# Noise-like analytic properties of imaginary chaos

Juhan Aru  École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
<EMAIL_ADDRESS>,  Guillaume Baverez  Institut für Mathematik, Rudower
Chaussee 25, 12489 Berlin<EMAIL_ADDRESS>,  Antoine Jego
École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland
<EMAIL_ADDRESS>and  Janne Junnila  University of Helsinki, Helsinki,
Finland<EMAIL_ADDRESS>
###### Abstract.

In this note we continue the study of imaginary multiplicative chaos
$\mu_{\beta}:=\exp(i\beta\Gamma)$, where $\Gamma$ is a two-dimensional
continuum Gaussian free field. We concentrate here on the fine-scale analytic
properties of $|\mu_{\beta}(Q(x,r))|$ as $r\to 0$, where $Q(x,r)$ is a square
of side-length $2r$ centred at $x$. More precisely, we prove monofractality of
this process, a law of the iterated logarithm as $r\to 0$ and analyse its
exceptional points, which have a close connection to fast points of Brownian
motion. Some of the technical ideas developed to address these questions also
help us pin down the exact Besov regularity of imaginary chaos, a question
left open in [JSW20]. All the mentioned properties illustrate the noise-like
behaviour of the imaginary chaos. We conclude by proving that the processes
$x\mapsto|\mu_{\beta}(Q(x,r))|^{2}$, when normalised additively and
multiplicatively, converge as $r\to 0$ in law, but not in probability, to
white noise; this suggests that all the information of the multiplicative
chaos is contained in the angular parts of $\mu_{\beta}(Q(x,r))$.

###### Contents

  1. 1 Introduction and main results
  2. 2 Preliminaries
  3. 3 Hölder regularity and monofractality: proof of Proposition 1.1 and Theorem 1.2
  4. 4 Law of the iterated logarithm: proof of Theorem 1.4
  5. 5 Exceptional points: proof of Theorem 1.5
  6. 6 Besov regularity of imaginary chaos: proof of Theorem 1.7
  7. 7 Convergence to white noise: proof of Theorem 1.8
  8. A A variant of Kolmogorov’s continuity theorem
  9. B A tail estimate for $|\mu^{*}(Q(0,1))|$

##  1\. Introduction and main results

Let $D\subset\mathbb{R}^{2}$ be an open bounded simply connected domain and
let $\Gamma$ be the Gaussian free field in $D$ with zero boundary condition.
That is, $\Gamma$ is a Gaussian, centred Schwartz distribution whose
covariance is given by the Green function $G_{D}$ of the Laplacian with
Dirichlet boundary condition, in the sense that for any smooth test functions
$\varphi$, $\psi$,

$\mathbb{E}[(\Gamma,\varphi)(\Gamma,\psi)]=\int_{D\times
D}\varphi(x)G_{D}(x,y)\psi(y)dx\,dy.$

In this paper, we normalise the Green function so that
$G_{D}(x,y)\sim-\log\left|x-y\right|$ when $x-y\to 0$. For
$\beta\in(0,\sqrt{2})$ the imaginary chaos $\mu(x)$ is formally defined as
$e^{i\beta\Gamma(x)}\mathbf{1}_{\left\\{x\in D\right\\}}.$ Because $\Gamma$ is
not defined pointwise, the exponential of $i\beta\Gamma$ is a priori not well
defined. A simple regularisation procedure helps out: for $\varepsilon>0$, let
$\Gamma_{\varepsilon}(x)$ be the $\varepsilon$-circle average of $\Gamma$
about $x\in D$; then $\mu$ is defined as the limit

$\mu(x)\qquad\coloneqq\qquad\lim_{\varepsilon\to
0}\varepsilon^{-\beta^{2}/2}e^{i\beta\Gamma_{\varepsilon}(x)}\mathbf{1}_{\left\\{x\in
D\right\\}}$ (1.1)

where the limit takes place in $L^{2}$ in any negative Sobolev space $H^{s}$
with $s<-1$; see [JSW20, Section 3.1]. In particular, $(\mu,\phi)$ are well-
defined random variables simultaneously for all smooth test functions $\phi$.
Further, if $f=\mathbf{1}_{A}$ is the indicator function of a Borel set
$A\subset\mathbb{C}$ (or indeed any measurable bounded function), one can
define $\mu(A)=(\mu,\mathbf{1}_{A})$ as the $L^{2}$ limit of
$\varepsilon^{-\beta^{2}/2}\int_{D}fe^{i\beta\Gamma_{\varepsilon}}$.

Imaginary multiplicative chaos seems to have first come up in the context of
directed polymers [DES93], though now we could name it rather the total mass
of imaginary multiplicative cascade. Imaginary multiplicative cascades, i.e.
imaginary chaos for log-correlated fields with a hierarchical structure, were
studied in [BJM10a] and subsequently in [BJ10, BJM10b]. A generalization to
more general log-correlated fields came about in the context of complex chaos
in [LRV15]. In [JSW20] the authors not only studied imaginary chaos as an
interesting analytic object, but made a direct connection to the scaling limit
of spins of the critical XOR-Ising model. Thereafter imaginary chaos has been
studied in several papers - in [AJJ22] it was shown that $(\mu,f)$ has a
bounded smooth density on $\mathbb{C}$, in [AJ21] it was shown that imaginary
chaos of the free field - in contrast to imaginary cascades - contains all the
information of the underlying log-correlated field. Finally, due to its
connections to Coloumb gases, imaginary chaos is now making appearance also in
probabilistic constructions of 2D conformal field theories [GKR23].

In this note we continue the study of imaginary multiplicative chaos. We
concentrate on the study of the process $(|\mu(Q(x,r))|)_{x\in D,r>0}$, where
$Q(x,r)=x+[-r,r]^{2}$ are squares of side length $2r$ centred at $x$. We will
see that this process has a very noise-like character that does not exhibit in
the $r\to 0$ limit any non trivial macroscopic properties. Notice that for
example the real part of $\mu(Q(x,r))$ does give rise to the interesting
macroscopic structure ([JSW20]), as probably does its angle. Imaginary
multiplicative chaos is considerably harder to study than the real chaos due
to the lack of monotonicity w.r.t. underlying field, and the cancellations
that need to be taken into account carefully. For example, to prove the
existence of density, one can use a simple Fourier argument in the case of
real chaos, but one relies on Malliavin calculus in the case of imaginary
chaos. In this note, we again need to combine (functional) analytic tools with
probabilistic arguments; several small technical ideas and interplays arise,
which we think might be of interest in studying random fields of similar
nature.

We will next explain what is contained in the paper in terms of results, and
also highlight some of the technical inputs going into proving these results.

###  1.1. Monofractality

Our first main result proves a certain monofractality of the imaginary chaos
$\mu$. To state this result, we first show that the process
$(\mu(Q(x,r)))_{x\in D,r\geq 0}$ has a version which is continuous. This is a
priori not clear since $\mu$ is not a random measure; see [JSW20, Theorem
1.2].

######  Proposition 1.1 (Hölder regularity).

The map

$\begin{array}[]{ccc}\\{(x,r)\in D\times[0,1],Q(x,r)\subset
D\\}&\longrightarrow&\mathbb{C}\\\ (x,r)&\longmapsto&\mu(Q(x,r))\end{array}$

possesses a continuous modification. Moreover, for all compact subsets $K$ of
$D$ and $\alpha<1-\beta^{2}/4$, there exists a random constant $C_{K,\alpha}$,
almost surely finite, such that for all
$Q(x,r),Q(x^{\prime},r^{\prime})\subset K$,

$\left|\mu(Q(x,r))-\mu(Q(x^{\prime},r^{\prime}))\right|\leq C_{K,\alpha}(r\vee
r^{\prime})^{\alpha}\left\|(x,r)-(x^{\prime},r^{\prime})\right\|^{\alpha}.$
(1.2)

As in the real case, also the imaginary chaos is believed to share a lot of
properties with complex multiplicative cascades which have been thoroughly
studied in [BJ10, BJM10a, BJM10b]. In particular, if we restrict ourselves to
cascades built using complex exponentials of a normal distribution, [BJM10a,
Theorem 2.9] establishes the monofractality of the cascade. Our first theorem
claims that the same result holds for the imaginary chaos $\mu$.

######  Theorem 1.2 (Monofractality).

With probability one, simultaneously for all $x\in D$,

$\liminf_{r\to 0}\frac{\log\left|\mu(Q(x,r))\right|}{\log r}=2-\beta^{2}/2.$
(1.3)

A key technical input here is the existence and boundedness of density for
imaginary chaos, i.e. the density of variables of the type $(\mu,f)$ where $f$
is a test function, first proved in [AJJ22] using Malliavin calculus. In
Theorem 3.2 we revisit the original proof to bring out some quantitative
bounds on the density that help us treat several simultaneous test functions
at once.

We remark that the monofractality result contrasts sharply with the case of a
two-dimensional real multiplicative $M_{\gamma}$ defined by

$M_{\gamma}(dx)=\lim_{\varepsilon\to
0}\varepsilon^{\gamma^{2}/2}e^{\gamma\Gamma_{\varepsilon}(x)}dx$

for $\gamma\in(0,2)$. Indeed, the quantity

$\liminf_{r\to 0}\frac{\log M_{\gamma}(Q(x,r))}{\log r}$

is known to depend heavily on the point $x$ and takes values in
$[(\sqrt{2}-\gamma/\sqrt{2})^{2},(\sqrt{2}+\gamma/\sqrt{2})^{2}]$ (see [RV14,
Section 4] for instance).

######  Remark 1.3.

To get some intuition on Theorem 1.2 and its difference with respect to real
chaos, we can consider near the origin the field $X$ with correlations given
by $\mathbb{E}\left[X(x)X(y)\right]=-\log\left|x-y\right|$. In that case, we
can decompose
$\mathbb{E}\left[X(x)X(y)\right]=-\log(\left|x-y\right|/r)+\log(1/r)$ so that
$X$ is the independent sum $X=X_{r}+N_{r}$ with

$(X_{r}(rx),x\in\mathbb{R}^{2})\overset{\mathrm{(d)}}{=}(X(x),x\in\mathbb{R}^{2})\qquad\mathrm{and}\qquad
N_{r}\sim\mathcal{N}(0,\log(1/r)).$

In particular,

$\displaystyle\int_{Q(0,r)}e^{i\beta X(y)}dy=r^{2-\beta^{2}/2}e^{i\beta
N_{r}}\int_{Q(0,1)}e^{i\beta X_{r}(ry)}dy,$

where the $r^{-\beta^{2}/2}$ comes from the renormalisation of $e^{i\beta X}$.
Therefore, in the case of imaginary chaos, the distribution of
$r^{-2+\beta^{2}/2}\left|\int_{Q(0,r)}e^{i\beta X(y)}dy\right|$ does not
depend on $r$. In the case of real chaos, on the contrary, the term $e^{\gamma
N_{r}}$ does not simplify and can be seen to be at the root of the
multifractal behaviour.

###  1.2. Law of the iterated logarithm and fast points

We next detail the study of the local behaviour of $\mu$ by providing a law of
the iterated logarithm for $\mu$.

######  Theorem 1.4 (Law of the iterated logarithm).

There exists $c^{*}(\beta)>0$ depending only on $\beta$ such that for all
given $x\in D$, we have

$\limsup_{r\to
0}\frac{\left|\mu(Q(x,r))\right|}{r^{2-\beta^{2}/2}(\log\left|\log
r\right|)^{\beta^{2}/4}}=c^{*}(\beta)^{-\beta^{2}/4}\quad\quad\mathrm{a.s.}$
(1.4)

To prove this result, we again turn to the analytic toolbox and start off by
making use of Sobolev embeddings to swap a uniform norm
$\left\|\cdot\right\|_{\infty}$ for a more tractable one. See the beginning of
Section 4.2 for more details.

As a next step we study exceptional points where $\mu$ has an atypical
behaviour and breaks the law of the iterated logarithm. These points are an
analogue of the fast points of Brownian motion [MP10, Section 10.1] \- indeed,
if one thinks of imaginary chaos as a noise-field, then integrating it over
squares would exactly give a 2D analogue for the definition of fast points.
Whereas different in nature, in terms of being atypical, one can also recall
here the notion of thick points of the GFF $\Gamma$ [HMP10].

In the following, we will denote by $\dim_{\mathcal{H}}(A)$ the Hausdorff
dimension of a set $A\subset\mathbb{C}$.

######  Theorem 1.5 (Exceptional points).

We have

$\sup_{x\in D}\limsup_{r\to
0}\frac{\left|\mu(Q(x,r))\right|}{r^{2-\beta^{2}/2}\left|\log
r\right|^{\beta^{2}/4}}=2^{\beta^{2}/4}c^{*}(\beta)^{-\beta^{2}/4}\quad\quad\mathrm{a.s.}$
(1.5)

Moreover, if we define for $a>0$ the set of $a$-fast points by

$\mathcal{T}(a)\coloneqq\left\\{x\in D:\limsup_{r\to
0}\frac{\left|\mu(Q(x,r))\right|}{r^{2-\beta^{2}/2}\left|\log
r\right|^{\beta^{2}/4}}\geq a\right\\},$ (1.6)

then we have for all $0\leq a\leq\left(2/c^{*}(\beta)\right)^{\beta^{2}/4}$,

$\dim_{\mathcal{H}}\left(\mathcal{T}(a)\right)=2-c^{*}(\beta)a^{4/\beta^{2}}\quad\quad\mathrm{a.s.}$
(1.7)

The constant $c^{*}(\beta)$ appearing in Theorems 1.4 and 1.5 comes from the
asymptotics of the tails of the imaginary chaos derived in [LSZ17, Corollary
A.2]. We now state this result precisely. Consider the whole plane GFF
$\Gamma^{*}$ and its associated imaginary chaos
$\mu^{*}(y)=\lim_{\varepsilon\to
0}\varepsilon^{-\beta^{2}/2}e^{i\beta\Gamma^{*}_{\varepsilon}(y)}$, where
$\Gamma_{\varepsilon}^{*}$ is the $\varepsilon$-regularisation of $\Gamma^{*}$
using circles averages. The whole plane GFF $\Gamma^{*}$ is a random
generalised function modulo global additive constants. These additive
constants disappear when one considers
$\left|\int_{Q(0,1)}\mu^{*}(y)dy\right|$ and so this latter object is a well-
defined nonnegative real random variable.

######  Theorem 1.6 ([LSZ17]).

There exists a constant $c^{*}(\beta)>0$ which may depend on $\beta$ such that

$\mathbb{P}\left(\left|\int_{Q(0,1)}\mu^{*}(y)dy\right|>t\right)=\exp\left(-c^{*}(\beta)t^{4/\beta^{2}}+o\left(t^{4/\beta^{2}}\right)\right)\mathrm{~{}as~{}}t\to\infty.$
(1.8)

The main result of [LSZ17] (Theorem 1 and Corollary 1.3 therein) actually
provides extremely precise asymptotics for the moments of
$\left|\int_{Q(0,1)}\mu^{*}(y)dy\right|^{2}$ (which are closely related to 2D
two-component plasma). The estimate (1.8) is then essentially shown to be a
consequence of their moment estimates; see Appendix B for further comments and
an alternative derivation.

###  1.3. Exact Besov regularity of imaginary chaos

The fractal properties of imaginary chaos are tightly linked to its Besov
regularity, a fact that can be interpreted in the broader context of the so-
called “Frisch-Parisi conjecture” which relates more generally multifractal
properties with Besov regularity (see e.g. [Jaf00] for an account).

In [JSW20, Theorem 3.16], it was shown that for all $1\leq p,q\leq\infty$ and
$s\in\mathbb{R}$, it a.s. holds that $\mu$ belongs to the Besov space
$B^{s}_{p,q}(D)$ if $s<-\frac{\beta^{2}}{2}$ and $\mu\not\in B^{s}_{p,q}(D)$
if $s>-\frac{\beta^{2}}{2}$. In this note, using ideas that we developed to
prove Theorems 1.2 and 1.4, we can settle the critical case
$s=-\frac{\beta^{2}}{2}$, i.e. we can decide for which values of
$p,q\in[1,\infty]$ it holds that $\mu\in B^{-\frac{\beta^{2}}{2}}_{p,q}(D)$.
We also note that these values are the same as the ones found for the
regularity of white noise [Ver10]. This is consistent with the fact that the
imaginary chaos converges to white noise when $\beta\to\sqrt{2}$ [JSW20].

######  Theorem 1.7 (Regularity of imaginary chaos).

Let $1\leq p,q\leq\infty$. If $p<\infty$ and $q=\infty$, then $\mu\in
B^{-\frac{\beta^{2}}{2}}_{p,q,\operatorname{loc}}(D)$ a.s. Otherwise, and
assuming further that $p\geq 2$, $\mu\not\in
B^{-\frac{\beta^{2}}{2}}_{p,q,\operatorname{loc}}(D)$ a.s.

###  1.4. The limiting field of $(|\mu(Q(x,r))|)_{x,r}$

We have seen that the absolute value of the imaginary chaos is monofractal.
Here we ask the question of whether any interesting macroscopic information is
left as $r\to 0$. For technical convenience, we work with the field
$|\mu(Q(x,r)|^{2}$ instead, and assume without loss of generality that $D$
contains $[0,1]^{2}+B(0,2)$.

First, just because of scaling, this field, converges to $0$ with no extra
normalization. Further, when one normalizes by $\delta^{\beta^{2}-4}$, it
converges to a constant field (this will actually follow from the proof below)
and thus contains no interesting information. A natural question is whether
one could regularize the process in a different manner and obtain an
interesting limiting field. One option would be to look for instance at its
oscillations around its mean.

We will show that, by subtracting the mean and renormalizing well, one indeed
gets a non-trivial limit; however, the limiting field turns out to be just
white noise. As the convergence takes place only in law and not in
probability, it seems plausible to say that in the limit most of the
information on the underlying GFF is lost. In fact, following the proof below,
one could make it even precise, by showing that the same result also holds
conditioned on any finite number of Fourier frequencies.

######  Theorem 1.8.

The processes

$W_{r}:=(r^{\beta^{2}-5}(|\mu(Q(x,r))|^{2}-\mathbb{E}(|\mu(Q(x,r))|^{2})))_{x\in[0,1]^{2}}$
(1.9)

converge as $r\to 0$ in distribution, but not in probability, in the space
$H^{-2}_{0}([0,1]^{2})$, to a constant multiple of the White noise in
$[0,1]^{2}$.

Although the path towards a proof is rather clear, given the independence
properties noticed in earlier parts of the paper, it is technically non-
trivial to push through. For example some symmetrization tricks are needed to
identify cancellations and obtain precise enough estimates; see the paragraph
following (7.1) for more details.

Finally, let us also mention a related result but with a different flavour
from [JSW20]: when $\beta\to\sqrt{2}$, one can renormalise the imaginary so
that it converges to white noise.

###  1.5. Organisation of the paper

The paper is organised as follows:

  * •

Section 2 is a preliminary section recalling a few facts about imaginary
multiplicative chaos. We also prove two lemmas that will be useful in the rest
of the paper (Lemmas 2.7 and 2.8).

  * •

Section 3 is dedicated to the study of the monofractality of $\mu$ and
contains the proofs of Proposition 1.1 and Theorem 1.2.

  * •

Section 4 is where we prove the law of the iterated logarithm (Theorem 1.4).

  * •

Section 5 is concerned with the study of the fast points and proves Theorem
1.5.

  * •

Section 6 is where we prove the optimal Besov regularity of $\mu$, i.e.
Theorem 1.7.

  * •

Section 7 is finally dedicated to the the proof of Theorem 1.8: the
convergence of $W_{r}$ to white noise.

  * •

Appendix A: we state and prove a variant of Kolmogorov’s continuity theorem
that we use in the proof of Proposition 1.1.

  * •

Appendix B contains a few comments on the tail estimate (1.8).

Acknowledgements The authors thank Nathanaël Berestycki for interesting
discussions during the very early stages of the project. J.A and A.J are
supported by Eccellenza grant 194648 of the Swiss National Science Foundation
and are members of NCCR Swissmap. J.J is supported by The Finnish Centre of
Excellence (CoE) in Randomness and Structures and was a member of NCCR
Swissmap.

##  2\. Preliminaries

###  2.1. Notational convention:

In this article, we will use two different normalisations for the exponential
of $i\beta\Gamma$:

$e^{i\beta\Gamma(x)}\quad=\quad\lim_{\varepsilon\to
0}\varepsilon^{-\beta^{2}/2}e^{i\beta\Gamma_{\varepsilon}(x)}\qquad\text{and}\qquad:e^{i\beta\Gamma(x)}:\quad=\quad\lim_{\varepsilon\to
0}e^{\beta^{2}\mathrm{Var}(\Gamma_{\varepsilon}(x))/2}e^{i\beta\Gamma_{\varepsilon}(x)},$
(2.1)

where $\Gamma_{\varepsilon}$ stands for the $\varepsilon$-circle average of
$\Gamma$. The second normalisation is the so-called Wick normalisation and is
distinguished from the first one by the two colons on both sides of the
exponential. These two normalisations lead to equivalent generalised functions
in the sense that they are related by

$e^{i\beta\Gamma(x)}\quad=\quad\operatorname{CR}(x,D)^{-\beta^{2}/2}:e^{i\beta\Gamma(x)}:.$
(2.2)

These normalisations have their own advantages. For instance,
$e^{i\beta\Gamma}$ has a neat spatial Markov property, whereas the moments of
$:e^{i\beta\Gamma}:$ have a simpler expression.

###  2.2. General moment bounds for imaginary chaos

We start this preliminary section by collecting some known estimates on the
moments of the imaginary chaos. Consider a log-correlated field $X$ defined on
some 2D open set $U\subset\mathbb{R}^{2}$. To be more specific and to be able
to apply the existing literature, we assume that the covariance of $X$ can be
written as

$-\log|x-y|+g(x,y)$

where $g\in C^{2}(U\times U)\times L^{1}(U\times U)$ is bounded from above.
For any $\mathbf{x}=(x_{1},\dots,x_{N})\in U^{N}$ and
$\mathbf{y}=(y_{1},\dots,y_{M})\in U^{M}$, denote by

$\mathcal{E}(X;\mathbf{x},\mathbf{y}):=-\sum_{1\leq i<j\leq
N}\mathbb{E}[X(x_{i})X(x_{j})]-\sum_{1\leq i<j\leq
M}\mathbb{E}[X(y_{i})X(y_{j})]+\sum_{\begin{subarray}{c}1\leq i\leq N\\\ 1\leq
j\leq M\end{subarray}}\mathbb{E}[X(x_{i})X(y_{j})].$ (2.3)

We will write $d\mathbf{x}=dx_{1}\dots dx_{N}$.

######  Lemma 2.1.

For any $f\in L^{\infty}(U)$,

$\mathbb{E}\left[\left|\int:e^{i\beta
X(x)}:f(x)dx\right|^{2N}\right]=\int_{U^{N}\times
U^{N}}\,d\mathbf{x}\,d\mathbf{y}\,e^{\beta^{2}\mathcal{E}(X;\mathbf{x};\mathbf{y})}\prod_{i=1}^{N}f(x_{i})\overline{f(y_{i})}.$

###### Proof.

See [JSW20, Section 3.2]. ∎

To bound the moments, we will moreover use the following Onsager inequality:

######  Lemma 2.2 ([JSW20], Proposition 3.6 (i)).

Let $K$ be a compact subset of $U$ and $(\mathbf{x},\mathbf{y})\in U^{N}\times
U^{N}$. Denote by $z_{i}=x_{i}$ and $z_{N+i}=y_{i}$ for $i=1,\dots,N$. Then

$\mathcal{E}(X;\mathbf{x};\mathbf{y})\leq-\frac{1}{2}\sum_{j=1}^{2N}\log\min_{i\neq
j}|z_{i}-z_{j}|+CN$

for some constant $C>0$ depending only on $g$ and $K$.

######  Lemma 2.3 ([JSW20], Lemma 3.10).

Let $B(0,1)$ be the unit ball in $\mathbb{R}^{2}$. For all
$\beta\in(0,\sqrt{2})$, there exists $C=C(\beta,d)>0$ such that for all $p\geq
1$,

$\int_{B(0,1)^{p}}d\mathbf{z}\exp\Big{(}-\frac{\beta^{2}}{2}\sum_{j=1}^{p}\log\min_{i\neq
j}|z_{i}-z_{j}|\Big{)}\leq C^{p}p^{\beta^{2}p/4}.$

######  Notation 2.4.

For $x,y\in D$, we will denote by

$G_{D}^{\prime}(x,y)=G_{D}(x,y)-\frac{1}{2}\log\operatorname{CR}(x,D)-\frac{1}{2}\log\operatorname{CR}(y,D).$
(2.4)

For any $\mathbf{x}=(x_{1},\dots,x_{N})\in D^{N}$ and
$\mathbf{y}=(y_{1},\dots,y_{M})\in D^{M}$, denote by

$\mathcal{E}^{\prime}(\Gamma;\mathbf{x},\mathbf{y}):=-\sum_{1\leq i<j\leq
N}G_{D}^{\prime}(x_{i},x_{j})-\sum_{1\leq i<j\leq
M}G_{D}^{\prime}(y_{i},y_{j})+\sum_{\begin{subarray}{c}1\leq i\leq N\\\ 1\leq
j\leq M\end{subarray}}G_{D}^{\prime}(x_{i},y_{j}).$ (2.5)

By Lemma 2.1 and the difference of normalisation (2.2) between $\mu$ and the
Wick normalisation of $e^{i\beta\Gamma}$, we have for all test function
$f:D\to\mathbb{R}$,

$\mathbb{E}\left|\int_{D}\mu(x)f(x)\,dx\right|^{2}=\int_{D\times
D}e^{\beta^{2}G_{D}^{\prime}(x,y)}f(x)f(y)\,dx\,dy$ (2.6)

and, more generally, for all $N\geq 1$,

$\mathbb{E}\left|\int_{D}\mu(x)f(x)\,dx\right|^{2N}=\int_{D^{N}\times
D^{N}}e^{\beta^{2}\mathcal{E}^{\prime}(\Gamma;\mathbf{x};\mathbf{y})}\prod_{j=1}^{N}f(x_{j})f(y_{j})\,d\mathbf{x}\,d\mathbf{y}.$
(2.7)

###  2.3. Moment bounds for rectangles

We will also need several times to control how the moments of imaginary chaos
depend on the side-lengths of a rectangle. This is given by the following
lemma:

######  Lemma 2.5.

Let $K$ be a compact subset of $D$. There exists $C=C(K,\beta)$ such that the
following holds. For any $a\times b$ rectangle $Q\subset K$ and any $N\geq 1$,
we have

$\mathbb{E}|\mu(Q)|^{2N}\leq
C^{N}(ab)^{(2-\frac{\beta^{2}}{2})N}N^{\frac{\beta^{2}}{2}N}.$

Before proving Lemma 2.5, we first recall the following recursive inequality
which works for any log-correlated field:

######  Lemma 2.6.

Let $\nu$ be an imaginary chaos defined on some domain $U$ (with the Wick
normalisation) and $A\subset U$ be a measurable set. Let $m\geq 1$ and
$A_{1},\dots,A_{m}$ be disjoint measurable subsets of $A$. Then for all
$N_{1},\dots,N_{m}\geq 1$, we have

$\frac{1}{(N!)^{2}}\mathbb{E}|\nu(A)|^{2N}\geq\prod_{j=1}^{m}\frac{1}{(N_{j}!)^{2}}\mathbb{E}|\nu(A_{j})|^{2N_{j}},\quad\text{where}\quad
N=N_{1}+\dots+N_{m}.$

###### Proof.

See the proof of [JSW20, Proposition 3.14]. ∎

###### Proof of Lemma 2.5.

It will be useful for us to compare the field $\Gamma_{D}$ in $K$ with a
stationary log-correlated field. Such a comparison is given by [AJJ22, Theorem
4.5], which allows us to write in $K$ the decomposition

$\Gamma_{D}=Z+Y,$ (2.8)

where $Z$ and $Y$ are two independent fields with the following properties.
The field $Y$ is Hölder-regular. On the other hand, $Z$ is a field defined on
the entire space $\mathbb{R}^{2}$ whose covariance is of the form

$\mathbb{E}Z(x)Z(y)=\int_{0}^{\infty}k(e^{u}(x-y))(1-e^{-\delta u})\,du$ (2.9)

for some $\delta>0$ and some rotationally invariant and compactly supported
seed covariance $k\colon\mathbb{R}^{2}\to\mathbb{R}$ with $k(0)=1$. In this
proof, the property that will be the most useful for us is the fact that the
law of $Z$ is invariant under translations.

Let $Q$ be a rectangle included in $K$. Let us first show that

$\mathbb{E}|\mu(Q)^{2N}|\leq C^{N}\mathbb{E}\big{|}\int_{Q}:e^{i\beta
Z(x)}:\,dx\big{|}^{2N}.$ (2.10)

By independence of $Z$ and $Y$ and by Lemma 2.1,

$\displaystyle\mathbb{E}|\mu(Q)^{2N}|$ $\displaystyle=\int_{Q^{N}\times
Q^{N}}d\mathbf{x}\,d\mathbf{y}e^{\beta^{2}\mathcal{E}(Z;\mathbf{x};\mathbf{y})}\prod_{j=1}^{N}\operatorname{CR}(x_{j},D)^{-\frac{\beta^{2}}{2}}\operatorname{CR}(y_{j},D)^{-\frac{\beta^{2}}{2}}$
$\displaystyle\qquad\times\mathbb{E}\left[e^{i\beta Y(x_{1})}\dots e^{i\beta
Y(x_{N})}e^{-i\beta Y(y_{1})}\dots e^{-i\beta Y(y_{N})}\right].$

By Jensen’s inequality, the expectation in the integrand is at most 1. The
remaining terms being nonnegative and $\sup_{x\in
K}\operatorname{CR}(x,D)^{-\beta^{2}/2}$ being finite, we deduce that
$\mathbb{E}|\mu(Q)^{2N}|$ is at most

$C^{N}\int_{Q^{N}\times
Q^{N}}d\mathbf{x}\,d\mathbf{y}e^{\beta^{2}\mathcal{E}(Z;\mathbf{x};\mathbf{y})}=C^{N}\mathbb{E}\big{|}\int_{Q}:e^{i\beta
Z(x)}:\,dx\big{|}^{2N},$

as stated in (2.10).

It is now enough to show that for any $a\times b$ rectangle
$Q\subset\mathbb{R}^{2}$,

$\mathbb{E}\big{|}\int_{Q}:e^{i\beta Z(x)}:\,dx\big{|}^{2N}\leq
C^{N}(ab)^{(2-\frac{\beta^{2}}{2})N}N^{\frac{\beta^{2}}{2}N}.$ (2.11)

We may cover $[0,1]^{2}$ with
$m=\lfloor\frac{1}{a}\rfloor\lfloor\frac{1}{b}\rfloor$ rectangles of size
$a\times b$. Then by Lemma 2.6 and stationarity we have

$\frac{1}{((mN)!)^{2}}\mathbb{E}\Big{|}\int_{[0,1]^{2}}:e^{i\beta
Z(x)}:\,dx\Big{|}^{2mN}\geq\Big{(}\frac{1}{(N!)^{2}}\mathbb{E}\Big{|}\int_{Q}:e^{i\beta
Z(x)}:\,dx\Big{|}^{2N}\Big{)}^{m},$

which gives us

$\mathbb{E}\Big{|}\int_{Q}:e^{i\beta
Z(x)}:\,dx\Big{|}^{2N}\leq\frac{(N!)^{2}}{((mN)!)^{2/m}}\Big{(}\mathbb{E}\Big{|}\int_{[0,1]^{2}}:e^{i\beta
Z(x)}:\,dx\Big{|}^{2Nm}\Big{)}^{1/m}.$ (2.12)

By applying [JSW20, Theorem 1.3] (or, equivalently, Lemmas 2.1, 2.2 and 2.3)
we have

$\Big{(}\mathbb{E}\Big{|}\int_{[0,1]^{2}}:e^{i\beta
Z(x)}:\,dx\Big{|}^{2Nm}\Big{)}^{1/m}\leq C^{N}(Nm)^{\frac{\beta^{2}}{2}N}$

for some constant $C>0$. One easily checks by using Stirling’s formula that
the remaining multiplicative factor on the right hand side of (2.12) is at
most $C^{N}m^{-2N}$, concluding the proof of (2.11). This finishes the proof
of Lemma 2.5. ∎

###  2.4. Contribution of the harmonic extension to imaginary chaos

The proofs of some of our main results are based on the spatial Markov
property which allows us to find a decomposition analogous to the one
presented in Remark 1.3. The difference here is that the field $N_{r}$
corresponding to the harmonic part of the GFF is no longer constant. However,
as it does not vary too rapidly we can recover an argument similar to Remark
1.3. This is the content of the next lemma. We will need this lemma for both
the GFF $\Gamma$ in $D$ with dirichlet boundary conditions and for the whole-
plane GFF $\Gamma^{*}$ (see [MS17, Proposition 2.8] for the spatial Markov
property for $\Gamma^{*}$).

######  Lemma 2.7.

Let $x\in D$ and $R_{0}>0$ such that $Q(x,R_{0})\subset D$. Let
$0<2r<R<R_{0}$. We decompose the GFF $\Gamma$ in $D$ and the whole-plane GFF
$\Gamma^{*}$ as the independent sums

$\Gamma=\Gamma^{(R)}+h^{(R)}\quad\mathrm{~{}and~{}}\quad\Gamma^{*}=\Gamma^{(R)}+{h^{*}}^{(R)}$

where $\Gamma^{(R)}$ is a GFF in $Q(x,R)$ and $h^{(R)}$ and ${h^{*}}^{(R)}$
are harmonic in $Q(x,R)$. By denoting $\mu^{(R)}$ the imaginary chaos
associated to $\Gamma^{(R)}$, there exist $c,c_{R_{0}}>0$ which may depend on
$\beta$ and on $\beta,R_{0}$ respectively such that for all
$\lambda\geq(r/R)^{1/2}$,

$\displaystyle\mathbb{P}\left(\left|\left|\mu(Q(x,r))\right|-\left|\mu^{(R)}(Q(x,r))\right|\right|>\lambda
r^{2-\frac{\beta^{2}}{2}}\right)\leq\frac{1}{c_{R_{0}}}\exp\Big{(}-c_{R_{0}}\Big{(}\frac{R}{r}\lambda^{2}\Big{)}^{\frac{2}{\beta^{2}+1}}\Big{)},$
(2.13)
$\displaystyle\mathbb{P}\left(\left|\left|\mu^{*}(Q(x,r))\right|-\left|\mu^{(R)}(Q(x,r))\right|\right|>\lambda
r^{2-\frac{\beta^{2}}{2}}\right)\leq\frac{1}{c}\exp\Big{(}-c\Big{(}\frac{R}{r}\lambda^{2}\Big{)}^{\frac{2}{\beta^{2}+1}}\Big{)}.$
(2.14)

###### Proof of Lemma 2.7.

We start by proving (2.13) whose proof is divided into two main steps.
Firstly, we will give quantitative estimates saying that $h^{(R)}$ does not
vary too much on $Q(x,r)$. We will then use Markov’s inequality and compute
moments of integrals of the imaginary chaos. This has the advantage of
transforming oscillating integrals into integrals of nonnegative functions
which are easier to handle.

We start by proving the following claim: there exists $C_{R_{0}}>0$ which may
depend on $R_{0}$ s.t.

$\sup_{y\in
Q(x,r)}\mathbb{E}\left[\left(h^{(R)}(y)-h^{(R)}(x)\right)^{2}\right]\leq
C_{R_{0}}\frac{r}{R}.$ (2.15)

An exact computation shows that for $y,z\in Q(x,r)$,

$\mathbb{E}\left[\left(h^{(R)}(y)-h^{(R)}(x)\right)^{2}\right]=2G_{Q(x,R)}(x,y)-2G_{D}(x,y)+\log\frac{\operatorname{CR}(x,D)\operatorname{CR}(y,D)}{\operatorname{CR}(x,Q(x,R))\operatorname{CR}(y,Q(x,R))}.$
(2.16)

Let $\varphi:D\to\mathbb{D}$ be a conformal map. By conformal invariance of
the Green function,

$\displaystyle 2G_{D}(x,y)-\log(\operatorname{CR}(x,D)\operatorname{CR}(y,D))$
$\displaystyle=2\log\frac{\left|1-\overline{\varphi(x)}\varphi(y)\right|}{\left|\varphi(x)-\varphi(y)\right|}+\log\left(\left|\varphi^{\prime}(x)\right|\left|\varphi^{\prime}(y)\right|\right)-\log\left(1-\left|\varphi(x)\right|^{2}\right)\left(1-\left|\varphi(y)\right|^{2}\right).$

In the following, we will make computations which are justified because
$\varphi\in C^{\infty}(D)$ and $\varphi^{-1}\in C^{\infty}(\mathbb{D})$. The
error terms will thus depend on $R_{0}$. We have

$\log\left|\varphi(x)-\varphi(y)\right|=\log\left|\varphi^{\prime}(x)\right|+\log\left|x-y\right|+O(r)$

and also,

$\log\left|1-\overline{\varphi(x)}\varphi(y)\right|=\log\left(1-\left|\varphi(x)\right|^{2}\right)+O\left(\frac{\left|y-x\right|}{1-\left|\varphi(x)\right|^{2}}\right)=\log\left(1-\left|\varphi(x)\right|^{2}\right)+O(r).$

Hence

$2G_{D}(x,y)-\log(\operatorname{CR}(x,D)\operatorname{CR}(y,D))=\log\frac{1}{\left|x-y\right|}+O(r).$

As $r<R/2$, $x$ and $y$ are in the bulk of $Q(x,R)$ and we can show similarly
that

$2G_{Q(x,R)}(x,y)-\log(\operatorname{CR}(x,Q(x,R))\operatorname{CR}(y,Q(x,R)))=\log\frac{1}{\left|x-y\right|}+O\left(\frac{r}{R}\right).$

Coming back to (2.16), it shows (2.15).

We now come back to the core of the proof of (2.13). In the rest of the proof,
the constants may depend on $\beta$ and on $R_{0}$. Let $\lambda>0$. By the
triangle inequality, the probability on the left hand side of (2.13) is at
most

$\mathbb{P}\left(\left|\int_{Q(x,r)}\mu^{(R)}(y)\left(e^{i\beta(h^{(R)}(y)-h^{(R)}(x))}-1\right)dy\right|>\lambda
r^{2-\frac{\beta^{2}}{2}}\right).$ (2.17)

Let $N\geq 1$. Recall the notation
$\mathcal{E}^{\prime}(\Gamma^{(R)};\mathbf{y};\mathbf{z})$ defined in (2.5).
As $\mu^{(R)}$ and $h^{(R)}$ are independent, we have by Lemma 2.1,

$\displaystyle\mathbb{E}\left[\left|\int_{Q(x,r)}\mu^{(R)}(y)\left(e^{i\beta(h^{(R)}(y)-h^{(R)}(x))}-1\right)dy\right|^{2N}\right]$
(2.18) $\displaystyle=\int_{Q(x,r)^{N}\times
Q(x,r)^{N}}d\mathbf{y}\,d\mathbf{z}\,e^{\beta^{2}\mathcal{E}^{\prime}(\Gamma^{(R)};\mathbf{y};\mathbf{z})}\mathbb{E}\left[\prod_{j=1}^{N}\left(e^{i\beta(h^{(R)}(y_{j})-h^{(R)}(x))}-1\right)\left(e^{-i\beta(h^{(R)}(z_{j})-h^{(R)}(x))}-1\right)\right].$

By a generalised Cauchy–Schwarz inequality, we can bound the expectation on
the right hand side of the above display by

$\max_{y\in
Q(x,r)}\mathbb{E}\left[\left|e^{i\beta(h^{(R)}(y)-h^{(R)}(x))}-1\right|^{2N}\right].$

Denoting $\Gamma^{Q(0,1)}$ the GFF in the unit square, we deduce by scaling
that (2.18) is at most

$\displaystyle C^{N}R^{(2-\beta^{2}/2)2N}\max_{y\in
Q(x,r)}\mathbb{E}\left[\left|e^{i\beta(h^{(R)}(y)-h^{(R)}(x))}-1\right|^{2N}\right]\int_{Q(0,\frac{r}{R})^{N}\times
Q(0,\frac{r}{R})^{N}}d\mathbf{y}\,d\mathbf{z}\,e^{\beta^{2}\mathcal{E}^{\prime}(\Gamma^{Q(0,1)};\mathbf{y};\mathbf{z})}.$

Using (2.15), we have

$\max_{y\in
Q(x,r)}\mathbb{E}\left[\left|e^{i\beta(h^{(R)}(y)-h^{(R)}(x))}-1\right|^{2N}\right]\leq
C^{N}\mathbb{E}\left[\left|h^{(R)}(y)-h^{(R)}(x)\right|^{2N}\right]\leq(N-1)!!\left(C\frac{r}{R}\right)^{N},$
(2.19)

where $(N-1)!!$ is the double factorial of $N-1$, i.e. the product of all
positive integers smaller than $N-1$ with the same parity as $N-1$. Moreover,
by Lemmas 2.2 and 2.3 and by scaling, we can bound

$\displaystyle\int_{Q(0,\frac{r}{R})^{N}\times
Q(0,\frac{r}{R})^{N}}d\mathbf{y}\,d\mathbf{z}\,e^{\beta^{2}\mathcal{E}^{\prime}(\Gamma^{Q(0,1)};\mathbf{y};\mathbf{z})}\leq
C^{N}\int_{Q(0,\frac{r}{R})^{2N}}d\mathbf{y}\prod_{j=1}^{2N}\left(\min_{k\neq
j}\left|y_{k}-y_{j}\right|\right)^{-\beta^{2}/2}$
$\displaystyle\qquad=C^{N}\left(\frac{r}{R}\right)^{(2-\beta^{2}/2)2N}\int_{Q(0,1)^{2N}}d\mathbf{y}\prod_{j=1}^{2N}\left(\min_{k\neq
j}\left|y_{k}-y_{j}\right|\right)^{-\beta^{2}/2}$ $\displaystyle\qquad\leq
C^{N}N^{N\beta^{2}/2}\left(\frac{r}{R}\right)^{(2-\beta^{2}/2)2N}.$

Putting things together, it implies that there exists $C>0$ such that the
expectation in (2.18) is not larger than

$(N-1)!!N^{N\beta^{2}/2}\left(C\frac{r}{R}r^{(2-\beta^{2}/2)2}\right)^{N}.$
(2.20)

By Markov’s inequality and using that $(N-1)!!\leq C^{N}N^{N/2}$, we deduce
that the probability on the left hand side of (2.13) is at most

$\Big{(}C\frac{r}{R}\frac{1}{\lambda^{2}}\Big{)}^{N}N^{(\beta^{2}+1)N/2}.$

We optimise in $N$ and choose $N=\left\lfloor
e^{-1}(\lambda^{2}\frac{R}{r})^{2/(\beta^{2}+1)}\right\rfloor$ which leads to
the estimate on the right hand side of (2.13). It concludes the proof of
(2.13).

The proof of (2.14) about the whole-plane GFF $\Gamma^{*}$ is simpler since
there is no boundary to deal with and follows from the same lines. ∎

######  Lemma 2.8.

Let $0<2r<R$ and denote by $\mu^{(R)}$ the imaginary chaos associated to the
GFF in $Q(0,R)$. There exists $c>0$ which may depend on $\beta$ s.t.

$\displaystyle\left|\mathbb{P}\left(\left|\mu^{(R)}(Q(0,r))\right|>r^{2-\beta^{2}/2}t\right)-\exp\left(-(1+o(1))c^{*}(\beta)t^{4/\beta^{2}}\right)\right|\leq\frac{1}{c}\exp\left(-c\left(\frac{R}{r}\right)^{\frac{2}{\beta^{2}+1}}\right),$
(2.21)

where $o(1)\to 0$ when $t\to\infty$, uniformly in $r,R$ as above.

###### Proof of Lemma 2.8.

Let $\varepsilon>0$. The probability in (2.21) being equal to

$\mathbb{P}\left(\left|\mu^{(R/r)}(Q(0,1))\right|>t\right),$

we can assume without loss of generality that $r=1$ and $R>2$. The tail
estimate (1.8) concerns the whole plane GFF $\Gamma^{*}$ that we write as the
independent sum $\Gamma^{(R)}+{h^{*}}^{(R)}$ where $\Gamma^{(R)}$ is the GFF
in $Q(0,R)$ with Dirichlet boundary condition and ${h^{*}}^{(R)}$ is harmonic
in $Q(0,R)$. By Lemma 2.7, there exists $c=c(\beta)>0$ s.t.

$\displaystyle\mathbb{P}\left(\left|\left|\mu^{(R)}(Q(0,1))\right|-\left|\mu^{*}(Q(0,1))\right|\right|>1\right)\leq\frac{1}{c}\exp\left(-cR^{\frac{2}{\beta^{2}+1}}\right).$

Hence we have the following upper bound

$\mathbb{P}\left(\left|\mu^{(R)}(Q(0,1))\right|>t\right)\leq\mathbb{P}\left(\left|\mu^{*}(Q(0,1))\right|>t-1\right)+\frac{1}{c}\exp\left(-cR^{\frac{2}{\beta^{2}+1}}\right)$

and an analogous lower bound. Together with (1.8), this shows (2.21). ∎

##  3\. Hölder regularity and monofractality: proof of Proposition 1.1 and
Theorem 1.2

We start this section by proving Proposition 1.1.

###### Proof of Proposition 1.1.

Fix a compact subset $K$ of $D$ and $N\geq 1$. If $A=Q(x,r)$ and
$A^{\prime}=Q(x^{\prime},r^{\prime})$ are two squares included in $K$, we have

$\displaystyle\mathbb{E}\left[\left|\mu(A)-\mu(A^{\prime})\right|^{2N}\right]$
$\displaystyle=\mathbb{E}\left[\left|\mu(A\backslash A\cap
A^{\prime})-\mu(A^{\prime}\backslash A\cap A^{\prime})\right|^{2N}\right]$
$\displaystyle\leq 2^{2N-1}\mathbb{E}\left[\left|\mu(A\backslash A\cap
A^{\prime})\right|^{2N}+\left|\mu(A^{\prime}\backslash A\cap
A^{\prime})\right|^{2N}\right].$

$A\backslash A\cap A^{\prime}$ and $A^{\prime}\backslash A\cap A^{\prime}$ can
be covered by finitely many $a\times b$ rectangles with either $a\leq 2(r\vee
r^{\prime})$ and $b\leq C\left\|(x,r)-(x^{\prime},r^{\prime})\right\|$, or the
bounds swapped for $a$ and $b$. By Lemma 2.5, we deduce that

$\displaystyle\mathbb{E}\left[\left|\mu(A)-\mu(A^{\prime})\right|^{2N}\right]$
$\displaystyle\leq C_{K,N}\left\\{(r\vee
r^{\prime})\left\|(x,r)-(x^{\prime},r^{\prime})\right\|\right\\}^{(2-\beta^{2}/2)N}.$

With the help of Lemma A.1 which is a slight modification of Kolmogorov’s
continuity theorem, it shows that $(x,r)\mapsto\mu(Q(x,r))$ restricted to the
set $\\{(x,r):Q(x,r)\subset K\\}$ possesses a continuous modification which
satisfies (1.2). It concludes the proof. ∎

We now turn to the proof of Theorem 1.2.

###### Proof of Theorem 1.2.

Thanks to Proposition 1.1, we only need to prove the upper bound of (1.3). Let
$\Gamma^{*}$ be the whole-plane GFF and let $\mu^{*}$ be the associated
imaginary chaos. As with the the GFF $\Gamma$ in the domain $D$, we can make
sense of $\mu^{*}(Q(x,r))$ simultaneously for all $x\in\mathbb{C}$ and $r>0$.
Moreover, for all compact sets $K\subset\mathbb{C}$ and
$\alpha<1-\beta^{2}/4$, there exists a random constant $C_{K,\alpha}$ almost
surely finite such that for all $Q(x,r),Q(x^{\prime},r^{\prime})\subset K$,

$\left|\mu^{*}(Q(x,r))-\mu^{*}(Q(x^{\prime},r^{\prime}))\right|\leq
C_{K,\alpha}(r\vee
r^{\prime})^{\alpha}\left\|(x,r)-(x^{\prime},r^{\prime})\right\|^{\alpha}.$
(3.1)

We are going to prove that almost surely, simultaneously for all $x\in D$,

$\liminf_{r\to 0}\frac{\log\left|\mu^{*}(Q(x,r))\right|}{\log r}\leq
2-\beta^{2}/2.$ (3.2)

It will be convenient to consider the whole-plane GFF instead of $\Gamma$
because of its translation and scaling invariance. We now explain how we
conclude from (3.2) the proof of the upper bound of (1.3). By spatial Markov
property ([MS17, Proposition 2.8]), we can couple the two GFFs $\Gamma^{*}$
and $\Gamma$ such that $\Gamma^{*}=\Gamma+h$ and such that $\Gamma$ and $h$
are independent and $h$ is harmonic in $D$. We claim that the map

$\begin{array}[]{ccc}\\{(x,r)\in D\times[0,1],Q(x,r)\subset
D\\}&\to&\mathbb{C}\\\ (x,r)&\mapsto&e^{-i\beta
h(x)}\mu^{*}(Q(x,r))-\mu(Q(x,r))\end{array}$

possesses a continuous modification. Moreover, for all compact subsets $K$ of
$D$ and $\alpha<3-\beta^{2}/2$, there exists a random constant $C_{K,\alpha}$
almost surely finite such that for all $Q(x,r)\subset K$,

$\left|e^{-i\beta h(x)}\mu^{*}(Q(x,r))-\mu(Q(x,r))\right|\leq
C_{K,\alpha}r^{\alpha}.$

The proof of this claim is very similar to the proof of Proposition 1.1 with
the additional ingredient (which is similar to (2.15)) that for all compact
subsets $K$ of $D$, there exists $C_{K}>0$ such that for all $Q(x,r)\subset
K$,

$\sup_{y\in Q(x,r)}\mathbb{E}\left[(h(x)-h(y))^{2}\right]\leq C_{K}r.$

We omit the details of the proof of this claim. Since $\alpha$ can be chosen
larger than $2-\beta^{2}/2$, we now see that this claim and (3.2) imply the
desired upper bound of (1.3).

The rest of the proof will consist in proving (3.2). Let us denote by

$\xi=2-\beta^{2}/2.$

Let $\delta>0$ and $K$ be an integer larger than $1/\delta-1$. We are going to
bound the probability of the event that there exists $x\in D$ such that

$\liminf_{r\to 0}\frac{\log\left|\mu^{*}(Q(x,r))\right|}{\log
r}\geq\xi+2\delta.$

For such an $x$, we have for small enough $r>0$,

$\left|\mu^{*}(Q(x,r))\right|\leq r^{\xi+\delta}.$ (3.3)

Let $p\geq 1$, $z\in 2^{-2p}\mathbb{Z}^{2}$ be such that $\left|z-x\right|\leq
2^{-2p}$. For all $j=0\dots 2K$, define $r_{j}:=2^{-p-j}$. By (3.1) with
$\alpha>(\xi+\delta)/3$, there exists a random constant $C(\omega)$ a.s.
finite such that for all $j=0\dots 2K$,

$\displaystyle\left|\mu^{*}(Q(z,r_{j}))-\mu^{*}(Q(x,r_{j}))\right|\leq
C(\omega)(r_{j}2^{-2p})^{\alpha}\leq C(\omega)r_{j}^{3\alpha}\leq
r_{j}^{\xi+\delta}$

if $p$ is large enough. With (3.3), it implies that if $p$ is large enough,
for all $j=0\dots 2K$,

$\left|\mu^{*}(Q(z,r_{j}))\right|\leq 2r_{j}^{\xi+\delta}.$

To put in a nutshell, we have shown that, up to an event of vanishing
probability,

$\displaystyle\left\\{\exists x\in D,\liminf_{r\to
0}\frac{\log\left|\mu^{*}(Q(x,r))\right|}{\log r}\geq\xi+2\delta\right\\}$
$\displaystyle\subset\bigcup_{n}\bigcap_{p\geq
n}\bigcup_{\begin{subarray}{c}z\in 2^{-2p}\mathbb{Z}^{2}\\\ z\in
D\end{subarray}}\left\\{\forall j=0\dots
2K,\left|\mu^{*}(Q(z,r_{j}))\right|\leq 2r_{j}^{\xi+\delta}\right\\}$
$\displaystyle\subset\bigcap_{n}\bigcup_{p\geq
n}\bigcup_{\begin{subarray}{c}z\in 2^{-2p}\mathbb{Z}^{2}\\\ z\in
D\end{subarray}}\left\\{\forall j=0\dots
2K,\left|\mu^{*}(Q(z,r_{j}))\right|\leq 2r_{j}^{\xi+\delta}\right\\}$ (3.4)

the last inclusion being elementary and not specific to the events considered
here. The goal is now to show that the probability of the event (3.4)
vanishes. Let $p$ be a large integer and $z\in 2^{-2p}\mathbb{Z}^{2}\cap D$.
By translation and scaling invariance,

$\displaystyle\mathbb{P}\left(\forall j=0\dots
2K,\left|\mu^{*}(Q(z,r_{j}))\right|\leq 2r_{j}^{\xi+\delta}\right)$
$\displaystyle=\mathbb{P}\left(\forall j=0\dots
2K,\left|\mu^{*}(Q(0,2^{-j}))\right|\leq 2\cdot
2^{-p\delta}2^{-(\xi+\delta)j}\right).$

In Lemma 3.1 below the current proof, we show that the above right hand side
term is at most $C_{K}2^{-2p\delta(K+1)}$ for some $C_{K}>0$ independent of
$p$. Since $K+1>1/\delta$, a union bound concludes that the probability of the
event (3.4) vanishes. It concludes the proof. ∎

######  Lemma 3.1.

Let $\mu^{*}$ be the imaginary chaos associated to the whole plane GFF. For
all integers $K\geq 1$, there exists $C_{K}>0$ such that for all
$\varepsilon>0$,

$\mathbb{P}\left(\forall
j=0,\dots,2K,|\mu^{*}(Q(0,2^{-j}))|\leq\varepsilon\right)\leq
C_{K}\varepsilon^{2(K+1)}.$

Before proving Lemma 3.1, we need to collect a density result following from
[AJJ22]. This result will feature a quantity $c_{1}(f)$ associated to some
complex-valued function $f:D\to\mathbb{C}$ that we define now. For $\delta>0$,
let $\mathcal{Q}_{\delta}$ be the collection of cubes of the form

$[4k_{1}\delta,(4k_{1}+1)\delta]\times[4k_{2}\delta,(4k_{2}+1)\delta]$ (3.5)

with $k_{1},k_{2}\in\mathbb{Z}$. Consider also the following cones of angle
$\pi/4$:

$\text{Cone}_{k}=\\{z\in\mathbb{C}:\arg z\in[k\pi/8,\pi/4+8k\pi/8]\\},$ (3.6)

for $k=0,\dots,15$. $c_{1}(f)$ is then defined by

$c_{1}(f):=\liminf_{\delta\to
0}\delta^{2}\\#\\{Q\in\mathcal{Q}_{\delta}:\exists
k\in\\{0,\dots,15\\},\forall x\in Q,f(x)\in\text{Cone}_{k}\text{ and
}|f(x)|\geq\left\|f\right\|_{\infty}/2\\}.$ (3.7)

Notice that $c_{1}(f)>0$ for any nonzero continuous function $f$.

######  Theorem 3.2.

Let $D$ be a bounded simply connected domain and $\Gamma$ be the GFF in $D$
with Dirichlet boundary conditions. Let $K$ be a compact subset of $D$ and
$f:D\to\mathbb{R}$ be a bounded measurable function supported in $K$. Let
$\mu(f)$ be the random variable defined by

$\mu(f)=\lim_{\varepsilon\to
0}\int_{D}\varepsilon^{-\beta^{2}/2}e^{i\beta\Gamma_{\varepsilon}(x)}f(x)dx,$

where $\Gamma_{\varepsilon}$ is the circle average approximation of $\Gamma$.
Then the law of $\mu(f)$ has a smooth density $\rho$ with respect to Lebesgue
measure on $\mathbb{C}$. Moreover, there exists $C>0$ that may depend on $D$,
$K$ and $\beta$ but not on $f$ such that $\left\|\rho\right\|_{\infty}\leq
Cc_{1}(f)^{-8}\left\|f\right\|_{\infty}^{-2}$.

###### Proof of Theorem 3.2.

This is a version of [AJJ22, Theorem 3.6] that is quantitative in terms of the
test function $f$. It follows directly from [AJJ22] by carefully recording the
dependence of $f$. We now give a few details. First of all, the imaginary
chaos considered in [AJJ22] is defined via a different approximation procedure
where the normalisation is done using
$e^{\frac{\beta^{2}}{2}\text{var}(\Gamma_{\varepsilon}(x))}$ instead of
$\varepsilon^{-\beta^{2}/2}$. This effectively changes the test function $f$
to $\tilde{f}:x\mapsto f(x)\text{CR}(x,D)^{-\beta^{2}/2}$ where
$\text{CR}(x,D)$ stands for the conformal radius of $D$ seen from $x$. Since
$c_{1}(\tilde{f})^{-8}\left\|\tilde{f}\right\|_{\infty}^{-2}$ differs from
$c_{1}(f)^{-8}\left\|f\right\|_{\infty}^{-2}$ by at most a multiplicative
constant depending only on $D$ and $K$, this will not affect the result.

In [AJJ22, Theorem 3.6], the test function $f$ was assumed to be real-valued
and continuous. As we are about to explain, complex-valued test functions can
be considered and the continuity assumption can be replaced by the assumption
that $c_{1}(f)>0$. We will give some details about this point while explaining
where the upper bound on $\left\|\rho\right\|_{\infty}$ comes from. By a
simple scaling argument, we can assume that $\left\|f\right\|_{\infty}=1$. All
the bounds obtained in the proof of [AJJ22, Theorem 3.6] are uniform in $f$
with $\left\|f\right\|_{\infty}=1$ and work in the exact same way for complex-
valued test functions, except the bounds in the proof of [AJJ22, Lemma 6.8].
In the proof of this lemma and with the notation therein, one has to bound
from below

$\displaystyle|\left\langle
DM,h_{\delta}\right\rangle_{H}|=\beta\sum_{Q\in\mathcal{Q}_{\delta}}\int_{Q\times
Q}f(x)\overline{f(y)}:e^{i\beta(\hat{Y}_{\delta}(x)+Z(x)}::e^{-i\beta(\hat{Y}_{\delta}(y)+Z(y)}:R_{\delta}(x,y)dxdy.$

$\mathcal{Q}_{\delta}$ is the collection of squares defined in (3.5). Since
each term in the above sum is a nonnegative real number, one can bound from
below $|\left\langle DM,h_{\delta}\right\rangle_{H}|$ by summing over specific
squares $Q\in\mathcal{Q}_{\delta}$, instead of summing over all squares in
$\mathcal{Q}_{\delta}$. In [AJJ22] and when $f$ is real-valued, we chose to
sum only on the squares $Q$ such that either
$f(x)\geq\left\|f\right\|_{\infty}/2$ for all $x\in Q$, or
$f(x)\leq-\left\|f\right\|_{\infty}/2$ for all $x\in Q$. For such a square
$Q$, we have $f(x)\overline{f(y)}\geq\left\|f\right\|_{\infty}^{2}/4$ for
$(x,y)\in Q\times Q$. In the proof of [AJJ22, Lemma 6.8], the total number of
squares $Q\in\mathcal{Q}_{\delta}$ with the above property is bounded from
below by some constant $c_{1}$ times $\delta^{-2}$. $|\left\langle
DM,h_{\delta}\right\rangle_{H}|$ is then successfully lower bounded and shown
to scale like $c_{1}$.

If $f$ is complex-valued and recalling the definition (3.6) of
$\text{Cone}_{k}$, we can instead sum over squares $Q$ such that there exists
$k\in\\{0,\dots,15\\}$ so that for all $x\in Q$, $f(x)\in\text{Cone}_{k}$ and
$|f(x)|\geq\left\|f\right\|_{\infty}/2$. For such a square $Q$, we now have
for all $(x,y)\in Q\times Q$,
$|f(x)\overline{f(y)}|\geq\left\|f\right\|_{\infty}^{2}/4$ and
$\arg(f(x)\overline{f(y)})\in[-\pi/4,\pi/4]$. In particular, the real part of
$f(x)\overline{f(y)}$ is at least
$\left\|f\right\|_{\infty}^{2}\cos(\pi/4)/4$. The rest of the argument goes
through. In particular, recalling the definition (3.7) of $c_{1}(f)$, if
$\delta$ is small enough, we can lower bound the number of appropriate squares
by $c_{1}(f)\delta^{-2}/2$ and give a lower bound on $|\left\langle
DM,h_{\delta}\right\rangle_{H}|$ by some quantity scaling like $c_{1}(f)$.
Together with Lemma 3.4 and Proposition 3.5 of [AJJ22], this shows that
$\left\|\rho\right\|_{\infty}\leq Cc_{1}(f)^{-8}$. ∎

###### Proof of Lemma 3.1.

Let $Q_{j}=Q(0,2^{-j})$, $j=0,\dots,2K$, and $Q_{2K+1}=\varnothing$. We start
with the elementary observation that

$\displaystyle\mathbb{P}\left(\forall j=0\dots
2K,\left|\mu^{*}(Q_{j})\right|\leq\varepsilon\right)\leq\mathbb{P}\left(\forall
j=0\dots 2K,\left|\mu^{*}(Q_{j}\backslash Q_{j+1})\right|\leq
2\varepsilon\right)$ $\displaystyle\leq\mathbb{P}\left(\forall
j=0,2,4,\dots,2K,\left|\mu^{*}(Q_{j}\backslash Q_{j+1})\right|\leq
2\varepsilon\right).$

In the last step, we neglected the constraints on odd values of $j$ so that
the sets $Q_{j}\setminus Q_{j+1}$, $i=0,2,4,\dots,2K$, are now disjoint and
well separated: the minimal distance between two such sets is at least
$2^{-2K}$. For $j=0,\dots,K$, let us define $A_{j}=Q_{2j}\setminus Q_{2j+1}$
and $B_{j}=Q(0,2^{-2j+1/2})\setminus Q(0,2^{-2(j+1)+1/2})$. By construction,
the $B_{j}$’s are disjoint and for all $j=0,\dots,K$, $A_{j}$ is contained in
$B_{j}$ and is at a positive distance to $\partial B_{j}$. We apply Markov’s
property inside each $B_{j}$: there exists a field $h$ such that for all
$j=0,\dots,K$, $\Gamma_{j}:=(\Gamma^{*}-h)\mathbf{1}_{B_{j}}$ has the law of a
GFF in $B_{j}$. Moreover, $\Gamma_{0},\dots,\Gamma_{K}$ and $h$ are mutually
independent and $h$ is harmonic inside each $B_{j}$. Conditionally on $h$, we
can now view $\mu^{*}(A_{j})$ as the imaginary chaos $\mu_{j}$ of $\Gamma_{j}$
integrated against the test function

$f_{j}(x):=e^{i\beta h(x)}\mathbf{1}_{\left\\{x\in A_{j}\right\\}}.$

Thus,

$\displaystyle\mathbb{P}\left(\forall
j=0,\dots,K,\left|\mu^{*}(A_{j})\right|\leq
2\varepsilon\right)=\mathbb{E}\left[\prod_{j=0}^{K}\mathbb{P}\left(|(\mu_{j},f_{j})|\leq
2\varepsilon|h\right)\right],$

where the expectation integrates out the randomness of $h$. By Theorem 3.2,
for all $j=0,\dots,K$ and conditionally given $h$, the law of the random
variable $(\mu_{j},f_{j})$ is absolutely continuous with respect to Lebesgue
measure on $\mathbb{C}$. Moreover, the associated densities are uniformly
bounded by some deterministic constant $C$ times $c_{1}(f_{j})^{-8}$. We
deduce that for all $j=0,\dots,K$, $\mathbb{P}\left(|(\mu_{j},f_{j})|\leq
2\varepsilon|h\right)\leq Cc_{1}(f_{j})^{-8}\varepsilon^{2}$, implying that

$\displaystyle\mathbb{P}\left(\forall
j=0,\dots,K,\left|\mu^{*}(A_{j})\right|\leq 2\varepsilon\right)$
$\displaystyle\leq
C\mathbb{E}\left[\prod_{j=0}^{K}c_{1}(f_{j})^{-8}\right]\varepsilon^{2(K+1)}$
$\displaystyle\leq
C\prod_{j=0}^{K}\mathbb{E}\left[c_{1}(f_{j})^{-8(K+1)}\right]^{\frac{1}{K+1}}\varepsilon^{2(K+1)},$

where we used Hölder’s inequality (more precisely, a generalisation to
finitely many variables) in the last inequality.

To conclude the proof, it remains to show that
$\mathbb{E}\left[c_{1}(f_{j})^{-8(K+1)}\right]$ is finite for all $j$. Let
$j\in\\{0,\dots,K\\}$ and $x_{j}\in A_{j}$ be at distance at least $2^{-10K}$
to the boundary of $A_{j}$. Let $\eta\in(0,2^{-10K})$. Note that if
$|h(x_{j})-h(y)|\leq 1/(100\beta)$ for all $y\in B(x_{j},\eta)$, then there
exists $k\in\\{0,\dots,15\\}$ such that $f_{j}(y)\in\text{Cone}_{k}$ for all
$y\in B(x_{j},\eta)$ and thus $c_{1}(f_{j})\geq c\eta^{2}$ for some constant
$c>0$. Thus,

$\displaystyle\mathbb{P}\left(c_{1}(f_{j})\leq
c\eta^{2}\right)\leq\mathbb{P}\Big{(}\sup_{y\in
B(x_{j},\eta)}|h(x_{j})-h(y)|>1/(100\beta)\Big{)}.$

Since $h$ is a smooth Gaussian field inside each $A_{j}$, then, by local
chaining inequalities (e.g. [VH14, Proposition 5.35]), the above right hand
side term is at most $Ce^{-c/\eta^{2}}$. This shows that the tails of
$1/c_{1}(f_{j})$ decays exponentially fast. In particular, all negative
moments of $c_{1}(f_{j})$ are finite, concluding the proof. ∎

##  4\. Law of the iterated logarithm: proof of Theorem 1.4

In this section, we will write

$\varphi(r)\coloneqq r^{2-\beta^{2}/2}(\log\left|\log
r\right|)^{\beta^{2}/4},\qquad r\in(0,1).$

The goal of this section is to establish the law of the iterated logarithm
(1.4) in Theorem 1.4. The proof of this result will be decomposed into two
main steps. First, we will show that (1.4) holds along any geometric sequence
of radii:

######  Lemma 4.1.

Let $x_{0}\in D$ and $\gamma\in(0,1)$ be fixed. Then

$\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},\gamma^{n}))\right|}{\varphi(\gamma^{n})}=c^{*}(\beta)^{-\beta^{2}/4}\quad\quad\mathrm{a.s.}$
(4.1)

We will then control the oscillations between two consecutive points in the
geometric sequence:

######  Lemma 4.2.

Let $x_{0}\in D$. For any $\varepsilon>0$ there exists $\gamma\in(0,1)$ such
that

$\sum_{n=1}^{\infty}\mathbb{P}\Big{[}\sup_{r\in[\gamma^{n+1},\gamma^{n}]}\left|\mu(Q(x_{0},\gamma^{n}))-\mu(Q(x_{0},r))\right|>\varepsilon\varphi(\gamma^{n+1})\Big{]}<\infty.$

Let us first see how the two lemmas imply Theorem 1.4.

###### Proof of Theorem 1.4.

Let $x_{0}\in D$. For any $\gamma\in(0,1)$ fixed, we can lower bound

$\limsup_{r\to
0}\frac{\left|\mu(Q(x_{0},r))\right|}{\varphi(r)}\geq\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},\gamma^{n}))\right|}{\varphi(\gamma^{n})}.$

By Lemma 4.1, the right hand side term is almost surely equal to
$c^{*}(\beta)^{-\beta^{2}/4}$ which provides the desired lower bound in
Theorem 1.4. It remains to prove the upper bound. By Lemma 4.1, to prove this
upper bound, it is enough to show that for any $\varepsilon>0$, there exists
$\gamma\in(0,1)$ such that we have

$\limsup_{n\to\infty}\sup_{r\in[\gamma^{n+1},\gamma^{n}]}\left|\frac{\left|\mu(Q(x_{0},r))\right|}{\varphi(r)}-\frac{\left|\mu(Q(x_{0},\gamma^{n}))\right|}{\varphi(\gamma^{n})}\right|\leq\varepsilon\quad\quad\text{a.s.}$
(4.2)

To show (4.2), we note that, by the triangle inequality and the fact that
$\varphi(r)$ is increasing near $0$, we get for
$r\in[\gamma^{n+1},\gamma^{n}]$ and $n$ large enough that

$\displaystyle\left|\frac{\left|\mu(Q(x_{0},r))\right|}{\varphi(r)}-\frac{\left|\mu(Q(x_{0},\gamma^{n}))\right|}{\varphi(\gamma^{n})}\right|$
$\displaystyle=\left|\Big{(}\frac{1}{\varphi(r)}-\frac{1}{\varphi(\gamma^{n})}\Big{)}\left|\mu(Q(x_{0},\gamma^{n}))\right|+\frac{\left|\mu(Q(x_{0},r))\right|-\left|\mu(Q(x_{0},\gamma^{n}))\right|}{\varphi(r)}\right|$
$\displaystyle\leq\Big{(}\frac{\varphi(\gamma^{n})}{\varphi(\gamma^{n+1})}-1\Big{)}\frac{\left|\mu(Q(x_{0},\gamma^{n}))\right|}{\varphi(\gamma^{n})}+\frac{\left|\mu(Q(x_{0},\gamma^{n}))-\mu(Q(x_{0},r))\right|}{\varphi(\gamma^{n+1})}.$

We conclude the proof by noticing that in the first term

$\lim_{n\to\infty}\frac{\varphi(\gamma^{n})}{\varphi(\gamma^{n+1})}-1=\gamma^{\frac{\beta^{2}}{2}-2}\lim_{n\to\infty}\frac{(\log\log\frac{1}{\gamma^{n}})^{\frac{\beta^{2}}{4}}}{(\log\log\frac{1}{\gamma^{n+1}})^{\frac{\beta^{2}}{4}}}-1=\gamma^{\frac{\beta^{2}}{2}-2}-1,$

which can be made as small as desired by choosing $\gamma$ close enough to
$1$. The second term is handled by Lemma 4.2 and Borel–Cantelli. ∎

The rest of this section is devoted to the proof of Lemmas 4.1 and 4.2.

###  4.1. Proof of Lemma 4.1

###### Proof of Lemma 4.1.

Fix $x_{0}\in D$, $\gamma\in(0,1)$ and let $r_{n}\coloneqq\gamma^{n}$. We
start by proving the lower bound which is the most difficult one. Consider the
subsequence $(r_{n}^{\prime})$ of $(r_{n})$ given by
$r_{n}^{\prime}=r_{k_{n}}$ with $k_{n}=\left\lfloor Kn\log\log n\right\rfloor$
for some $K>0$. In particular, if $K$ is large enough, it satisfies

$\liminf_{n\to\infty}\frac{\log(r^{\prime}_{n-1}/r^{\prime}_{n})}{\log\log
n}>\beta^{2}+1\qquad\mathrm{and}\qquad\lim_{n\to\infty}\frac{\log\left|\log
r^{\prime}_{n}\right|}{\log n}=1.$ (4.3)

As

$\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},r_{n}))\right|}{\varphi(r_{n})}\geq\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},r_{n}^{\prime}))\right|}{\varphi(r_{n}^{\prime})},$

it is enough to prove the lower bound along the subsequence
$(r^{\prime}_{n})$. In the following, we will decompose the GFF $\Gamma$ as
the independent sum of a GFF $\Gamma^{Q(x_{0},r^{\prime}_{n})}$ in
$Q(x_{0},r^{\prime}_{n})$ plus a harmonic function
$h^{Q(x_{0},r^{\prime}_{n})}$ in $Q(x_{0},r^{\prime}_{n})$. We will denote
$\mu^{(n)}$ the imaginary chaos associated to
$\Gamma^{Q(x_{0},r^{\prime}_{n})}$. Lemma 2.7 together with (4.3) implies that

$\displaystyle\sum_{n}\mathbb{P}\left(\left|\left|\mu(Q(x_{0},{r^{\prime}_{n}}))\right|-\left|\mu^{(n-1)}(Q(x_{0},{r^{\prime}_{n}}))\right|\right|>{r^{\prime}_{n}}^{2-\beta^{2}/2}\right)$
$\displaystyle\leq
C\sum_{n}\exp\left(-c\left(\frac{r_{n-1}^{\prime}}{r_{n}^{\prime}}\right)^{\frac{2}{\beta^{2}+1}}\right)$
$\displaystyle\leq C\sum_{n}\exp\left(-c(\log n)^{2}\right)<\infty.$

By Borel-Cantelli lemma, it implies that a.s.

$\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},{r^{\prime}_{n}}))\right|}{\varphi(r_{n}^{\prime})}=\limsup_{n\to\infty}\frac{\left|\mu^{(n-1)}(Q(x_{0},{r^{\prime}_{n}}))\right|}{\varphi(r_{n}^{\prime})}.$
(4.4)

If $m\in\\{n,n+1\\}$ and $t\in(1/2,2)$, Lemma 2.8 implies that

$\mathbb{P}\left(\frac{\left|\mu^{(n-1)}(Q(x_{0},{r^{\prime}_{m}}))\right|}{{r^{\prime}_{m}}^{2-\beta^{2}/2}}>t^{\beta^{2}/4}c^{*}(\beta)^{-\beta^{2}/4}(\log
n)^{\beta^{2}/4}\right)=n^{-t+o(1)}.$ (4.5)

In particular, Borel-Cantelli lemma implies that a.s.

$\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},{r^{\prime}_{n}}))\right|}{\varphi(r_{n}^{\prime})}=\limsup_{n\to\infty}\frac{\left|\mu^{(n-1)}(Q(x_{0},{r^{\prime}_{n}})\backslash
Q(x_{0},{r^{\prime}_{n+1}}))\right|}{\varphi(r_{n}^{\prime})}.$

Now, the random variables appearing on the right hand side are independent
when we restrict ourselves to even or odd integers. Moreover, by (4.5) and the
triangle inequality, they satisfy for $t\in(1/2,2)$,

$\mathbb{P}\left(\frac{\left|\mu^{(n-1)}(Q(x_{0},{r^{\prime}_{n}})\backslash
Q(x_{0},{r^{\prime}_{n+1}}))\right|}{{r^{\prime}_{n}}^{2-\beta^{2}/2}}>t^{\beta^{2}/4}c^{*}(\beta)^{-\beta^{2}/4}(\log
n)^{\beta^{2}/4}\right)=n^{-t+o(1)}.$

With this and a simple use of the first and second Borel-Cantelli lemmas, we
deduce that a.s.

$\limsup_{\begin{subarray}{c}n\to\infty\\\
n\mathrm{~{}even}\end{subarray}}\frac{\left|\mu^{(n-1)}(Q(x_{0},{r^{\prime}_{n}})\backslash
Q(x_{0},{r^{\prime}_{n+1}}))\right|}{\varphi(r_{n}^{\prime})}=c^{*}(\beta)^{-\beta^{2}/4},$

concluding the proof of the lower bound of (1.4).

The upper bound is much easier because for this bound we use the version of
Borel Cantelli lemma which does not require any independence. Consider
$R_{0}>0$ s.t. $Q(x_{0},R_{0})\subset D$. Since the ratio $r_{n}/R_{0}$ decays
much faster than $r^{\prime}_{n}/r^{\prime}_{n-1}$, we can repeat the previous
arguments to show the analogue of (4.4):

$\limsup_{n\to\infty}\frac{\left|\mu(Q(x_{0},r_{n}))\right|}{\varphi(r_{n})}=\limsup_{n\to\infty}\frac{\left|\mu^{(R_{0})}(Q(x_{0},r_{n}))\right|}{\varphi(r_{n})}.$

With Lemma 2.8, we have for $t\in(1/2,2)$,

$\mathbb{P}\left(\frac{\left|\mu^{(R_{0})}(Q(x_{0},r_{n}))\right|}{r_{n}^{2-\beta^{2}/2}}>t^{\beta^{2}/4}c^{*}(\beta)^{-\beta^{2}/4}(\log
n)^{\beta^{2}/4}\right)=n^{-t+o(1)}.$

The previous probabilities are thus summable if $t>1$ and Borel-Cantelli lemma
concludes the proof of the upper bound of (1.4). ∎

###  4.2. Proof of Lemma 4.2

This section is dedicated to the proof of Lemma 4.2. Let $\gamma\in(0,1)$ and
$n\geq 1$. Define the function

$F_{n}:r\in I_{n}\mapsto\mu(Q(x_{0},\gamma^{n})\setminus
Q(x_{0},r)),\qquad\text{where}\qquad I_{n}:=[\gamma^{n+1},\gamma^{n}].$ (4.6)

In Lemma 4.2, we want to control the uniform norm of $F_{n}$. This norm is
difficult to study directly. For instance, we do not know a priori how to
compute its expectation. Our proof will be based on Sobolev embeddings which
allow us to control this norm via more tractable norms. Let us first recall
some definitions and some results on fractional Sobolev spaces. For more
background on Sobolev spaces, we refer to [DNPV12].

Sobolev spaces. Let $I\subset\mathbb{R}$ be an interval. For $s\in(0,1)$ and
$p\geq 1$, the Sobolev space $W^{s,p}(I)$ is defined as

$W^{s,p}(I):=\left\\{f\in
L^{p}(I):\left\|f\right\|_{W^{s,p}(I)}^{p}:=\left\|f\right\|_{L^{p}(I)}^{p}+\int_{I\times
I}\frac{|f(x)-f(y)|^{p}}{|x-y|^{sp+1}}dxdy<\infty\right\\}.$ (4.7)

Recall also that for $\alpha\in(0,1)$, the space of $\alpha$-Hölder continuous
functions is defined as

$C^{\alpha}(I):=\left\\{f\in
C^{0}(I):\left\|f\right\|_{C^{\alpha}(I)}:=\sup_{x\in I}|f(x)|+\sup_{x\neq
y\in I}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}<\infty\right\\}.$ (4.8)

The Sobolev embedding we will rely on is the following (see e.g. [DNPV12,
Theorem 8.2]): when $sp>1$, then $W^{s,p}(I)\subset C^{\alpha}(I)$ for
$\alpha=s-1/p$. Moreover, there exists $C(s)>0$ that may depend on $s$ but not
on $p$, nor on $I$, such that for all intervals $I$ and for all $f\in
W^{s,p}(I)$,

$\sup_{x\neq y\in I}\frac{|f(x)-f(y)|}{|x-y|^{\alpha}}\leq
C(s)\left(\int_{I\times
I}\frac{|f(x)-f(y)|^{p}}{|x-y|^{sp+1}}dxdy\right)^{1/p}\quad\quad(\alpha=s-1/p).$
(4.9)

The fact that the constant $C(s)$ can be chosen independently of $p$ can be
checked by following the proof of [DNPV12, Theorem 8.2]. The fact that the
constant can be chosen independently of the interval $I$ follows directly by a
scaling argument.

With the Sobolev inequality (4.9) at hand, the proof of Lemma 4.2 will be
based on the following estimate:

######  Lemma 4.3.

Let $F_{n}$ be the function defined in (4.6). For all $s\in(0,1-\beta^{2}/4)$,
there exists $C>0$ that may depend on $\beta$ and $s$ such that for all $n\geq
1$ and even integers $p\geq 2$,

$\mathbb{E}\left[\int_{I_{n}\times
I_{n}}\frac{|F_{n}(r)-F_{n}(r^{\prime})|^{p}}{|r-r^{\prime}|^{sp+1}}drdr^{\prime}\right]\leq
C^{p}p^{\frac{\beta^{2}}{4}p}(1-\gamma)^{(1-\beta^{2}/4-s)p}\gamma^{(2-\beta^{2}/2)np}\gamma^{-spn+n}.$
(4.10)

###### Proof of Lemma 4.3.

Let $s\in(0,1-\beta^{2}/4)$, $n\geq 1$ and $p\geq 2$ be an even integer. By
Lemma 2.5, we have for all $r,r^{\prime}\in I_{n}$,

$\displaystyle\mathbb{E}\left[|F_{n}(r)-F_{n}(r^{\prime})|^{p}\right]=\mathbb{E}\left[\mu(Q(x_{0},r\vee
r^{\prime})\setminus Q(x_{0},r\wedge r^{\prime}))\right]\leq C^{p}((r\vee
r^{\prime})|r-r^{\prime}|)^{(1-\beta^{2}/4)p}p^{\frac{\beta^{2}}{4}p}.$

By Fubini, we deduce that the left hand side of (4.10) is at most

$\displaystyle C^{p}p^{\frac{\beta^{2}}{4}p}\int_{I_{n}\times I_{n}}(r\vee
r^{\prime})^{(1-\beta^{2}/4)p}|r-r^{\prime}|^{(1-\beta^{2}/4-s)p-1}drdr^{\prime}.$

Recalling that $I_{n}=[\gamma^{n+1},\gamma^{n}]$, we bound $(r\vee
r^{\prime})^{(1-\beta^{2}/4)p}$ in the last integral above by
$\gamma^{(1-\beta^{2}/4)np}$. The integral over $I_{n}\times I_{n}$ of
$|r-r^{\prime}|^{(1-\beta^{2}/4-s)p-1}$ is then equal to

$\frac{2}{(1-\beta^{2}/4-s)p((1-\beta^{2}/4-s)p+1)}(\gamma^{n}-\gamma^{n+1})^{(1-\beta^{2}/4-s)p+1}.$

This concludes the proof. ∎

###### Proof of Lemma 4.2.

Let $\varepsilon>0$ be fixed. Let $\gamma>0$ be close to 1 and $n\geq 1$. Let
$s\in(0,1-\beta^{2}/4)$ be a fixed parameter ($s=1/2$ would do) and $p\geq 2$
be a large even integer that we will choose precisely later. Let
$\alpha=s-1/p$. We have,

$\displaystyle\mathbb{P}\left(\sup_{r\in
I_{n}}|F_{n}(r)|\geq\varepsilon\varphi(\gamma^{n+1})\right)$
$\displaystyle=\mathbb{P}\left(\sup_{r\in
I_{n}}|F_{n}(r)-F_{n}(\gamma^{n})|\geq\varepsilon\varphi(\gamma^{n+1})\right)$
$\displaystyle\leq\mathbb{P}\left(\sup_{r\neq r^{\prime}\in
I_{n}}\frac{|F_{n}(r)-F_{n}(r^{\prime})|}{|r-r^{\prime}|^{\alpha}}\geq\varepsilon\varphi(\gamma^{n+1})(\gamma^{n}-\gamma^{n+1})^{-\alpha}\right).$

By Markov’s inequality, this is at most

$(\varepsilon^{-1}(1-\gamma)^{\alpha})^{p}\varphi(\gamma^{n+1})^{-p}\gamma^{\alpha
pn}\mathbb{E}\left[\left(\sup_{r\neq r^{\prime}\in
I_{n}}\frac{|F_{n}(r)-F_{n}(r^{\prime})|}{|r-r^{\prime}|^{\alpha}}\right)^{p}\right].$

Combining the Sobolev inequality (4.9) and Lemma 4.3, we can bound the above
expectation by

$C^{p}p^{\frac{\beta^{2}}{4}p}(1-\gamma)^{(1-\beta^{2}/4-s)p}\gamma^{(2-\beta^{2}/2)np}\gamma^{-spn+n}.$

Putting things together and recalling that
$\varphi(\gamma^{n+1})=\gamma^{(2-\beta^{2}/2)np}\log((n+1)|\log\gamma|)^{\beta^{2}/4}$,
we obtain that

$\mathbb{P}\left(\sup_{r\in
I_{n}}|F_{n}(r)|\geq\varepsilon\varphi(\gamma^{n+1})\right)\leq(1-\gamma)^{-1}(C\varepsilon^{-1}(1-\gamma)^{1-\beta^{2}/4}(\log
n)^{-\beta^{2}/4})^{p}p^{\frac{\beta^{2}}{4}p}.$

We can now optimise in $p$. We choose
$p=e^{-1}(C\varepsilon^{-1}(1-\gamma)^{1-\beta^{2}/4}(\log
n)^{-\beta^{2}/4})^{-\frac{4}{\beta^{2}}}$ (or rather an even integer close to
this value), leading to the following bound:

$\mathbb{P}\left(\sup_{r\in
I_{n}}|F_{n}(r)|\geq\varepsilon\varphi(\gamma^{n+1})\right)\leq(1-\gamma)^{-1}\exp\left(-\frac{\beta^{2}}{4}e^{-1}(C\varepsilon^{-1}(1-\gamma)^{1-\beta^{2}/4})^{-\frac{4}{\beta^{2}}}\log
n\right).$

If $\gamma$ is close enough to $1$, then the coefficient in front of $\log n$
is larger than 1 making the right hand side term of the above display summable
in $n$. This concludes the proof. ∎

##  5\. Exceptional points: proof of Theorem 1.5

###### Proof of Theorem 1.5.

We can restrict ourselves to the bulk of $D$ by considering squares which are
at distance at least $\eta>0$ from the boundary and then let $\eta\to 0$. We
start by proving the upper bound of (1.5) and (1.7). Let $a>0$,
$d>2-c^{*}(\beta)a^{4/\beta^{2}}$ and $\delta,\varepsilon>0$ be small enough
so that

$\delta(1-\beta^{2}/4-\varepsilon)-2\varepsilon>\varepsilon\mathrm{~{}and~{}}-2-2\delta+c^{*}(\beta)a^{4/\beta^{2}}+(1+\delta)d>\delta.$
(5.1)

In this part, the constants may depend on $\delta,\varepsilon$ and $\eta$.
With the help of the Hölder estimate (1.2), we are going to discretise the
space $D$ and the set of radii. We will consider the sequence
$r_{n}=n^{-1/\delta}$. Let $x\in D$ and $r>0$. Let $n\geq 1$ such that
$r_{n+1}\leq r\leq r_{n}$ and notice that $r_{n}-r_{n+1}\leq
Cr_{n+1}^{1+\delta}$. Take $y\in r_{n+1}^{1+\delta}\mathbb{Z}^{2}\cap D$ such
that $\left|x-y\right|\leq r_{n+1}^{1+\delta}$. If
$\left|\mu(Q(x,r))\right|\geq ar^{2-\beta^{2}/2}\left|\log
r\right|^{\beta^{2}/4}$, then by the Hölder estimate (1.2) applied to
$\alpha=1-\beta^{2}/4-\varepsilon$, we have

$\displaystyle\left|\mu(Q(y,r_{n+1}))\right|$ $\displaystyle\geq
ar^{2-\beta^{2}/2}\left|\log
r\right|^{\beta^{2}/4}-C(r\max(r-r_{n+1},\left|y-x\right|))^{1-\beta^{2}/4-\varepsilon}$
$\displaystyle\geq ar_{n+1}^{2-\beta^{2}/2}\left|\log
r_{n+1}\right|^{\beta^{2}/4}-Cr_{n+1}^{2-\beta^{2}/2-2\varepsilon+\delta(1-\beta^{2}/4-\varepsilon)}$
$\displaystyle\geq ar_{n+1}^{2-\beta^{2}/2}\left|\log
r_{n+1}\right|^{\beta^{2}/4}-Cr_{n+1}^{2-\beta^{2}/2+\varepsilon},$

where we used (5.1) in the last inequality. In particular, if we denote

$\mathcal{T}_{n}(a)\coloneqq\left\\{y\in r_{n}^{1+\delta}\mathbb{Z}^{2}\cap
D:\left|\mu(Q(y,r_{n}))\right|\geq ar_{n}^{2-\beta^{2}/2}\left|\log
r_{n}\right|^{\beta^{2}/4}-Cr_{n}^{2-\beta^{2}/2+\varepsilon}\right\\},$

it implies that

$\mathcal{T}(a)\subset\bigcap_{p\geq 1}\bigcup_{n\geq
p}\bigcup_{y\in\mathcal{T}_{n}(a)}D(y,r_{n}^{1+\delta}).$ (5.2)

To conclude that $\dim(\mathcal{T}(a))\leq d$ a.s., it is thus enough to show
that

$\sum_{n\geq 1}\\#\mathcal{T}_{n}(a)(r_{n}^{1+\delta})^{d}$ (5.3)

is a.s. finite. Lemmas 2.7 and 2.8 that we apply with $R_{0}=\eta/10$ imply
that if $d(x,\partial D)>\eta$,

$\mathbb{P}\left(\left|\mu(Q(x,r_{n}))\right|\geq
ar_{n}^{2-\beta^{2}/2}\left|\log
r_{n}\right|^{\beta^{2}/4}\right)=\exp\left(-(1+o(1))c^{*}(\beta)a^{4/\beta^{2}}\left|\log
r_{n}\right|\right)$

and

$\mathbb{E}\left[\\#\mathcal{T}_{n}(a)\right]\leq
r_{n}^{-2-2\delta+c^{*}(\beta)a^{4/\beta^{2}}+o(1)}.$

Hence, recalling that $r_{n}=n^{-1/\delta}$, the expected value of (5.3) is at
most

$\sum_{n\geq
1}n^{-(-2-2\delta+c^{*}(\beta)a^{4/\beta^{2}}+(1+\delta)d)/\delta+o(1)}$

which is finite thanks to (5.1). In particular it shows that (5.3) is finite
a.s. which concludes the proof of the upper bound of (1.7). The upper bound of
(1.5) follows as well: if $a>\left(2/c^{*}(\beta)\right)^{\beta^{2}/4}$, we
have shown that

$\sum_{n\geq 1}\mathbb{P}\left(\\#\mathcal{T}_{n}(a)\geq
1\right)\leq\sum_{n\geq 1}\mathbb{E}\left[\\#\mathcal{T}_{n}(a)\right]$

is finite. Borel-Cantelli lemma therefore implies that a.s. if $n$ is large
enough $\mathcal{T}_{n}(a)$ is empty. (5.2) then concludes that
$\mathcal{T}(a)$ is almost surely empty. Since this is true for all
$a>\left(2/c^{*}(\beta)\right)^{\beta^{2}/4}$, it shows the upper bound of
(1.5).

Before we move on to the proof of the lower bound, let us recall the notion of
limsup fractal (see [MP10, Section 10.1]). Suppose $D=(0,1)^{2}$, let
$\mathfrak{Q}_{n}\coloneqq\\{(k2^{-n},(k+1)2^{-n})\times(l2^{-n},(l+1)2^{-n}),\,0\leq
k,l\leq 2^{n}-1\\}$ be the collection of dyadic squares of sidelength $2^{-n}$
for each $n\in\mathbb{N}$, and
$\mathfrak{Q}\coloneqq\cup_{n\in\mathbb{N}}\mathfrak{Q}_{n}$. Let
$Z=(Z(Q))_{Q\in\mathfrak{Q}}$ be a collection of Bernoulli random variables
indexed by $\mathfrak{Q}$ such that $p_{n}\coloneqq\mathbb{P}(Z(Q)=1)$ is
independent of $Q\in\mathfrak{Q}_{n}$. The limsup fractal associated to the
process $Z$ is by definition

$A\coloneqq\bigcap_{n\in\mathbb{N}}\bigcup_{k\geq
n}\bigcup_{Q\in\mathfrak{Q}_{k}:\,Z(Q)=1}Q.$

For each $Q\in\mathfrak{Q}_{m}$ and $n\geq m$, we also write

$M_{n}(Q)\coloneqq\sum_{Q^{\prime}\in\mathfrak{Q}_{n},\,Q^{\prime}\subset
Q}Z(Q^{\prime}).$

[MP10] provides a lower bound for the Hausdorff dimension of a limsup fractal:

######  Proposition 5.1.

[MP10, Proposition 10.6] Suppose that there exists $\zeta(n)\geq 1$ and
$\gamma\in(0,2)$ such that the following holds for each $m\in\mathbb{N}$.

  1. (1)

For each $Q\in\mathfrak{Q}_{m}$, and $n\geq m$ we have

$\mathrm{Var}(M_{n}(Q))\leq\zeta(n)\mathbb{E}[M_{n}(Q)].$

  2. (2)

$\underset{n\to\infty}{\lim}\zeta(n)\frac{2^{n(\gamma-2)}}{p_{n}}=0$.

Then almost surely $\dim_{\mathcal{H}}A\geq\gamma$.

For $n\in\mathbb{N}$, let $r_{n}\coloneqq 2^{-n}$ and $H_{n}\coloneqq
r_{n}\mathbb{Z}^{2}\cap D^{\prime}$ where $D^{\prime}\Subset D$. Fix
$\delta\in(0,\frac{1}{2})$ and set:

$\mathcal{C}_{n}(a)\coloneqq\left\\{x\in
H_{n},\,\left|\mu^{(Q(x,r_{n}^{1-2\delta}))}(Q(x,r_{n}^{1-\delta}))\right|\geq
ar_{n}^{(1-\delta)(2-\frac{\beta^{2}}{2})}\left|\log
r_{n}^{1-\delta}\right|^{\frac{\beta^{2}}{4}}\right\\}.$

The limsup fractal we will be interested in is defined by:

$A(a)\coloneqq\bigcap_{n\in\mathbb{N}}\bigcup_{k\geq
n}\bigcup_{x\in\mathcal{C}_{k}(a)}Q(x,r_{k}).$

Let us first show that $A(a)\subset\mathcal{T}(a)$. Let $x\in A(a)$. By
definition, there exist an increasing sequence of integers $(k_{n})_{n\geq 1}$
and a sequence of points $(x_{k_{n}})_{n\geq 1}$ such that
$x_{k_{n}}\in\mathcal{C}_{k_{n}}(a)$ and $x\in Q(x_{k_{n}},r_{k_{n}})$ for all
$n\in\mathbb{N}$. Now by Proposition 1.1 applied to any $\alpha$ satisfying
$\frac{1-\delta}{1-\frac{\delta}{2}}(1-\frac{\beta^{2}}{4})<\alpha<1-\frac{\beta^{2}}{4}$,
we find that

$|\mu(Q(x,r_{k_{n}}^{1-\delta}))-\mu(Q(x_{k_{n}},r_{k_{n}}^{1-\delta}))|=O(r_{k_{n}}^{\alpha(1-\delta)}\times
r_{k_{n}}^{\alpha})=o\left(r_{k_{n}}^{(1-\delta)(2-\beta^{2}/2)}\left|\log
r_{k_{n}}^{1-\delta}\right|^{4/\beta^{2}}\right).$

Thus, we have that

$\underset{k\to\infty}{\limsup}\frac{|\mu(Q(x,r_{k_{n}}^{1-\delta}))|}{r_{k_{n}}^{(1-\delta)(2-\beta^{2}/2)}\left|\log
r_{k_{n}}^{1-\delta}\right|^{4/\beta^{2}}}\geq a,$

showing that $x\in\mathcal{T}(a)$.

To obtain a lower bound on the Hausdorff dimension of $\mathcal{T}(a)$, it is
sufficient to obtain one for $A(a)$. To this end, we will show that we can
apply Proposition 5.1. Let $M_{n}(Q)\coloneqq\\#(\mathcal{C}_{n}(a)\cap Q)$ as
above. Notice first that, by Lemma 2.8,
$p_{n}\coloneqq\mathbb{P}(x\in\mathcal{C}_{n}(a))$ is independent of $x$ and
equal to $r_{n}^{(1-\delta)c^{*}(\beta)a^{4/\beta^{2}}+o(1)}$ as $n\to\infty$.
Second, we bound the second moment of $M_{n}(Q)$. Assume $Q=[0,1]^{2}$ for
simplicity. Notice that the events $\\{x\in\mathcal{C}_{n}(a)\\}$ and
$\\{y\in\mathcal{C}_{n}(a)\\}$ are independent as soon as $x,y\in H_{n}$
satisfy $|x-y|>2\sqrt{2}r_{n}^{1-2\delta}$. Therefore,

$\displaystyle\mathbb{E}\left[M_{n}(Q)^{2}\right]$
$\displaystyle=\sum_{\begin{subarray}{c}x,y\in H_{n}\\\
|x-y|>2\sqrt{2}r_{n}^{1-2\delta}\end{subarray}}\mathbb{P}\left(x\in\mathcal{C}_{n}(a),y\in\mathcal{C}_{n}(a)\right)+\sum_{\begin{subarray}{c}x,y\in
H_{n}\\\ |x-y|\leq
2\sqrt{2}r_{n}^{1-2\delta}\end{subarray}}\mathbb{P}\left(x\in\mathcal{C}_{n}(a),y\in\mathcal{C}_{n}(a)\right)$
$\displaystyle\leq\sum_{\begin{subarray}{c}x,y\in H_{n}\\\
|x-y|>2\sqrt{2}r_{n}^{1-2\delta}\end{subarray}}\mathbb{P}\left(x\in\mathcal{C}_{n}(a)\right)\mathbb{P}\left(y\in\mathcal{C}_{n}(a)\right)+\sum_{\begin{subarray}{c}x,y\in
H_{n}\\\ |x-y|\leq
2\sqrt{2}r_{n}^{1-2\delta}\end{subarray}}\mathbb{P}\left(x\in\mathcal{C}_{n}(a)\right)$
$\displaystyle\leq\mathbb{E}[M_{n}(Q)]^{2}+Cr_{n}^{-4\delta}\mathbb{E}[M_{n}(Q)].$

Thus, with $\zeta(n)\coloneqq Cr_{n}^{-4\delta}$, we have:

$\mathrm{Var}(M_{n}(Q))\leq\zeta(n)\mathbb{E}[M_{n}(Q)].$

Recalling that $p_{n}=r_{n}^{(1-\delta)c^{*}(\beta)a^{4/\beta^{2}}+o(1)}$, we
conclude by Proposition 5.1 that a.s. $\dim_{\mathcal{H}}(A(a))\geq
2-(1-\delta)c^{*}(\beta)a^{4/\beta^{2}}-4\delta$. The lower bound of (1.7)
follows since $\delta$ can be chosen arbitrarily small. The lower bound of
(1.5) is then a consequence of (1.7) which finishes the proof of Theorem 1.5.
∎

##  6\. Besov regularity of imaginary chaos: proof of Theorem 1.7

In this section, we will prove Theorem 1.7 on the optimal Besov regularity of
the imaginary chaos. The proof is based on wavelet analysis and similar ideas
were also used in [JSV19]. As a new ingredient in the boundary case where the
smoothness index $s=-\beta^{2}/2$, we employ a law of large numbers type
argument to show that the quantities $A_{j}$ (see below) appearing in the
wavelet version of the Besov norm converge almost surely to their nonzero
mean. On the other hand, the proof in the case $p=+\infty$ will rely on what
we have developed in the earlier sections of the current paper, especially
Theorem 1.4. Let us first recall from [Mey93, Chapter 6] basic properties of
wavelet bases and the characterisation of Besov spaces using them.

Wavelet bases and Besov spaces. The wavelet bases we will use in 2D are
defined in terms of a father wavelet $\phi\in C^{2}(\mathbb{R}^{2})$ and three
mother wavelets $\psi^{(\tau)}\in C^{2}(\mathbb{R}^{2})$ for
$\tau\in\\{0,1\\}^{2}\setminus\\{(0,0)\\}$. For $j\geq 0$, let

$\Lambda_{j}:=\\{\lambda=2^{-j}k+2^{-j-1}\tau:k\in\mathbb{Z}^{2},\tau\in\\{0,1\\}^{2}\setminus\\{(0,0)\\}\\}$

and for $\lambda=2^{-j}k+2^{-j-1}\tau\in\Lambda_{j}$, let
$\psi_{\lambda}(x)=2^{j}\psi^{(\tau)}(2^{j}x-k)$. The father and mother
wavelets are such that
$\\{\phi(\cdot-k),k\in\mathbb{Z}^{2}\\}\cup\bigcup_{j\geq
0}\\{\psi_{\lambda},\lambda\in\Lambda_{j}\\}$ forms an orthonormal basis of
$L^{2}(\mathbb{R}^{n})$.

The usefulness of wavelet bases stems from the fact that the wavelets can be
chosen to have several desirable properties at the same time. In particular we
will assume that
$\operatorname{supp}\phi,\operatorname{supp}\psi^{\tau}\subset K$ for some
square $K\subset\mathbb{R}^{n}$.

The localisation and smoothness properties allows one to characterise various
function spaces in terms of the wavelet coefficients. The characterisation for
Besov spaces that we will use is the following: If
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$ is a distribution with the wavelet
series

$f(x)=\sum_{k\in\mathbb{Z}}\gamma(k)\phi(x-k)+\sum_{j=0}^{\infty}\sum_{\lambda\in\Lambda_{j}}\alpha(\lambda)\psi_{\lambda}(x),$
(6.1)

then a norm equivalent to the Besov norm
$\|\cdot\|_{B_{p,q}^{s}(\mathbb{R}^{2})}$, assuming $|s|<2$,111Here $2$ comes
from the fact that we chose our wavelets to be $C^{2}$. More generally one
could choose them to be $C^{r}$ and expand the valid range of $s$ to $|s|<r$.
is given by

$\|f\|_{B_{p,q}^{s}(\mathbb{R}^{2})}\sim\|(\gamma(k))_{k\in\mathbb{Z}}\|_{\ell^{p}}+\Big{\|}\Big{\\{}2^{j(1-2/p+s)}\Big{(}\sum_{\lambda\in\Lambda_{j}}|\alpha(\lambda)|^{p}\Big{)}^{1/p}\Big{\\}}_{j\geq
1}\Big{\|}_{\ell^{q}}.$

If $U$ is an open subset of $\mathbb{R}^{2}$, the space
$B_{p,q,\operatorname{loc}}^{s}(U)$ is defined as the space of distributions
$f\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$ such that for all $u\in
C^{\infty}_{c}(U)$, $fu\in B_{p,q}^{s}(\mathbb{R}^{2})$.

We are now ready to prove Theorem 1.7.

###### Proof of Theorem 1.7.

Recall that we are interested in whether $\mu$ belongs to
$B_{p,q,\operatorname{loc}}^{s}(D)$ where $s$ equals the critical value
$s=-\beta^{2}/2$. Let $u\in C^{\infty}_{c}(D)$ and denote by $\alpha(\lambda)$
and $\gamma(k)$ the coefficients appearing in the wavelet decomposition (6.1)
of $u\mu$. We will denote by

$A_{j}:=2^{j(1-2/p-\beta^{2}/2)}\Big{(}\sum_{\lambda\in\Lambda_{j}}|\alpha(\lambda)|^{p}\Big{)}^{1/p}.$
(6.2)

The proof will deal with three cases separately.

Case 1: $p<\infty,q=\infty$. We want to show that $\mu\in
B^{-\beta^{2}/2}_{p,\infty,\operatorname{loc}}(D)$ a.s. Since the function
$u\in C^{\infty}_{c}(D)$ we fixed at the beginning of the proof is arbitrary,
it is enough to show that $\mu u\in
B^{-\beta^{2}/2}_{p,\infty}(\mathbb{R}^{2})$ a.s. Note that the sequence
$(\gamma(k))_{k\in\mathbb{Z}}$ appearing in the wavelet decomposition (6.1) of
$\mu u$ contains only finitely many nonzero terms. Its $\ell^{p}$-norm is
therefore finite and the goal now is to show that the sequence $(A_{j})_{j\geq
1}$, defined in (6.2), is almost surely bounded. Note also that, for all
$j\geq 1$, the number of nonzero $\alpha(\lambda)$ for $\lambda\in\Lambda_{j}$
is at most $C2^{2j}$ which implies by Hölder’s inequality that, for all
$p^{\prime}>p$:

$2^{-2p/j}\Big{(}\sum_{\lambda\in\Lambda_{j}}|\alpha(\lambda)|^{p}\Big{)}^{1/p}\leq
C2^{-2p^{\prime}/j}\Big{(}\sum_{\lambda\in\Lambda_{j}}|\alpha(\lambda)|^{p^{\prime}}\Big{)}^{1/p^{\prime}}.$
(6.3)

This shows that it is actually enough to show that the sequence
$(A_{j})_{j\geq 1}$ is a.s. bounded when $p$ is an even integer. In the rest
of Case 1, we will assume that $p=2N$ for some integer $N\geq 1$. We will
proceed in two steps by showing that

$\displaystyle(\mathbb{E}[A_{j}^{p}])_{j\geq 1}\text{ is bounded,}$ (6.4) and
$\displaystyle A_{j}^{p}-\mathbb{E}[A_{j}^{p}]\to 0\qquad\text{a.s.}$ (6.5)

Let us start with the proof of (6.4). By Fubini, we have for all $j\geq 1$,

$\displaystyle\mathbb{E}[A_{j}^{p}]=2^{j(1-2/p-\beta^{2}/2)p}\sum_{\lambda\in\Lambda_{j}}\mathbb{E}[|\alpha(\lambda)|^{p}]$

and for all $\lambda=2^{-j}k+2^{-j-1}\tau\in\Lambda_{j}$,

$\displaystyle\mathbb{E}[|\alpha(\lambda)|^{p}]=2^{jp}\mathbb{E}\left|\int\mu(x)u(x)\psi^{\tau}(2^{j}x-k)dx\right|^{p}.$

Recall that the support of $\psi$ is contained in some compact rectangle $K$.
As in the proof of the claim (2.10), and recalling that $p$ is an even
integer, we can bound

$\mathbb{E}[|\alpha(\lambda)|^{p}]\leq\left\|u\right\|_{\infty}^{p}\left\|\psi^{\tau}\right\|_{\infty}^{p}2^{jp}\mathbb{E}[|\mu(2^{-j}(k+K))|^{p}]\mathbf{1}_{\left\\{(2^{-j}(k+K))\cap\operatorname{supp}u\neq\varnothing\right\\}}.$

For the integers $k$ such that
$(2^{-j}(k+K))\cap\operatorname{supp}u\neq\varnothing$, we can bound by Lemma
2.5, $\mathbb{E}[|\mu(2^{-j}(k+K)|^{p}]\leq C_{p}2^{-j(2-\beta^{2}/2)p}$ for
some constant $C_{p}$ independent of $k$ or $j$. We deduce that there exists
$C>0$ (possibly depending on $p$, $u$ and $\psi$) such that for all $j\geq 1$
and $\lambda\in\Lambda_{j}$,

$\mathbb{E}[|\alpha(\lambda)|^{p}]\leq C2^{-j(1-\beta^{2}/2)p}.$ (6.6)

Since $\mathbb{E}[|\alpha(\lambda)|^{p}]$ vanishes for all
$\lambda\in\Lambda_{j}$ except for at most $O(2^{2j})$ of them, this concludes
that $(\mathbb{E}[A_{j}^{p}])_{j\geq 1}$ is bounded as stated in (6.4).

We now move to the proof of (6.5). Recall that $p=2N$ is an even integer and
define

$S_{j}:=\sum_{\lambda\in\Lambda_{j}}|\mu(u\psi_{\lambda})|^{2N},\quad\quad
j\geq 1.$

Let $\lambda_{1},\lambda_{2}\in\Lambda_{j}$ and set
$D_{i}:=\operatorname{supp}\psi_{\lambda_{i}}$, $i=1,2$. Suppose that
$D_{1}\cap D_{2}=\emptyset$, set $\Delta:=\sup\\{|x-y|,\,x\in D_{1},y\in
D_{2}\\}$ and $\delta:=\inf\\{|x-y|,\,x\in D_{1},y\in D_{2}\\}$. Recall the
notations $\mathcal{E}(\Gamma;\mathbf{x};\mathbf{y})$ and
$\mathcal{E}^{\prime}(\Gamma;\mathbf{x};\mathbf{y})$ defined in (2.3) and
(2.5). By (2.7), for $m=1,2$, $\mathbb{E}[|\mu(u\psi_{\lambda_{m}})|^{p}]$
equals

$\displaystyle\int_{D_{m}^{N}\times
D_{m}^{N}}d\mathbf{x}^{m}\,d\mathbf{y}^{m}e^{\beta^{2}\mathcal{E}^{\prime}(\Gamma;\mathbf{x}^{m};\mathbf{y}^{m})}\prod_{k=1}^{N}u(x^{m}_{k})u(y^{m}_{k})\psi_{\lambda_{m}}(x^{m}_{k})\psi_{\lambda_{m}}(y^{m}_{k})$

and $\mathbb{E}[|\mu(u\psi_{\lambda_{1}})|^{p}|\mu(u\psi_{\lambda_{2}})|^{p}]$
equals

$\displaystyle\int_{D_{1}^{N}\times D_{1}^{N}\times D_{2}^{N}\times
D_{2}^{N}}$ $\displaystyle
d\mathbf{x}^{1}\,d\mathbf{y}^{1}\,d\mathbf{x}^{2}\,d\mathbf{y}^{2}e^{\beta^{2}\mathcal{E}^{\prime}(\Gamma;\mathbf{x}^{1}\mathbf{x}^{2};\mathbf{y}^{1}\mathbf{y}^{2})}\prod_{m=1,2}\prod_{k=1}^{N}u(x^{m}_{k})u(y^{m}_{k})\psi_{\lambda_{m}}(x^{m}_{k})\psi_{\lambda_{m}}(y^{m}_{k}).$

In the last display we denoted by $\mathbf{x}^{1}\mathbf{x}^{2}\in
D_{1}^{N}\times D_{2}^{N}$ the concatenation of the vectors $\mathbf{x}^{1}\in
D_{1}^{N}$ and $\mathbf{x}^{2}\in D_{2}^{N}$. We deduce that
$\mathrm{Cov}(|\mu(u\psi_{\lambda_{1}})|^{p};|\mu(u\psi_{\lambda_{2}})|^{p})$
is at most

$\displaystyle
C^{N}\left\|u\right\|_{\infty}^{4N}\left\|\psi_{\lambda}\right\|_{\infty}^{4N}\int_{D_{1}^{N}\times
D_{1}^{N}\times D_{2}^{N}\times
D_{2}^{N}}d\mathbf{x}^{1}\,d\mathbf{y}^{1}\,d\mathbf{x}^{2}\,d\mathbf{y}^{2}\Big{|}e^{\beta^{2}\mathcal{E}(\Gamma;\mathbf{x}^{1}\mathbf{x}^{2};\mathbf{y}^{1}\mathbf{y}^{2})}-\prod_{m=1,2}e^{\beta^{2}\mathcal{E}(\Gamma;\mathbf{x}^{m};\mathbf{y}^{m})}\Big{|}.$

Expanding
$\mathcal{E}(\Gamma;\mathbf{x}^{1}\mathbf{x}^{2};\mathbf{y}^{1}\mathbf{y}^{2})$,
we have for all
$(\mathbf{x}^{1},\mathbf{y}^{1},\mathbf{x}^{2},\mathbf{y}^{2})\in
D_{1}^{N}\times D_{1}^{N}\times D_{2}^{N}\times D_{2}^{N}$,

$\mathcal{E}(\Gamma;\mathbf{x}^{1}\mathbf{x}^{2};\mathbf{y}^{1}\mathbf{y}^{2})=\sum_{m=1,2}\mathcal{E}(\Gamma;\mathbf{x}^{m};\mathbf{y}^{m})+\sum_{1\leq
k,l\leq
N}G(x_{k}^{1},y_{l}^{2})+G(y_{k}^{1},x_{l}^{2})-G(x_{k}^{1},x_{l}^{2})-G(y_{k}^{1},y_{l}^{2}).$

By definition of $\Delta$ and $\delta$ and approximating the Green function by
the log of the distance, the absolute value of the second sum on the right
hand side is at most $4N^{2}\log\Delta/\delta$. We thus obtain that
$\mathrm{Cov}(|\mu(u\psi_{\lambda_{1}})|^{p};|\mu(u\psi_{\lambda_{2}})|^{p})$
is at most

$\displaystyle
C^{N}\Big{(}\Big{(}\frac{\Delta}{\delta}\Big{)}^{4N^{2}\beta^{2}}-1\Big{)}\left\|u\right\|_{\infty}^{4N}\left\|\psi_{\lambda}\right\|_{\infty}^{4N}\prod_{m=1,2}\int_{D_{m}^{N}\times
D_{m}^{N}}d\mathbf{x}^{m}\,d\mathbf{y}^{m}e^{\beta^{2}\mathcal{E}(\Gamma;\mathbf{x}^{m};\mathbf{y}^{m})}.$

By scaling, the above product of integrals is at most
$C_{N}2^{-(2-\beta^{2}/2)4jN}$. Since we also have
$\left\|\psi_{\lambda}\right\|_{\infty}^{4N}\leq
2^{4jN}\left\|\psi\right\|_{\infty}^{4N}$, we have obtained that

$\mathrm{Cov}(|\mu(u\psi_{\lambda_{1}})|^{p};|\mu(u\psi_{\lambda_{2}})|^{p})\leq
C_{N}2^{-(1-\beta^{2}/2)4jN}\Big{(}\Big{(}\frac{\Delta}{\delta}\Big{)}^{4N^{2}\beta^{2}}-1\Big{)}.$
(6.7)

This is the main estimate we will need in order to bound:

$\mathrm{Var(S_{j})}=\sum_{\lambda_{1},\lambda_{2}\in\Lambda_{j}}\mathrm{Cov}(|\mu(u\psi_{\lambda_{1}})|^{p};|\mu(u\psi_{\lambda_{2}})|^{p}).$
(6.8)

Indeed, if $|\lambda_{1}-\lambda_{2}|\geq 2^{-j/2}$, then the associated
$\Delta$ and $\delta$ satisfy $\frac{\Delta}{\delta}=1+O(2^{-j/2})$ and (6.7)
shows that

$\mathrm{Cov}(|\mu(u\psi_{\lambda_{1}})|^{p};|\mu(u\psi_{\lambda_{2}})|^{p})\leq
C_{N}2^{-(1-\beta^{2}/2)4jN-j/2}.$

The contribution of such $\lambda_{1},\lambda_{2}$ to the sum in (6.8) is thus
at most
$C_{N}2^{-j/2-4j(2-\beta^{2}/2)N+4j}=C_{p}2^{-j/2-2j(2-2/p-\beta^{2}/2)p}$,
recalling that $p=2N$. On the other hand, if $|\lambda_{1}-\lambda_{2}|\leq
2^{-j/2}$, then we simply use Cauchy–Schwarz inequality and (6.6) to bound
$\mathrm{Cov}(|\mu(u\psi_{\lambda_{1}})|^{p};|\mu(u\psi_{\lambda_{2}})|^{p})\leq
C2^{-2pj(2-\frac{\beta^{2}}{2})}$. Since there are at most $C2^{3j}$ couples
$(\lambda_{1},\lambda_{2})$ such that $|\lambda_{1}-\lambda_{2}|\leq
2^{-j/2}$, this shows that the contribution of such couples to the sum in
(6.8) is at most $C_{p}2^{-j-2j(2-2/p-\beta^{2}/2)p}$. Altogether, we have
shown that

$\mathrm{Var}(S_{j})\leq C2^{-j/2-2j(2-2/p-\beta^{2}/2)p}.$

Applying the Chebychev inequality, and recalling that $A_{j}$ is defined in
(6.2), we get that

$\displaystyle\mathbb{P}\left(|A_{j}^{p}-\mathbb{E}[A_{j}^{p}]|\geq
2^{-j/8}\right)$ $\displaystyle\leq
2^{j/4}\mathrm{Var}(A_{j}^{p})=2^{j/4}2^{2j(2-2/p-\beta^{2}/2)p}\mathrm{Var}(S_{j})\leq
C2^{-j/4}.$

Borel–Cantelli lemma then concludes the proof of (6.5). Together with (6.4),
this shows that $(A_{j})_{j\geq 1}$ is a.s. bounded, and thus $\mu u\in
B^{-\beta^{2}/2}_{p,\infty}(\mathbb{R}^{2})$ a.s., finishing the proof in this
case.

Case 2: $2\leq p<\infty,q<\infty$. We want to show that $\mu\not\in
B_{p,q,\operatorname{loc}}^{-\beta^{2}/2}(D)$ a.s. It is sufficient to prove
that $\mu u\not\in B_{p,q}^{-\beta^{2}/2}(\mathbb{R}^{2})$ a.s. for some
specific choice of $u$. Any nonzero $C_{c}^{\infty}(D)$ function $u$ will do.
Arguing as in (6.3), it is enough to show that $\mu u\not\in
B_{2,q}^{-\beta^{2}/2}(\mathbb{R}^{2})$ where $p$ has been replaced by 2 (some
even integer not larger than $p$). By (6.5), we know that
$A_{j}^{2}-\mathbb{E}A_{j}^{2}\to 0$ a.s. To conclude that $(A_{j})_{j\geq 1}$
does not belong to $\ell^{q}$ a.s., it is thus enough to show that
$(\mathbb{E}A_{j}^{2})_{j\geq 1}$ is bounded away from 0. Let $j\geq 1$ and
$\lambda=2^{-j}k+2^{-j-1}\tau\in\Lambda_{j}$ be such that the support of
$\psi_{\lambda}$ is included in $\\{u\neq 0\\}$. There are at least $c2^{2j}$
such $\lambda$. Recalling the notation $G_{D}^{\prime}$ defined in (2.4) and
by (2.6), $\mathbb{E}[|\alpha(\lambda)|^{2}]$ is equal to

$\displaystyle 2^{2j}\int_{K\times
K}dx\,dye^{\beta^{2}G_{D}^{\prime}(x,y)}u(x)u(y)\psi^{\tau}(2^{j}x-k)\psi^{\tau}(2^{j}y-k).$

$(x,y)\mapsto G_{D}^{\prime}(x,y)+\log|x-y|$ is smooth and vanishes on the
diagonal. Therefore, for $(x,y)\in K\times K$ such that
$\psi^{\tau}(2^{j}x-k)\psi^{\tau}(2^{j}y-k)\neq 0$,

$G_{D}^{\prime}(x,y)=\log 2^{j}-\log(2^{j}|x-y|)+O(2^{-j}).$

Similarly, for such $(x,y)$,

$u(x)u(y)=u(2^{-j}k)^{2}+O(2^{-j}).$

Arguing as in the proof of the upper bound of $\mathbb{E}A_{j}^{p}$ (see
(6.4)) to bound the integral containing the error terms, we deduce that
$\mathbb{E}[|\alpha(\lambda)|^{2}]$ is equal to

$\displaystyle
u(2^{-j}k)^{2}2^{2j(1+\frac{\beta^{2}}{2})}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}dx\,dy\frac{\psi^{\tau}(2^{j}x-k)\psi^{\tau}(2^{j}y-k)}{(2^{j}|x-y|)^{\beta^{2}}}+o(2^{-j(1-\beta^{2}/2)p}).$

Doing the change of variables $x^{\prime}=2^{j}x-k$ and $y^{\prime}=2^{j}y-k$,
we obtain that

$\displaystyle
u(2^{-j}k)^{2}2^{-2j(1-\frac{\beta^{2}}{2})}\int_{\mathbb{R}^{2}\times\mathbb{R}^{2}}dx^{\prime}\,dy^{\prime}\frac{\psi^{\tau}(x^{\prime})\psi^{\tau}(y^{\prime})}{|x^{\prime}-y^{\prime}|^{\beta^{2}}}+o(2^{-j(1-\beta^{2}/2)p}).$

The integral above is simply the second moment of $|\int
e^{i\beta\Gamma^{*}}\psi^{\tau}|$ where $\Gamma^{*}$ is the whole plane GFF.
This is some positive constant independent of $j$, so we have obtained that
$\mathbb{E}[|\alpha(\lambda)|^{2}]\geq c2^{-2j(1-\beta^{2}/2)}$. Recalling the
definition (6.2) of $A_{j}$, this shows that $\mathbb{E}A_{j}^{2}\geq c$ as
desired. This concludes the proof that $\mu u\not\in
B_{p,q}^{-\beta^{2}/2}(\mathbb{R}^{2})$ in this case.

Case 3: $p=+\infty$. We want to show that $\mu\not\in
B_{p,q,\operatorname{loc}}^{-\beta^{2}/2}(D)$ a.s. It is sufficient to prove
that $\mu u\not\in B_{p,q}^{-\beta^{2}/2}(\mathbb{R}^{2})$ a.s. for some
specific choice of $u$. Consider for instance some nonnegative $u\in
C^{\infty}_{c}(D)$ such that $u=1$ on some compact set $K\subset D$ whose
interior is not empty. Let $p^{\prime}\in[1,\infty]$ and
$q^{\prime}\in[1,\infty]$ be the conjugates of $p$ and $q$ given by
$\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $\frac{1}{q}+\frac{1}{q^{\prime}}=1$.
By Besov duality (see e.g. [Tri83, Section 2.11]), we have for any $f\in
B_{p^{\prime},q^{\prime}}^{\beta^{2}/2}(\mathbb{R}^{2})$,

$\left\|\mu
u\right\|_{B_{p,q}^{-\beta^{2}/2}(\mathbb{R}^{2})}\geq|\mu(uf)|/\left\|f\right\|_{B_{p^{\prime},q^{\prime}}^{\beta^{2}/2}(\mathbb{R}^{2})}.$
(6.9)

Since we currently consider the case $p=+\infty$, we will have $p^{\prime}=1$.
Let $x_{0}$ be any given point in the interior of $K$ and $\delta>0$ be small
enough so that $Q(x_{0},2\delta)\subset K$. Let $f\in
C^{\infty}_{c}(\mathbb{R}^{2})$ be a nonnegative function which is equal to 1
in $Q(0,1)$ and which vanishes outside of $Q(0,2)$. Define
$f_{\delta}(x)=f((x-x_{0})/\delta)$, $x\in\mathbb{R}^{2}$. We apply (6.9) to
$f_{\delta}$. We first claim that

$\left\|f_{\delta}\right\|_{B_{p^{\prime},q^{\prime}}^{\beta^{2}/2}(\mathbb{R}^{2})}\leq
C\delta^{2/p^{\prime}-\beta^{2}/2}.$ (6.10)

Let us first assume that (6.10) holds and see how we conclude from this.
Recall that $p^{\prime}=1$. Since $u=1$ on the support of $f_{\delta}$, the
duality estimate (6.9) applied to $f_{\delta}$ yields

$\left\|\mu u\right\|_{B_{p,q}^{-\beta^{2}/2}(\mathbb{R}^{2})}\geq
c\delta^{-2+\beta^{2}/2}|\mu(f_{\delta})|.$

Using a very similar approach to what we did in the proof of Theorem 1.4, we
can show that

$\limsup_{\delta\to
0}\delta^{-2+\beta^{2}/2}|\mu(f_{\delta})|=+\infty\qquad\text{a.s.}$

The difference with the setup of Theorem 1.4 is that we are not integrating
$\mu$ against the uniform measures of concentric small squares, but against
scaled versions of the nonnegative smooth test function $f$. The same line of
argument can be applied in this case as well. This shows that $\left\|\mu
u\right\|_{B_{p,q}^{-\beta^{2}/2}(\mathbb{R}^{2})}=+\infty$ a.s. We see from
this argument the special role played by $p=+\infty,p^{\prime}=1$. Because of
the scaling $\delta^{2/p^{\prime}}$ in (6.10), the same reasoning would not
have worked for any other value of $p$.

It remains to show (6.10). It is very likely that this is a standard fact, but
we will derive it for the reader’s convenience. Recall that we denote by
$\phi$ and $\psi$ the father and mother wavelets. For $j\geq 1$ and
$\lambda\in\Lambda_{j}$, let us denote by

$\alpha_{\delta}(\lambda)=\int\psi_{\lambda}f_{\delta}.$

Let $j\geq 1$. The number of $\lambda\in\Lambda_{j}$ such that the supports of
$\psi_{\lambda}$ and $f_{\delta}$ have a nonempty intersection is at most
$2^{2j}\delta^{2}$. Let $\lambda\in\Lambda_{j}$ be such. Because the integral
of $\psi_{\lambda}$ vanishes,

$|\alpha_{\delta}(\lambda)|=\left|\int\psi_{\lambda}(f_{\delta}-f_{\delta}(x_{0}))\right|\leq\frac{\left\|f^{\prime}\right\|_{\infty}}{\delta}\int|\psi_{\lambda}(x)|\,|x-x_{0}|\,dx\leq
C\frac{2^{-j}}{\delta}\int|\psi_{\lambda}|\leq C\frac{2^{-2j}}{\delta}.$

If $j$ is such that $2^{-j}\geq\delta$, we simply bound
$|\alpha_{\delta}(\lambda)|\leq\left\|f\right\|_{\infty}\left\|\psi_{\lambda}\right\|_{L^{1}}\leq
C2^{-j}$. Overall, this shows that

$\Big{(}\sum_{\lambda\in\Lambda_{j}}|\alpha_{\delta}(\lambda)|^{p^{\prime}}\Big{)}^{1/p^{\prime}}\leq
C2^{j(2/p^{\prime}-1)}\delta^{2/p^{\prime}}\min(2^{-j}\delta^{-1},1).$

We conclude that

$\displaystyle\Big{\|}\Big{\\{}2^{j(1-2/p^{\prime}+\beta^{2}/2)}\Big{(}\sum_{\lambda\in\Lambda_{j}}|\alpha_{\delta}(\lambda)|^{p^{\prime}}\Big{)}^{1/p^{\prime}}\Big{\\}}_{j\geq
1}\Big{\|}_{\ell^{q^{\prime}}}$
$\displaystyle\leq\Big{(}\delta^{2q^{\prime}/p^{\prime}}\sum_{j=1}^{\left\lfloor|\log_{2}\delta|\right\rfloor}2^{jq^{\prime}\beta^{2}/2}+\delta^{2q^{\prime}/p^{\prime}-q^{\prime}}\sum_{j>\left\lfloor|\log_{2}\delta|\right\rfloor}2^{-jq^{\prime}(1-\beta^{2}/2)}\Big{)}^{1/q^{\prime}}\leq
C\delta^{2/p^{\prime}-\beta^{2}/2}.$

One can similarly control the coefficients coming from the father wavelet
$\phi$ to obtain the bound (6.10). This concludes the proof of Theorem 1.7. ∎

##  7\. Convergence to white noise: proof of Theorem 1.8

Throughout the section, we assume without loss of generality that
$[0,1]^{2}+B(0,2)\subset D$. Recall that, as defined in (1.9), we consider for
all $r\in(0,1)$ the process

$W_{r}(z)\coloneqq
r^{\beta^{2}-5}\big{(}|\mu(Q(z,r))|^{2}-\mathbb{E}|\mu(Q(z,r))|^{2}\big{)},\quad
z\in[0,1]^{2}.$

The goal of this section is to prove the convergence of $W_{r}$ as $r\to 0$ to
the white noise as stated in Theorem 1.8. It is in fact technically more
convenient to work with an integrated version of $W_{r}$ that is defined, for
$(x,y)\in[0,1]^{2}$, by

$B_{r}(x,y)\coloneqq\sqrt{A}\int_{[0,x]\times[0,y]}W_{r}(z)\,dz$

where $A>0$ is the normalising constant whose inverse is defined in (7.1)
below. The next result states that $(B_{r})_{r\in(0,1)}$ converges to the
Brownian sheet:

######  Theorem 7.1.

The processes $(B_{r})_{r\in(0,1)}$ converge in distribution in
$C^{0}([0,1]^{2})$, but not in probability, to the Brownian sheet in
$[0,1]^{2}$ as $r\to 0$.

Let us first see how the convergence of $(W_{r})_{r\in(0,1)}$ follows:

###### Proof of Theorem 1.8, assuming Theorem 7.1.

By definition, for all $x,y\in[0,1]^{2}$,
$W_{r}(x,y)=A^{-1/2}\partial_{x}\partial_{y}B_{r}(x,y)$. Because
$(B_{r})_{r\in(0,1)}$ converges in distribution in $C^{0}([0,1]^{2})$, it also
converges in distribution in $L^{2}([0,1]^{2})$. By duality, it implies that
$(W_{r})_{r\in(0,1)}$ converges in distribution in $H^{-2}_{0}([0,1]^{2})$ to
$A^{-1/2}$ times the distributional derivative $\partial_{x}\partial_{y}$ of
the Brownian sheet. The latter is a constant multiple of the white noise. ∎

The lemmas quoted in the following proof will be stated and proved in the
subsequent sections (Sections 7.1, 7.2 and 7.3).

###### Proof of Theorem 7.1.

By Lemma 7.2 we see that $(B_{r})_{r\in(0,1)}$ is tight in $C([0,1]^{2})$. To
show convergence in distribution it is therefore enough to show the uniqueness
of subsequential limits. Assume thus that $B_{r_{k}}\to B$ in $C([0,1]^{2})$
for some sequence $r_{k}\to 0$. We note that by Lemma 7.5
$B(s,t)-B(s^{\prime},t^{\prime})$ is independent of $B(s^{\prime},t^{\prime})$
whenever $s\geq s^{\prime}$ and $t\geq t^{\prime}$. Together with the variance
estimate $\mathbb{E}B(s,t)^{2}=st$ established in Lemma 7.6 this is enough to
characterise the Brownian sheet by [Zak81, Proposition 5.4].

If the convergence were true in probability, then in particular $B_{r}(1,1)$
would have to converge in probability, and indeed even in $L^{2}(\Omega)$ by
the uniform integrability coming from the moment bound obtained in Lemma 7.4.
This, however, is ruled out by Lemma 7.7. ∎

Before stating and proving the intermediate lemmas mentioned in the previous
proof, we would like to highlight a key feature of the proofs of this section.

Symmetrization trick. In several places in this section, we will use
“symmetrization tricks” that allow us to bound efficiently integrals with
integrands whose signs are alternating. We give here a concrete simple
example. For $x,y,u,v\in\mathbb{R}^{2}$, let

$\mathbf{r}(x,y,u,v)=\frac{|x-u|^{\beta^{2}}|y-v|^{\beta^{2}}}{|x-v|^{\beta^{2}}|y-u|^{\beta^{2}}}.$

Let us consider the following integral:

$A^{-1}:=\int_{\mathbb{R}^{2}}dw\int_{Q(0,1)^{2}}dx\,dy\int_{Q(w,1)^{2}}du\,dv\frac{\mathbf{r}(x,y,u,v)-1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}.$
(7.1)

We wish to show that it is absolutely convergent as an integral over
$w\in\mathbb{R}^{2}$. The only possible source of divergence is when
$|w|\to\infty$. Expanding $\mathbf{r}(x,y,u,v)-1$, we can show that for
$x,y,u,v$ as in the integral above, $|\mathbf{r}(x,y,u,v)-1|=O(|w|^{-2})$ as
$|w|\to\infty$. This estimate is not sufficient for the absolute integrability
of the integral (we would get a logarithmic divergence), but we can exploit
the symmetry of the integrand to show that it is in fact $O(|w|^{-4}$).
Indeed, notice that swapping $x$ and $y$ changes $\mathbf{r}$ to
$1/\mathbf{r}$: $\mathbf{r}(y,x,u,v)=1/\mathbf{r}(x,y,u,v)$. Therefore, for
all $w\in\mathbb{R}^{2}$,

$\displaystyle f(w)$
$\displaystyle:=\int_{Q(0,1)^{2}}dx\,dy\int_{Q(w,1)^{2}}du\,dv\frac{\mathbf{r}(x,y,u,v)-1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}$
$\displaystyle=\frac{1}{2}\int_{Q(0,1)^{2}}dx\,dy\int_{Q(w,1)^{2}}du\,dv\frac{1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}\Big{(}\mathbf{r}(x,y,u,v)+\frac{1}{\mathbf{r}(x,y,u,v)}-2\Big{)}.$

If $|w|$ is large enough and if we denote by $\mathbf{r}=\mathbf{r}(x,y,u,v)$
for $x,y,u,v$ as in the above integral,
$\mathbf{r}+1/\mathbf{r}-2=(\mathbf{r}-1)^{2}/\mathbf{r}$, so we get
$f(w)=O(|w|^{-4})$ as claimed. The fact that the integral is positive also
follows from the fact $(\mathbf{r}-1)^{2}/\mathbf{r}$ is nonnegative.

###  7.1. Tightness of $(B_{r})_{r\in(0,1)}$

In this section, we show the following tightness result:

######  Lemma 7.2.

The family $(B_{r})_{r\in(0,1)}$ is tight in $C^{0}([0,1]^{2})$.

Before proving this lemma, we first state and prove two intermediate results.
The first one is a simple estimate on the Green function. The second one, on
the other hand, provides a precise upper bound on the correlations of $W_{r}$
which is the main technical ingredient needed in the proof of Lemma 7.2. In
both of these lemmas we need to bound integrals whose integrands do not have
constant signs. The cancellations are important to take into account and we
will do so using symmetrization tricks similar in flavour to the one below
(7.1).

######  Lemma 7.3.

There exists $C>0$ such that for any $z_{1},w_{1},z_{2},w_{2}\in[0,1]^{2}$ we
have

$|G(z_{1},z_{2})+G(w_{1},w_{2})-G(z_{1},w_{2})-G(w_{1},z_{2})|\leq
C\frac{|z_{1}-w_{1}||z_{2}-w_{2}|}{r^{2}},$

where $r=\min(|z_{1}-z_{2}|,|z_{1}-w_{2}|,|w_{1}-z_{2}|,|w_{1}-w_{2}|)$.

###### Proof.

Let $r$ be as in the statement of lemma and denote by $\Gamma_{r}$ the
mollification of $\Gamma$ with respect to some radially symmetric non-negative
mollifier $\varphi$ whose support is included in the unit ball. Since the
Green function is harmonic, we have

$\displaystyle|G(z_{1},z_{2})+G(w_{1},w_{2})-G(z_{1},w_{2})-G(w_{1},z_{2})|=|\mathbb{E}[(\Gamma(z_{1})-\Gamma(w_{1}))(\Gamma(z_{2})-\Gamma(w_{2}))]|$
$\displaystyle=|\mathbb{E}[(\Gamma_{r}(z_{1})-\Gamma_{r}(w_{1}))(\Gamma_{r}(z_{2})-\Gamma_{r}(w_{2}))]|\leq\sqrt{\mathbb{E}(\Gamma_{r}(z_{1})-\Gamma_{r}(w_{1}))^{2}}\sqrt{\mathbb{E}(\Gamma_{r}(z_{2})-\Gamma_{r}(w_{2}))^{2}}.$

By Jensen’s inequality we further have

$|\Gamma_{r}(z)-\Gamma_{r}(w)|^{2}\leq|z-w|^{2}\Big{(}\frac{1}{|z-w|}\int_{0}^{|z-w|}\Big{|}\nabla\Gamma_{r}\big{(}w+t\frac{z-w}{|z-w|}\big{)}\Big{|}^{2}\,dt\Big{)}.$

Lemma 7.3 will then follow from the following bound: there exists $C>0$ such
that for all $x\in[0,1]^{2}$ and $r\in(0,\sqrt{2})$,

$\mathbb{E}|\nabla\Gamma_{r}(x)|^{2}\leq Cr^{-2}.$ (7.2)

We now prove (7.2). Let $x\in[0,1]^{2}$ and $r\in(0,\sqrt{2})$. We have

$\mathbb{E}|\partial_{1}\Gamma_{r}(x)|^{2}=\int_{B(0,r)^{2}}G(x-y,x-z)\partial_{1}\varphi_{r}(y)\partial_{1}\varphi_{r}(z)dydz.$

We want to bound the integral on the right hand side whose integrand does not
have a constant sign. In order to take into account some cancellations, we
will use the following _symmetrization trick_. Since $\varphi$ is radially
symmetric, $2\mathbb{E}|\partial_{1}\Gamma_{r}(x)|^{2}$ can be written as

$\displaystyle\int_{B(0,r)^{2}}(G(x-y,x-z)-G(x+y,x-z))\partial_{1}\varphi_{r}(y)\partial_{1}\varphi_{r}(z)dydz$
(7.3) $\displaystyle\leq
r^{-6}\left\|\partial_{1}\varphi\right\|_{\infty}^{2}\int_{B(0,r)^{2}}|G(x-y,x-z)-G(x+y,x-z)|dydz.$

Recall that we can write $G(y,z)=-\log|y-z|+g(y,z)$ where $g$ is bounded in
$[0,1]^{2}+B(0,\sqrt{2})$. Since

$\int_{B(0,r)^{2}}\left|\log\frac{|y+z|}{|y-z|}\right|dydz=O(r^{4})$

by scaling, this shows that $\mathbb{E}|\partial_{1}\Gamma_{r}(x)|^{2}$ is
bounded by $Cr^{-2}$ for some $C>0$. One can obtain a similar bound for
$\mathbb{E}|\partial_{2}\Gamma_{r}(x)|^{2}$ concluding the proof of (7.2).
Note that without the symmetrization trick, we would have obtained an extra
$|\log r|$ factor on the right hand side of (7.2). This finishes the proof of
the lemma. ∎

######  Lemma 7.4.

For any $N\geq 1$ there exists $C_{N}>0$ such that for any measurable subset
$U\subset[0,1]^{2}$ we have

$\int_{U^{N}}|\mathbb{E}[W_{r}(z_{1})\dots W_{r}(z_{N})]|\,dz_{1}\dots
dz_{N}\leq C_{N}|U|^{N/2}.$ (7.4)

###### Proof.

We can split the integral (7.4) in several parts depending on which points are
further than $4r$ away from their closest neighbours. Without loss of
generality it is therefore enough to check that for any $0\leq n\leq N$ we
have

$\int_{\mathcal{D}_{n}}|\mathbb{E}[W_{r}(z_{1})\dots
W_{r}(z_{n})W_{r}(w_{1})\dots W_{r}(w_{N-n})]|\lesssim|U|^{N/2},$

where

$\displaystyle\mathcal{D}_{n}$
$\displaystyle=\\{(z_{1},\dots,z_{n},w_{1},\dots,w_{N-n})\in
U^{N}:|w_{j}-z_{k}|\geq 4r\text{ for all }j,k$
$\displaystyle\wedge|w_{j}-w_{k}|\geq 4r\text{ for all }j\neq k$
$\displaystyle\wedge\text{for all }j\text{ we have }|z_{j}-z_{k}|\leq 4r\text{
for some }k\neq j\\}.$

Recalling the notation $G_{D}^{\prime}$ defined in (2.4), we compute

$\displaystyle|\mathbb{E}[W_{r}(z_{1})\dots W_{r}(z_{n})W_{r}(w_{1})\dots
W_{r}(w_{N-n})]|$
$\displaystyle=r^{N\beta^{2}-5N}\Big{|}\mathbb{E}\int\prod_{j=1}^{n}\big{(}\mu(x_{j})\overline{\mu(y_{j})}-e^{\beta^{2}G^{\prime}(x_{j},y_{j})}\big{)}\prod_{j=1}^{N-n}\big{(}\mu(u_{j})\overline{\mu(v_{j})}-e^{\beta^{2}G^{\prime}(u_{j},v_{j})}\big{)}\Big{|},$

where the integral is over $x_{j},y_{j}\in Q(z_{j},r)$ and $u_{k},v_{k}\in
Q(w_{k},r)$ with $1\leq j\leq n$, $1\leq k\leq N-n$. Recall the notation
$\mathcal{E}^{\prime}(\Gamma;\cdot;\cdot)$ defined in (2.5) and the formula
(2.7). In the display below, we denote by $\mathbf{x}_{\pi}=(x_{j})_{j\in\pi}$
and similar notations for $\mathbf{x}_{\tau}$,
$\mathbf{y}_{\pi},\mathbf{y}_{\tau}$. Expanding the two products and computing
the expected value gives

$\displaystyle
r^{N\beta^{2}-5N}\Big{|}\int\sum_{\pi\subset\\{1,\dots,n\\}}\sum_{\tau\subset\\{1,\dots,N-n\\}}\mathbb{E}\Big{[}\prod_{j\in\pi}(\mu(x_{j})\overline{\mu(y_{j})})\prod_{j\notin\pi}(-e^{\beta^{2}G^{\prime}(x_{j},y_{j})})$
$\displaystyle\quad\cdot\prod_{j\in\tau}(\mu(u_{j})\overline{\mu(v_{j})})\prod_{j\notin\tau}(-e^{\beta^{2}G^{\prime}(u_{j},v_{j})})\Big{]}\Big{|}$
$\displaystyle=r^{N\beta^{2}-5N}\Big{|}\int\sum_{\pi\subset\\{1,\dots,n\\}}e^{\beta^{2}\mathcal{E}^{\prime}(\mathbf{x}_{\pi},\mathbf{y}_{\pi})}\prod_{j\notin\pi}e^{\beta^{2}G^{\prime}(x_{j},y_{j})}\sum_{\tau\subset\\{1,\dots,N-n\\}}(-1)^{|\pi|+|\tau|}e^{\beta^{2}\mathcal{E}^{\prime}(\mathbf{u}_{\tau},\mathbf{v}_{\tau})}$
$\displaystyle\quad\cdot
e^{\beta^{2}\sum_{j\in\pi,k\in\tau}(G^{\prime}(x_{j},v_{k})+G^{\prime}(y_{j},u_{k})-G^{\prime}(x_{j},u_{k})-G^{\prime}(y_{j},v_{k}))}\prod_{j\notin\tau}e^{\beta^{2}G^{\prime}(u_{j},v_{j})}\Big{|}.$

Note next that

$\displaystyle\mathcal{E}^{\prime}(\mathbf{u}_{\tau},\mathbf{v}_{\tau})+\sum_{j\in\pi,k\in\tau}(G^{\prime}(x_{j},v_{k})+G^{\prime}(y_{j},u_{k})-G^{\prime}(x_{j},u_{k})-G^{\prime}(y_{j},v_{k}))+\sum_{j\notin\tau}G^{\prime}(u_{j},v_{j})$
$\displaystyle=\sum_{j=1}^{N-n}G^{\prime}(u_{j},v_{j})+\sum_{j,k\in\tau,j<k}\Delta_{j,k}+\sum_{j\in\pi,k\in\tau}\widetilde{\Delta}_{j,k},$

where

$\displaystyle\widetilde{\Delta}_{j,k}$ $\displaystyle\coloneqq
G^{\prime}(x_{j},v_{k})+G^{\prime}(y_{j},u_{k})-G^{\prime}(x_{j},u_{k})-G^{\prime}(y_{j},v_{k})\quad\text{for
all }1\leq j\leq n,1\leq k\leq N-n$ $\displaystyle\Delta_{j,k}$
$\displaystyle\coloneqq
G^{\prime}(u_{j},v_{k})+G^{\prime}(u_{k},v_{j})-G^{\prime}(u_{j},u_{k})-G^{\prime}(v_{j},v_{k})\quad\text{for
all }1\leq j<k\leq N-n.$

Thus we have

$\displaystyle|\mathbb{E}[W_{r}(z_{1})\dots W_{r}(z_{n})W_{r}(w_{1})\dots
W_{r}(w_{N-n})]|$
$\displaystyle=r^{N\beta^{2}-5N}\Big{|}\int\sum_{\pi\subset\\{1,\dots,n\\}}(-1)^{|\pi|}e^{\beta^{2}\mathcal{E}^{\prime}(\mathbf{x}_{\pi},\mathbf{y}_{\pi})+\beta^{2}\sum_{j\notin\pi}G^{\prime}(x_{j},y_{j})}e^{\beta^{2}\sum_{j=1}^{N-n}G^{\prime}(u_{j},v_{j})}$
$\displaystyle\quad\cdot\sum_{\tau\subset\\{1,\dots,N-n\\}}(-1)^{|\tau|}e^{\beta^{2}\sum_{j,k\in\tau,j<k}\Delta_{j,k}+\beta^{2}\sum_{j\in\pi,k\in\tau}\widetilde{\Delta}_{j,k}}\Big{|}.$

We now use once more a symmetrization trick by observing that the integrand is
symmetric upon swapping the variables $u_{k}\leftrightarrow v_{k}$. Moreover
doing the swap simply switches the sign of $\Delta_{j,k}$ and
$\widetilde{\Delta}_{j,k}$. Hence we can write the above as

$\displaystyle|\mathbb{E}[W_{r}(z_{1})\dots W_{r}(z_{n})W_{r}(w_{1})\dots
W_{r}(w_{N-n})]|$
$\displaystyle=2^{-(N-n)}r^{N\beta^{2}-5N}\Big{|}\int\sum_{\pi\subset\\{1,\dots,n\\}}(-1)^{|\pi|}e^{\beta^{2}\mathcal{E}^{\prime}(\mathbf{x}_{\pi},\mathbf{y}_{\pi})+\beta^{2}\sum_{j\notin\pi}G^{\prime}(x_{j},y_{j})}e^{\beta^{2}\sum_{j=1}^{N-n}G^{\prime}(u_{j},v_{j})}$
$\displaystyle\quad\cdot\sum_{\sigma\in\\{0,1\\}^{N-n}}\sum_{\tau\subset\\{1,\dots,N-n\\}}(-1)^{|\tau|}e^{\beta^{2}\sum_{j,k\in\tau,j<k}(-1)^{\sigma_{j}+\sigma_{k}}\Delta_{j,k}+\beta^{2}\sum_{j\in\pi,k\in\tau}(-1)^{\sigma_{k}}\widetilde{\Delta}_{j,k}}\Big{|}.$

We will next focus on estimating the two inner sums in terms of the variables
$\Delta_{j,k},\widetilde{\Delta}_{j,k}$, which by Lemma 7.3 are bounded by
$C\frac{r^{2}}{|w_{j}-w_{k}|^{2}}\leq R$ (resp. by
$C\frac{r^{2}}{|z_{j}-w_{k}|^{2}}\leq R$) for some $R>0$. To this end we
define the analytic function

$\displaystyle f((\Delta_{j,k}),(\widetilde{\Delta}_{j,k}))\coloneqq$
$\displaystyle\sum_{\sigma\in\\{0,1\\}^{N-n}}\sum_{\tau\subset\\{1,\dots,N-n\\}}(-1)^{|\tau|}e^{\beta^{2}\sum_{j,k\in\tau,j<k}(-1)^{\sigma_{j}+\sigma_{k}}\Delta_{j,k}+\beta^{2}\sum_{j\in\pi,k\in\tau}(-1)^{\sigma_{k}}\widetilde{\Delta}_{j,k}}$

and consider its Taylor series around $0$. We have by the multinomial theorem
that

$\displaystyle f((\Delta_{j,k}),(\widetilde{\Delta}_{j^{\prime},k^{\prime}}))$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{\sigma\in\\{0,1\\}^{N-n}}\sum_{\tau\subset\\{1,\dots,N-n\\}}(-1)^{|\tau|}\frac{\beta^{2\ell}}{\ell!}\big{(}\sum_{j,k}(-1)^{\sigma_{j}+\sigma_{k}}\Delta_{j,k}+\sum_{j^{\prime},k^{\prime}}(-1)^{\sigma_{k^{\prime}}}\widetilde{\Delta}_{j^{\prime},k^{\prime}}\big{)}^{\ell}$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{\sigma\in\\{0,1\\}^{N-n}}\sum_{\tau\subset\\{1,\dots,N-n\\}}\sum_{\sum_{j,k}p_{j,k}+\sum_{j^{\prime},k^{\prime}}q_{j^{\prime},k^{\prime}}=\ell}\frac{(-1)^{|\tau|}\beta^{2\ell}}{\ell!}{\ell\choose(p_{j,k})_{j,k},(q_{j^{\prime},k^{\prime}})_{j^{\prime},k^{\prime}}}$
$\displaystyle\quad\cdot\prod_{j,k}(-1)^{p_{j,k}(\sigma_{j}+\sigma_{k})}\Delta_{j,k}^{p_{j,k}}\prod_{j^{\prime},k^{\prime}}(-1)^{q_{j^{\prime},k^{\prime}}\sigma_{k^{\prime}}}\widetilde{\Delta}_{j^{\prime},k^{\prime}}^{q_{j,k}},$

where the sums and products and index sets of variables are over
$j,k\in\tau,j<k$ and $j^{\prime}\in\pi,k^{\prime}\in\tau$. We may equivalently
rephrase this condition by saying that the variables $p_{j,k}$ and
$q_{j^{\prime},k^{\prime}}$ range over the whole index set with $1\leq j<k\leq
N-n$ and $1\leq j^{\prime}\leq n$, $1\leq k^{\prime}\leq N-n$, that their sum
is $\ell$ and with the extra condition that $p_{j,k}=0$ if one of $j,k$ is not
in $\tau$ and similarly $q_{j^{\prime},k^{\prime}}=0$ if $j^{\prime}\notin\pi$
or $k^{\prime}\notin\tau$. Exchanging the two inner most sums then gives us

$\displaystyle\sum_{\ell=0}^{\infty}\sum_{\sigma\in\\{0,1\\}^{N-n}}\sum_{\sum_{j,k}p_{j,k}+\sum_{j^{\prime},k^{\prime}}q_{j^{\prime},k^{\prime}}=\ell}\sum_{\begin{subarray}{c}\tau\subset\\{1,\dots,N-n\\}\\\
p_{j,k}\neq 0\Rightarrow j,k\in\tau\\\ q_{j^{\prime},k^{\prime}}\neq
0\Rightarrow
k^{\prime}\in\tau\end{subarray}}\frac{(-1)^{|\tau|}\beta^{2\ell}}{\ell!}{\ell\choose(p_{j,k})_{j,k},(q_{j^{\prime},k^{\prime}})_{j^{\prime},k^{\prime}}}$
$\displaystyle\quad\cdot\prod_{j,k}(-1)^{p_{j,k}(\sigma_{j}+\sigma_{k})}\Delta_{j,k}^{p_{j,k}}\prod_{j^{\prime},k^{\prime}}(-1)^{q_{j^{\prime},k^{\prime}}\sigma_{k^{\prime}}}\widetilde{\Delta}_{j^{\prime},k^{\prime}}^{q_{j,k}}$
$\displaystyle=\sum_{\ell=0}^{\infty}\sum_{\sigma\in\\{0,1\\}^{N-n}}\sum_{\begin{subarray}{c}\sum_{j,k}p_{j,k}+\sum_{j^{\prime},k^{\prime}}q_{j^{\prime},k^{\prime}}=\ell\\\
\forall 1\leq r\leq N-n\text{ exists non-zero }p_{j,r},p_{r,k}\text{ or
}q_{j^{\prime},r}\end{subarray}}\frac{(-1)^{N-n}\beta^{2\ell}}{\ell!}{\ell\choose(p_{j,k})_{j,k},(q_{j^{\prime},k^{\prime}})_{j^{\prime},k^{\prime}}}$
$\displaystyle\quad\cdot\prod_{j,k}(-1)^{p_{j,k}(\sigma_{j}+\sigma_{k})}\Delta_{j,k}^{p_{j,k}}\prod_{j^{\prime},k^{\prime}}(-1)^{q_{j^{\prime},k^{\prime}}\sigma_{k^{\prime}}}\widetilde{\Delta}_{j^{\prime},k^{\prime}}^{q_{j,k}},$

where we have used the fact that if the indices $j,k,k^{\prime}$ of those
$p_{j,k}$, $q_{j^{\prime},k^{\prime}}$ that are non-zero span a set
$S\subsetneq\\{1,\dots,N-n\\}$, then the sum over $\tau$ can be written as a
sum over the subsets of $\\{1,\dots,N-n\\}\setminus S$ and for any non-empty
finite set $A$ we have $\sum_{\tau\subset A}(-1)^{\tau}=0$.

We note next that the sum over $\sigma$ will be $0$ unless every $\sigma_{j}$
appears at least twice in the two products. Because the series is absolutely
summable this means that we have the bound

$|f((\Delta_{j,k}),(\widetilde{\Delta}_{j^{\prime},k^{\prime}}))|\leq
C\sum_{p_{j,k},q_{j^{\prime},k^{\prime}}}\prod_{j,k}\frac{r^{2p_{j,k}}}{|w_{j}-w_{k}|^{2p_{j,k}}}\prod_{j^{\prime},k^{\prime}}\frac{r^{2q_{j,k}}}{|z_{j^{\prime}}-w_{k^{\prime}}|^{2q_{j,k}}},$

where $C>0$ is some constant that depends on $N$ and $R$ and the sum is over
such $p_{j,k}$ and $q_{j^{\prime},k^{\prime}}$ for which

  * •

for all $1\leq r\leq N-n$ we have that there exist at least two different
pairs $(j,r),(r,k)$ or $(j^{\prime},r)$ such that the corresponding variables
are non-zero, and

  * •

the variables $p_{j,k}$ and $q_{j^{\prime},k^{\prime}}$ are minimal in the
sense that the above condition is not anymore satisfied after decreasing any
of the variables $p_{j,k}$ and $q_{j^{\prime},k^{\prime}}$ by one.

One easily checks that there are only finitely many such configurations
$p_{j,k}$, $q_{j^{\prime},k^{\prime}}$. In light of the above bound we see
that it is enough to check that

$\displaystyle r^{N\beta^{2}-5N}\int_{\mathcal{D}_{n}}\int
e^{\beta^{2}\mathcal{E}^{\prime}(\mathbf{x}_{\pi},\mathbf{y}_{\pi})+\beta^{2}\sum_{j\notin\pi}G^{\prime}(x_{j},y_{j})}e^{\beta^{2}\sum_{j=1}^{N-n}G^{\prime}(u_{j},v_{j})}$
$\displaystyle\quad\cdot\prod_{j,k}\frac{r^{2p_{j,k}}}{|w_{j}-w_{k}|^{2p_{j,k}}}\prod_{j^{\prime},k^{\prime}}\frac{r^{2q_{j^{\prime},k^{\prime}}}}{|z_{j^{\prime}}-w_{k^{\prime}}|^{2q_{j^{\prime},k^{\prime}}}}\lesssim|U|^{N/2}$

where we have fixed $\pi\subset\\{1,\dots,n\\}$ and a configuration
$p_{j,k},q_{j^{\prime},k^{\prime}}$ satisfying the conditions given above. As
a reminder, here the first integral is over those points
$z_{1},\dots,z_{n},w_{1},\dots,w_{N-n}\in U^{N}$ that satisfy the nearest
neighbour distance requirements given in the beginning and the second integral
is again over $x_{j},y_{j}\in Q(z_{j},r)$ and $u_{j},v_{j}\in Q(w_{j},r)$. The
integral over $u_{j},v_{j}$ is $O(r^{(N-n)(4-\beta^{2})})$ and the integral
over those $x_{j},y_{j}$ for which $j\notin\pi$ is similarly
$O(r^{(n-|\pi|)(4-\beta^{2})})$. The integral over those $x_{j},y_{j}$ for
which $j\in\pi$ is by a standard application of Onsager’s inequality and
scaling $O(r^{|\pi|(4-\beta^{2})})$. Altogether we hence have an upper boundd

$r^{-N}\int_{\mathcal{D}_{n}}\prod_{j,k}\frac{r^{2p_{j,k}}}{|w_{j}-w_{k}|^{2p_{j,k}}}\prod_{j^{\prime},k^{\prime}}\frac{r^{2q_{j^{\prime},k^{\prime}}}}{|z_{j^{\prime}}-w_{k^{\prime}}|^{2q_{j^{\prime},k^{\prime}}}}.$

We will next bound the integrals over the $w_{j}$. It is helpful to view the
dependencies as a graph with vertices $z_{1},\dots,z_{n},w_{1},\dots,w_{N-n}$
such that there are $p_{j,k}$ edges between $w_{j}$ and $w_{k}$ and
$q_{j^{\prime},k^{\prime}}$ edges between $z_{j^{\prime}}$ and
$w_{k^{\prime}}$. The minimality condition of the configuration implies that
for each of the edges one of the endpoints must have degree at most $2$. We
will then proceed integrating over all of the vertices of the graph using the
following operations:

  1. (1)

Pick a vertex $w_{j}$ with degree $2$ that has a double edge to its neighbour
$u$. Integrate $\frac{r^{4}\mathbf{1}_{|w_{j}-u|\geq 4r}}{|w_{j}-u|^{4}}$ over
$w_{j}$ to obtain an upper bound of order $r^{2}$ and remove $w_{j}$ from the
graph.

  2. (2)

Pick a vertex $w_{j}$ with degree $2$ that has a double edge to its neighbour
$w_{k}$. Integrate $\frac{r^{4}\mathbf{1}_{|w_{j}-u|\geq 4r}}{|w_{j}-u|^{4}}$
over $w_{j}$ and use Jensen’s inequality to obtain an upper bound of order
$r\sqrt{|U|}$. Remove $w_{j}$ from the graph.

  3. (3)

Pick a vertex $w_{j}$ with degree $2$ that has two neighbours $u$ and $v$ and
which is such that removing $w_{j}$ does not increase the number of connected
components of the graph when restricted to the $w$-vertices. Use the AM-GM
inequality to replace the term $\frac{r^{4}}{|w_{j}-u|^{2}|w_{j}-v|^{2}}$ with
the term $\frac{r^{4}}{|w_{j}-u|^{4}}+\frac{r^{4}}{|w_{j}-v|^{4}}$,
effectively replacing the graph with two new graphs where instead of two
single edges $w_{j}$ now has a double edge between either $u$ or $v$.

  4. (4)

Pick a vertex $w_{j}$ with degree $1$ and neighbour $u$. Integrate
$\frac{r^{2}\mathbf{1}_{|w_{j}-w_{k}|}}{|w_{j}-u|^{2}}$ over $u$ and use
Jensen’s inequality to obtain an upper bound of order $r\sqrt{|U|}$. Remove
$w_{j}$ from the graph.

  5. (5)

Pick a vertex $w_{j}$ with degree $0$. Integrating over $w_{j}$ gives a bound
of order $|U|$. Remove $w_{j}$ from the graph.

We will now exhaust all the vertices $w_{j}$ in the graph by using the above
operations in the following manner: Start by picking a connected component of
the graph. If this connected component contains any $z$-vertices, we simply
use operations (2), (3) and (4), thereby obtaining an upper bound of
$(r\sqrt{|U|})^{m}$, where $m$ is the number of vertices $w_{j}$ in the
connected component. If the component does not contain any $z$-vertices, we
will use one operation (1) (possibly after using (3) first) and then do
operations (2), (3) and (4) until only a degree $0$-vertex remains so that we
finish with one operation (5). The operations (1) and (5) together give us
$r^{2}|U|$ and the rest of the operations give $(r\sqrt{|U|})^{m-2}$, so we
again get $(r\sqrt{|U|})^{m}$ in total. Taking the product over all the
components gives an upper bound of order $r^{N-n}|U|^{\frac{N-n}{2}}$.

Finally we will integrate over the $z_{j}$ under the constraint that every
$z_{j}$ has another $x_{k}$ no more than $4r$ away from it. Here one may split
the integral according to the nearest neighbour function as in the proof of
[JSW20, Lemma 3.10]. Fixing a nearest neighbour function, recall that the
nearest neighbour graph of $z_{j}$ consists of components which are built from
two trees whose roots are joined in a 2-cycle. We then get for each such
component in a straightforward manner an upper bound of order
$\min(r^{2},|U|)^{m-2}r^{2}|U|$, where $m\geq 2$ is the number of vertices in
the component. Since $\min(r^{2},|U|)\leq r\sqrt{|U|}$, we in fact have an
upper bound of $r^{m}|U|^{\frac{m}{2}}$ for the given component and taking the
product over all the components gives an upper bound of order
$r^{n}|U|^{\frac{n}{2}}$, which finishes the proof. ∎

Now that we have proved Lemma 7.4, Lemma 7.2 will follow quickly:

###### Proof of Lemma 7.2.

By Kolmogorov’s tightness criterion (see e.g. [SV06, Theorem 2.1.6]) it is
enough to check e.g. that there exists a constant $C>0$ such that

$\mathbb{E}|B_{r}(z)-B_{r}(w)|^{6}\leq C|z-w|^{3}$

for all $z,w\in[0,1]^{2}$ and $r>0$. This on the other hand follows from Lemma
7.4 since

$\mathbb{E}|B_{r}(z)-B_{r}(w)|^{6}\leq
A^{3}\int_{U^{6}}|\mathbb{E}[W_{r}(z_{1})\dots W_{r}(z_{6})]|dz_{1}\dots
dz_{6}\lesssim|U|^{3},$

where $U$ is the symmetric difference between
$[0,\operatorname{Re}(z)]\times[0,\operatorname{Im}(z)]$ and
$[0,\operatorname{Re}(w)]\times[0,\operatorname{Im}(w)]$ and thus
$|U|\lesssim|z-w|$. ∎

###  7.2. Study of subsequential limits

Thanks to Lemma 7.2, we now know that $(B_{r})_{r\in(0,1)}$ is tight. In this
section, we will study the properties of its subsequential limits. We will
show in Lemma 7.5 the independence of their increments and compute in Lemma
7.6 the value of the variance.

######  Lemma 7.5.

Let $s\geq s^{\prime}$ and $t\geq t^{\prime}$ and let $B$ be a subsequential
limit in distribution of $(B_{r})_{r}$. Then $B(s,t)-B(s^{\prime},t^{\prime})$
is independent of $B(s^{\prime},t^{\prime})$.

###### Proof.

We first note that by continuity it is enough to check that for any
$0<\varepsilon<\min(s^{\prime},t^{\prime})$ we have
$(B(s,t)-B(s^{\prime},t^{\prime}))\bot
B(s^{\prime}-\varepsilon,t^{\prime}-\varepsilon)$. Let us fix such
$\varepsilon$ and set $a\coloneqq s^{\prime}-\varepsilon$, $b\coloneqq
t^{\prime}-\varepsilon$. Let us decompose our GFF as
$\Gamma=\widetilde{\Gamma}+h$, where $\widetilde{\Gamma}$ is a zero-boundary
GFF in
$\widetilde{D}:=[0,s^{\prime}-\frac{\varepsilon}{2}]\times[0,t^{\prime}-\frac{\varepsilon}{2}]$
and $h$ is harmonic in $\widetilde{D}$ and independent of
$\widetilde{\Gamma}$. We then define for all $r>0$ the random variables

$\widetilde{B}_{r}\coloneqq\sqrt{A}\int_{[0,a]\times[0,b]}\tilde{h}_{r}(z)\,dz,$

where

$\tilde{h}_{r}(z)\coloneqq
r^{\beta^{2}-5}\Big{(}\big{|}\int_{Q(z,r)}e^{i\beta\widetilde{\Gamma}(x)}\,dx\big{|}^{2}-\mathbb{E}\big{|}\int_{Q(z,r)}e^{i\beta\widetilde{\Gamma}(x)}\,dx\big{|}^{2}\Big{)}.$

Notice that $\widetilde{B}_{r}$ is independent of
$B_{r}(s,t)-B_{r}(s^{\prime},t^{\prime})$ for all $r<\varepsilon/2$. We will
therefore be done if we can show that
$\mathbb{E}|\widetilde{B}_{r}-B_{r}(a,b)|^{2}\to 0$ as $r\to 0$. In this
proof, we will write for $x,y,u,v\in\widetilde{D}$,

$\widetilde{G}^{\prime}(x,y)=\mathbb{E}[\widetilde{\Gamma}(x)\widetilde{\Gamma}(y)]-\frac{1}{2}\log\operatorname{CR}(x,\widetilde{D})-\frac{1}{2}\log\operatorname{CR}(y,\widetilde{D})$

and

$\widetilde{\mathcal{E}}^{\prime}(x,u;y,v)=\widetilde{G}^{\prime}(x,y)+\widetilde{G}^{\prime}(x,v)+\widetilde{G}^{\prime}(u,y)+\widetilde{G}^{\prime}(u,v)-\widetilde{G}^{\prime}(x,u)-\widetilde{G}^{\prime}(y,v).$

We have

$\mathbb{E}|\widetilde{B}_{r}-B_{r}(a,b)|^{2}=A\int_{([0,a]\times[0,b])^{2}}dz\,dw\mathbb{E}[(\tilde{h}_{r}(z)-W_{r}(z))(\tilde{h}_{r}(w)-W_{r}(w))].$

We will be done by dominated convergence theorem if we can show that
$\mathbb{E}[(\tilde{h}_{r}(z)-W_{r}(z))(\tilde{h}_{r}(w)-W_{r}(w))]$ is
bounded and that for $z\neq w$ it tends to $0$ as $r\to 0$.

We will start by showing the second claim. Here it is enough to consider the
four terms $\mathbb{E}\tilde{h}_{r}(z)\tilde{h}_{r}(w)$,
$\mathbb{E}\tilde{h}_{r}(z)W_{r}(w)$, $\mathbb{E}W_{r}(z)\tilde{h}_{r}(w)$ and
$\mathbb{E}W_{r}(z)W_{r}(w)$ separately. By looking at the proof of Lemma 7.4
we see that $\mathbb{E}\tilde{h}_{r}(z)\tilde{h}_{r}(w)$ and
$\mathbb{E}W_{r}(z)W_{r}(w)$ will for small enough $r$ have an upper bound of
the form $\frac{r^{2}}{|z-w|^{4}}$. The mixed case
$\mathbb{E}W_{r}(z)\tilde{h}_{r}(w)$ can be reduced to the
$\mathbb{E}\tilde{h}_{r}(z)\tilde{h}_{r}(w)$ case as follows: We have

$\displaystyle\mathbb{E}W_{r}(z)\tilde{h}_{r}(w)=r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dv\mathbb{E}\big{[}(e^{i\beta\Gamma(x)}e^{-i\beta\Gamma(y)}-e^{\beta^{2}G^{\prime}(x,y)})$
$\displaystyle\hskip
70.0pt\cdot(e^{i\beta\widetilde{\Gamma}(u)}e^{-i\beta\widetilde{\Gamma}(v)}-e^{\beta^{2}\widetilde{G}^{\prime}(u,v)})\big{]}$
$\displaystyle=r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dve^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]}\mathbb{E}\big{[}(e^{i\beta\widetilde{\Gamma}(x)}e^{-i\beta\widetilde{\Gamma}(y)}-e^{\beta^{2}\widetilde{G}^{\prime}(x,y)})$
$\displaystyle\quad\cdot(e^{i\beta\widetilde{\Gamma}(u)}e^{-i\beta\widetilde{\Gamma}(v)}-e^{\beta^{2}\widetilde{G}^{\prime}(u,v)})\big{]}$
$\displaystyle=r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dve^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]}e^{\beta^{2}\widetilde{G}^{\prime}(x,y)+\beta^{2}\widetilde{G}^{\prime}(u,v)}$
$\displaystyle\quad\cdot\big{(}e^{\beta^{2}\widetilde{G}^{\prime}(x,v)+\beta^{2}\widetilde{G}^{\prime}(y,u)-\beta^{2}\widetilde{G}^{\prime}(x,u)-\beta^{2}\widetilde{G}^{\prime}(y,v)}-1\big{)}.$

Notice that the integral won’t change if we swap $x$ and $y$ in the integrand.
Thus the above equals

$\displaystyle\frac{1}{2}r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dve^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]}e^{\beta^{2}\widetilde{G}^{\prime}(x,y)+\beta^{2}\widetilde{G}^{\prime}(u,v)}$
$\displaystyle\quad\cdot\big{(}p(x,y,u,v)+p(x,y,u,v)^{-1}-2\big{)},$

where

$p(x,y,u,v)=e^{\beta^{2}\widetilde{G}^{\prime}(x,v)+\beta^{2}\widetilde{G}^{\prime}(y,u)-\beta^{2}\widetilde{G}^{\prime}(x,u)-\beta^{2}\widetilde{G}^{\prime}(y,v)}.$

For any real number $r>0$ we have $r+r^{-1}-2\geq 0$ and
$\mathbb{E}[(h(x)-h(y))^{2}]$ is uniformly bounded in $[0,a]\times[0,b]$, so
we see that

$\mathbb{E}W_{r}(z)\tilde{h}_{r}(w)\lesssim\mathbb{E}\tilde{h}_{r}(z)\tilde{h}_{r}(w).$

It remains to show the boundedness of
$\mathbb{E}[(\tilde{h}_{r}(z)-W_{r}(z))(\tilde{h}_{r}(w)-W_{r}(w))]$. By
Cauchy–Schwarz it is enough to show that
$\mathbb{E}(\tilde{h}_{r}(z)-W_{r}(z))^{2}$ is bounded. A direct but somewhat
tedious computation shows that $\mathbb{E}(\tilde{h}_{r}(z)-W_{r}(z))^{2}$ is
equal to

$\displaystyle
r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(z,r)^{2}}du\,dv(e^{\beta^{2}\widetilde{\mathcal{E}}^{\prime}(x,u;y,v)}A+e^{\beta^{2}\widetilde{G}^{\prime}(u,v)+\beta^{2}\widetilde{G}^{\prime}(x,y)}B)$

where

$\displaystyle
A=1-e^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]}-e^{\frac{\beta^{2}}{2}\mathbb{E}[(h(u)-h(v))^{2}]}$
$\displaystyle+e^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(u))^{2}]+\frac{\beta^{2}}{2}\mathbb{E}[(h(y)-h(v))2]-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]-\frac{\beta^{2}}{2}\mathbb{E}[(h(u)-h(v))2]-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(v))^{2}]-\frac{\beta^{2}}{2}\mathbb{E}[(h(y)-h(u))^{2}]},$
$\displaystyle B$
$\displaystyle=e^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]}+e^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(u)h(v))^{2}]}-e^{-\frac{\beta^{2}}{2}\mathbb{E}[(h(x)-h(y))^{2}]-\frac{\beta^{2}}{2}\mathbb{E}[(h(u)-h(v))^{2}]}-1.$

One easily checks that both in $A$ and $B$ all the exponents are $O(r)$. Thus
by expanding the exponentials as series we see that $B=O(r^{2})$. For $A$ we
get

$\beta^{2}\mathbb{E}h(x)h(v)+\beta^{2}\mathbb{E}h(y)h(u)-\beta^{2}\mathbb{E}h(x)h(u)-\beta^{2}\mathbb{E}h(y)h(v)+O(r^{2}),$

but one also easily checks that

$\displaystyle\mathbb{E}h(x)h(v)+\mathbb{E}h(y)h(u)-\mathbb{E}h(x)h(u)-\mathbb{E}h(y)h(v)=-\mathbb{E}(h(x)-h(y))(h(u)-h(v))=O(r^{2}),$

so in total $\mathbb{E}(\tilde{h}_{r}(z)-W_{r}(z))^{2}$ is bounded by a
constant times

$\displaystyle
r^{2\beta^{2}-8}\int_{B(z,r)^{2}}dx\,dy\int_{B(w,r)^{2}}du\,dv(e^{\beta^{2}\widetilde{\mathcal{E}^{\prime}}(x,y;u,v)}+e^{\beta^{2}\widetilde{G}^{\prime}(x,y)+\beta^{2}\widetilde{G}^{\prime}(u,v)})\lesssim
1.$

∎

######  Lemma 7.6.

Let $B$ be a subsequential limit in distribution of $B_{r}$. Then
$\mathbb{E}B(s,t)^{2}=st$.

###### Proof.

Let $U=[0,s]\times[0,t]$. In this proof, we will denote by

$\mathbf{r}(x,y,u,v)=\frac{|x-u|^{\beta^{2}}|y-v|^{\beta^{2}}}{|x-v|^{\beta^{2}}|y-u|^{\beta^{2}}}$

and

$g^{\prime}(x,y)=G(x,y)+\log|x-y|-\frac{1}{2}\log\operatorname{CR}(x,D)-\frac{1}{2}\log\operatorname{CR}(y,D),$

for $x,y,u,v\in D$. $g^{\prime}$ is smooth and vanishes on the diagonal. We
will use several times the fact that swapping $x$ and $y$ changes $r$ into
$1/r$. A direct computation shows that

$\displaystyle\mathbb{E}B_{r}(s,t)^{2}$
$\displaystyle=A\int_{U^{2}}dz\,dw\,r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dv\frac{e^{\beta^{2}(g^{\prime}(x,y)+g^{\prime}(u,v))}}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}$
$\displaystyle\quad\cdot\big{(}\mathbf{r}(x,y,u,v)e^{\beta^{2}(g^{\prime}(x,v)+g^{\prime}(y,u)-g^{\prime}(x,u)-g^{\prime}(y,v))}-1\big{)}.$

It is easy to see that in the limit as $r\to 0$ we may replace
$g^{\prime}(x,y)+g^{\prime}(u,v)$ by $g^{\prime}(z,z)+g^{\prime}(w,w)=0$ since
the difference between the two is $O(r)$ and the rest of the integral is
bounded by arguments similar to Lemma 7.4. Doing this substitution and by
applying the symmetry upon swapping $x$ and $y$ (see the paragraph following
(7.1) for more details) we get

$\displaystyle\frac{A}{2}\int_{U^{2}}dz\,dw\,r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dv\frac{1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}\frac{((1+a)\mathbf{r}(x,y,u,v)-1)^{2}}{(1+a)\mathbf{r}(x,y,u,v)},$

where
$a=e^{\beta^{2}(g^{\prime}(x,v)+g^{\prime}(y,u)-g^{\prime}(x,u)-g^{\prime}(y,v))}-1=O(r^{2})$.
Writing $\mathbf{r}=\mathbf{r}(x,y,u,v)$, we note that

$\displaystyle\frac{(\mathbf{r}(1+a)-1)^{2}}{\mathbf{r}(1+a)}$
$\displaystyle=\frac{(\mathbf{r}-1)^{2}}{\mathbf{r}}-\frac{(\mathbf{r}-1)^{2}a}{\mathbf{r}(1+a)}+\frac{2a(\mathbf{r}-1)}{1+a}+\frac{\mathbf{r}a^{2}}{1+a}.$

As $a=O(r^{2})$ and $|\mathbf{r}-1|\lesssim\frac{r^{2}}{|z-w|^{2}}$, we see
that only the first term $\frac{(\mathbf{r}-1)^{2}}{\mathbf{r}}$ will remain
in the limit. Undoing the symmetry trick, we thus have reduced the task to
finding the limit of $A\int_{U^{2}}dz\,dwu_{r}(z,w)$, where

$\displaystyle u_{r}(z,w)\coloneqq
r^{2\beta^{2}-10}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dv\frac{\mathbf{r}(x,y,u,v)-1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}.$

We note that by the above considerations $|u_{r}(z,w)|=O(r)$ when $|z-w|\geq
r^{1/4}$. We may therefore restrict our focus to the case where
$|z-w|<r^{1/4}$. Let us consider a fixed $z$ and do the integral over $w$. By
translation we may assume that $z=0$. We have by scaling

$\displaystyle\int_{Q(0,r^{1/4})}dwr^{2\beta^{2}-10}\int_{Q(0,r)^{2}}dx\,dy\int_{Q(w,r)^{2}}du\,dv\frac{\mathbf{r}(x,y,u,v)-1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}$
$\displaystyle=\int_{Q(0,r^{-3/4})}dw\int_{Q(0,1)^{2}}dx\,dy\int_{Q(w,1)^{2}}du\,dv\frac{\mathbf{r}(x,y,u,v)-1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}.$

As we explained in the paragraph following (7.1), the integral w.r.t. $w$
converges absolutely as $r\to 0$ due to symmetry properties. Therefore, the
right hand side of the above display converges as $r\to 0$ to

$\displaystyle\int_{\mathbb{R}^{2}}dw\int_{Q(0,1)^{2}}dx\,dy\int_{Q(w,1)^{2}}du\,dv\frac{\mathbf{r}(x,y,u,v)-1}{|x-y|^{\beta^{2}}|u-v|^{\beta^{2}}}.$

This integral agrees with the inverse of the normalising constant $A$; see
(7.1). The integral over $z$ then gives $|U|=st$ as required. ∎

###  7.3. The convergence does not hold in probability

######  Lemma 7.7.

The sequence $(B_{r}(1,1))_{r\in(0,1)}$ is not Cauchy in $L^{2}(\Omega)$ as
$r\to 0$.

###### Proof.

Let $0<r<s$. We have

$\displaystyle\mathbb{E}|B_{r}(1,1)-B_{s}(1,1)|^{2}=A\int_{([0,1]^{2})^{2}}dz\,dw\mathbb{E}[(W_{r}(z)-W_{s}(z))(W_{r}(w)-W_{s}(w))],$

and thus we see that it is enough to show that for fixed $s$ we have

$\lim_{r\to 0}\int_{([0,1]^{2})^{2}}dz\,dw\mathbb{E}[W_{r}(z)W_{s}(w)]=0.$

The expectation $\mathbb{E}[W_{r}(z)W_{s}(w)]$ is equal to

$\displaystyle
r^{\beta^{2}-5}s^{\beta^{2}-5}\int_{Q(z,r)^{2}}dx\,dy\int_{Q(w,s)^{2}}du\,dve^{\beta^{2}G^{\prime}(x,y)+\beta^{2}G^{\prime}(u,v)}\big{(}e^{\beta^{2}(G^{\prime}(x,v)+G^{\prime}(y,u)-G^{\prime}(x,u)-G^{\prime}(y,v))}-1\big{)}.$

We will consider separately the cases where $|z-w|<r^{1/4}$ and $|z-w|\geq
r^{1/4}$. In the first case we note that since
$G^{\prime}(x,v)-G^{\prime}(y,v)=O(r)$ and the same holds for
$G^{\prime}(y,u)-G^{\prime}(x,u)$, we have

$|\mathbb{E}[W_{r}(z)W_{s}(w)]|\lesssim s^{-1}.$

This gets multiplied by the measure of the set $|z-w|<r^{1/4}$, which is
$O(\sqrt{r})$, and we get the claim. In the second case we will use the bound
from Lemma 7.3 and the symmetry trick of swapping $x$ and $y$ to get

$|\mathbb{E}[W_{r}(z)W_{s}(w)]|\lesssim\frac{rs}{|z-w|^{4}},$

whose integral over the set $|z-w|\geq r^{1/4}$ is of the order $\sqrt{r}s$,
which again goes to $0$ as $r\to 0$. ∎

##  Appendix A A variant of Kolmogorov’s continuity theorem

The following lemma is a variant of Kolmogorov’s continuity theorem. We
provide a proof for the convenience of the reader.

######  Lemma A.1.

Let $\left(B(x,r),x\in D,0\leq r\leq 1\right)$ be a random process indexed by