm{Loc}}(f,p_{0}+\rho))$
$\displaystyle+\sum_{u\in\mathbb{Z}^{d}}\mathbb{P}\Big{(}\\{u\in\mathcal{U}_{\mathrm{Loc}}(f,p_{0}+\rho)\\}\cap\big{\\{}B_{w}\overset{\\{f\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{u}\big{\\}}\Big{)}$
$\displaystyle\quad+\sum_{u\in\mathbb{Z}^{d}}\mathbb{P}\Big{(}\\{u\in\mathcal{U}_{\mathrm{Loc}}(f,p_{0}+\rho)\\}\cap\big{\\{}B_{w}\overset{\\{\widetilde{f}_{0}+\rho\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{u}\big{\\}}\Big{)}.$

By Lemma 3.2 the first term on the right hand side is bounded by $c(1+\lvert
w\rvert)^{-\beta+\delta}$. For any $\epsilon\in(0,\delta/2d)$, the second term
can be bounded by

(3.3) $\sum_{\lvert u-w\rvert\leq\lvert
w\rvert^{\epsilon}}\mathbb{P}(u\in\mathcal{U}_{\mathrm{Loc}}(f,p_{0}+\rho))+\sum_{\lvert
u-w\rvert>\lvert
w\rvert^{\epsilon}}\mathbb{P}\Big{(}B_{w}\overset{\\{f\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{u}\Big{)}.$

By Lemma 3.2 the first sum in (3.3) is at most

$\displaystyle\sum_{\lvert u-w\rvert\leq\lvert w\rvert^{\epsilon}}c(1+\lvert
u\rvert)^{-\beta+\delta/2}\leq c^{\prime}\lvert w\rvert^{d\epsilon}(1+\lvert
w\rvert)^{-\beta+\delta/2}\leq c^{\prime}(1+\lvert w\rvert)^{-\beta+\delta}$

for constants $c,c^{\prime}>0$. By Assumption 1.6 and Lemma 2.4, the second
sum in (3.3) decays super-polynomially in $\lvert w\rvert$. If $\rho\equiv 0$,
then the third term of (3.2) is identical to the second term (since
$\widetilde{f}_{0}$ has the same distribution as $f$). Hence we conclude that

$\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{0}))\leq c(1+\lvert
w\rvert)^{-\beta+\delta},$

and the first statement of the lemma follows by stationarity.

If $\rho$ is not identically zero, then we can bound the third term of (3.2)
using the same argument as for the second term, except that we use Proposition
3.4 to control

$\sum_{\lvert u-w\rvert>\lvert
w\rvert^{\epsilon}}\mathbb{P}\Big{(}B_{w}\overset{\\{\widetilde{f}_{0}+\rho\geq\ell\\}_{<\infty}}{\longleftrightarrow}B_{u}\Big{)}$

rather than Assumption 1.6. ∎

##  4\. Volume, surface area and Euler characteristic of the unbounded
component

To summarise our progress so far: by Theorem 2.2 and Proposition 2.7 our
functionals $\mu_{\star}$ will each satisfy a CLT if we can obtain appropriate
moment bounds for the change in the value of the functional under
perturbation. In the previous section we obtained bounds on the probability of
topological contributions to this change. It remains to control the magnitude
of such contributions as well as the geometric contributions. In the next
three subsections, which can be read independently, we do this using arguments
specific to each of the three functionals.

###  4.1. Volume

Controlling the magnitude of topological contributions to $\Delta_{v}$ is
trivial for the volume functional since the volume of any unit cube is bounded
deterministically. It is also straightforward to bound the geometric
contributions to $\Delta_{v}$ in this case, as the following lemma shows.

For a $C^{2}$ function $g:\mathbb{R}^{d}\to\mathbb{R}$ let
$\mathrm{Vol}_{\infty}[D,g]=\mathrm{Vol}[\\{g\geq\ell\\}_{\infty}\cap D]$.

######  Lemma 4.1.

Let $g,g+p:\mathbb{R}^{d}\to\mathbb{R}$ be $C^{2}$ functions with at most a
finite number of stratified critical points on any unit cube (i.e. on any
$B_{v}$ for $v\in\mathbb{Z}^{d}$).

  1. (1)

For any distinct $v_{1},\dots,v_{n}\in\mathbb{Z}^{d}$

$\mathrm{Vol}_{\infty}\left[\bigcup_{i=1}^{n}B_{v_{i}},g\right]=\sum_{i=1}^{n}\mathrm{Vol}_{\infty}[B_{v_{i}},g].$

  2. (2)

For any $v\in\mathbb{Z}^{d}$

$\lvert\mathrm{Vol}_{\infty}[B_{v},g]-\mathrm{Vol}_{\infty}[B_{v},g+p]\rvert\leq
1.$

  3. (3)

For any $v\notin\mathcal{U}_{\mathrm{FR}}(g,p)$

$\lvert\mathrm{Vol}_{\infty}[B_{v},g]-\mathrm{Vol}_{\infty}[B_{v},g+p]\rvert\leq\mathrm{Vol}\big{[}B_{v}\cap\\{\lvert
g-\ell\rvert\leq\lvert p\rvert\\}\big{]}.$

###### Proof.

The first statement follows immediately from the fact that
$\mathrm{Vol}_{\infty}[\cdot,g]$ is a measure such that
$\mathrm{Vol}_{\infty}[\partial B_{v},g]=0$ for any $v\in\mathbb{Z}^{d}$. The
second statement is trivial since
$0\leq\mathrm{Vol}_{\infty}[B_{v},g],\mathrm{Vol}_{\infty}[B_{v},g+p]\leq 1$.

Now assume $v\notin\mathcal{U}_{\mathrm{FR}}(g,p)$ and let $H$ denote the
stratified isotopy on $B_{v}$ specified by Lemma 3.3. We claim that if $A$ is
a component of $\\{g\geq\ell\\}\cap B_{v}$ or of $\\{g\leq\ell\\}\cap B_{v}$
then

$A\cap\\{\lvert g-\ell\rvert>\lvert p\rvert\\}\subseteq H(A,1).$

Applying the claim to all components of $\\{g\geq\ell\\}_{\infty}\cap B_{v}$
we see that on $B_{v}$

(4.1)
$\\{g\geq\ell\\}_{\infty}\subseteq\\{g+p\geq\ell\\}_{\infty}\cup\\{\lvert
g-\ell\rvert\leq\lvert p\rvert\\}.$

Applying the same logic to the components of $\\{g\geq\ell\\}_{<\infty}\cap
B_{v}$ or $\\{g\leq\ell\\}\cap B_{v}$ we obtain

$(\\{g\geq\ell\\}_{\infty})^{c}\cap\\{\lvert g-\ell\rvert>\lvert
p\rvert\\}\subseteq(\\{g+p\geq\ell\\}_{\infty})^{c},$

or equivalently

$\\{g+p\geq\ell\\}_{\infty}\subseteq\\{g\geq\ell\\}_{\infty}\cup\\{\lvert
g-\ell\rvert\leq\lvert p\rvert\\}.$

Combined with (4.1), this shows that

$\lvert\mathrm{Vol}_{\infty}[B_{v},g]-\mathrm{Vol}_{\infty}[B_{v},g+p]\rvert\leq\mathrm{Vol}\big{[}B_{v}\cap\\{\lvert
g-\ell\rvert\leq\lvert p\rvert\\}\big{]}$

which verifies the third statement of the lemma.

It remains to prove the claim. Assume that $A$ is a component of
$\\{g\geq\ell\\}\cap B_{v}$ (the proof for the remaining case is near
identical). We define

$\epsilon=\min_{A_{1}\neq
A_{2}}\min_{t\in[0,1]}\mathrm{dist}(H(A_{1},t),H(A_{2},t))$

where $A_{1}$ and $A_{2}$ are components of $\\{g\geq\ell\\}\cap B_{v}$. In
other words $\epsilon$ is the minimum distance between the images under $H$ of
any two distinct excursion components. Since $H$ is continuous and
$H(\cdot,t)$ is a homeomorphism for each $t$, we see that $\epsilon>0$.

Let $x\in A\cap\\{\lvert g-\ell\rvert>\lvert p\rvert\\}$ so that
$x\in\\{g+tp\geq\ell\\}=H(\\{g\geq\ell\\},t)$ for all $t\in[0,1]$. Suppose
that $x\in H(A^{\prime},t^{\prime})$ for some excursion component
$A^{\prime}\neq A$ and $t^{\prime}\in(0,1]$. Then
$\mathrm{dist}(x,H(A,t^{\prime}))\geq\epsilon$ and so by the intermediate
value theorem there exists $t^{\prime\prime}\in(0,t^{\prime})$ such that
$\mathrm{dist}(x,H(A,t^{\prime\prime}))=\epsilon/2>0$. But then $x$ is not
contained in $H(\\{g\geq\ell\\},t^{\prime\prime})$ which yields a
contradiction. We conclude that $x\in H(A,t)$ for all $t\in[0,1]$, proving the
claim. ∎

Armed with this lemma, we can easily obtain moment bounds on the geometric
contributions to $\Delta_{v}$ using Gaussian tail inequalities:

######  Lemma 4.2.

Let $f$ satisfy Assumption 1.3 and let $\rho:\mathbb{R}^{d}\to\mathbb{R}$ be a
deterministic $C^{2}$ function. Then there exists an absolute constant $c>0$
such that for $w\in\mathbb{Z}^{d}$

$\mathbb{E}\Big{[}\mathrm{Vol}[B_{w}\cap\\{\lvert
f-\ell\rvert\leq\lvert\rho\rvert\\}]\Big{]}\leq c\|\rho\|_{C(B_{w})}.$

If $\rho$ also satisfies the conditions in Lemma 3.5, then for each $\delta>0$
there exists $c>0$ such that for $w\in\mathbb{Z}^{d}$

$\mathbb{E}\Big{[}\mathrm{Vol}[B_{w}\cap\\{\lvert f-\ell\rvert\leq\lvert
p_{0}+\rho\rvert\\}]\Big{]}\leq c(1+\lvert w\rvert)^{-\beta+\delta}.$

In particular, by stationarity, for any $v\in\mathbb{Z}^{d}$,

$\mathbb{E}\Big{[}\mathrm{Vol}[B_{w}\cap\\{\lvert f-\ell\rvert\leq\lvert
p_{v}\rvert\\}]\Big{]}\leq c(1+\lvert v-w\rvert)^{-\beta+\delta}.$

###### Proof.

By Fubini’s theorem

$\displaystyle\mathbb{E}\Big{[}\mathrm{Vol}[B_{w}\cap\\{\lvert
f-\ell\rvert\leq\lvert\rho\rvert\\}]\Big{]}=\int_{B_{w}}\mathbb{P}(\lvert
f(x)-\ell\rvert\leq\lvert\rho(x)\rvert)\;dx$
$\displaystyle\leq\int_{B_{w}}\sqrt{\frac{2}{\pi}}\lvert\rho(x)\rvert\;dx$

using the (standard Gaussian) density of $f(x)$. The right hand side is at
most $\sqrt{2/\pi}\|p\|_{C(B_{w})}$, which proves the first statement of the
lemma.

Applying Fubini’s theorem once more yields

(4.2) $\mathbb{E}\Big{[}\mathrm{Vol}[B_{w}\cap\\{\lvert f-\ell\rvert\leq\lvert
p_{0}+\rho\rvert\\}]\Big{]}=\int_{B_{w}}\mathbb{P}(\lvert
f(x)-\ell\rvert\leq\lvert p_{0}(x)+\rho(x)\rvert)\;dx.$

By the union bound, for any $\eta>0$

(4.3) $\displaystyle\mathbb{P}(\lvert f(x)-\ell\rvert\leq\lvert
p_{0}(x)+\rho(x)\rvert)$ $\displaystyle\leq\mathbb{P}(\lvert
f(x)-\ell\rvert<\eta)+\mathbb{P}(\lvert p_{0}(x)+\rho(x)\rvert>\eta)$
$\displaystyle\leq\sqrt{2/\pi}\eta+\mathbb{P}(\lvert
p_{0}(x)\rvert>\eta-\lvert\rho(x)\rvert).$

From the definition of $p_{0}$ (in (1.4)) and the decay of $q$ (i.e.
Assumption 1.3), $p_{0}(x)$ is normally distributed with mean zero and
variance

$2\int_{B_{0}}q^{2}(x-u)\;du\leq c(1+\lvert x\rvert)^{-2\beta}$

for some $c>0$. Setting $\eta=c(1+\lvert w\rvert)^{-\beta+\delta}$ (and
recalling that $\lvert\rho(x)\rvert<c(1+\lvert x\rvert)^{-\beta}$) we obtain
for $x\in B_{w}$

(4.4) $\mathbb{P}(\lvert p_{0}(x)\rvert>\eta-\lvert\rho(x)\rvert)\leq
c_{1}\exp\left(-c_{2}(1+\lvert w\rvert)^{2\delta}\right)$

for constants $c_{1},c_{2}>0$. Combining (4.2)-(4.4) proves the second
statement of the lemma. ∎

###### Proof of Theorem 1.9 (for $\star=\mathrm{Vol}$).

Taking $\mu:=\mu_{\mathrm{Vol}}$ it is enough to verify the four conditions of
Theorem 2.2. It is trivial that $\mu_{\mathrm{Vol}}$ has finite second moments
since $\mu_{\mathrm{Vol}}(v+\Lambda_{n})\leq(2n)^{d}$ and the remaining
conditions will follow if we can justify an application of Proposition 2.7.

Condition (2.15) follows immediately from Lemma 4.1. Given
$v,w\in\mathbb{Z}^{d}$ we define

$Y_{v}(w)=\mathrm{Vol}\big{[}B_{w}\cap\\{\lvert f-\ell\rvert\leq\lvert
p_{v}\rvert\\}\big{]}+\mathds{1}_{w\in\mathcal{U}_{\mathrm{FR}}(f,p_{v})}$

then by Lemma 4.1 we have $\lvert\Delta_{v}(B_{w})\rvert\leq Y_{v}(w)$,
verifying (2.16). By Lemma 4.2

$\mathbb{E}\left[\mathrm{Vol}\big{[}B_{w}\cap\\{\lvert f-\ell\rvert\leq\lvert
p_{v}\rvert\\}\big{]}^{2+\epsilon}\right]\leq\mathbb{E}\left[\mathrm{Vol}\big{[}B_{w}\cap\\{\lvert
f-\ell\rvert\leq\lvert p_{v}\rvert\\}\big{]}\right]\leq c(1+\lvert
v-w\rvert)^{-\beta+\delta}$

for any $\delta>0$ and some $c$ depending on $\delta$. Combined with Lemma 3.5
we see that

$\mathbb{E}[Y_{v}(w)+Y_{v}(w)^{2+\epsilon}]\leq c(1+\lvert
v-w\rvert)^{-\beta+\delta}.$

Since $\beta>3d$, by taking $\delta>0$ sufficiently small we verify (2.17) and
so we may apply Proposition 2.7 which completes the proof. ∎

###  4.2. Surface area

We now move on to consider the surface area functional. Our first order of
business is to control the magnitude of topological and geometric
contributions to $\Delta_{v}(B_{w})$. As in the previous subsection, we begin
with some deterministic arguments.

Let $g,p:\mathbb{R}^{d}\to\mathbb{R}$ be $C^{2}$ such that $g$ and $g+p$ have
no stratified critical points at level $\ell$ on any unit cube $B_{v}$ for
$v\in\mathbb{Z}^{d}$. For any finite union of unit cubes $D$, we define

$\mathrm{SA}_{\infty}[g,D]=\mathcal{H}^{d-1}[D\cap\partial(\\{g\geq\ell\\}_{\infty})]$

where $\mathcal{H}^{d-1}$ denotes $(d-1)$-dimensional Hausdorff measure (and
we define $\mathrm{SA}_{\infty}[g+p,D]$ analogously). We assume that the level
sets of $g$ and $g+p$ restricted to the boundaries of unit cubes have zero
surface area, that is

$\mathcal{H}^{d-1}[\\{g=\ell\\}\cap\partial
B_{v}]=\mathcal{H}^{d-1}[\\{g+p=\ell\\}\cap\partial B_{v}]=0\qquad\text{for
all }v\in\mathbb{Z}^{d}.$

This assumption will allow us to ignore such boundary components when using
additivity of $\mathrm{SA}_{\infty}$ over unions of cubes below and will of
course be satisfied with probability one for our intended application to the
Gaussian fields $f$ and $\widetilde{f}_{v}$.

The topological contribution to $\Delta_{v}(B_{w})$ is simple to control, as
we can bound the change in surface area for unstable unit cubes by the total
surface area for the original and perturbed functions:

######  Lemma 4.3.

Let $g,p$ be as above and $w\in\mathbb{Z}^{d}$, then

$\displaystyle\lvert\mathrm{SA}_{\infty}[g,B_{w}]-\mathrm{SA}_{\infty}[g+p,B_{w}]\rvert\leq\mathcal{H}^{d-1}[\\{g=\ell\\}\cap
B_{w}]+\mathcal{H}^{d-1}[\\{g+p=\ell\\}\cap B_{w}].$

###### Proof.

This follows from the triangle inequality along with the fact that the
boundaries of $\\{g\geq\ell\\}_{\infty}$ and $\\{g+p\geq\ell\\}_{\infty}$ are
subsets of $\\{g=\ell\\}$ and $\\{g+p=\ell\\}$ respectively. ∎

It remains to control the geometric contribution to changes in the surface
area. Since we are only interested in the boundary of
$\\{g\geq\ell\\}_{\infty}$ we cannot simply use the total change in area of
level sets. Instead we need to compare components of $\\{g=\ell\\}$ and
$\\{g+p=\ell\\}$ which are both contained in unbounded excursion sets. The
isotopy constructed in Lemma 3.3 allows us to do this.

Consider a unit cube $B_{w}$ which is finite-range stable for $(g,p)$ (i.e.
$w\notin\mathcal{U}_{\mathrm{FR}}(g,p)$). Let $\mathrm{Comp}$ denote the set
of components of $\\{g=\ell\\}\cap B_{w}$ and let $H$ be the stratified
isotopy of $B_{w}$ specified in Lemma 3.3.

######  Lemma 4.4.

Let $g,p$ be as above and $w\notin\mathcal{U}_{\mathrm{FR}}(g,p)$, then

$\lvert\mathrm{SA}_{\infty}[g,B_{w}]-\mathrm{SA}_{\infty}[g+p,B_{w}]\rvert\leq\sum_{L\in\mathrm{Comp}}\lvert\mathcal{H}^{d-1}[H(L,1)]-\mathcal{H}^{d-1}[L]\rvert.$

###### Proof.

By Lemma 3.3 we know that $L\in\mathrm{Comp}$ is a subset of
$\\{g\geq\ell\\}_{\infty}$ if and only if $H(L,1)$ is a subset of
$\\{g+p\geq\ell\\}_{\infty}$. Therefore

$\mathrm{SA}_{\infty}[g,B_{w}]-\mathrm{SA}_{\infty}[g+p,B_{w}]=\sum_{L\in\mathrm{Comp}\;:\;L\subseteq\\{g\geq\ell\\}_{\infty}}\mathcal{H}^{d-1}[L]-\mathcal{H}^{d-1}[H(L,1)].$

The statement of the lemma then follows from the triangle inequality and
adding the non-negative terms which correspond to
$L\subseteq\\{g\geq\ell\\}_{<\infty}$. ∎

Hence the geometric contribution to $\Delta_{v}(B_{w})$ is bounded by the
change in surface area for each component of the level set. Our strategy for
controlling the latter is the following: assuming that the level set
$\\{g=\ell\\}\cap B$ is somewhat stable compared to the perturbation $p$, we
show that $\\{g=\ell\\}\cap B$ can be projected normally onto
$\\{g+p=\ell\\}\cap B$ to yield a bijection (up to boundary effects) that
identifies components which correspond under $H$ (i.e. the bijection maps $L$
to $H(L,1)$). The area formula and the implicit function theorem then allow us
to bound the change in surface area for each component.

The stability assumptions we need are the following:

######  Assumption 4.5.

Let $w\in\mathbb{Z}^{d}$. We assume that $g,p:\mathbb{R}^{d}\to\mathbb{R}$ are
$C^{2}$ and satisfy

  1. (1)

$(g,p)$ is finite-range stable on $B_{w}$ at level $\ell$,

  2. (2)

for $x\in B_{w}$, if $g(x)=\ell$ then $\lvert\nabla g(x)\rvert>A_{1}$,

  3. (3)

$\lvert\nabla^{2}g(x)\rvert<A_{2}$ for all $x\in B_{w}$,

  4. (4)

$\|p\|_{C^{1}(B)}<A_{0}$.

for $A_{1}\in(0,1)$, $A_{2}\geq 1$ and $A_{0}<A_{1}^{2}/(C_{d}A_{2})$ where
$C_{d}>10$ depends only on the dimension $d$ and will be specified in the
proof of Lemma 6.3.

Given functions $g,p$ and constants $A_{0},A_{1},A_{2}$ as above, we define

$\mathcal{U}_{\mathrm{SA}}(g,p):=\left\\{w\in\mathbb{Z}^{d}\;\middle|\;\inf_{x\in
B_{w}:g(x)=\ell}\lvert\nabla g(x)\rvert<A_{1}\text{ or
}\sup_{B_{w}}\lvert\nabla^{2}g\rvert>A_{2}\text{ or
}\|p\|_{C^{1}(B_{w})}>A_{0}\right\\}.$

The geometric contribution to the change in the surface area functional can
then be bounded (using the argument described above) as follows:

######  Lemma 4.6.

Let $g,p:\mathbb{R}^{d}\to\mathbb{R}$ be $C^{2}$ functions such that $g$ and
$g+p$ have no stratified critical points at level $\ell$ on any unit cube
$B_{v}$ for $v\in\mathbb{Z}^{d}$. If
$w\notin\mathcal{U}_{\mathrm{FR}}(g,p)\cup\mathcal{U}_{\mathrm{SA}}(g,p)$,
then

$\displaystyle\lvert\mathrm{SA}_{\infty}[g,B]-\mathrm{SA}_{\infty}[g+p,B]\rvert\leq$
$\displaystyle
C_{d}\frac{A_{2}A_{0}}{A_{1}^{2}}\mathcal{H}^{d-1}[\\{g=\ell\\}\cap B]$
$\displaystyle+2\mathcal{H}^{d-1}\Big{[}\\{g=\ell\\}\cap(\partial
B)_{+\frac{7A_{0}}{A_{1}}}\Big{]}$
$\displaystyle+2\mathcal{H}^{d-1}\Big{[}\\{g+p=\ell\\}\cap(\partial
B)_{+\frac{7A_{0}}{A_{1}}}\Big{]}$

where $(\partial B)_{+a}:=\\{x\in B\;|\;\mathrm{dist}(x,\partial B)\leq a\\}$.

The proof of this lemma is given in Section 6.

We will require one more ingredient to control the magnitude of
topological/geometric contributions: a moment bound for (total) surface area.

######  Proposition 4.7.

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a stationary $C^{k}$ Gaussian field
($k\geq 2$) with spectral density and let $\rho:\mathbb{R}^{d}\to\mathbb{R}$
be a deterministic $C^{k}$ function, then

$\mathbb{E}\Big{[}\mathcal{H}^{d-1}[B_{0}\cap\\{f+\rho=\ell\\}]^{k^{\prime}}\Big{]}\leq
C<\infty$

where $k^{\prime}=\frac{k(k+1)}{2}-2$ and $C\in(0,\infty)$ depends only on
$\|\rho\|_{C^{k}(B_{0})}$ and the distribution of $f$.

Furthermore for any compact set $E$

$\mathbb{E}\Big{[}\mathcal{H}^{d-1}[E\cap\\{f+\rho=\ell\\}]\Big{]}\leq
C(1+\|\rho\|_{C^{1}(E)})\mathrm{Vol}[E]$

where $C\in(0,\infty)$ depends only on the distribution of $f$.

###### Proof.

The first statement is a direct application of [2, Theorem 5.2].

By the Kac-Rice theorem [3, Theorem 6.8 and Proposition 6.12]

(4.5)
$\mathbb{E}\Big{[}\mathcal{H}^{d-1}[E\cap\\{f+\rho=\ell\\}]\Big{]}=\int_{E}\mathbb{E}\Big{[}\lvert\nabla(f+\rho)(x)\rvert\;\Big{|}\;(f+\rho)(x)=\ell\Big{]}p_{(f+\rho)(x)}(\ell)\;dx$

where $p_{(f+\rho)(x)}$ denotes the density of the Gaussian random variable
$(f+\rho)(x)$. Since $f$ is stationary, this density is bounded above
uniformly in $x$ by a constant depending only on $\mathrm{Var}[f(0)]$. By the
triangle inequality and the fact that $\rho$ is deterministic

(4.6)
$\mathbb{E}\Big{[}\lvert\nabla(f+\rho)(x)\rvert\;\Big{|}\;(f+\rho)(x)=\ell\Big{]}\leq\|\rho\|_{C^{1}(E)}+\mathbb{E}\Big{[}\lvert\nabla
f(x)\rvert\;\Big{|}\;f(x)=\ell-\rho(x)\Big{]}.$

For any $C^{1}$ Gaussian field with constant variance, the value of the field
at a point is independent of its gradient at the same point (this is a
standard fact in the literature, see [1, Section 5.6]). Therefore

(4.7) $\mathbb{E}\Big{[}\lvert\nabla
f(x)\rvert\;\Big{|}\;f(x)=\ell-\rho(x)\Big{]}=\mathbb{E}\Big{[}\lvert\nabla
f(x)\rvert\Big{]}=\mathbb{E}\Big{[}\lvert\nabla f(0)\rvert\Big{]}.$

Combining (4.5)-(4.7) proves the second statement of the proposition. ∎

We can now combine all of these estimates to justify the conditions of
Proposition 2.7 for $\mu=\mu_{\mathrm{SA}}$. In the following two lemmas we
will give slightly more general statements which will be useful later when
proving positivity of the limiting variance (Theorem 1.10).

######  Lemma 4.8.

Let $f$ satisfy Assumption 1.3 and $\rho:\mathbb{R}^{d}\to\mathbb{R}$ be a
deterministic $C^{2}$ function. The following holds with probability one: for
any distinct $v_{1},\dots,v_{n}\in\mathbb{Z}^{d}$

$\mathrm{SA}_{\infty}\left[f+\rho,\bigcup_{i=1}^{n}B_{v_{i}}\right]=\sum_{i=1}^{n}\mathrm{SA}_{\infty}\left[f+\rho,B_{v_{i}}\right].$

###### Proof.

Since $\mathrm{SA}_{\infty}[f+\rho,\cdot]$ is a measure, it is enough to show
that $\mathrm{SA}_{\infty}[f+\rho,\partial B_{v}]=0$ almost surely for any
$v\in\mathbb{Z}^{d}$. Fixing such a $v$, consider a stratum $S\subset B_{v}$
of dimension $d^{\prime}\leq d-1$. By the Kac-Rice theorem ([3, Theorem 6.8
and Proposition 6.12])

$\mathcal{H}^{d^{\prime}-1}[\\{f+\rho=\ell\\}\cap S]<\infty$

almost surely, and hence

$\mathrm{SA}_{\infty}[f+\rho,S]\leq\mathcal{H}^{d-1}[\\{f+\rho=\ell\\}\cap
S]=0.$

Summing over the different strata of $\partial B_{v}$ completes the proof. ∎

######  Lemma 4.9.

Let $f$ satisfy Assumptions 1.3 and 1.6 and let
$p:\mathbb{R}^{d}\to\mathbb{R}$ be a (possibly degenerate) $C^{k}$ Gaussian
field such that $f+p-\mathbb{E}[p]$ is stationary with a spectral density. For
$w\in\mathbb{Z}^{d}$ let

$M_{2}(w)=\sup_{\lvert\alpha\rvert,\lvert\gamma\rvert\leq
2}\sup_{x,y\in(w+\Lambda_{2})}\left\lvert\partial_{x}^{\alpha}\partial_{y}^{\gamma}\mathrm{Cov}[p(x),p(y)]\right\rvert.$

We assume that $M_{2}(w)$ and $\|\mathbb{E}[p]\|_{C^{1}(B_{w})}$ are bounded
uniformly over $w$. Then there exists a collection of random variables $Y(w)$
such that

(4.8)
$\lvert\mu_{\mathrm{SA}}(B_{w},f,\ell)-\mu_{\mathrm{SA}}(B_{w},f+p,\ell)\rvert\leq
Y(w)$

almost surely and for any $\delta>0$ there exists $C_{\delta}>0$ (independent
of $w$) such that

(4.9) $\mathbb{E}[Y(w)]\leq
C_{\delta}\mathbb{P}\left(w\in\mathcal{U}_{\mathrm{FR}}(f,p)\right)^{\frac{k^{\prime}-1}{k^{\prime}}}+C_{\delta}\left(\sqrt{M_{2}(w)}+\|\mathbb{E}[p]\|_{C^{1}(B_{w})}\right)^{\frac{k^{\prime}-1}{3k^{\prime}-1}-\delta}.$

Moreover if $p=p_{v}$ and $\beta>\beta_{\mathrm{SA}}(k)$ then there exists
$\epsilon,\delta,C>0$ (independent of $w$) such that

(4.10) $\mathbb{E}[Y(w)]+\mathbb{E}[Y(w)^{2+\epsilon}]\leq C(1+\lvert
v-w\rvert)^{-3d-\delta}$

######  Remark 4.10.

The assumption that $f+p-\mathbb{E}[p]$ be stationary with a spectral density
is much stronger than what is required for the conclusion of this lemma to
hold. We use this assumption only in order to apply Proposition 4.7 but [2,
Theorem 5.2] gives much weaker (albeit less concise) sufficient conditions for
the same result. We apply this lemma with $p$ equal to $p_{v}$, $p_{v}+\rho$
or $\rho$ where $\rho$ is some deterministic function. Therefore the current
statement is most convenient for our purposes.

###### Proof of Lemma 4.9.

Given $w\in\mathbb{Z}^{d}$, we define

$\sigma^{2}_{w,p}:=\sup_{\lvert\alpha\rvert\leq 1}\sup_{x\in
B_{w}}\mathrm{Var}[\partial^{\alpha}p(x)]\leq M_{2}(w)$

and in Assumption 4.5 set

(4.11) $\displaystyle
A_{0}=\|\mathbb{E}[p]\|_{C^{1}(B_{w})}+\mathbb{E}\left[\|p-\mathbb{E}[p]\|_{C^{1}(B_{w})}\right]+\sigma_{w,p}^{1-\delta},$
$\displaystyle
A_{1}=A_{0}^{\frac{k^{\prime}}{3k^{\prime}-1}-\delta}\qquad\text{and}\qquad
A_{2}=c_{d}^{\prime}A_{0}^{-2\delta}$

where $\delta,c_{d}^{\prime}>0$ will be specified later. By Kolmogorov’s
theorem [44, Appendix A.9] there exists an absolute constant $C>0$ such that

$\mathbb{E}\left[\max_{\lvert\alpha\rvert\leq 1}\sup_{x\in
B_{w}}\lvert\partial^{\alpha}p(x)-\mathbb{E}[\partial^{\alpha}p(x)]\rvert\right]\leq
C\sqrt{M_{2}(w)}.$

By assumption, $M_{2}(w)$ and $\|\mathbb{E}[p]\|_{C^{1}(B_{w})}$ are bounded
uniformly over $w$ by some constant $\widetilde{C}$, hence there exists
$C^{\prime}>0$ depending only on $\widetilde{C}$ such that

(4.12)
$A_{0}\leq\|\mathbb{E}[p]\|_{C^{1}(B_{w})}+C^{\prime}M_{2}(w)^{\frac{1-\delta}{2}}.$

In particular $A_{0}$ is bounded uniformly over $w$ by some constant
(depending only on $\widetilde{C}$). We note that
$A_{1}^{2}/(C_{d}A_{2})=A_{0}^{2k^{\prime}/(3k^{\prime}-1)}/(C_{d}c_{d}^{\prime})$
and therefore, since $A_{0}$ is bounded, by choosing $c_{d}^{\prime}$
sufficiently small we can ensure that this expression is greater than $A_{0}$
for all $w$ (as required in Assumption 4.5).

We next define

$Y(w)=\mathcal{L}(w)\mathds{1}_{w\in\mathcal{U}_{\mathrm{FR}}(f,p)\cup\mathcal{U}_{\mathrm{SA}}(f,p)}+\mathcal{L}_{\partial}(w)$

where

$\displaystyle\mathcal{L}(w)$
$\displaystyle:=\mathcal{H}^{d-1}[B_{w}\cap\\{f=\ell\\}]+\mathcal{H}^{d-1}[B_{w}\cap\\{f+p=\ell\\}]$
$\displaystyle\mathcal{L}_{\partial}(w)$
$\displaystyle:=s_{w}\mathcal{H}^{d-1}[B_{w}\cap\\{f=\ell\\}]+2\mathcal{H}^{d-1}[(\partial
B_{w})_{+t_{w}}\cap\\{f=\ell\\}]$
$\displaystyle\qquad\qquad+2\mathcal{H}^{d-1}[(\partial
B_{w})_{+t_{w}}\cap\\{f+p=\ell\\}]$

and

$s_{w}:=C_{d}\frac{A_{2}A_{0}}{A_{1}^{2}}=C_{d}c_{d}^{\prime}A_{0}^{\frac{k^{\prime}-1}{3k^{\prime}-1}}\qquad\text{and}\qquad
t_{w}:=\frac{7A_{0}}{A_{1}}=7A_{0}^{\frac{2k^{\prime}-1}{3k^{\prime}-1}+\delta}.$

Combining Lemmas 4.3 and 4.6 shows that (4.8) holds for our definition of
$Y(w)$. It remains to prove the moment bounds (4.9) and (4.10)

First we bound the probability that $w\in\mathcal{U}_{\mathrm{SA}}(f,p)$. The
Borell-TIS inequality [1, Theorem 2.1.1] states that for a continuous centred
Gaussian field $h(t)$ defined on a compact set $T$

$\mathbb{P}\left(\sup_{t\in T}h(t)-\mathbb{E}\Big{[}\sup_{t\in
T}h(t)\Big{]}>u\right)\leq e^{-\frac{u^{2}}{2\sigma_{T}^{2}}}$

for all $u>0$, where $\sigma_{T}^{2}:=\sup_{t\in T}\mathrm{Var}[h(t)]$.
Applying this to $\pm\partial^{\alpha}p$ for each $\lvert\alpha\rvert\leq 1$
and using the union bound, we have

$\displaystyle\mathbb{P}(\|p\|_{C^{1}(B_{w})}>A_{0})$
$\displaystyle\leq\mathbb{P}\left(\|p-\mathbb{E}[p]\|_{C^{1}(B_{w})}>\mathbb{E}\left[\|p-\mathbb{E}[p]\|_{C^{1}(B_{w})}\right]+\sigma_{w,p}^{1-\delta}\right)$
$\displaystyle\leq 2(d+1)e^{-\sigma_{w,p}^{-2\delta}/2}\leq
2(d+1)e^{-A_{0}^{-2\delta/(1-\delta)}/2}.$

A similar application of the Borell-TIS inequality to the second derivatives
of $f$ yields that

$\mathbb{P}\left(\sup_{x\in B_{w}}\lvert\nabla^{2}f(x)\rvert>A_{2}\right)\leq
C_{1}e^{-c(A_{0}^{-\delta}-C_{2})^{2}}$

provided that $A_{0}^{\delta}>C_{2}$, where $c,C_{1},C_{2}>0$ are constants
depending only on the distribution of $f$. By increasing $C_{1}$ if necessary,
we can relax the requirement that $A_{0}^{\delta}>C_{2}$.

Next we note that by Lemma 7 of [44] for any $\delta^{\prime}>0$ there exists
$C_{\delta^{\prime}}>0$ such that

$\mathbb{P}\left(\inf_{x\in B_{w}}\max\\{\lvert f(x)-\ell\rvert,\lvert\nabla
f(x)\rvert\\}<\tau\right)<C_{\delta^{\prime}}\tau^{1-\delta^{\prime}}$

for all $\tau>0$. Applying this with $\tau=A_{1}$ and $\delta^{\prime}$
sufficiently small ensures that the right-hand side is at most
$C_{\delta^{\prime}}A_{1}^{1-\delta^{\prime}}\leq
C^{\prime}_{\delta}A_{0}^{k^{\prime}/(3k^{\prime}-1)-2\delta}$. Combining the
three bounds above with the definition of $\mathcal{U}_{\mathrm{SA}}$, we have
that for all $w\in\mathbb{Z}^{d}$

(4.13) $\mathbb{P}(w\in\mathcal{U}_{\mathrm{SA}}(f,p))\leq
C_{\delta}A_{0}^{\frac{k^{\prime}}{3k^{\prime}-1}-2\delta}.$

We now work towards (4.9). By Hölder’s inequality, Proposition 4.7 and (4.13)

$\displaystyle\mathbb{E}\left[\mathcal{L}(w)\mathds{1}_{w\in\mathcal{U}_{\mathrm{FR}}\cup\mathcal{U}_{\mathrm{SA}}}\right]$
$\displaystyle\leq\mathbb{E}\left[\mathcal{L}(w)^{k^{\prime}}\right]^{\frac{1}{k^{\prime}}}\left(\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}})+\mathbb{P}(w\in\mathcal{U}_{\mathrm{SA}})\right)^{\frac{k^{\prime}-1}{k^{\prime}}}$
$\displaystyle\leq
C\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}})^{\frac{k^{\prime}-1}{k^{\prime}}}+C_{\delta}A_{0}^{\frac{k^{\prime}-1}{3k^{\prime}-1}-2\delta}$

where we have abbreviated
$\mathcal{U}_{\mathrm{FR}}:=\mathcal{U}_{\mathrm{FR}}(f,p)$ and
$\mathcal{U}_{\mathrm{SA}}:=\mathcal{U}_{\mathrm{SA}}(f,p)$. By the second
statement of Proposition 4.7

$\mathbb{E}[\mathcal{L}_{\partial}(w)]\leq C\max\\{s_{w},t_{w}\\}\leq
C^{\prime}A_{0}^{\frac{k^{\prime}-1}{3k^{\prime}-1}}.$

Since $\delta>0$ can be chosen arbitrarily small, combining the last two
equations with (4.12) (and the definition of $Y(w)$) proves (4.9).

Let us now assume that $p=p_{v}$ and $\beta>\beta_{\mathrm{SA}}(k)$. By
definition of $p_{v}$

$\mathrm{Cov}[p_{v}(x),p_{v}(y)]=2\int_{B_{v}}q(x-z)q(y-z)\;dz,$

so by Assumption 1.3 and (4.12)

$A_{0}\leq C^{\prime}M_{2}(w)^{\frac{1-\delta}{2}}\leq
C^{\prime\prime}(1+\lvert v-w\rvert)^{-\beta(1-\delta)}$

for all $w\in\mathbb{Z}^{d}$. By Lemma 3.5

$\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{v}))\leq c_{\delta}(1+\lvert
v-w\rvert)^{-\beta+\delta}$

for any $\delta>0$. Hence taking $\delta>0$ sufficiently small and using (4.9)
we have

$\displaystyle\mathbb{E}[Y(w)]$ $\displaystyle\leq C_{\delta}(1+\lvert
v-w\rvert)^{-\beta\left(\frac{k^{\prime}-1}{3k^{\prime}-1}-\delta\right)}\leq
C_{\delta}(1+\lvert v-w\rvert)^{-4d}$

where the final inequality follows upon taking $\delta>0$ sufficiently small
since $\beta>\beta_{\mathrm{SA}}(k)=6d\frac{3k^{\prime}-1}{k^{\prime}-2}$.
This verifies the first part of (4.10).

Bounding the expectation of $Y(w)^{2+\epsilon}$ uses very similar arguments:
by Hölder’s inequality, Proposition 4.7, Lemma 3.5 and (4.13)

(4.14)
$\displaystyle\mathbb{E}[\mathcal{L}(w)^{2+\epsilon}\mathds{1}_{w\in\mathcal{U}_{\mathrm{SA}}\cup\mathcal{U}_{\mathrm{FR}}}]$
$\displaystyle\leq\mathbb{E}[\mathcal{L}(w)^{k^{\prime}}]^{\frac{2+\epsilon}{k^{\prime}}}\left(\mathbb{P}(w\in\mathcal{U}_{\mathrm{FR}})+\mathbb{P}(w\in\mathcal{U}_{\mathrm{SA}})\right)^{\frac{k^{\prime}-2-\epsilon}{k^{\prime}}}$
$\displaystyle\leq C_{\delta}(1+\lvert
v-w\rvert)^{-(\beta-\delta)\frac{k^{\prime}-2-\epsilon}{k^{\prime}}}+C_{\delta}A_{0}^{\frac{k^{\prime}-2-\epsilon}{3k^{\prime}-1}-2\delta}$
$\displaystyle\leq C_{\delta}^{\prime}(1+\lvert
v-w\rvert)^{-\beta\left(\frac{k^{\prime}-2-\epsilon}{3k^{\prime}-1}-2\delta\right)(1-\delta)}.$

Once again, since $\beta>\beta_{\mathrm{SA}}(k)$ we can ensure that the above
exponent is strictly less than $-3d$ by taking $\epsilon$ and $\delta$
sufficiently small. By Proposition 4.7

(4.15)
$\displaystyle\mathbb{E}[(s_{w}\mathcal{H}^{d-1}[B_{w}\cap\\{f=\ell\\}])^{2+\epsilon}]\leq
Cs_{w}^{2+\epsilon}$ $\displaystyle\leq C^{\prime}(1+\lvert
v-w\rvert)^{-\beta(1-\delta)(2+\epsilon)\frac{k^{\prime}-1}{3k^{\prime}-1}}$
$\displaystyle\leq C^{\prime\prime}(1+\lvert v-w\rvert)^{-4d}.$

We temporarily denote $X:=\mathcal{H}^{d-1}[(\partial
B_{w})_{+t_{w}}\cap\\{f=\ell\\}]$ and define
$\theta=\frac{k^{\prime}-2-\epsilon}{(k^{\prime}-1)(2+\epsilon)}$. Then by
Littlewood’s inequality for $L^{p}$ spaces and Proposition 4.7

(4.16)
$\mathbb{E}\left[X^{2+\epsilon}\right]\leq\mathbb{E}\left[X^{k^{\prime}}\right]^{\frac{(1-\theta)(2+\epsilon)}{k^{\prime}}}\mathbb{E}\left[X\right]^{\theta(2+\epsilon)}\leq
Ct_{w}^{\theta(2+\epsilon)}\leq C^{\prime}(1+\lvert
v-w\rvert)^{-\beta\left(\frac{k^{\prime}-2-\epsilon}{3k^{\prime}-1}-2\delta\right)}.$

As before we can ensure that the exponent of the right-most term is strictly
less than $-3d$ by taking $\epsilon,\delta>0$ sufficiently small. Combining
(4.14)-(4.16) shows that

$\mathbb{E}[Y(w)^{2+\epsilon}]\leq C(1+\lvert v-w\rvert)^{-3d-\delta}$

for some $\epsilon,\delta>0$ and all $w\in\mathbb{Z}^{d}$, which verifies the
second part of (4.10) and hence completes the proof of the lemma. ∎

###### Proof of Theorem 1.9 (for $\star=\mathrm{SA}$).

The surface area functional $\mu_{\mathrm{SA}}$ is dominated by the total area
of the level set and so has finite second moments courtesy of Proposition 4.7.
Lemmas 4.8 and 4.9 verify the assumptions of Proposition 2.7 so we see that
the stabilisation, bounded moment and moment decay conditions hold. Hence an
application of Theorem 2.2 yields the desired result. ∎

###  4.3. Euler characteristic

Turning to the Euler characteristic, we start by describing some fundamental
properties of this functional which we make use of below. Readers who are
unfamiliar with the Euler characteristic may find it helpful to consult [1,
Chapter 6] for an overview (in the setting of smooth Gaussian fields).

The Euler characteristic is well defined for a class of sets known as ‘basic
complexes’. The precise definition is given in [1, Chapter 6], however for our
purposes it will be sufficient to know that this class includes the excursion
sets of functions satisfying certain regularity conditions:

######  Lemma 4.11.

Let $D\subset\mathbb{R}^{d}$ be the union of a finite number of closed faces
(of any dimension) of unit cubes. If $g:D\to\mathbb{R}$ is suitably regular
(as defined in [1, Chapter 6]) on $D$ at level $\ell$ then any union of
components of $\\{g\geq\ell\\}\cap D$ is a basic complex.

###### Proof.

This result is proven for the closed face of a cube in [1, Theorem 6.2.2], the
proof generalises trivially to a union of such faces. ∎

######  Lemma 4.12.

Let $f$ satisfy Assumption 1.3 then with probability one $f$ is suitably
regular on any union of closed faces of unit cubes at level $\ell$.

###### Proof.

This is given for a single face by [1, Theorem 11.3.3]. The lemma holds since
the number of faces of unit cubes is countable. ∎

The Euler characteristic is well known to be additive and topologically
invariant:

######  Lemma 4.13.

Let $A_{1}$ and $A_{2}$ be basic complexes.

  1. (1)

If $A_{1}\cup A_{2}$ and $A_{1}\cap A_{2}$ are also basic complexes then

$\mathrm{EC}[A_{1}\cup
A_{2}]=\mathrm{EC}[A_{1}]+\mathrm{EC}[A_{2}]-\mathrm{EC}[A_{1}\cap A_{2}]$

  2. (2)

If $A_{1}$ is homotopy equivalent to $A_{2}$ then
$\mathrm{EC}[A_{1}]=\mathrm{EC}[A_{2}]$.

###### Proof.

The first property is proven in [1, Theorem 6.1.1] the second property in [26,
Theorem 2.44]. ∎

We will also make use of the fact that the Euler characteristic of an
excursion set can be bounded in terms of the number of critical points of the
underlying function:

######  Lemma 4.14.

Let $D$ be a union of closed faces of the unit cubes
$B_{v_{1}},\dots,B_{v_{n}}$ and $g:D\to\mathbb{R}$ be suitably regular on $D$
at level $\ell$. Let $U$ be a union of connected components of
$\\{g\geq\ell\\}\cap D$, then there exists $c_{d}>0$ depending only on $d$
such that

$\lvert\mathrm{EC}[U]\rvert\leq
c_{d}\sum_{i=1}^{n}\overline{N}_{\mathrm{Crit}}(B_{v_{i}},g)$

where $\overline{N}_{\mathrm{Crit}}(A,g)$ denotes the number of stratified
critical points of $g$ in $A$.

###### Proof.

Let $F$ be one of the faces which makes up $D$, then [1, Section 9.4] gives an
expression for $\mathrm{EC}[\\{g\geq\ell\\}\cap F]$ as an alternating sum of
stratified critical points of $g$ contained in $\\{g\geq\ell\\}\cap F$. In
particular taking this sum over the components of $\\{g\geq\ell\\}\cap F$ in
$U$, this implies that $\lvert\mathrm{EC}[U\cap
F]\rvert\leq\overline{N}_{\mathrm{Crit}}(F,g)$.

Now let $F_{1},\dots,F_{m}$ denote the faces which make up $D$ (we assume that
the $F_{i}$ are distinct but not necessarily disjoint). Iterating the additive
property of the Euler characteristic (Lemma 4.13) we have

(4.17) $\displaystyle\mathrm{EC}[U]=\mathrm{EC}\left[\bigcup_{i=1}^{m}U\cap
F_{i}\right]=\sum_{i=1}^{m}\mathrm{EC}[U\cap F_{i}]-$
$\displaystyle\sum_{i\neq j}\mathrm{EC}[U\cap F_{i}\cap F_{j}]+\dots$
$\displaystyle\quad+(-1)^{m-1}\mathrm{EC}[U\cap F_{1}\cap\dots\cap F_{m}].$

The $k$-th term of this expression is equal to

$\displaystyle(-1)^{k-1}\sum_{j_{1},\dots,j_{k}}\mathrm{EC}[U\cap
F_{j_{1}}\cap\dots\cap
F_{j_{k}}]=(-1)^{k-1}\sum_{i=1}^{n}\sum_{F_{j_{1}}\subseteq
B_{v_{i}}}\sum_{j_{2},\dots,j_{k}}\mathrm{EC}[U\cap F_{j_{1}}\cap\dots\cap
F_{j_{k}}]$

where $j_{1},\dots,j_{k}$ are summed over distinct values. Since
$F_{j_{1}}\cap\dots\cap F_{j_{k}}$ is a closed face (possibly empty) which is
a subset of $B_{v_{i}}$, the absolute value of the above term is at most

(4.18) $\sum_{i=1}^{n}\sum_{F_{j_{1}}\subset
B_{v_{i}}}\sum_{j_{2},\dots,j_{k}}\overline{N}_{\mathrm{Crit}}(B_{v_{i}},g)\leq
c_{d}\sum_{i=1}^{n}\overline{N}_{\mathrm{Crit}}(B_{v_{i}},g)$

using the fact that the number of distinct faces of unit cubes which intersect
$B_{v_{i}}$ is bounded by a constant depending only on $d$. For the same
reason, if $k>c_{d}$ then

$\mathrm{EC}[U\cap F_{j_{1}}\cap\dots\cap F_{j_{k}}]=\mathrm{EC}[\emptyset]=0$

and so at most $c_{d}$ of the sums on the right hand side of (4.17) are non-
zero. Combining this with the bound in (4.18) proves the lemma. ∎

With these preliminary results we can now decompose the change in the Euler
characteristic under perturbation. Since the Euler characteristic of a set
depends only on its topology (Lemma 4.13) there will be no geometric
contributions to $\Delta_{v}(\cdot)$, which simplifies our arguments somewhat
compared to the previous subsection.

######  Lemma 4.15.

If $g$ and $g+p$ are suitably regular on $\Lambda_{n}$ then

$\lvert\mathrm{EC}_{\infty}[\Lambda_{n},g]-\mathrm{EC}_{\infty}[\Lambda_{n},g+p]\rvert\leq
c_{d}\sum_{w\in\mathbb{Z}^{d}\cap\Lambda_{n}}\big{(}\overline{N}_{\mathrm{Crit}}(B_{w},g)+\overline{N}_{\mathrm{Crit}}(B_{w},g+p)\big{)}\mathds{1}_{w\in\mathcal{U}_{\mathrm{FR}(g,p)}}.$

###### Proof.

We define the unstable subset of $\Lambda_{n}$ as

$Q:=\Lambda_{n}\cap\bigcup_{w\in\mathcal{U}_{\mathrm{FR}}(g,p)}B_{w}$

then by the first point of Lemma 4.13, for $h=g,g+p$

(4.19)
$\mathrm{EC}_{\infty}[\Lambda_{n},h]=\mathrm{EC}_{\infty}[\overline{\Lambda_{n}\setminus
Q},h]+\mathrm{EC}_{\infty}[Q,h]-\mathrm{EC}_{\infty}[\overline{\Lambda_{n}\setminus
Q}\cap Q,h].$

By Lemma 3.3, $\\{g\geq\ell\\}_{\infty}\cap\overline{\Lambda_{n}\setminus Q}$
is homeomorphic to
$\\{g+p\geq\ell\\}_{\infty}\cap\overline{\Lambda_{n}\setminus Q}$ and so by
Lemma 4.13

(4.20) $\mathrm{EC}_{\infty}[\overline{\Lambda_{n}\setminus
Q},g]=\mathrm{EC}_{\infty}[\overline{\Lambda_{n}\setminus Q},g+p].$

By Lemma 4.14, for $h=g,g+p$

$\lvert\mathrm{EC}_{\infty}[Q,h]\rvert,\lvert\mathrm{EC}_{\infty}[\overline{\Lambda_{n}\setminus
Q}\cap Q,h]\rvert\leq
c_{d}\sum_{w\in\Lambda_{n}\cap\mathcal{U}_{\mathrm{FR}}(g,p)}\overline{N}_{\mathrm{Crit}}(B_{w},h).$

Combining this with (4.19) and (4.20) proves the lemma. ∎

With these deterministic tools we can now prove the stabilisation property for
$\mu=\mu_{\mathrm{EC}}$.

######  Lemma 4.16.

Let $f$ satisfy Assumptions 1.3 and 1.6, then $\mu_{\mathrm{EC}}$ satisfies
the stabilisation condition in Theorem 2.2.

###### Proof.

Let $D_{n}=v_{n}+\Lambda_{n}$ be given such that $D_{n}\to\mathbb{R}^{d}$. By
Lemma 3.5, with probability one we may choose $m$ and $N$ such that
$\mathcal{U}_{\mathrm{FR}}(f,p_{0})\subseteq\Lambda_{m-1}$ and
$\Lambda_{m}\subseteq\cap_{n>N}D_{n}$.

For $h=f,f+p_{0}$ by Lemma 4.13

(4.21)
$\mathrm{EC}_{\infty}[D_{n},h]=\mathrm{EC}_{\infty}[\Lambda_{m},h]+\mathrm{EC}_{\infty}[\overline{D_{n}\setminus\Lambda_{m}},h]-\mathrm{EC}_{\infty}[\partial\Lambda_{m},h].$

Since $\mathcal{U}_{\mathrm{FR}}(f,p_{0})\subseteq\Lambda_{m-1}$, by Lemmas
3.3 and 4.13

$\mathrm{EC}_{\infty}[\overline{D_{n}\setminus\Lambda_{m}},f]=\mathrm{EC}_{\infty}[\overline{D_{n}\setminus\Lambda_{m}},f+p_{0}]\quad\text{and}\quad\mathrm{EC}_{\infty}[\partial\Lambda_{m},f]=\mathrm{EC}_{\infty}[\partial\Lambda_{m},f+p_{0}].$

Substituting into (4.21) we see that

$\Delta_{0}(D_{n})=\mathrm{EC}_{\infty}[D_{n},f]-\mathrm{EC}_{\infty}[D_{n},f+p_{0}]=\mathrm{EC}_{\infty}[\Lambda_{m},f]-\mathrm{EC}_{\infty}[\Lambda_{m},f+p_{0}]=:\Delta_{0}$

which is constant for all $n>N$, proving convergence. Moreover, for any other
sequence $D_{n}^{\prime}$ we will have $\Delta_{0}(D^{\prime}_{n})=\Delta_{0}$
for sufficiently large $n$, which verifies the stabilisation condition. ∎

To prove the the bounded moment and moment decay properties, via Proposition
2.7, we will require control of the moments of the number of critical points.

######  Proposition 4.17.

Let $f:\mathbb{R}^{d}\to\mathbb{R}$ be a stationary $C^{k}$ Gaussian field
($k\geq 2$) such that the support of its spectral measure contains an open
set. For $v\in\mathbb{Z}^{d}$ let $N_{\mathrm{Crit}}(B_{v})$ denote the number
of critical points of $f$ in $B_{v}$, then
$\mathbb{E}[N_{\mathrm{Crit}}(B_{v})^{k-1}]<\infty$.

###### Proof.

This is a special case of [22, Theorem 1.2]. ∎

Observe that the conditions of this result are satisfied for any field
satisfying Assumption 1.3 with the same value of $k$ since $K\in C^{2k+}$
ensures that $f\in C^{k}$ almost surely.

###### Proof of Theorem 1.9 (for $\star=\mathrm{EC}$).

Since the Euler characteristic of an excursion set is dominated by the number
of critical points (Lemma 4.14) which have finite second moments (Proposition
4.17) we see that $\mu_{\mathrm{EC}}$ has finite second moments. Lemma 4.16
verifies the stabilisation condition for $\mu_{\mathrm{EC}}$ so if we can
prove the bounded moments and moment decay conditions then an application of
Theorem 2.2 will yield Theorem 1.9 for $\star=\mathrm{EC}$.

Given $v,w\in\mathbb{Z}^{d}$ we let

$Y_{v}(w):=c_{d}\big{(}\overline{N}_{\mathrm{Crit}}(B_{w},f)+\overline{N}_{\mathrm{Crit}}(B_{w},f+p_{v})\big{)}\mathds{1}_{w\in\mathcal{U}_{\mathrm{FR}}(f,p_{v})}$

where $c_{d}>0$ is taken from the statement of Lemma 4.15. This lemma then
implies that

$\lvert\Delta_{v}(\Lambda_{n})\rvert\leq\sum_{w\in\mathbb{Z}^{d}\cap\Lambda_{n}}Y_{v}(w)$

for any $n\in\mathbb{N}$. Adopting the notation
$\overline{N}_{\mathrm{Crit}}(w):=\overline{N}_{\mathrm{Crit}}(B_{w},f)+\overline{N}_{\mathrm{Crit}}(B_{w},f+p_{v})$,
by Hölder’s inequality, Proposition 4.17 and Lemma 3.5

$\mathbb{E}\left[Y_{v}(w)^{2+\epsilon}\right]\leq
c^{\prime}_{d}\mathbb{E}\left[\overline{N}_{\mathrm{Crit}}(w)^{k-1}\right]^{\frac{2+\epsilon}{k-1}}\mathbb{P}\left(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{v})\right)^{\frac{k-3-\epsilon}{k-1}}\leq
C_{\delta}(1+\lvert v-w\rvert)^{-(\beta-\delta)\frac{k-3-\epsilon}{k-1}}$

for any $\delta>0$. Since $\beta>\frac{3(k-1)}{k-3}d$, taking $\epsilon$ and
$\delta$ sufficiently small ensures that the final exponent is less than
$-3d$. A similar argument yields the stronger bound

$\mathbb{E}\left[Y_{v}(w)\right]\leq
c^{\prime}_{d}\mathbb{E}\left[\overline{N}_{\mathrm{Crit}}(w)^{k-1}\right]^{\frac{1}{k-1}}\mathbb{P}\left(w\in\mathcal{U}_{\mathrm{FR}}(f,p_{v})\right)^{\frac{k-2}{k-1}}\leq
C_{\delta}(1+\lvert v-w\rvert)^{-(\beta-\delta)\frac{k-2}{k-1}}.$

Together the last three equations allow us to apply Proposition 2.7 which
shows that $\mu_{\mathrm{EC}}$ satisfies the bounded moment and moment decay
conditions as required. ∎

###  4.4. First-order behaviour and finitary CLT

Our characterisation of the first-order behaviour of the functionals
$\mu_{\star}$ (i.e. Proposition 1.8) follows from an ergodic argument.

###### Proof of Proposition 1.8 (for
$\star\in\\{\mathrm{Vol},\mathrm{SA}\\}$).

We recall the white noise process $W=(W_{v})_{v\in\mathbb{Z}^{d}}$ which
defines our field via $f=q\ast W$. Given $u\in\mathbb{Z}^{d}$ we let
$\tau_{u}$ denote translation by $u$ (i.e. $(\tau_{u}W)_{v}=W_{v-u}$). we also
define $T_{1},\dots,T_{d}$ to be translations by distance one in the positive
direction of each coordinate axis respectively. Since the $W_{v}$ are
independent and identically distributed, $T_{1},\dots,T_{d}$ are measure
preserving transformations of $W$ and the invariant $\sigma$-algebra for each
$T_{i}$ is trivial.

For $\star\in\\{\mathrm{Vol},\mathrm{SA}\\}$, we define
$F(W)=\mu_{\star}([0,1]^{d})=\mu_{\star}(B_{0})$. If $\star=\mathrm{Vol}$,