# Gaussian fields and percolation

Dmitry<EMAIL_ADDRESS>[

###### Abstract

In the last two decades there was a lot of progress in understanding the
geometry of smooth Gaussian fields. This survey aims to cover one particular
line of research: the large scale behaviour of level and excursion sets and
their (conjectured) connection to the percolation theory.

##  1 Introduction

Gaussian fields appear naturally in many areas of science (e.g. in quantum
chaos [43], medical imaging [89], oceanography [30, 54], cosmology [5] etc.).
They are also very interesting objects from a purely mathematical point of
view.

Our goal is to survey recent results aiming at understanding the large scale
structure of level sets of Gaussian fields and how it is connected to the
percolation theory.

###  1.1 Gaussian fields

We start with a very brief introduction to Gaussian fields, mostly in the
stationary case. We do not aim at comprehensive coverage. Moreover, most of
this section is not even completely rigorous it should be treated more like a
heuristics that is ultimately correct but requires some extra work. Our goal
here is just to explain how one could and should think of these fields. We
refer to proper books [1, 4, 20, 53] for more information about Gaussian
fields and their basic properties.

There are different ways to think about Gaussian fields. The first one is
‘probabilistic’. One says that a Gaussian field on an index set $X$ is a
collection of jointly Gaussian random variables $F_{x}=F(x)$ indexed by
elements of $X$. It is a well known fact that such collections can be uniquely
determined by their mean values $m(x)=\mathbb{E}[F(x)]$ and pairwise
covariances $K(x,y)=\mathbb{E}[F(x)F(y)]$. Since the mean value can be changed
to $0$ by subtracting the deterministic function $m(x)$, for the rest of this
paper we assume that all fields are centred i.e. $m(x)=0$ for all $x$. In this
definition of the field the nature of the index set $X$ is completely
irrelevant, but it becomes important when we ask whether such fields exist and
whether they could be constructed from $K(x,y)$. Kolmogorov’s heorem gives the
existence of such a field. It can be shown that if $K$ is sufficiently ‘nice’
then the field is also ‘nice’. In particular, the modulus of continuity of $F$
can be explicitly written in terms of $K$. We refer to [1, Section 1.3] for
precise statements and complete proofs. Another exposition that we really like
is the Appendix in [67] where Nazarov and Sodin gave a short exposition in
terms of the standard metric in $\mathbb{R}^{n}$ instead of the _canonical
metric_ $d(x,y)=(\mathbb{E}\left[(F(x)-F(y))^{2}\right])^{1/2}$. The canonical
metric has its own advantages in more general settings and it is clear how to
compare $d(x,y)$ and $|x-y|$ but we think that is more natural to use the
Euclidean metric for fields defined on domains in $\mathbb{R}^{n}$.

The other way of thinking, that we call ‘analytic’ emphasises how the field
depends on $x$. Roughly speaking, we think that $F(x)$ is a function drawn at
random from some space of functions. Alternatively, we can think that the
probability measure is identified with a Gaussian measure on some function
space. Probably the simplest way to construct a Gaussian function using this
approach is to consider a random series

$F(x)=\sum a_{i}\phi_{i}(x)$

where $a_{i}$ are i.i.d. standard normal variables and $\\{\phi_{i}(x)\\}$ is
a collection of linearly independent functions. Unless the collection is
finite it is not immediately obvious that such series converges in any
meaningful sense. Clearly, the answer depends on the collection of functions
$\phi_{i}$. We will discuss this question a bit later.

On a more formal level, one can think that a Gaussian field is a map
$\Omega\times X\rightarrow\mathbb{R}$, where $\Omega$ is the corresponding
probability space. The ‘probabilistic’ point of view emphasises the fact that
for each $x\in X$ the maps $F(\cdot,x):\Omega\rightarrow\mathbb{R}$ is a
Gaussian random variable. The ‘analytic’ viewpoint emphasises properties of
the function $F(\omega,\cdot):X\rightarrow\mathbb{R}$.

These two descriptions are related and it should be possible to switch from
one point of view to another depending on a particular question. Assuming that
the series $F=\sum a_{i}\phi_{i}$ converges and we can swap summation and
expectation, it is easy to see that

$K(x,y)=\mathbb{E}[F(x)F(y)]=\sum\phi_{i}(x)\phi_{i}(y).$ (1.1)

The other direction is less straightforward but still very well understood. In
this paper, we are interested only in the case $X=\mathbb{R}^{n}$ which
significantly simplifies things. We also restrict our attention to the
stationary case. This means that we want fields $F(\cdot)$ and $F(\cdot+z)$ to
have the same distribution for every $z\in\mathbb{R}^{n}$. In terms of the
covariance kernel $K$ it means

$K(x,y)=K(x+z,y+z),\quad\forall x,y,z\in\mathbb{R}^{n}.$

In particular, this means that we can write $K(x,y)=K(x-y)$ where, abusing
notations, we use $K$ for both functions of one and two variables. Since $K$
is a covariance kernel, it is a non-negative definite function. By Bochner’s
theorem [1, Theorem 5.4.1] there is a symmetric Borel measure $\rho$ (i.e.
$\rho(A)=\rho(-A)$ for every measurable $A$) such that

$K(x)=\int_{\mathbb{R}^{n}}e^{-2\pi ix\cdot t}\rho(dt).$

The measure $\rho$ is called the _spectral measure_ of the field $F$. Given
this measure, we can construct a Hilbert space

$L^{2}_{sym}(\rho)=\\{\psi(t)\in L^{2}(\rho):\ \psi(-t)=\bar{\psi}(t)\\}.$

This symmetry condition ensures that the corresponding Fourier transform

$\mathcal{F}(\psi)(x)=\int_{\mathbb{R}^{n}}\psi(t)e^{-2\pi ix\cdot t}\rho(dt)$

is a real-valued function for every $\psi\in L^{2}_{sym}$. We define the
Hilbert space

$H=\mathcal{F}L^{2}_{sym}(\rho)=\left\\{\int_{\mathbb{R}^{n}}e^{-2\pi ix\cdot
t}\psi(t)\rho(dt),\psi\in L^{2}_{sym}(\rho)\right\\}$

where the scalar product is inherited from $L^{2}$. This is a reproducing
kernel Hilbert space (RKHS) with the reproducing kernel $K$. Indeed, if
$\phi=\mathcal{F}\psi$ then

$\displaystyle\langle K_{y}(x),\phi(x)\rangle_{H}=$ $\displaystyle\langle
K(y-x),\phi(x)\rangle_{H}=\langle e^{2\pi iy\cdot
t},\psi(t)\rangle_{L^{2}(\rho)}$ $\displaystyle=$
$\displaystyle\int_{\mathbb{R}^{n}}e^{-2\pi iy\cdot
t}\psi(t)\rho(dt)=\phi(y).$

Given the Hilbert space $H$ one can construct the field as an isonormal
process or white noise in $H$. We refer to [70, Section 2.1] for more
information about isonormal processes. Let $\\{\psi_{n}\\}$ be any orthonormal
basis in $L^{2}_{sym}(\rho)$. We can formally define the white noise as

$W=W_{\rho}=W_{L^{2}_{sym}(\rho)}=\sum_{k=1}^{\infty}a_{k}\psi_{k}(t),$

where $a_{k}$ are independent standard normal variables. This series obviously
diverges in $L^{2}$, so one should treat it as a formal expression. If one
wants to be rigorous, then the white noise is a collection of centred Gaussian
random variables $Wh$ indexed by $h\in L^{2}_{sym}(\rho)$ such that

$\mathrm{Cov}(Wh_{1},Wh_{2})=\langle
h_{1},h_{2}\rangle_{L^{2}(\rho)}=\int_{\mathbb{R}^{n}}h_{1}(t)\bar{h}_{2}(t)\rho(dt).$

On the level of formal series, if $h_{j}=\sum_{k}b_{j,k}\psi_{k}$ for $j=1,2$,
then

$Wh_{j}=\langle W,h_{j}\rangle=\sum a_{k}b_{j,k},\qquad j=1,2$

and

$\displaystyle\mathrm{Cov}(Wh_{1},Wh_{2})=$ $\displaystyle\mathbb{E}\left[\sum
a_{k}b_{1,k}\sum a_{k}b_{2,k}\right]$ $\displaystyle=$ $\displaystyle\sum
b_{1,k}b_{2,k}=\langle h_{1},h_{2}\rangle.$

Using this language we can formally write

$F(x)=(\mathcal{F}W)(x)=We^{-2\pi ix\cdot t}=\int_{\mathbb{R}^{n}}e^{-2\pi
ix\cdot t}\sum a_{k}\psi_{k}(t)\rho(dt)=\sum a_{k}\phi_{k}(x),$

where $\phi_{k}=\mathcal{F}\psi_{k}$ form an orthonormal basis in $H$. As in
(1.1) we can see that $F$ is a Gaussian field with the covariance kernel
$K(x,y)$. In particular, we can see that it does not depend on our choice of
the orthonormal basis.

There is a particularly important case when the spectral measure $\rho$ is
continuous with respect to the Lebesgue measure. In this case, we can write
$\rho(dt)=\rho(t)dt$ and it is easy to see that

$W_{\rho}=\sqrt{\rho(t)}W_{dt}.$

In particular, in this case, we can rewrite everything in terms of the
standard $L^{2}$ space and obtain the following representation:

$F(x)=W*q=Wq(x-\cdot)$

where $*$ is the convolution, $W$ is the canonical white noise in $L^{2}$ and

$q=\int\exp(-2\pi ix\cdot t)\sqrt{\rho(t)}dt$

is a convolution square root of $K$ i.e. $K=q*q$. This representation is also
known as the _moving averages_ representation [1, Section 5.3].

Let us conclude this section by emphasising one more time that explanations in
this section are correct only on the ‘physics level of rigour’, but they could
be written absolutely rigorously.

###  1.2 Percolation

In this section, we give a _very_ brief introduction to percolation theory. A
proper introduction would require writing a whole book. Our goal is to
introduce the main notations and terms that we will be using throughout this
survey. More information about percolation can be found in the classical book
of Grimmett [40], Bollobás and Riordan present a different take on the subject
that also includes some results about the scaling limit of the critical
percolation. There are also good lecture notes by Werner [87] and Duminil-
Copin [31].

There are two main versions of the Bernoulli percolation: _bond_ percolation
and _site_ percolation. In both cases, we start with a graph. For the purpose
of this survey we restrict the attention to regular lattices in
$\mathbb{R}^{2}$. Two main examples are the square lattice and the triangular
lattice. We choose some parameter $p$ and independently declare edges (bond
percolation) or vertices (site percolation) open with probability $p$ or
closed with probability $1-p$. Percolation clusters are connected subgraphs
made of open bonds or sites. In the context of fields, the most relevant is
the bond percolation.

Quite often it is also convenient to consider simultaneously the graph and its
dual. When we declare a bond in the primary graph to be open, we declare the
dual bond closed and vice versa. This dual picture is particularly nice for
the square lattice since its dual graph is also the square lattice. See Figure
1 for an example.

Figure 1: Left: a part of the square lattice and its dual. Right: an example
of the critical percolation ($p=1/2$) on both lattices and curves separating
primary and dual clusters.

One of the first questions about percolation models is whether there is an
infinite cluster. We define $\theta(p)$ to be the probability that a fixed
vertex, say the origin, belongs to an infinite cluster. The critical
probability is

$p_{c}=\inf_{p\in[0,1]}\\{p:\theta(p)>0\\}.$

A priori, it is not obvious that for a given graph the critical $p_{c}$ is
non-trivial i.e. $0<p_{c}<1$. It is also not immediately clear whether there
is an infinite cluster at $p=p_{c}$ i.e whether $\theta(p_{c})=0$ or not.

For the square lattice $\mathbb{Z}^{2}$ the critical probability $p_{c}=1/2$.
The simplest way to argue that this must be the case is by self-duality. One
can argue that at $p=1/2$ percolations on the primal and dual lattices are the
same, hence at $1/2$ the model is self-dual and so it must be critical. There
are several other heuristics giving $p_{c}=1/2$.

######  Theorem 1.1 (Kesten [46]).

For the Bernoulli percolation on $\mathbb{Z}^{2}$ the critical probability
$p_{c}=1/2$. For $p\leq 1/2$ there is no infinite cluster with probability
one, for $p>1/2$ there is an infinite cluster with probability one.

By now there are several rigorous proofs of this result. The first proof is
due to Kesten. A simpler and shorter proof was given by Bollobás and Riordan
in [24].

The critical probability is lattice dependent, but it is believed that under
mild assumptions on the regularity of the lattice the large scale behaviour at
criticality is universal i.e. lattice independent. It is also conjectured that
critical percolation has a conformally invariant scaling limit. In 1999
Schramm [79] introduced a one-parameter family of random curves defined in
terms of the Loewner Evolution. They are called Stochastic or Schramm Loewner
Evolution (SLE) curves. It is conjectured that interfaces between clusters
(see Figure 1) converge to SLE(6) curves. Since then, the proof has been
simplified [7, 49] and the entire argument is very short and elegant. It has
been extended to the convergence of the complete collection of the percolation
interfaces [28] to a generalization of SLE that is called the _Conformal Loop
Ensemble_ or CLE. So far all proofs rely heavily on the structure of the
triangular lattice and all attempts to generalize it to different lattices
were futile. Some partial results were obtained by Beffara[9]. See also a
discussion of universality in [8].

##  2 Main examples

There are two main examples that we should keep in mind: the Random Plane Wave
(RPW) and the Bargmann-Fock field. Here we discuss their two-dimensional
versions, but they have natural higher-dimensional generalizations.

###  2.1 Random Plane Wave

Let us start with the formal definition. The RPW field can be defined as a
stationary Gaussian field in $\mathbb{R}^{2}$ with the covariance kernel
$K(x)=J_{0}(|x|)$, where $J_{0}$ is the $0$-th Bessel function. From this
description we can see that point-wise covariances go to zero as points move
away from each other, but the rate is quite slow since

$J_{0}(t)=\sqrt{\frac{2}{\pi t}}\cos\left(t-\frac{\pi}{4}\right)+O(t^{-3/2}).$

Another important observation is that $K$ is an oscillating function.

The corresponding spectral measure is the normalized arc-length on the unit
circle. This is a very singular measure, which is yet another indication that
covariance is slowly decaying. On the other hand, it is compactly supported,
hence the field is real analytic. Since the support is on the unit circle, the
field is an eigenfunction of the Laplacian. If one thinks naively of the white
noise in $L^{2}_{sym}(\rho)$ as a collection of independent Gaussians at each
point of the unit circle, then one can think that the field is a random linear
combination of plane waves with the same energy travelling in all possible
directions. In this sense, one can think that the RPW is a natural notion of a
‘typical’ eigenfunction of the Laplacian in $\mathbb{R}^{2}$.

Finally, the RPW has the following series representation

$F(x)=\sum_{-\infty}^{\infty}C_{n}J_{|n|}(|x|)e^{in\theta},$ (2.1)

where $C_{k}$ are independent _complex_ Gaussian random variables subject to
the condition $C_{-k}=\bar{C}_{k}$. See Figure 2 (left) for a sample of the
random plane wave.

Figure 2: The random plane wave (left) and the random spherical harmonic of
degree $100$ (right). Black and white are regions where the fields are
negative or positive.

This field also appears as the scaling limit of ‘narrow-band’ functions [90].
In the case of the sphere, eigenfunctions of the Laplacian are spherical
harmonics. For a fixed degree $n$ the space of spherical harmonics is of
dimension $2n+1$ and the corresponding eigenvalue is $-n(n+1)$. The eigenspace
is equipped with the natural $L^{2}$ metric. The random spherical harmonic of
degree $n$ is defined as the standard Gaussian vector in the eigenspace. To be
more precise, if we choose any orthonormal basis, for example, the standard
$Y_{n}^{k}$, then the random spherical harmonic of degree $n$ is

$F_{n}(\theta,\phi)=\sum_{k=-n}^{n}a_{n}^{k}Y_{n}^{k}(\theta,\phi),$

where $(\theta,\phi)$ are the standard spherical coordinates and $a_{n}^{k}$
are independent $N(0,1/(2n+1))$ random variables. With this normalization the
expected $L^{2}$ norm of $F$ is $1$. Figure 2 (right) shows an example of a
random spherical harmonic.

From the addition formula for the Legendre polynomials one can see that the
covariance kernel of this field is

$K_{n}(x,y)=\mathbb{E}[F_{n}(x)F_{n}(y)]=\frac{1}{2n+1}\sum_{k=-n}^{n}Y_{n}^{k}(x)Y_{n}^{k}(y)=P_{n}(\cos\theta(x,y)),$

where $x,y$ are two points on the unit sphere, $P_{n}$ is the Legendre
polynomial of degree $n$ and $\theta(x,y)$ is the angle (i.e spherical
distance) between $x$ and $y$.

From this formula one can immediately see that this is a (spherically)
stationary field. Alternatively, this could be seen from the invariance of the
eigenspace and $L^{2}$ norm under rotations.

The random spherical harmonics converge to the random plane wave as
$n\to\infty$. To be more precise, let $x_{0}$ be any point on the sphere and
$\exp_{x_{0}}:T_{x_{0}}S^{2}\rightarrow S^{2}$ be the exponential map from the
tangent plane to the sphere. Then we can define

$f_{n}(x)=F_{n}(\exp_{x_{0}}(x/n)).$

Note that we rescale by $n$ which is approximately the square root of the
eigenvalue. This means that we rescale so that the wavelength becomes of order
$1$. Clearly, the covariance of this field is

$P_{n}(\cos\theta(\exp_{x_{0}}(x/n),\exp{x_{0}}(y/n))).$

Since the exponential map is almost isometry near $x_{0}$, the spherical
distance between images is almost the distance between the points, hence
uniformly in $x$ and $y$ the covariance behaves as

$P_{n}\left(\cos\frac{|x-y|}{n}\right)\to J_{0}(|x-y|),$

where the limit follows from Hilb’s asymptotics for Legendre polynomials.

It is possible to define a similar field on any compact manifold. In the
generic case there are no large eigenspaces like in the spherical case.
Instead, we take $L^{2}$ normalised eigenfunctions of the Laplacian $\phi_{k}$
and consider

$f_{n}(x)=\sum_{k=n^{2}}^{n^{2}+n}a_{k}\phi_{k}(x)$

where $a_{k}$ are i.i.d $N(0,1/n)$. That is, like in the spherical case, we
take a linear combination of order $n$ eigenfunctions around $n^{2}$-th
eigenfunction. It is possible to show, see [90] that under some mild
assumptions of the manifold, a similarly rescaled field converges to the
random plane wave.

###  2.2 Bargmann-Fock Feild

We define the Bargmann-Fock field in $\mathbb{R}^{2}$ as

$F(x)=F(x_{1},x_{2})=e^{-|x|^{2}/2}\sum_{m,n=0}^{\infty}a_{n,m}\frac{1}{\sqrt{n!m!}}x_{1}^{n}x_{2}^{m}.$
(2.2)

It is easy to see that the covariance kernel is

$\displaystyle\mathbb{E}\left[F(x)F(y)\right]=$ $\displaystyle
e^{-|x|^{2}/2}e^{-|y|^{2}/2}\sum\frac{(x_{1}y_{1})^{n}(x_{2}y_{2})^{m}}{n!m!}$
$\displaystyle=$ $\displaystyle e^{-|x|^{2}/2}e^{-|y|^{2}/2}e^{x\cdot
y}=e^{-|x-y|^{2}/2}.$

Since the covariance depends on $|x-y|$ only, the field is stationary and
isotropic. Since the kernel is the Gaussian density function, the spectral
measure has also Gaussian-type density. It is possible to give a more direct
description of the corresponding Hilbert space, but at least formally, it
could be described as the only Hilbert space for which functions in (2.2) form
an orthonormal basis. This, space is very closely related to the space of
holomorphic functions known as Bargmann, Segal-Bargmann or Bargmann-Fock space
that appears in quantum mechanics [6]. See Figure 3 for a sample of the
Bargmann-Fock field.

Figure 3: Bargmann-Fock (left) and Kostlan ensemble of degree $300$ (right).
Black and white are regions where the fields are negative or positive.

Similarly to the RPW, the Bargmann-Fock field is a scaling limit of a certain
family of fields on the unit sphere (see the right plot in Figure 3). Let
$H_{n}$ be the space of all homogeneous polynomials of degree $n$ in
$\mathbb{R}^{3}$. We equip it with the scalar product such that the following
functions form an orthonormal basis

$\sqrt{\frac{n!}{\alpha_{1}!\alpha_{2}!\alpha_{3}!}}x^{\alpha},$

where $x^{\alpha}=x_{1}^{\alpha_{1}}x_{2}^{\alpha_{2}}x_{3}^{\alpha_{3}}$ and
$|\alpha|=\alpha_{1}+\alpha_{2}+\alpha_{3}=n$. Up to a constant factor the
corresponding scalar product is the same as the Bargmann-Fock product

$\langle
f_{1},f_{2}\rangle=\int_{\mathbb{C}^{3}}f_{1}(z)\bar{f}_{2}(z)e^{-\|z\|^{2}}dz.$

This is the real trace of the only structure on complex polynomials that is
invariant under the unitary group (although there are many real ensembles that
are invariant with respect to the orthogonal transformations [51]). Thus it is
possible to claim that this is the most natural Hilbert space structure on
$H_{n}$.

The scalar product on $H_{n}$ gives rise to a canonical Gaussian measure. The
corresponding field

$F_{n}(x)=\sum_{|\alpha|=n}a_{\alpha}\sqrt{\frac{n!}{\alpha_{1}!\alpha_{2}!\alpha_{3}!}}x^{\alpha}$

where $a_{\alpha}$ are independent standard Gaussian random variables. This
field is known as the Kostlan ensemble or real Fubini-Study ensemble. Direct
computation shows that the covariance kernel for this field is

$K_{n}(x,y)=\cos^{n}\theta(x,y).$

As before, we can use the exponential map to project this field to a tangent
plane

$f_{n}(x)=F_{n}(\exp_{x_{0}}(x/\sqrt{n})).$

Similarly, we can compute the asymptotic behaviour of the covariance

$\cos^{n}\theta(\exp_{x_{0}}(x/\sqrt{n}),\exp_{x_{0}}(y/\sqrt{n}))\approx\cos^{n}\left(\frac{|x-y|}{\sqrt{n}}\right)\to
e^{-|x-y|^{2}/2}.$

This implies that the scaling limit of the Kostlan ensemble is the Bargmann-
Fock field.

######  Remark 2.1.

As we have explained above, the Kostlan ensemble is the natural candidate for
a model of a ‘typical’ homogeneous polynomial of degree $n$. Hence, its zero
or nodal lines are the natural model of a ‘typical’ real projective variety of
degree $n$. Thus the results about these nodal lines could be interpreted as
the results about ‘typical’ projective varieties.

##  3 Number of level sets

We can think that a Gaussian field describes a landscape where $f(x)$ is the
height function. Using this analogy, it is very natural to represent the field
via the collection of its level sets $\\{f=\ell\\}$ or excursion sets
$\\{f\geq\ell\\}$. This is exactly how we use isolines on topographic maps.
Under mild regularity assumptions a typical level set is a collection of
smooth hypersurfaces since $f$ and $\nabla f$ can not vanish simultaneously.
This is known as Bulinskaya’s lemma. It was originally proved by Bulinskaya in
the one-dimensional case [27], see [1, Lemma 11.2.10] for a proof of the
general statement.

###  3.1 One-dimensional case

The one-dimensional case is particularly simple. In this case, level sets are
just isolated points and excursion sets are intervals between them. In the
context of Gaussian processes, the level sets are known as _level crossings_.
Quite often they are separated into up- and down-crossings. The study of level
sets, especially nodal sets, where $\ell=0$ has a long history and goes back
to the works of Kac [44] and Rice [74]. One of the main advantages is that the
number of level crossings is a _local_ observable. This means that the number
of level crossings in a disjoint union of two sets is equal to the sum of
level crossings inside these sets. In particular, this allows writing an
integral formula for moments of the number of level crossings. This formula is
now known as the Kac-Rice formula. It holds in much higher generality, but we
formulate only the simplest version. A detailed discussion of this formula and
its proof in a very general setting can be found in [1, Sections 11.2 and
11.5].

######  Theorem 3.1 (Kac-Rice Formula).

Let $f$ be a Gaussian process on an interval $I$. We assume that $f$ has
$C^{1}$-paths, that it $f\in C^{1}$ with probability one. We also assume that
for pairwise distinct points $t_{1},\dots,t_{k}\in I$ the joint distribution
of $(f(t_{1}),\dots,f(t_{k}))$ is non-degenerate. Let $N_{\ell}$ be the number
of points where $f(t)=\ell$, then the $k$th factorial moment

$\mathbb{E}[N_{\ell}^{[k]}]=E[N_{\ell}(N_{\ell}-1)\cdots(N_{\ell}-k+1)]$

is given by the following multiple integral

$\iint_{I^{k}}\mathbb{E}\left[\prod_{i=1}^{k}|f^{\prime}(t_{i})|\mid
f(t_{1})=\cdots=f(t_{k})=\ell\right]p_{t_{1},\dots,t_{k}}(\ell,\dots,\ell)\prod_{i=1}^{k}dt_{i},$
(3.1)

where $p_{t_{1},\dots,t_{k}}$ is the joint density of
$(f(t_{1}),\dots,f(t_{k}))$.

In particular,

$\mathbb{E}\left[N_{\ell}\right]=\int_{I}\mathbb{E}\left[|f^{\prime}(t)|\mid
f(t)=\ell\right]p_{t}(\ell)dt.$

One of the nice things about this formula is that for Gaussian processes the
conditional expectation can be computed in terms of the covariance kernel.
Indeed,

$(f^{\prime}(t_{1}),\dots,f^{\prime}(t_{k}),f(t_{1}),\dots,f(t_{k}))$

is a Gaussian vector. Its covariance can be written explicitly in terms of the
covariance kernel of $f$ and its derivatives. Using the Gaussian regression
formula we can explicitly compute the conditional expectation. Although in
theory it can be done, in practice, even with $k=2$ the resulting expression
could be quite complicated and the analysis of the multiple integral becomes
non-trivial.

There is one case when this formula becomes particularly simple. Let us assume
that $f$ is a centred stationary process with covariance kernel $K$, then the
law of $(f^{\prime}(t),f(t))$ does not depend on $t$. It is centred and its
covariance matrix is

$\begin{pmatrix}-K^{\prime\prime}(0)&0\\\ 0&K(0)\end{pmatrix}.$

We can immediately see that $f^{\prime}(0)$ and $f(0)$ are independent. So the
integral becomes

$\displaystyle|I|\mathbb{E}\left[|f^{\prime}(0)|\right]p_{f(0)}(\ell)=$
$\displaystyle|I|\sqrt{\frac{-2K^{\prime\prime}(0)}{\pi}}\frac{1}{\sqrt{2\pi
K(0)}}e^{-\ell^{2}/2}$ $\displaystyle=$
$\displaystyle\frac{|I|}{\pi}\sqrt{\frac{-K^{\prime\prime}(0)}{K(0)}}e^{-\ell^{2}/2}.$

###  3.2 Higher dimensions

In higher dimensions the situation is much more complicated. First of all, the
level sets now can have non-trivial geometry and topology. Moreover, even a
simple observable like the number of the connected components of a level set
is more complicated. The main difference is that it is no longer an additive
quantity, so there is no integral formula similar to the Kac-Rice formula to
compute it. One can still write a Kac-Rice formula in a higher dimensional
case, but it will not count the number of components, instead, it will compute
the volume of the level set. Since this is not the main focus of our survey,
we will not discuss it at length, but a lot is known about additive quantities
like volume. Their expected volume can be easily computed in the stationary
case. For some fields it is even possible to write good estimates on the
variance and even prove (central) limit theorems. Since it is only
tangentially related to our main question about the large scale behaviour of
level sets, we will not discuss this line of research any further. Instead, we
refer to a survey by Wigman [88] and lecture notes by Berzin, Latour and León
[18] and references wherein.

For a long time the question about the expected number of level sets was
completely out of reach. The breakthrough came in 2009 when Nazarov and Sodin
made an observation that the number of level sets is semi-local, that is the
number of ‘large’ level sets is small. Initially, they proved the law of large
numbers for random spherical harmonics.

######  Theorem 3.2 (Nazarov-Sodin [65]).

Let $f$ be the random spherical harmonic of degree $n$. Then there exists
$c>0$ such that for every $\epsilon>0$ we have

$\mathbb{P}\left[\left|\frac{N(f)}{n^{2}}-c\right|>\epsilon\right]\leq
C(\epsilon)e^{-c(\epsilon)n},$

where $N(f)$ is the number of nodal domains and $c(\epsilon)$ and
$C(\epsilon)$ are constants that depend on $\epsilon$ only.

In particular, this strong concentration theorem implies the law of large
numbers, namely that $N(f)/n^{2}\to a$ a.s.

Later, they extended this result to stationary fields in $\mathbb{R}^{n}$ that
satisfy rather mild assumptions:

  1. 1.

The spectral measure has finite $4$-th moment. This assumption implies that
the field almost surely is $C^{2-\epsilon}$ for every $\epsilon>0$.

  2. 2.

The spectral measure has no atoms. This implies ergodicity with respect to the
shifts.

  3. 3.

The spectral measure is not supported on a linear hyperplane. This implies
that the distribution of $\nabla f$ is non-degenerate.

######  Theorem 3.3 (Nazarov-Sodin [67]).

Let $f$ be a stationary field in $\mathbb{R}^{n}$ such that its spectral
measure satisfies the three assumptions above. Let $S$ be a bounded open
convex set containing the origin and $N_{S}(R,f)$ be the number of nodal
domains of $f$ that are contained in the rescaled set $R\cdot S$, then there
is $c\geq 0$ such that

$\frac{N_{S}(R,f)}{R^{n}\mathrm{Vol}(S)}\to c.$

The convergence is a.s. and in $L^{1}$. Moreover, under additional mild
assumptions $c>0$.

######  Remark 3.4.

Although the theorem above was stated for the number of nodal domains, the
same proof works for the number of level/excursion sets for all levels.

It is also possible to estimate the variance of the number of level/excursion
sets. There are two complementary results. In [68] Nazarov and Sodin linked
nodal lines of Gaussian fields to a percolation model on a certain degree four
graph and obtained a low bound on the number of nodal domains from the
variance of the number of percolation clusters. Their argument could be viewed
as a step towards justification of the Bogomolny-Schmit heuristics. As a
result, they have shown that if $f$ is a $C^{3}$ stationary field and its
covariance kernel decays polynomially with some positive exponent, then the
variance of the number of nodal domains in a ball of radius $R$ grows faster
than $R^{\sigma}$ for some positive $\sigma$. On one hand, the conclusion is
rather weak since $\sigma$ is not specified, on the other hand, the
assumptions of the field are absolutely minimal. In particular, the result is
applicable to the RPW.

At about the same time we obtained a complementary result. In [13] we have
shown that if $f$ satisfies some regularity assumptions, its covariance is
non-negative and decays sufficiently fast and the spectral density near the
origin is bounded away near the origin, then for a generic level the variance
of the number of level sets inside a ball of radius $R$ grows at least as
$R^{2}$. We refer to [13, Theorem 2.7] for the precise list of assumptions and
the precise explanation for which levels it holds. We would like to point out
that the proof _does not hold_ for $\ell=0$, our assumptions are much stronger
than that of [68] but our conclusion is also much stronger and it is sharp for
a wide class of fields.

Imposing even stronger assumption it is possible to prove a Central Limit
Theorem that is valid for all levels and in all dimensions [14].

######  Theorem 3.5.

Let $f$ be a stationary field in $\mathbb{R}^{d}$ with the covariance $K=q*q$.
Assume that $q\in C^{5}$, $q$ and its derivatives decay faster that
$|x|^{\beta}$ for some $\beta>9d$, then the variance of the number of
level/excursion sets $N(R,\ell)$ inside the cube $\Lambda_{R}=[-R,R]^{d}$
grows like the volume

$\frac{\mathrm{Var}\left[N(R,\ell)\right]}{\mathrm{Vol}(\Lambda_{R})}\to\sigma^{2},\qquad
R\to\infty$

and

$\frac{N(R,\ell)-\mathbb{E}\left[N(R,\ell)\right]}{\sqrt{\mathrm{Vol}(\Lambda_{R})}}\to
Z,$

where $Z$ is a normal random variable with variance $\sigma$. Moreover, if
$\int K>0$, then $\sigma>0$ for all $\ell$.

We do not believe that our assumptions are necessary. For example, the natural
bound on all moments of the number of critical points would allow to reduce
the decay rate to $\beta>3d$. There is no reason to believe that it is the
best possible result, most probably, the result should hold for $\beta>0$.
Similarly, we do not think that $C^{5}$ assumption is optimal.

##  4 Percolation Conjecture

###  4.1 Number of nodal domains for the random plane wave

The laws of large numbers discussed above led naturally to the question about
the nature of the constant $c$ mentioned above. Sometimes this constant is
referred to as the _Nazarov-Sodin_ constant, but it is important to remember
that it is really a functional that depends on the spectral measure and the
level.

The question about the number of nodal domains also appears in quantum chaos.
Physicists Blum, Gnutzmann and Smilansky [19] proposed that the nodal domain
statistics could be used as a criterion for quantum chaos. Same year Bogomolny
and Schmit [22] proposed a percolation model for the number of nodal domains
of the random plane wave. Later in [23] they used (non-rigorous) Harris
criterion [41] to strengthen the percolation conjecture. In [21] Bogomolny,
Dubertrand and Schmit gave further numerical support to the percolation
conjecture by showing numerically that nodal domains of the random plane waves
converge to SLE(6).

Roughly speaking in [22] Bogomolny and Schmit have argued that on average
nodal lines form a square lattice, but since nodal lines do not intersect
almost surely, these intersections should be resolved one way or another. By
symmetry, both possibilities are equally likely, so the nodal structure could
be identified with bond percolation on the square lattice with $p=1/2$. In the
case of the square lattice, the number of percolation clusters per vertex is
known explicitly. This gives a prediction for the number of nodal domains per
unit volume. This is essentially the constant appearing in Theorems 3.2 and
3.3. To be more precise, they have conjectured that the expected number of
nodal domains in a rescaled domain $R\cdot S$ is

$\frac{\mathrm{Area}(S)R^{2}}{2\pi}\frac{3\sqrt{3}-5}{\pi}\approx\frac{\mathrm{Area}(S)R^{2}}{2\pi}0.0624373.$

Moreover, they also claimed that the variance of the number of nodal domains
also scales as $R^{2}$. Their conjecture was supported by numerical
simulations.

Later, more careful computations [64, 50, 12] have shown that the number
$0.06243$ is not quite right, the real value is approximately $0.0589$. This
shows that the prediction of Bogomolny and Schmit is not quite right. On the
other hand, the number of percolation clusters per vertex is not a universal
quantity, it is lattice-dependent. Nodal lines are clearly not a small
perturbation of the square lattice, so there is no reason to believe that the
square lattice percolation would have exactly the same density of clusters.
Moreover, there is one more scaling in this argument. To compare the number of
clusters per vertex and the number of domains per unit volume, we have to find
the right relation between vertices and volume. Bogomolny-Schmit proposed to
relate the wavelength and the lattice mesh. One can also argue that it is
equally reasonable to relate the number of vertices and the number of critical
points (after all lattice faces play the role of local minima and maxima and
bond intersections are related to saddle points). This scaling gives a close,
but a different prediction. We would go even further and claim that one of the
really interesting questions is why the Bogomolny-Schmit prediction is off
just by $5\%$.

It is possible to interpret nodal domains of the random plane wave in terms of
a percolation-type model.

Figure 4: Graph of flow-lines (left), corresponding percolation (center) and
nodal domains (right).

Since the random plane wave is an eigenfunction of the Laplacian, all local
minima are negative and local maxima are positive. Let us consider gradient
flow lines from a saddle that go along eigenvectors of the Hessian. Two
ascending lines will terminate at two local maxima and two descending lines
will terminate at local minima. This way we have two planar graphs. The
‘positive’ graph has its vertices at local maxima and two maxima are connected
by a bond if there is a saddle such that its ascending flow lines terminate at
these maxima. Similarly, we define the ‘negative’ graph. These two graphs are
dual to each other and intersections of their bonds are saddles. See Figure 4
for a sample of these graphs. We do the following percolation on these graphs:
if the field is positive at a saddle, then we keep the ‘positive’ (or
ascending) bond and remove the negative one, if it is negative, then we keep
the ‘negative’ bond and remove the ‘positive’ one. Clearly, by symmetry, we
choose one or the other with the probability $1/2$. It is easy to see that the
resulting percolation cluster corresponds exactly to nodal domains.

On one hand, this is a bond percolation-type model, but it is a horrible one.
First of all, we do percolation on a random graph, secondly, signs of the
functions at different saddles are not independent (moreover, correlation
decays slowly since the Bessel function decays slowly) and finally, how we
choose bonds is correlated with the graph itself.

So far there was no progress in analysing this percolation model. On the other
hand, it is plausible, that this picture explains why Bogomolny-Schmit
prediction is so close to the true value. Since the ‘positive’ and ‘negative’
graphs have the same distribution, they are self-dual. In particular, on
average a lot of their parameters are the same as that of the square lattice.
For example, the average face is a quadrangle and the average degree is $4$.
Moreover, simulations suggest that the probability that a vertex has a large
degree has very light tails. Most of the percolation clusters are small, so
the number of percolation clusters is mostly defined by the local structure of
the graph. Probably this is the reason why cluster density for the square
lattice and for the flow-line graph are so close, but it is not clear whether
it is possible to make rigorous sense out of this hand-waving argument.

###  4.2 Universal observable

As we have explained above, the number of percolation clusters per vertex is
lattice-dependent and hence it is very hard to get it right. On the other
hand, $p=1/2$ is the critical probability for the bond percolation on the
square lattice. It is conjectured that all ‘global’ observable in critical
percolation models are universal, i.e. independent of the lattice. The
percolation interpretation of the nodal domains is clearly self-dual, so it is
very natural to assume that it is critical as well. It is quite natural to re-
interpret the claim of Bogomolny and Schmit and to conjecture that the nodal
domains of the random plane wave are in the same ‘universality class’ as the
critical percolation. In particular, to conjecture that many global
observables have the same behaviour.

In the critical percolation context, one of the most natural observables is
the crossing probability. That is the probability that in an $n\times\lambda
n$ rectangle there is an open cluster connecting the left side to the right
side. It is conjectured that this probability has a scaling limit as
$n\to\infty$ and the limit does not depend on the lattice. The limiting
probability is given by an explicit formula in terms of $\lambda$. This
formula was first predicted by Cardy [29] using the Conformal Field Theory. It
has been proven only in the case of the critical percolation on the triangular
lattice [83].

The same observable can be defined for the RPW or any other smooth field in
the plane. We ask whether there is a positive nodal domain connecting the left
and right sides of an $n\times\lambda n$ rectangle. Equivalently, this is a
probability that there is a curve crossing the rectangle such that the field
is positive along this curve. This crossing probability for the random plane
wave has been numerically computed in [12] and it matches Cardy’s formula very
well. This supports the following form of the Bogomolny-Schmit conjecture

######  Conjecture 4.1 (Bogomolny-Schmit conjecture).

All large scale connectivity properties of the RPW nodal lines and domains are
the same as for the critical percolation. In particular, all crossing events
have the same scaling limits. The collection of all nodal lines has a scaling
limit which is conformally invariant and given by the Conformal Loop Ensemble
CLE($6$).

It is also very natural to conjecture that the relationship between the RPW
and percolation goes beyond the nodal domains. If we consider some other level
$\ell\neq 0$, then the large scale properties of excursion sets should
correspond to the percolation with non-critical $p$. We call this statement
the _off-critical Bogomolny-Schmit conjecture_.

These two conjectures are still wide open and there was almost no progress in
understanding the connection between the RPW and percolation. There are some
numerical evidence and heuristics in support of this conjecture but there are
no rigorous results supporting it. On the other hand, there is one result that
is somewhat contrary to the off-critical conjecture. In [13] it was shown that
for non-zero $\ell$ the variance of the number of excursion sets in a square
of size $R$ grows at least as $R^{3}$ whereas for percolation the variance of
the number of clusters grows as $R^{2}$ both in critical and off-critical
case. It is important to note that the number of clusters is non-universal
(although the variance growth rate is) and is not directly related to the
large scale connectivity properties, so this result does not immediately
disprove the percolation conjecture, but it shows that _some_ large scale
observables behaves differently.

###  4.3 Generalized Bogomolny-Schmit conjecture

Over the last decade it became clear that the connection between the RPW and
percolation is not unique to the RPW, it seems to be true for a wide class of
smooth Gaussian fields. It is still not clear what is exactly the class of
fields for which the correspondence should hold, but we believe that some
statement along the following lines should hold.

######  Conjecture 4.2.

Let $f$ be a centred stationary Gaussian field in $\mathbb{R}^{2}$ satisfying
the following assumptions

  * •

With probability one $f$ is sufficiently smooth.

  * •

The distribution of $f$ and its derivatives is non-degenerate.

  * •

The covariance kernel is sufficiently symmetric.

  * •

The covariance kernel decays sufficiently fast.

  * •

The covariance kernel is (strongly) positive.

Then the nodal domains of $f$ should have the same large scale properties as
critical percolation clusters and excursion sets for non-zero levels should
behave like off-critical percolation clusters.

Some smoothness is clearly needed at least to guarantee that level sets are
smooth curves. A priori, it is not clear why the joint distribution of $f$ and
its derivatives at various points should be non-degenerate for the conjecture
to hold, but so far all partial results require some form of non-degeneracy.
On the other hand, it is not a very strong restriction. For example, if the
support of the spectral measure contains an open set, then one can usually
prove the required non-degeneracy without too much effort. It is more
complicated with fields like the RPW that have singular spectral measure.

Some symmetry is also clearly necessary, without it there are obvious
counterexamples. The exact requirement is not obvious. Currently, most of
results are valid under the assumption that $K(x_{1},x_{2})=K(x_{1},-x_{2})$
which is probably very close to the minimal requirement.

The decay of correlation is very natural: percolation is completely local,
different bonds are open or closed completely independent, so for the field it
is natural to require that distant parts are weakly correlated. What is the
minimal decay of correlation is not clear, but but there are reasons to
believe that kernel decaying faster than $|x|^{-\beta}$ for $\beta>2$ is
enough. This requirement will be discussed later.

So far, many results require either that $K\geq 0$ of that $K=q*q$ with $q\geq
0$ (strong positivity). The last assumption is also far from obvious and it is
not obvious whether it is required. In particular, the covariance kernel of
the RPW is not positive. This assumption is needed in many partial results
since it guaranteed that the field is positively associated (see Theorem 4.5
below) which allows using many percolation techniques. The strong positivity
assumption works very well with the moving averages representation $f=W*q$.

The fact that nodal lines should correspond to the critical percolation is not
really surprising. By symmetry of the normal distribution, $f$ and $-f$ have
the same distribution. Hence the distributions of $\\{f\leq 0\\}$ and
$\\{f\geq 0\\}$ are the same. This means that the model is self-dual, so it
should be critical. This is very similar to the heuristics implying that
$p_{c}=1/2$ for the bond percolation on $\mathbb{Z}^{2}$ (Kesten’s theorem
1.1).

Finally, we would like to point out that from this point of view the
Bogomolny-Schmit conjecture for the random plane wave looks even more
surprising. As we have mentioned above, its kernel is not positive, it decays
rather slowly and some of its derivatives are deterministically related since
it is an eigenfunction of the Laplacian. On the other hand, the Bargmann-Fock
field is real analytic, the joint distributions of various values and
derivatives are non-degenerate, and the covariance kernel is positive and
decays super-exponentially fast.

Let us conclude this section with two important non-examples.

######  Example 4.3.

Let us consider the field that is a version of the discrete white noise. We
tile the plane by unit squares. The field is constant within each square and
values at different squares are independent standard Gaussians. This field is
not stationary, but if we randomly shift tiling it will be stationary. The
nodal domains of this field are exactly the same as site percolation clusters
on the square lattice with $p=1/2$ which is _not critical_. So in this case
nodal domains do not behave like critical percolation.

######  Example 4.4.

The other example is arguable the most well-studied Gaussian field: the
Gaussian Free Field. This is a $\log$-correlated field. Since the covariance
kernel blows up on the diagonal, the field is not even a function, it is a
random distribution. Despite it being a distribution, there is a way to make
sense of its nodal lines and it has been shown [80] that these nodal lines are
Schramm-Loewner Evolution SLE($4$) curves while the percolation interfaces are
conjectured to converge to SLE($6$) curves, so in this case the nodal lines
_are conformally invariant_ but belong to a different universality class.

###  4.4 Previous results related to percolation

Long before the work of Bogomolny and Schmit there were several results about
percolating properties of the level sets of Gaussian fields. To be more
precise, one can ask whether the level/excursion sets of a Gaussian field are
almost surely bounded or not.

Probably the first result in this direction is a series of works of Molchanov
and Stepanov [57, 58, 59]. They were looking at a wide class of fields both on
$\mathbb{Z}^{d}$ and $\mathbb{R}^{d}$, not necessary Gaussian, and discussed
when the set $\\{f\leq\ell\\}$ contains almost surely a unique unbounded
component. In particular, in [58, Theorem 3.5] they have shown that if $f$ is
a continuous stationary Gaussian field with the covariance kernel decaying
sufficiently fast (essentially faster than $|x|^{-d}$), then there is a finite
level $\ell$ such that $\\{f\leq\ell\\}$ percolates a.s.

The second important result that we want to mention is a more recent one by
Alexander. Theorem 3.4 of [3] states that if $f$ is a stationary $C^{1}$
Gaussian field in $\mathbb{R}^{2}$ such that its covariance kernel is non-
negative and decays to zero at infinity, then all level lines are almost
surely bounded. In other words level lines never percolate.

Finally, we would like to mention one result that is seemingly unrelated. We
say that random variables $x_{i}$ are positively associated if

$\mathrm{Cov}\left[f(x_{1},\dots,x_{n}),g(x_{1},\dots,x_{n})\right]\geq 0$

for all functions $f$ and $g$ that are non-decreasing in each variable. In
[73] Pitt have proved the following theorem:

######  Theorem 4.5.

Jointly Gaussian random variables $x_{i}$ are positively associated if and
only if $\mathrm{Cov}\left[x_{i},x_{j}\right]\geq 0$ for all $i$ and $j$.

In other words, positively correlated Gaussian variables are positively
associated. Despite its simplicity, this is a surprisingly non-trivial
statement.

In the context of Gaussian fields we have the notion of increasing events.
These are the events such that if it holds for a field $f$, then it holds for
all larger fields. In particular, all connectivity events for excursion sets
are increasing events. Pitt’s theorem implies that if the covariance kernel is
non-negative, then all increasing events are positively correlated. Pitt’s
theorem is stated for finite-dimensional Gaussian vectors, but it is not very
hard to show that standard approximation techniques extend it to the field
setting (see [76, Lemma A12] for details).

This result is a field equivalent of the Harris [42] or Fortuin, Kasteleyn,
Ginibre (FKG) [36] inequalities that give positive correlations of increasing
events in percolation. Such inequalities play a very important role in
percolation theory since they allow to combine various crossing events in a
controlled way. Although there are some approaches to the percolation that do
not rely on Harris-FKG, it is still a fundamental part of the theory. This is
the main reason why the positive kernel is one of the assumptions in our
generalised Bogomolny-Schmit conjecture 4.2.

##  5 Many small steps towards the percolation conjecture

###  5.1 Locality

One of the defining features of percolation models is locality: different
bonds are open or closed independently. In particular, this means that
percolation pictures within disjoint domains are independent. Locality plays
an important role in the percolation theory and in the study of its scaling
limit. SLE($6$) was initially identified as the (conjectured) scaling limit of
percolation interfaces because it is the only SLE that has some form of the
locality property.

Naively, the locality in random fields should follow from the decay of
correlation, but the question is more subtle. Let’s consider two disjoint
domains that are far apart. Although point-wise values are weakly correlated,
there are uncountably many points, so collectively, they could be strongly
correlated.

Indeed, if we look at our main examples, they are real analytic functions, so
if we know the field in any open set, then we know it everywhere.
Equivalently, the sigma-algebra generated by the restriction of the field to
any open set is the full sigma-algebra. This means that level lines in
disjoint domains can not be really independent. Fortunately, it turns out that
relevant events, such as crossing events, indeed decorrelate when domains are
well separated.

The relevant notion in the case of fields is quasi-independence of crossing
and other similar events. For crossing events we would like to obtain a
statement of this kind: let $B_{1}$ and $B_{2}$ be two disjoint rectangles,
then the covariance between crossing events in $R\cdot B_{1}$ and $R\cdot
B_{2}$ goes to $0$ as $R$ goes to infinity. Sometimes we also need to control
the rate of convergence. More generally, we would like to have a similar
estimate for all events describing the topology of nodal level sets.

Below we will describe a range of such results. In this section we always
implicitly assume that all fields are centred, stationary, sufficiently smooth
(usually $C^{3}$ is enough) and non-degenerate (different theorems use
different notions of non-degeneracy, but in all examples it is usually easy to
check these conditions). Below we will only concentrate on the more important
assumption on the decay of the covariance kernel.

One of the first results in this direction was obtained by Nazarov, Sodin and
Volberg in [69, Theorem 3.2] (see also [66, Theorem 3.1]). They worked in the
context of the Gaussian entire function (GEF) which is complex-valued rather
than real-valued, but its correlations also decay fast similarly to the
Bargmann-Fock field. They have shown that the restriction of GEF to two well
separated compacts could be written as restrictions of two independent copies
plus a small perturbation. Essentially the same argument would work for the
Bargmann-Fock and other fields with very fast decaying covariance. This is not
exactly what we need, one also has to show that the small perturbation does
not change the structure of the nodal set with high probability.

The first, very natural approach to such results is discretization. The main
idea is rather straightforward. Let us evaluate the field inside the box
$R\cdot B$ on a lattice with mesh $\epsilon$. This gives us a _finite_
Gaussian vector with dimension of order $R^{2}\epsilon^{-2}$. If the mesh is
fine enough, then it should be possible to argue that knowledge of the field
in these points captures all relevant information with very high probability.
For example, if we know the field on a mesh then we know whether nodal domains
percolate or not with very high probability. Now if we have two boxes, then we
have two finite-dimensional vectors. If boxes are far apart, then correlations
between each component of one vector and each component of the other vector
are small.

One of the first results in this direction is due to Mischaikow and Wanner
[56] who used discretization of the field to compute the homology of the nodal
set. A better result was obtained by Beffara and Gayet in [10]. They were
studying the crossing events (we will discuss it in more detail a bit later)
and used exactly the approach mentioned above. They have shown [10, Theorem
1.6] that if $\epsilon=o(R^{-8-\delta})$ for some small $\delta>0$, then with
probability almost one the nodal structure of the discretized field is the
same as the structure of the original field. After that, they gave an estimate
of the dependence of Gaussian vectors.

######  Theorem 5.1 (Theorem 4.3 [10]).

Let $X$ and $Y$ be two Gaussian vectors of dimensions $n$ and $m$
correspondingly. We consider two joint distributions $\mu$ and $\nu$ such that
under $\mu$ they are independent and under $\nu$ pairwise covariances
$\mathrm{Cov}(X_{i},Y_{j})\leq\delta$ for all $i$ and $j$, then we have the
following bound on the total variation distance:

$d_{\mathrm{TV}}(\mu,\nu)\leq\mathrm{const}(m+n)^{8/5}\delta^{1/5}.$ (5.1)

This means that two correlated vectors are essentially independent as long as
$\delta=o((m+n)^{-8})$. Combining this with an estimate on the mesh size we
get that nodal structures in two domains of size $R$ that are $R$ apart are
essentially independent if the covariance kernel decays faster than
$|x|^{-144}$.

Both of these estimates can be improved. In [15, Theorem 1] we have shown that
discretization gives a faithful representation as long as
$\epsilon=o(R^{-2-\delta})$ and in [15, Proposition 16] we have improved an
estimate (5.1) by showing that the covariance of all events depending only on
signs of $X$ and $Y$ correspondingly (all events describing nodal sets are of
this type) is bounded by

$\mathrm{const}(m+n)^{4/3}\delta^{1/3}\log^{1/3}(1/\delta).$ (5.2)

The best to day result in this direction was obtained by Rivera and
Vanneuville [76, Theorem 1.12]. There they have shown that crossing events are
quasi-independent if the covariance kernel $K(x)=o(|x|^{-4})$. Their argument
is quite different. Although they do use discretization as an intermediate
step, their estimates are _uniform_ in the lattice step.

Inspired by the work of Rivera and Vanneuville as well as earlier work of
Piterbarg [72] we have obtained in [16] an explicit formula for the covariance
of a certain class of ‘topological’ events. Unlike all previous results, our
approach is dimension-independent so we are able to obtain quasi-independence
in higher dimensions. We also do not assume that the field is centred, hence,
without loss of generality, we consider nodal sets instead of general level
sets.

Before explaining the main result and the main ideas behind the proof, we have
to introduce some notations. To simplify the explanation, we do it in the
simplest possible setting and we will skip some technicalities. Full details
can be found in the original publication.

Let $B=[a_{1},b_{1}]\times\dots\times[a_{n},b_{n}]$ be a rectangular box in
$\mathbb{R}^{n}$. We treat it as a stratified $n$-dimensional manifold. We
decompose it as the union of its $n$-dimensional interior ($n$-dimensional
strata), open $(n-1)$-dimensional sides etc up to all corners that are
$0$-dimensional strata. We will be considering isotopies that preserve this
stratification, that is isotopies of $B$ that send each stratum to itself. We
say that two compact subsets of $B$ are equivalent if there is a stratified
isotopy of $B$ that sends one set to another. Elementary topological events
are events that the nodal sets/domains of a field inside $B$ belong to a
certain equivalence class. Topological events are events from the
$\sigma$-algebra generated by elementary topological events. This is a
rigorous way of saying that the nodal set has a certain structure. This notion
on one hand allows enough flexibility so that the analytic rigidity is not a
problem anymore. On the other hand, this $\sigma$-algebra is rich enough to
describe all events that we are interested in such as the number of components
or various crossing events.

Next, we need a field equivalent of the notion of a _pivotal_ point in
percolation. Let us fix some topological event $A$. We say that a point $x$ is
_positively pivotal_ for a function $f$ if small positive local perturbations
of $f$ near $x$ belong to $A$ and small negative local perturbations do not
and we say that it is _negatively pivotal_ if it is the other way round. One
should think that $f$ is on the boundary of a topological event and
positivity/negativity give us a notion of ‘orientation’. Note that not all
boundary functions have pivotal points. Up to a small exceptional set of
boundary functions pivotal points are non-degenerate stratified critical
points of $f$ with $f(x)=0$. See Figure 5 for an illustration.

Figure 5: Central panels show functions with pivotal points for the lef-right
crossing event. Left and right show small perturbations. In the top row the
pivotal point is a saddle inside the domain, in the bottom row the pivotal
point is an extremum of the field restricted to a one-dimensional stratum.

Let $f$ be a Gaussian field with the covariance kernel $K(x,y)$111We do not
assume that $f$ is stationary. We assume that $f\in C^{2}$ and non-degenerate.
Let $B_{1}$ and $B_{2}$ be two boxes and $A_{1}$, $A_{2}$ be two topological
events in these boxes. Then there are two positive _pivotal measures_
$\pi^{\pm}$ such that

$\mathbb{P}[A_{1}\cap
A_{2}]-\mathbb{P}[A_{1}]\mathbb{P}[A_{2}]=\int_{B_{1}\times
B_{2}}K(x,y)(d\pi^{+}(x,y)-d\pi^{-}(x,y)).$ (5.3)

These measures are also stratified. For each pair of strata in $B_{1}$ and
$B_{2}$ these measures have a component that is continuous with respect to the
product of the corresponding Lebesgue measures on these strata. The density at
$(x,y)$ is very closely related to the joint density of two stratified
critical points at $x$ and $y$ subject that they are pivotal for $A_{1}$ and
$A_{2}$ correspondingly. The positive measure $\pi^{+}$ corresponds to the
case when either both points are positively pivotal or negatively pivotal and
the negative measure $\pi^{-}$ corresponds to the case when one is positively
pivotal and the other is negatively pivotal. These densities are essentially
the same as the conditional expectations that appear in Kac-Rice formula 3.1.

The precise formula for the densities is rather long, complicated and involves
an interpolation between a pair of identical copies of $f$ and a pair of
independent copies. All details can be found in [16].

This formula has a couple of important corollaries. When both events are
increasing, there are no negatively pivotal points, hence the negative term
$\pi^{-}$ disappears. If we also assume that $K\geq 0$ then the integrand in
(5.3) is non-negative and we recover Pitt’s positive association result.
Densities appearing in the definition of $\pi^{\pm}$ could be uniformly
bounded. If $B_{i}$ are two boxes of the size of order $R$ then the covariance
can be uniformly bounded by a term comparable to

$R^{2d}\sup_{x\in B_{1},y\in B_{2}}K(x,y).$

This means that if the covariance kernel decays is $o(|x-y|^{-2d})$ then all
topological events are quasi-independent. When $d=2$ this is exactly the
quasi-independence result from [76].

Finally, (5.3) leads to some form of Harris criterion or rather to a necessary
condition. Let $R\cdot B_{1}$ and $R\cdot B_{2}$ be two disjoint boxes
rescaled by a large factor $R$. If we assume the ‘percolation hypothesis’,
then, as $R$ tends to infinity, crossing events in the rescaled boxes should
decorrelate. Pivotal densities appearing in the definition of $\pi^{+}$ (there
is no $\pi^{-}$ since events are increasing) are similar to densities of
‘four-arm saddles at scale $R$’ i.e. saddle points $x$ such that four level
lines $\\{f=f(x)\\}$ stay separated within a ball of radius $R$. So the
covariance of crossing events should be comparable to

$I(R)^{2}\int_{R\cdot B_{1}\times R\cdot B_{2}}K(x-y)dxdy,$

where $I(R)$ is the density of four-arm saddles at scale $R$. These ‘four-arm’
saddles are the field equivalent of ‘four-arm’ events in the percolation
theory. Assuming the percolation hypothesis $I(R)$ should behave like
$R^{-\zeta_{4}}$, where $\zeta_{4}$ is the so-called _four-arm exponent_. By
Kesten scaling relations [47] $\zeta_{4}=d-1/\nu$ where $\nu$ is the
_correlation length exponent_. Roughly speaking it is defined so that a
Bernoulli percolation with non-critical $p$ is indistinguishable from the
critical percolation on scale $R$ as long as $|p-p_{c}|\ll R^{-1/\nu}$. So,
assuming the percolation hypothesis, we should have

$R^{2/\nu-2d}\int_{R\cdot B_{1}\times R\cdot B_{2}}K(x-y)dxdy\to 0.$

This is almost the same criterion as the field version of the Harris criterion
that appeared in [23] (see also [86]).

It is believed that in dimension $d=2$ the correlation length exponent
$\nu=4/3$ for all lattices. So far, it has been proven only for the triangular
lattice [84]. Assuming this exponent, the Harris criterion becomes

$R^{-5/2}\int_{R\cdot B_{1}\times R\cdot B_{2}}K(x-y)dxdy\to 0.$ (5.4)

Roughly speaking, for positive $K$ this corresponds to $K$ decaying faster
than $|x|^{-3/2}$. Note that the covariance kernel for the random plane wave
is decaying like $|x|^{-1/2}$, but it is also _oscillating_ , so it could be
argued that it satisfies this form of the Harris criterion. This gives a
(rather weak) support to the percolation conjecture for the random plane wave.

Although this is not completely rigorous and is only a necessary condition,
the fact that it is essentially the same as the Harris criterion derived by
Bogomolny and Schmit using a completely different approach gives some support
to this heuristic.

###  5.2 Russo-Seymour-Welsh inequality

Crossing probability plays a very important role in percolation theory.
Proving Cardy’s formula was the most important step in Smirnov’s proof of the
convergence of critical percolation interfaces on the triangular lattice to
SLE($6$). So far, we are very far from proving Cardy’s formula for any
Gaussian field.

The first step in understanding the scaling properties of percolation is the
Russo-Seymour-Welsh (RSW) inequality that was first obtained independently by
Russo [78] and by Seymour and Welsh [82].

######  Theorem 5.2 (Russo-Seymour-Welsh).

Let us consider the Bernoulli percolation with $p=1/2$ on the square lattice.
Then for every $\rho>1$ there is $c>0$ which depends on $\rho$ but not on $n$
such that

$c<\mathbb{P}\left[\text{there is a horizontal crossing in }[0,\rho
n]\times[0,n]\right]<1-c,$

Similar statements have been proved for other percolation models as well. We
use the term RSW for all such statements. Other, shorter and more general
proofs were obtained by Bollobás and Riordan in [24, Corollary 7] and Tassion
[85].

RSW immediately implies a bound on a ‘one-ar’ event:

######  Theorem 5.3.

For the Bernoulli percolation with $p=1/2$ on the square lattice there is
$\delta>0$ such that for all $s<t$ the probability that there is a percolation
cluster connecting the boundary of $[-s,s]^{2}$ to the boundary of $[-t,t]$ is
bounded by $(s/t)^{\delta}$.

######  Remark 5.4.

Theorem 5.3 immediately implies that a.s. there are no infinite clusters.
Hence RSW type inequalities can be used to identify the critical probability
$p_{c}$. One should expect that for $p<p_{c}$ the crossing probability goes to
zero as the scale goes to infinity. For $p>p_{c}$ we expect it to go to $1$.
We will discuss this in more detail in the next section.

The first RSW result for nodal domains of Gaussian fields was obtained by
Beffara and Gayet [10]. They have combined Tassion’s very general approach to
RSW with quasi-independence estimates that we have discussed in the previous
section to obtain the following RSW-type result:

######  Theorem 5.5 (Beffara-Gayet Theorem 1.1 [10]).

Let $f$ be a stationary centred Gaussian field in $\mathbb{R}^{2}$ with the
covariance kernel $K$. We assume that $f\in C^{4}$ a.s.,
$K(x_{1},x_{2})=K(x_{1},-x_{2})$, $K(x)\geq 0$ and $K(x)=o(|x|^{-325})$ as
$|x|\to\infty$. Then for any quad (i.e. a smooth bounded open domain
$\Omega\subset\mathbb{R}^{2}$ with two marked disjoint boundary arcs
$\gamma_{1}$ and $\gamma_{2}$) there is $c>0$ such that for all $R>0$

$\mathbb{P}\left[\\{x\in R\cdot\Omega:f(x)>0\\}\text{ connects
}R\cdot\gamma_{1}\text{ and }R\cdot\gamma_{2}\right]\geq c.$

If $R$ is sufficiently large, then the same is true for $\\{f=0\\}$.

######  Remark 5.6.

Their main aim was to prove RSW for the Bargmann-Fock field, which clearly
satisfies all these conditions. Since its covariance decays faster than any
polynomial, there was no need to optimize the covariance decay rate.

The main contribution to the exponent $325$ is due to strong assumptions in
their quasi-independence argument. Better quasi-independence results give
better assumptions on the covariance. In [15, Theorem] the exponent was
reduced to $16$ and a bit later it was reduced to $4$ in [76, Theorem 1.1]. It
seems that $4$ is the best exponent that could be obtained using quasi-
independence methods. Later, Muirhead and Vanneuville [63] reduced the
exponent to $2$ using very different ideas. We will discuss it in the next
section.

Although [12] gives strong numerical evidence that there is RSW for the random
plane wave, current methods are not sufficient to tackle this case. The
covariance kernel decays like $|x|^{-1/2}$ which is too slow for quasi-
independence results. Moreover, it is not positive, so there is no FKG
inequality (see Theorem 4.5) which is heavily used in the proof of RSW.

A bit later the argument of Beffara and Gayet was extended to the Kostlan
ensemble.

######  Theorem 5.7 (Beliaev-Muirhead-Wigman [17]).

Let $f_{n}$ be the Kostlan ensemble of degree $n$. For any quad on the unit
sphere that does not contain antipodal points, there is $c>0$ such that the
probability that $\\{f_{n}>0\\}$ crosses it is at least $c$ for all $n$. The
same is true for $\\{f=0\\}$ is $n$ is sufficiently large.

Moreover, the constant $c$ is uniform in the quad, provided that we can
control its ‘shape’. The precise statement can be found in [17, Definition
1.4].

We would like to point out a few important differences between Theorems 5.5
and 5.7. The second theorem (i) deals with percolation on the sphere which is
a bit more complicated than the planar one (ii) the estimate is uniform over a
class of different fields (iii) the covariance kernel of the Kostlan ensemble
can be negative for odd $n$ (iv) Tassion-style argument is applied not to a
discretized field but directly to the continuous field.

######  Remark 5.8.

As we have mentioned in the Remark 2.1 the Kostlan ensemble is a natural model
for a ‘typical’ homogeneous polynomial of degree $n$ and its nodal lines are
‘typical’ real algebraic varieties. Theorem 5.7 gives strong support to the
percolation conjecture for the Kostlan ensemble. This allows us to conjecture
that a ‘typical’ projective variety converges to the Conformal Loop Ensemble
CLE($6$) as $n$ tends to infinity. This gives an interesting connection
between algebraic geometry and Conformal Field Theory.

In [11] Beffara and Gayet used perturbation techniques to obtain RSW for a
class of models without FKG. They have argued that any model which in a
certain sense close to a model with RSW has RSW as well. In particular, this
can be applied to Gaussian fields that decorrelate quickly. In a sense this is
close to an idea used in [17]: if negative correlations are very small, then
one can still obtain good low bounds on the intersection of crossing events.

###  5.3 Phase transition

One can think that a field $f$ defines a landscape and the level $\ell$
defines a water level. When $\ell$ is a very large negative number, we should
expect that there is a huge ‘continent’ with small ‘lakes’. As the water level
is rising, these lakes are getting larger and larger. At $\ell=0$ there is a
balance between the land and the water and after that we switch to an ‘ocean
with islands’ regime. When $\ell\to\infty$, these islands are getting smaller
and eventually disappear. A similar picture happens in percolation when $p$ is
increasing from $0$ to $1$. The phenomenon of switching from ‘land with lakes’
to ‘ocean with islands’ is called a phase transition.

Figure 6: Three different level sets of the same sample of the Bargmann-Fock
field. In all figures grey is the set above the level, white is below the
level and black is the largest component of the excursion set. The levels are
$\ell=0.1$ (left), $\ell=0$ (center) and $\ell=-0.1$ (right). As the level is
decreasing, the largest component changes from small, to thin but comparable
to the size of the domain, to a dense one.

It should not be a surprise that the first result of this type has been proved
for the Bargmann-Fock field. See Figure 6 for an example of a phase
transition.

######  Theorem 5.9 (Rivera-Vanneuville [77]).

Let $f$ be the Bargmann-Fock field, then a.s the set $\\{f\geq\ell\\}$ has a
unique unbounded component when $\ell<0$ and a.s. has no unbounded components
if $\ell\geq 0$. Moreover, when $\ell<0$ the crossing probability at scale $R$
(i.e. the probability that there is a curve connecting left and right sides of
$[0,\rho R]\times[0,R]$ such that $f>\ell$ along this curve) is at least
$1-\exp(-cR)$, where $c>0$ depends of $\rho$ and $\ell$ but not on $R$.

We would like to make a few remarks:

  * •

We use slightly different notations, Rivera and Vanneuville considered sets
$\\{f\geq-\ell\\}$ that are increasing in $\ell$, whereas our sets
$\\{f\geq\ell\\}$ are decreasing.

  * •

This result was stated for the Bargmann-Fock field, but in fact the proof of
the first part of the theorem is also valid for ‘nice’ fields with
sufficiently fast decay of correlations (faster than $|x|^{-5}$). See [76,
Theorem 2.11] for the complete list of assumptions. The last part of Theorem
5.9 requires super-exponential decay of correlations.

  * •

The fact that all nodal domains are bounded follows from RSW. By monotonicity
the same is true for all $\ell>0$.

  * •

The first part of the theorem is similar to Kesten’s theorem about phase
transition for Bernoulli percolation on $\mathbb{Z}^{2}$ (see Theorem 1.1).
The second part is similar to Kesten’s exponential decay, see [46, Theorem 2].

This paper uses all the standard techniques and assumptions that we have
discussed in the previous section. They are needed to have RSW, to control box
crossings and to have FKG. Beyond this, Rivera and Vanneuville used one more
tool that originates in theoretical computer science: analysis of influences.

To explain the main ideas we first have to introduce some notations. Let
$\mathbb{P}_{p}^{n}$ be the product $p$-Bernoulli measure on $X=\\{0,1\\}^{n}$
(clearly the Bernoulli percolation on a graph with $n$ bonds can be identified
with such measure). Let $A$ be an event, i.e. $A\subset X$. The influence of
$i$-th coordinate $\mathrm{Infl}_{i}^{p}(A)$ is the probability that changing
the value of $i$-th coordinate will change $\mathds{1}_{A}$. In other terms,
this is the probability that the $i$-th coordinate is _pivotal_ for the event
$A$. The first important ingredient is the Russo differential formula

$\frac{d}{dp}\mathbb{P}^{n}_{p}\left[A\right]=\sum_{i=1}^{n}\mathrm{Infl}_{i}^{p}(A),$
(5.5)

where $A$ is any increasing event (typical example: any crossing event). The
right hand side can be also interpreted as the expected number of pivotal
coordinates. This formula was obtained independently by Russo in [78, Lemma 3]
and Margulis [55, Lemma 2.6]. See also [40, Section 2.4] for a more detailed
discussion. It has been noted even in the early work of Margulis that this
result could be used to show sharp transition.

The second ingredient is the Kahn–Kalai–Linial (KKL) theorem [45, 26] that
gives a lower bound on the sum of influences in terms of the maximal
influence. Roughly speaking, it tells us that if all influences are small,
then their sum is large. This approach to proving Kesten’s theorem was used in
[24]. To be more precise, they have used a sharp threshold result of Friedgut
and Kalai [37] based on KKL. See also [25] where these ideas are used in the
more relevant context of Voronoi percolation.

For more information about the harmonic analysis of Boolean functions, noise
sensitivity and other related results we refer to very good lecture notes by
Garban and Steif [38].

One of the main contributions of River and Vanneuville is the adaptation of
these ideas to the field setting (in particular, they extend the Russo formula
and KKL from the Bernoulli product measure to Gaussian measures.

A bit later Muirhead and Vanneuville [63] extended this result to a much
larger class of fields. They work in the setting of fields having spectral
density, hence representable as a convolution of the white noise with a
certain kernel $q$ satisfying $K=q*q$. They obtained the following result:

######  Theorem 5.10 (Theorems 1.6 and 1.11 [63]).

Let $f=W*q$ be a stationary Gaussian field satisfying natural smoothness,
symmetry and regularity assumptions. We also assume that $q\geq 0$ and that
$q$ and its derivatives of order up to $3$ decay faster than $|x|^{-\beta}$
for some $\beta>2$, then there is sharp phase transition for the excursion
sets $\\{f\geq\ell\\}$. Namely, (i) for $\ell>0$ all components are bounded
and crossing probabilities on scale $R$ decays exponentially fast as
$R\to\infty$ (ii) there is an RSW estimate for $\ell=0$ (in particular, all
components are bounded) (iii) for $\ell<0$ there is a unique unbounded
component and crossing probabilities are exponentially close to $1$.

Note that the decay rates of $K$ and $q$ are essentially the same, so the
second part gives RSW under a weaker decay assumption than [76, Theorem 1.1].

There are few important improvements compared to all previous results. First,
is that they do not use discretization of the field, instead, similar to [16],
they approximate the field by a smooth field such that the corresponding
Hilbert space is finite-dimensional. Secondly, instead of quasi-independence,
they use a ‘sprinkled’ version. Namely, for crossing events $A$ and $B$ they
compare $\mathbb{P}[f\in A]\mathbb{P}[f\in B]$ to $\mathbb{P}[f-\epsilon\in
A\cap B]$ instead of $\mathbb{P}[f\in A\cap B]$. Sprinkling was also used in
[77]. For use of sprinkling in percolation we refer to the discussion in
Section 2.6 of [40] and [40, Theorem 2.46]. See also [2, Lemma 4.2]. Finally,
they adapted some new ideas that were recently used by Duminil-Copin, Raoufi
and Tassion to study lattice models and different versions of percolation [33,
34, 35].

To be more precise, they have used a version of OSSS inequality [71] that
gives a low bound on the sum of influences. Without going into technicalities,
it can give a bound on the sum of influences in terms of _revealments_.

Let $A\subset X$, where $X=E^{n}$ is a product space equipped with a
probability product measure. We consider a _random_ algorithm $\mathcal{A}$.
The algorithm reveals coordinates of $\omega$ until it is able to determine
whether $\omega\in A$ or not. The revealment of $i$-th coordinate is the
probability that this coordinate will be revealed by $\mathcal{A}$.
Essentially OSSS inequality tells us that if all revealments are small, then
the sum of influences is large, hence the derivative in the Russo formula
(5.5) is large.

Muirhead and Vanneuville approximate the field by a convolution of the
truncated kernel $q$ with a discrete white noise. The random algorithm works
approximately like this: we start with the white noise in a rectangle. We
first choose a horizontal line and reveal all white noise coordinates around
it so that the values of $f$ on the line are known (there are finitely many of
them since $q$ is compactly supported). After that, we explore all negative
domains intersecting the line. If one of them crosses vertically, then there
is no positive horizontal crossing, otherwise there is a positive crossing.
See Figure 7 for an example of what is revealed by this algorithm. A white
noise coordinate is revealed if it is close to one of these negative domains.
This probability is controlled by the one-arm exponent, so RSW gives an upper
bound on revealments.

Figure 7: An example of the revealment algorithm. Light grey is where $f>0$,
dark grey is where $f<0$. All negative components intersecting random
horizontal lines are black. Only the part of the field revealed by the
algorithm is shown. To be more precise, to reveal this part of the field we
have to reveal all white noise coordinates within a finite distance.

The assumption in the main result of [63] were slightly improved by Rivera
[75]. In particular, the strong positivity assumption $q\geq 0$ which is not
very natural and hard to verify was replaced by a more natural weak positivity
assumption $K\geq 0$. What is probably more important, the argument of Rivera
uses a Gaussian Talagrand inequality. This is arguably a much more natural
tool in the case of continuous fields than KKL or OSSS inequalities used
before.

The assumptions needed for the phase transition were weakened further in [61,
Theorem 1.3]. Probably the most amazing thing is that their result uses
extremely weak assumptions on the covariance decay rate. They only assume that
$K$ decays faster than $(\log\log(|x|))^{-\beta}$ for some $\beta>2$. In
particular, this result is valid for the random plane wave where the
covariance kernel decay so slowly that we are unable to establish quasi-
independence and the kernel is not positive, so there is no FKG and negative
correlations are not negligible.

To deal with some of these issues the authors have modified Tassion’s argument
[85] (see also [48, 10, 17]) so that it uses sprinkling to glue together
crossing events instead of FKG. The resulting argument is much more robust and
uses weaker assumptions but it does not work at all at criticality. So [61]
establishes the sharp phase transition but not RSW at criticality.

The second ingredient is the use of Chatterjee’s approach to
supercontractivity. This is somewhat similar to the use of Talagrand’s
inequality in [75]. The authors used it to show the following criterion for
the sharp threshold. For an increasing event $A$ one can define the threshold
height $T_{A}$, which is the first level at which the event happens. For
example, let $A$ be the crossing event. Clearly, for very large $\ell$ the
excursion set $\\{f>\ell\\}$ does not cross and for a very large negative
$\ell$ it does cross. By monotonicity, there is the threshold $\ell=T_{A}$ at
which the excursion set switches between crossing and non-crossing. At the
level $T_{A}$ there is a pivotal place where the change in the topology
happens. This place is called the _threshold location_. The criterion states
(in a quantified way) that the sharp threshold is equivalent to the
delocalization of the threshold location. On a heuristic level, it is similar
to the revealment approach. If there is a place that where the threshold is
localized with high probability, then this place should be pivotal for
crossing and any algorithm discovering the event $A$ should reveal it with
high probability.

Severo extended these results to higher dimensions. Theorem 1.2 [81] states
that there is a sharp phase transition for a stationary field in
$\mathbb{R}^{n}$ satisfying the usual regularity, symmetry and smoothness
assumptions provided that its covariance kernel is strongly positive ($q\geq
0$) and decays faster than $|x|^{\beta}$ for some $\beta>n$. The argument is
based on the comparison between the field and its approximation by a finite-
range field.

There are several recent results that are not directly related to the
percolation conjecture, but they also deal with phase transitions and
connectivity probabilities in sub- and supercritical regimes in Gaussian
fields. They consider these questions for a large class of _strongly
correlated_ Gaussian fields in $\mathbb{Z}^{n}$ and $\mathbb{R}^{n}$. A
typical example would be the discrete Gaussian Free Field in dimensions $n\geq
3$ where the covariance kernel decays like $|x|^{d-2}$. It was also shown that
there is a RSW-type estimate for very strongly but positively correlated
fields. A typical example would be a field with the Cauchy covariance kernel
$1/(1+|x|^{2})^{\alpha/2}$ with $\alpha<2$. This gives some indication that
there are scaling limits beyond the Bernoulli percolation universality class.
For more information we refer to [32, 39, 62, 60] and references wherein.

##  6 What we do not know

Let us start with a technical question that we still find interesting and
important. It could be thought of as questions about reproducing kernel
Hilbert spaces (RKHS). As we have seen before, quite often it is useful to
have quantified finite-dimensional approximations. This leads to the following
question

######  Question 6.1.

Let $H$ be an RKHS made of $C^{k}$ functions. What is the best finite-
dimensional approximation of $H$? That is, we would like to find an
$n$-dimensional Hilbert space $H_{n}$ such that functions from $H$ could be
well approximated in $C^{k}(B(R))$ norm by functions from $H_{n}$. One can try
to optimize the dimension $n$ and how good is the approximation in terms of
the radius $R$.

There is also a similar question about the localization of orthonormal bases
of an RKHS $H$. Recall that the corresponding Gaussian field (isonormal
process) can be written as $\sum a_{k}\phi_{k}$ where $\\{\phi_{k}\\}$ is an
orthonormal basis and $a_{k}$ are independent Gaussian random variables. Its
covariance kernel is the reproducing kernel $K$ of $H$. Truncating the series
after $n$ terms we obtain a field $f_{n}$ with a covariance kernel
$K_{n}(x,y)=\sum_{k=1}^{n}\phi_{k}(x)\phi_{k}(y)$. Clearly, $K_{n}\to K$ as
$n\to\infty$. We would like to understand the rate of convergence.

######  Question 6.2.

Given the ball or radius $R$ and $\epsilon>0$, what is the best $n$ such that
$f_{n}$ is $\epsilon$-close to $f$ in $C^{k}(B(R))$ norm? Equivalently, what
is the best $n$ such that all basis functions except $n$ are very small in
$B(R)$?

Here, we can look at our two main examples: the random plane wave and the
Bargmenn-Fock field. For the random plane wave we have series (2.1). Since
Bessel functions $J_{n}$ decay exponentially fast in $n$ it can be shown that
truncating the series at $cR$ terms gives an exponentially good approximation
in the disc of radius $R$. So here the number of terms grows linearly in $R$.
Similarly, for the Bargmann-Fock field, we can remove all terms where
$n,m>cR$, but the number of such terms is of order $R^{2}$. So the random
plane wave admits much better finite approximations. One can think that this
is a manifestation of the rigidity of the RPW. One consequence of this better
approximation is that the variance of the number of excursion sets (for a
generic non-zero level) in a ball of radius $R$ grows like222Only the low bond
on the growth rate is known rigorously but it is believed to be sharp $R^{3}$
[13] for the random plane wave and like $R^{2}$ for the Bargmann-Fock field
[13, 14]. So this difference is not superficial, it tells us something
important about fields.

######  Question 6.3.

Is there a way to see this difference by considering the corresponding
covariance kernels without using an explicit orthonormal basis?

Now, let us return to percolation. Despite all the progress in the last five
years, there are many questions that are still open, in particular, it is
still not quite clear under what assumptions the percolation conjecture should
hold. It is also not clear whether a Harris criterion heuristics could be made
rigorous. Here we will list several open problems/conjectures where we think
that the answer is positive.

######  Conjecture 6.4 (Bargman-Fock percolation).

For the Bargmann-Fock field in $\mathbb{R}^{2}$

  * •

Show the existence of one of the universal exponents and that they are equal
to the percolation ones.

  * •

Prove the existence of a conformally invariant scaling limit of any
observable. For example, prove Cardy’s formula for crossing probability.

  * •

Show that nodal lines converge to the conformal loop ensemble CLE($6$).

Although there are very strong reasons to believe that this conjecture is
true, we probably have to understand the Bernoulli percolation better before
tackling it. After all, so far these results are only known for the
percolation on the triangular lattice. Their proofs are based on Cardy’s
formula and its proof relies very heavily on the specifics of the triangular
lattice. Available proofs are rather simple and elegant (especially the one in
[49]) but they are very hard to generalize and extend.

######  Conjecture 6.5 (Random Plane Wave percolation).

For the RPW in $\mathbb{R}^{2}$

  * •

Show that there is no percolation at $\ell=0$ i.e. that all nodal lines are
bounded.

  * •

Show that the variance of the number of nodal domains in a box of size $R$
grows as $R^{2}$ (possibly up to a polylogarithmic factor). Note that for the
non-zero level it is not true.

  * •

Prove some form of quasi-independence of crossing events for nodal lines.

  * •

Prove RSW.

  * •

Prove the same statements as in Conjecture 6.4.

This conjecture is a bit more debatable, but we believe that there is good
numerical support for the conjecture. Proving it will obviously require even
more new ideas since even fewer percolation ideas can be adapted to the random
plane wave setting since its covariance kernel is slowly decaying and
oscillating.

Beyond these particular forms of the percolation conjecture, there are
questions about natural extensions:

######  Question 6.6.

It is easy to believe that the Gaussianity of the fields is not paramount.
Clearly, it is a very strong property that allows us to use many tools, but it
is easy to imagine that some percolation properties are more universal. For
example, if we replace independent normal coefficients in the definition of
the Bargmann-Fock field with some other independent random variable, then the
series will converge provided the tails are not heavy. Will the nodal domains
of such fields behave like the critical percolation?

The other related class of fields are shot noise fields. Let $\mathcal{P}$ be
a Poisson point process. We define a random measure $\mu=\sum
a_{i}\delta_{y_{i}}$ where the sum is over all points $y_{i}$ in $\mathcal{P}$
and the coefficients $a_{i}$ are i.i.d random variables. For an integrable
function $g$ we define the shot noise field as the convolution $\mu*g$. It is
known [52] that under some assumptions there is a phase transition at $0$ for
the excursion sets of such fields.

######  Question 6.7.

Are nodal domains of shot noise fields behave like the critical percolation?

[Acknowledgments] I would like to thank M. Lifshits for encouraging me to
write this survey and S. Muirhead for valuable comments on the first draft.

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