Harmonic balls are domains which satisfy the mean-value property for harmonic functions.
We establish the existence and uniqueness of harmonic balls on Liouville quantum gravity (LQG) surfaces using the obstacle problem formulation of Hele-Shaw flow. We show that LQG harmonic balls are neither Lipschitz domains nor LQG metric balls, and that the boundaries of their complementary connected components are Jordan curves.

We conjecture that LQG harmonic balls are the scaling limit of internal diffusion limited aggregation (IDLA) on random planar maps.
In a companion paper, we prove this in the special case of mated-CRT maps.

§ INTRODUCTION

§.§ Overview

Simulations of $\gamma$-LQG harmonic balls with respect to the same GFF instance for $\gamma = 0.5, 1.0, 1.8$. The colors distinguish harmonic balls of different mass. From the figures, it appears that the complement of an LQG harmonic ball is not necessarily connected. We expect that this is the case for a `typical' LQG harmonic ball.

Let $\mu$ be a locally finite Radon measure on $\C$ . A harmonic ball for $\mu$ is an open set $\Lambda \subset \C$ containing the origin which satisfies the mean-value property

\begin{equation} \label{eq:harmonic-ball}
f(0) = \frac{1}{\mu(\Lambda)} \int_{\Lambda} f(z) \mu(d z),
\end{equation}

for all functions $f$ which are harmonic in a neighborhood of the closure of $\Lambda$.

§.§.§ Liouville quantum gravity

In this article, we will construct and study harmonic balls in the setting of Liouville quantum gravity (LQG). LQG is a canonical one-parameter family of random fractal surfaces which were introduced by Polyakov in the 1980s in the context of bosonic string theory <cit.>. One sense in which these surfaces are canonical is that they are known or conjectured to describe the scaling limits of various types of random planar maps (see Section <ref> for more details).

Heuristically, for $\gamma \in (0,2)$ and a domain $U\subset\C$, a $\gamma$-LQG surface parameterized by $U$ is the two-dimensional Riemannian manifold with Riemannian metric tensor $e^{\gamma h} \, (dx^2+dy^2)$, where $dx^2+dy^2$ is the Euclidean metric tensor and $h$ is a variant of the Gaussian free field (GFF) on $U$. This metric tensor does not make literal sense since $h$ is a random generalized function, not a true function. Nevertheless, it is still possible to define the associated volume form, a.k.a. the $\gamma$-Liouville measure. This is a random, locally finite Radon measure on $U$ which is informally given by

\begin{equation} \label{eq:lqg-measure}
\mu_h(dz) = e^{\gamma h(z)} \, dz  ,
\end{equation}

where $dz$ denotes Lebesgue measure.

The expression (<ref>) can be made rigorous as part of a general theory
of regularized random measures called Gaussian multiplicative chaos <cit.>. This theory shows that the Liouville measure is a well-defined (random) Radon measure which is measurable with
respect to $h$ (the converse is also true <cit.>). However, the Liouville measure is quite irregular: $\mu_h$ is supported on the `thick' points of the GFF, a dense fractal set of Hausdorff dimension $2- \frac{\gamma^2}{2}$, and hence is mutually singular with respect to Lebesgue measure <cit.>.

There is a vast literature on LQG: see <cit.> for introductory survey articles and <cit.> for a more detailed introduction. However, only minimal prior knowledge of this literature is needed to read this paper.
The necessary background will be reviewed in Section <ref>.

§.§.§ LQG harmonic balls

We will be interested in harmonic balls, as defined in (<ref>), in the case when $\mu$ is the $\gamma$-Liouville measure, $\mu_h$, for some $\gamma\in (0,2)$.
We call these domains $\gamma$-LQG harmonic balls.
One of our main motivations for studying LQG harmonic balls is that we expect them to be the scaling limits of internal diffusion limited aggregation (IDLA)  <cit.> on random planar maps.
See Section <ref> for further discussion.

We will construct $\gamma$-LQG harmonic balls via a certain partial differential equation involving $\mu_h$. In particular, this PDE, Hele-Shaw flow,
describes the movement of viscous fluid on a $\gamma$-LQG surface, see, , <cit.>.
The irregularity of $\mu_h$ precludes applying the classical theory of existence and uniqueness of harmonic balls <cit.> in the LQG setting.
Much of the existing technology requires $\mu$ to be bounded from above and below by a multiple of the Lebesgue measure — a constraint
too strict to be satisfied, even approximately, by $\mu_h$. Consequently, even the existence of $\gamma$-LQG harmonic balls is far from obvious.
Indeed, nontrivial harmonic balls do not exist in general — take, for instance,  $\mu$ to be Dirac measure supported away from the origin.

We address this difficulty by following a different path, inspired by arguments from discrete Laplacian growth <cit.>. Roughly speaking, we show that it is unlikely for Brownian motion to avoid regions of large $\mu_h$-mass
and then use this to show harmonic balls exist. This argument also leads to
geometric requirements harmonic balls satisfy. We later use these requirements to show that `typical' $\gamma$-LQG harmonic balls are neither Lipschitz domains nor LQG metric balls.

In contrast to the constructions of other objects associated with LQG, such as the LQG measure, the LQG metric, and Liouville Brownian motion, our construction of LQG harmonic balls does not use any approximation or regularization procedure. Rather, LQG harmonic balls are constructed directly as the solutions of an optimization problem involving the LQG measure (see Section <ref>).

An LQG surface is a certain type of random fractal (albeit not one defined as a subset of $\mathbb R^d$ for some $d$). Analysis on fractals is a well-studied topic, see, , <cit.>.
One program of research in this area is to construct the Laplacian on a fractal and then use this to develop a theory of elliptic PDE on the fractal.
As discussed further in Section <ref>, Brownian motion, and hence the Laplacian, has been constructed on LQG surfaces <cit.>.
The present paper may be thought of as an initial step in the study of PDE on LQG surfaces.

§.§ Statement of the main result

Fix the LQG parameter $\gamma\in (0,2)$. Let $h$ be a whole-plane Gaussian free field, or more generally a whole-plane Gaussian free field plus the function $-\boldsymbol{\alpha}_0 \log|\cdot|$, where $\boldsymbol{\alpha}_0 < Q := 2/\gamma+\gamma/2$. Let $\mu_h$ be the $\gamma$-LQG area measure associated with $h$.
The precise definitions of $h$ and $\mu_h$ will be reviewed in Section <ref>.
For now, the unfamiliar reader can think of $\mu_h$ as a random, non-atomic, locally finite Borel measure on $\C$ which assigns positive
mass to every open subset of $\C$.

Our main result concerns existence and uniqueness of a family of $\gamma$-LQG harmonic balls. To prove uniqueness, we enlarge the class of harmonic functions in the definition to include those of the form,

\begin{equation} \label{eq:harmonic-fns}
\tilde H_O(D) = \{ \int_O G_O(\cdot,y) \,\nu(dy) : \mbox{$\nu$ is a signed Radon measure with support in $O \backslash D$}\},
\end{equation}

where $O, D \subset \C$ are bounded open sets and $G_O$ is the Green's function for Brownian motion in the domain $O$
(defined in Section <ref> below).
That is, $\Lambda$ is a harmonic ball if (<ref>) is
satisfied for all functions $f = f_1 + f_2$ where $f_1$ is harmonic in a neighborhood of the closure of $\Lambda$
and $f_2 \in \tilde H_O(\Lambda)$ for some $O \Supset \Lambda$.

We are now ready to state our main existence and uniqueness result.

On an event of probability one, there exists a unique family of harmonic balls $\{\Lambda_t\}_{t > 0}$
satisfying the following properties:

* For each $t > 0$, $\mu_h(\Lambda_t) = t$, $\mu_h(\partial \Lambda_t) = 0$, and $\Lambda_t$ is equal to the interior of its closure.

* The domains $\Lambda_t$ are bounded, connected, contain the origin, increase continuously in $t$ (in the Hausdorff topology), and satisfy $\cap_{t > 0} \Lambda_t = \{0\}$.

In some literature on harmonic balls, , <cit.> or <cit.>, uniqueness is only proven up to sets of zero mass. In our setting, we get exact uniqueness thanks to the requirement that $\Lambda_t$ is equal to the interior of its closure.

We also show that typical harmonic balls are `novel'; that is, they are
neither Euclidean balls nor LQG metric balls. We also show that the boundaries of their complementary connected components are Jordan curves. Compare Figures <ref> and <ref>.
The precise definition of the LQG metric ball will be given in Section <ref>; for now the reader may think of
it as the natural notion of metric ball on an LQG surface.

The following is true on an event of probability 1. For Lebesgue-a.e. t, $\Lambda_t$, constructed in Theorem <ref>,
is neither a Lipschitz domain nor an LQG-metric ball. Moreover, for each $t>0$ the boundaries of the connected components of $\C\setminus \overline{\Lambda}_t$ are Jordan curves.

We remark that the harmonic balls given by Theorem <ref> are locally determined by $\mu_h$, in the sense of the following statement.

For each fixed $t\geq 0$, the closed LQG harmonic ball $\overline{\Lambda}_t$ is a local set for $h$ in the sense of <cit.>, , for each deterministic open set $U\subset\C$, the event $\{\overline{\Lambda}_t \subset U\}$ is measurable with respect to $\sigma(h|_U)$.

Simulations of $\gamma$-LQG metric balls with respect to the same GFF instance as Figure <ref> for $\gamma = 0.5, 1.0, 1.8$.

§.§ Background

We provide some context and motivation for the study of harmonic balls on LQG surfaces.

§.§.§ Harmonic balls

The term `harmonic ball' was coined by Shahgholian-Sjödin in <cit.>
and is a special case of quadrature domains for harmonic functions.
A quadrature domain is a subset of $\mathbb{C}$ for which the integral of a harmonic function can be expressed
as a sum of simpler functionals (such as point evaluations).
Quadrature domains have a long history and are closely related to classical balayage (sweeping) <cit.>, Hele-Shaw
flow <cit.>, obstacle problems <cit.>, and Laplacian growth <cit.>.
We direct the interested reader to <cit.>
and the introductions of the theses of Roos <cit.> and Sjödin <cit.> for
excellent expositions.

Of particular relevance to our work are the papers of Hedenmalm-Shimorin <cit.> and Gustafsson-Roos <cit.> which construct and analyze harmonic balls on Riemannian manifolds. Hedenmalm-Shimorin, building upon the work of Sakai <cit.>, show that harmonic balls on sufficiently smooth hyperbolic surfaces have boundaries which are the unions of a finite number of real analytic simple curves.
Gustafsson-Roos show that harmonic balls and geodesic balls coincide on Riemannian surfaces
if and only if the Gaussian curvature of the manifold is constant. We emphasize that while some of the
basic constructions in these works may be adapted to our setting, an LQG surface is not a Riemannian
manifold in the literal sense and so the results do not apply.

§.§.§ Internal DLA

One of our main motivations for studying LQG harmonic balls stems from a connection with internal diffusion limited aggregation (IDLA) on random planar maps.
IDLA was introduced as a toy model for chemical corrosion in <cit.> and is a special case of a growth model studied by Diaconis-Fulton in <cit.>. IDLA is a random aggregation model defined as follows: start with $n$ walkers at the origin in $\Z^2$ and let each walker evolve according to a simple random walk until it reach a site in $\Z^2$ not occupied by any previous walker. This rule
generates a growing sequence of sets $A_n \subset \Z^2$ indexed by the number of walkers $n \in \N$.

In a foundational work, Lawler-Bramson-Griffeath proved that $A_n$, suitably rescaled, converges to a Euclidean ball in $\R^2$
as $n$ goes to infinity <cit.>. Later, Asselah-Gaudillère <cit.> and independently Jerison-Levine-Sheffield established logarithmic fluctuations of $A_n$ around its limit <cit.>.

Implicit in the proof of Lawler-Bramson-Griffeath is that harmonic balls are Euclidean balls when $\mu$ is the Lebesgue measure —
a simple proof of this was given by Ülkü Kuran in 1972 <cit.>. Interestingly, the connection between IDLA and quadrature domains generalizes. Levine-Peres showed in <cit.> that the scaling limit of IDLA for any initial condition (for example multiple point sources) is given by a corresponding quadrature domain.

IDLA has also been studied on several other graphs including: Cayley graphs of groups with polynomial <cit.> and  exponential <cit.> growth, supercritical percolation clusters <cit.>, Sierpinski gasket graphs <cit.>, and cylinders <cit.>— see <cit.> for a thorough survey of IDLA. In each of these cases the limit shape is either a Euclidean ball or a metric ball. On the other hand, Asselah-Rahmani showed that IDLA on the comb lattice has a limit shape which is neither a Euclidean ball nor a metric ball but rather a domain which satisfies a certain mean-value property <cit.>.
In a similar vein, Lucas has shown that IDLA with biased random walkers on $\Z^d$ converges under parabolic scaling to a domain which satisfies the mean-value property for caloric functions <cit.>.

§.§.§ Random planar maps

A planar map is a graph embedded in $\C$ in such a way that no two edges cross, viewed modulo orientation preserving homeomorphisms $\C \rightarrow \C$.
Various types of random planar maps are expected, and in some cases proven, to converge to $\gamma$-LQG surfaces. For example, uniform random planar maps (including uniform triangulations, quadrangulations, etc.) converge to $\sqrt{8/3}$-LQG surfaces in the Gromov-Hausdorff sense <cit.> and, at least in the case of triangulations, when embedded into $\C$ via the so-called Cardy embedding <cit.>. Similar convergence results are expected to hold for various types of non-uniform random planar maps toward $\gamma$-LQG with $\gamma\not=\sqrt{8/3}$. For example, random planar maps sampled with probability proportional to the number of spanning trees they admit are expected to converge to $\sqrt 2$-LQG. We refer to <cit.> for a survey of work relating random planar maps and LQG.

The aforementioned results on IDLA suggest that the scaling limits of IDLA on random planar maps are described by harmonic balls on LQG surfaces.
Random walks on (reasonably embedded) random planar maps are also expected to converge to (time changes of) Brownian motions on LQG surfaces
— this has recently been proven for a one-parameter family of random planar maps called mated-CRT maps in <cit.>.
In a companion work we verify that the scaling limit of IDLA on mated-CRT maps is given by LQG harmonic balls. It is still an open problem to prove this for other random planar map models, e.g., uniform random planar maps (see Problem <ref>).

§.§ Open problems

We collect some questions suggested by this work. The first has been mentioned previously
and is arguably the most important question here.

Show that the scaling limit of IDLA on random planar maps in the appropriate $\gamma$-LQG universality class, other than mated-CRT maps, is described by $\gamma$-LQG harmonic balls.
For example, on a uniform planar map show that the scaling limit of IDLA is a $\sqrt{8/3}$-LQG harmonic ball.

Possible topologies of convergence in Problem <ref> include a version of the Gromov-Hausdorff distance for metric spaces decorated by compact sets; or convergence of the IDLA clusters w.r.t. the Hausdorff distance when the random planar map is embedded into $\C$ appropriately.
There are also some purely continuum directions one could pursue — the following is an example.

Compute the Hausdorff dimension of the boundary of an LQG harmonic ball, with respect to the Euclidean metric and with respect to the LQG metric.

We expect that the Euclidean and LQG dimensions of the harmonic ball boundary are each strictly greater than one. We note that the Hausdorff dimensions of the boundary of an LQG metric ball with respect to the Euclidean and LQG metrics have been computed in <cit.>.

It is also of interest to determine the analogue of LQG metric geodesics in
the setting of harmonic balls. In particular, we are interested in extending the theory of `Hele-Shaw geodesics', in the sense of
<cit.>, to our setting. As mentioned previously, Hedenmalm-Shimorin in <cit.> investigated harmonic balls on smooth Riemannian surfaces
and showed that their boundaries are piece-wise smooth curves. Because of this smoothness, they were able to define
Hele-Shaw geodesics as a family of curves originating from a fixed point which are orthogonal to the boundary of a harmonic ball at any point.
One may think of these geodesics as describing the trajectory of a single fluid particle started at a fixed point on a Riemannian surface.
As LQG harmonic balls do not have smooth boundaries, it is unclear how to adapt this to our setting, but
a weaker version of these objects may exist.

Construct and analyze the analogue of `Hele-Shaw geodesics' <cit.> on LQG surfaces.

A helpful intermediate step would be to show some additional regularity of harmonic balls. For example, the harmonic balls
we construct are monotone in $t$ but we are unable to show strict monotonicity in $t$.

Prove or disprove that the family of harmonic balls $\{\Lambda_t\}_{t>0}$, given by Theorem <ref>,  is strictly monotone in $t$, , $\overline{\Lambda}_s \subset \Lambda_t$ whenever $s < t$.

§.§ Paper and proof outline

We start in Section <ref> by reviewing the definition of the GFF, the definition of LQG, and some results about Liouville Brownian motion.  We then introduce the fundamental obstacle problem  which we use to construct candidate harmonic balls,  clusters, in Section <ref>
and establish some basic properties of the constructed clusters in Section <ref>.
The results of Sections <ref> and <ref> are straightforward extensions of those appearing in the obstacle problem literature — the only property of the LQG measure which is used there is that it is a Radon measure with certain volume growth bounds, Lemma <ref> below.

Several of the results of Sections <ref> and <ref> (including the statement that the clusters are harmonic balls, see Section <ref>) are conditional on two properties which are not proven in these sections: namely, that the clusters do not attain a large Euclidean diameter in an arbitrarily small amount of time and that the boundaries of the clusters have zero LQG mass. Establishing these properties is the main goal of Sections <ref> through <ref>.

In Section <ref> we outline a strategy for showing that the clusters are harmonic balls; that is, they grow continuously and their boundaries have zero LQG mass.
Our approach for verifying these properties is completely new and relies on a novel Harnack-type estimate, Proposition <ref>,
which clusters must satisfy. In particular, this estimate forces clusters to have `no thin-tentacles', as in <cit.>. Roughly speaking, our Harnack-type estimate says that there is a constant $\alpha > 0$ such that if $A \subset \C$ is an annulus on which the LQG measure $\mu_h$ is reasonably well-behaved, then if the cluster $\Lambda_t$ crosses between the inner and outer boundaries of $A$ we must have $\mu_h(A \cap \Lambda_t) \geq \alpha \mu_h(A)$.
Our proof of the Harnack-type estimate in Section <ref> combines potential theoretic techniques
with methods from LQG theory. See the beginning of Section <ref> for an outline of the argument.

In Sections <ref> and <ref>, respectively, we use the Harnack-type estimate to prove that the clusters grow continuously in time, Proposition <ref>, and that boundaries of the clusters have measure zero,
Proposition <ref>. As demonstrated in Theorem <ref>, these properties are enough to ensure that
clusters are LQG harmonic balls.

Having constructed LQG harmonic balls, we show, by adapting ideas from the obstacle problem literature <cit.>, their uniqueness,
Proposition <ref>, in Section <ref>.

The Harnack-type estimate imposes strong geometric constraints on LQG harmonic balls. For instance,
it disallows LQG harmonic balls from `crossing' annuli too many times, Lemma <ref>. We use this in Section <ref> to show
that the boundaries of complementary connected components of LQG harmonic balls are Jordan curves, Proposition <ref>.

These geometric constraints may also be translated into a strong relationship between LQG harmonic balls and the underlying LQG area measure,
Lemma <ref>.
Since the LQG measure is quite variable, this imposes an irregularity on LQG harmonic balls. We use this to show
that LQG harmonic balls do not satisfy the cone condition, Lemma <ref>, in Section <ref>.
Consequently, LQG harmonic balls cannot be Lipschitz domains.

Another feature of the Harnack-type estimate is that it precludes LQG harmonic balls from having `approximate pinch points' which have small Euclidean diameter but which come close to disconnecting sets of large LQG mass from the origin within the cluster. On the other hand, an LQG metric ball has such approximate pinch points, as we show in Section <ref>.
This shows that LQG harmonic balls are not LQG metric balls.
A key technical input in the proof is Proposition <ref>, which shows that a region in the plane can have small LQG diameter but large LQG mass simultaneously with positive probability.

§.§ Notation and conventions

* Inequalities/equalities between functions/scalars are interpreted pointwise.

* Differential inequalities/equalities are interpreted in the distributional sense.

* For a set $A \subset \C$, $\partial A$ denotes its topological boundary, $\overline{A} = A \cup \partial A$ its closure,
and $\mathrm{int}(A)$ its interior.

* For two sets $A, B \subset \C$, say that $A \Subset B$ if $\overline{A} \subset B$.

* $B_r(x)$ denotes the open ball of Euclidean radius $r > 0$ centered at $x \in \C$, when $x$ is omitted, the ball is centered at 0.

* For a set $A \subset \C$, we denote the $r$-neighborhood of $A$ by $B_r(A) = A + B_r$.

* For $0 < r_1 < r_2$, denote an open annulus centered at $z$ by

\begin{equation} \label{eq:annulus}
\A_{r_1, r_2}(z) = B_{r_2}(z) \backslash \overline{B}_{r_1}(z)
\end{equation}

and $\A_{r_1, r_2} := \A_{r_1,r_2}(0)$.

* Let $\{E^{r}\}_{z > 0}$ be a one-parameter family of events.
We say that $E^r$ occurs with polynomially high probability as $r \to 0$
if there exists $p>0$ such that $\P[E^r] \geq 1 - O(r^p)$.

* For two sets $A, B \subset \C$, $\dist(A,B) = \inf_{a \in A, b \in b} \dist(a,b)$ where $\dist$ denotes the Euclidean distance between two points.

§.§ Acknowledgments

A.B. thanks Charlie Smart and Bill Cooperman for useful discussions.
A.B. was partially supported by NSF grant DMS-2202940 and a Stevanovich fellowship.
E.G. was partially supported by a Clay research fellowship.

§ PRELIMINARIES

In this section we review the definitions and basic properties of the Gaussian Free Field (GFF),
the Liouville Quantum Gravity (LQG) area measure, and the LQG metric.  We present just enough exposition for the purposes of this paper;
the book <cit.> and surveys <cit.> provide more details.

§.§ Gaussian free field

The whole-plane GFF $h^{\C}$ is the centered Gaussian random generalized
function on $\C$ with covariances

\begin{equation} \label{eq:cov-kernel}
\Cov(h^\C(z), h^\C(w)) := \log \frac{\max(|z|,1) \max(|w|,1)}{|z-w|} ,\quad\forall z,w\in\C .
\end{equation}

The GFF $h^{\C}$ is not well-defined pointwise since the covariance kernel in (<ref>) diverges to $\infty$ as $z\to w$.
Nevertheless, for $z\in\C$ and $r>0$, one can define the average of $h^{\C}$ over the circle of radius $r$ centered at $z$, which we denote by $h^{\C}_r(z)$ <cit.>.

The whole plane GFF is usually defined modulo additive constant. Our choice of covariance in (<ref>) corresponds to fixing this additive constant so that $h^\C_1(0) = 0$ (see, e.g., <cit.>).

The law of the whole-plane GFF, viewed modulo additive constant, is invariant under complex affine transformations of $\C$. This translates into the following
invariance property for $h^{\C}$,

\begin{equation} \label{eq:gff-scaling}
h^{\C} \overset{d}{=} h^{\C}(a\cdot +b) - h^{\C}_{|a|}(b), \quad \forall a \in \C \backslash \{0\}, \quad \forall b \in \C.
\end{equation}

Fix $\gamma \in (0,2)$ and $\boldsymbol{\alpha}_0 \in (-\infty, Q)$, where

\begin{equation} \label{eq:Q}
Q := \frac{2}{\gamma} + \frac{\gamma}{2}.
\end{equation}

Throughout this paper we take $h$ to be the whole-plane GFF with an $\boldsymbol{\alpha}_0$ log singularity at the origin.

Specifically, let $h^{\C}$ denote the whole-plane GFF normalized so that its circle average over the unit disk is
zero and set

\begin{equation} \label{eq:gff}
h = h^{\C} - \boldsymbol{\alpha}_0 \log |\cdot|.
\end{equation}

It is immediate from (<ref>) that

\begin{equation} \label{eq:h-coordinate-change}
h \overset{d}{=} h(a\cdot) - h_{|a|}, \quad \forall a \in \C \backslash \{0\}.
\end{equation}

§.§ Liouville Quantum Gravity

Let $\mu_h$ denote the $\gamma$-LQG area (Liouville) measure associated to $h$. One of the (many) possible ways of defining $\mu_h$ is as the a.s. weak limit

\begin{equation} \label{eq:measure-lim}
\mu_h = \lim_{\epsilon \rightarrow 0} \epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)} \,dz
\end{equation}

where $dz$ denotes Lebesgue measure and $h_\epsilon(z)$ is the circle average <cit.>.
In fact, the measure $\mu_{\tilde h}$ exists for any random generalized function $\tilde h$ of the form  $h + f$ where $f$ is a possibly random continuous function.
[LQG measure]
The LQG area measure $\mu_h$ satisfies the following properties.

* Radon measure. A.s., $\mu_h$ is a non-atomic Radon measure.

* Locality. For every deterministic open set $U \subset \C$,
$\mu_h(U)$ is given by a measurable function of
$h \vert_{U}$.

* Weyl scaling. A.s., $e^{\gamma f} \cdot \mu_h = \mu_{h + f}$
for every continuous function $f: \C \to \R$.

* Conformal covariance. A.s., the following is true. Let $U , \tilde U \subset \C$ be open and let $\phi$ be a conformal
map from $\tilde U$ to $U$. Then, with $Q$ as in (<ref>),

\begin{equation} \label{eq:measure-coord}
\mu_{h \circ \phi + Q \log |\phi'|}(A) = \mu_h(\phi(A)) \quad \mbox{for all Borel measurable $A \subset \tilde U$}.
\end{equation}

The first three properties in Fact <ref> are immediate from the definition (<ref>). The conformal covariance property was proven to hold a.s. for a fixed conformal map in <cit.> and extended to all conformal maps simultaneously in <cit.>.

It was shown in <cit.> that one can define also the LQG metric $D_h$,
which is the limit of regularized versions of the Riemannian distance function associated with the Riemannian metric tensor $e^{\gamma h}(d x^2 + dy^2)$.
Like the LQG measure, the LQG metric is a fractal-type object. It induces the same topology on $\C$ as the Euclidean metric, but the Hausdorff dimension of the metric space $(\C , D_h)$ is a.s. given by a deterministic number $d_\gamma > 2$ <cit.>. The value of $d_\gamma$ is not known explicitly except that $d_{\sqrt{8/3}}=4$.

In order to state an analog of Fact <ref> for the LQG metric, we make the following definitions. For a Euclidean-continuous path $P$ in $\C$, we write $\mathrm{len}(P; D_h)$ for its length with respect to $D_h$.
For an open set $U \subset \C$, the internal metric of $D_h$ on $U$
is defined by

\begin{equation} \label{eq:internal-metric}
D_h(z,w; U) = \inf \{ \mathrm{len}(P; D_h) : \mbox{$P$ is a path from $z$ to $w$ in $U$}\}, \quad \mbox{$\forall z,w, \in U$}.
\end{equation}

As in the case of the measure, the metric $D_{\tilde h}$ exists whenever $\tilde h = h + f$, where $f$ is a possibly random continuous function.

[LQG metric]
The LQG metric $D_h$ has the following properties.

* Euclidean topology and length metric. A.s., $D_h$ induces the same topology on $\C$ as the Euclidean metric and is a length metric, that is, $D_h(z,w)$ is the infimum of the $D_h$-length of paths from $z$ to $w$.

* Locality. For every deterministic open set $U \subset \C$, the $D_h$-internal metric on $U$ is given by a measurable function of
$h \vert_{U}$.

* Weyl scaling. Let

\begin{equation} \label{eq:xi}
\xi = \frac{\gamma}{d_{\gamma}} ,
\end{equation}

where $d_\gamma$ is the Hausdorff dimension of the $\gamma$-LQG metric as above.
Almost surely, for every continuous function $f: \C \to \R$,

\[
D_{h + f}(u, v) = \inf_{P: u \to v} \int_0^{\mathrm{len}(P; D_h)} e^{\xi f(P(t))} d t, \quad \forall u,v \in \C.
\]

* Coordinate change for scaling and translation. Let $r > 0$ and $z \in C$. Almost surely, with $Q$ as in (<ref>),

\[
D_h(r u + z, r v + z) = D_{h(r\cdot +z) + Q \log r}(u,v), \quad \forall u,v, \in C.
\]

The properties listed in Fact <ref> were verified for the LQG metric in <cit.>. In fact, it is shown in <cit.> that these properties uniquely characterize $D_h$.

In what follows, for sets $A,B\subset \C$, we write

\begin{equation} \label{eq:set-dist}
D_h(A,B) = \inf_{x\in A, y\in B} D_h(x,y) .
\end{equation}

For disjoint compact sets $K_1,K_2\subset \C$, a $D_h$-geodesic from $K_1$ to $K_2$ is a path from $K_1$ to $K_2$ of minimal $D_h$-length. It is easily seen from the length metric property and a compactness argument that $D_h$-geodesics always exist (see, e.g., <cit.>).

§.§ Green's function

Let $G_{O}:\overline{O} \times \overline{O} \to \R \cup \{\infty\}$ denote the Green's function for standard Brownian motion killed upon exiting a bounded open set $O \subset \C$.

We make use of the following standard properties of the Green's function of a (sufficiently nice) set.

The Green's function of a ball, $B_R$ of radius $R > 0$, has the following properties for every $x \in B_R$.

* Fundamental solution: $\Delta G_{B_R}(x,\cdot) = -\delta_x(\cdot)$ on $B_R$.

* Positive: $G_{B_R}(x, \cdot) > 0$ on $B_R$.

* Zero boundary: $G_{B_R}(x, \cdot) = 0$ on $\partial B_R$.

* Smooth away from the pole: $G_{B_R}(x, \cdot)$ is infinitely differentiable away from $x$.

§.§ Liouville Potential Theory

In this section we collect well known potential theoretic estimates on the LQG measure.
We first note bounds on the LQG mass of annuli and balls.

For each $\beta^+ \in (0, (2 -\gamma)^2/2)$ and $\beta^- > (2+\gamma)^2/2$
it holds with polynomially high probability as $\epsilon \to 0$ that

\begin{equation} \label{eq:ball-volume}
\epsilon^{\beta^-} \leq \mu_h(B_\epsilon(z)) \leq \epsilon^{\beta^+} ,\quad\forall z \in B_1 .
\end{equation}

Furthermore, for each $0 < r_1 < r_2$ there exists constants $C_1, C_2$ so that for all $z \in B_1$

\begin{equation} \label{eq:ann-volume}
C_2 \epsilon^{\beta^-} \leq \mu_h(\A_{r_1 \epsilon, r_2 \epsilon}(z)) \leq C_1 \epsilon^{\beta^+}
\end{equation}

with polynomially high probability as $\epsilon \to 0$, where here we use the notation for annuli from (<ref>).

Exactly the same argument as in <cit.> shows that (<ref>) holds with polynomially high probability as $\epsilon \to 0$. The estimate (<ref>) follows from (<ref>) and the fact that

\[
B_{\frac{r_2-r_1}{4} \epsilon}(z + \frac{r_2\epsilon+r_1\epsilon}{2}  e_1) \subset \A_{r_1 \epsilon, r_2 \epsilon}(z) \subset B_{r_2 \epsilon}(z)
\]

where $e_1 = (1,0)$.

Liouville Brownian motion (LBM) is the natural diffusion associated with $\gamma$-LQG. Roughly speaking, LBM is obtained from ordinary Brownian motion (sampled independently from $h$) by changing time so that the process has `constant $\gamma$-LQG speed'. LBM was constructed in <cit.>. It was shown in <cit.> to describe the scaling limit of random walk on a certain family of random planar maps.

The volume growth bounds given by Lemma <ref> lead to control
on the expected exit time of LBM from balls.

Let $O$ denote a smooth bounded open set.
The expected exit time of LBM from $O$ started at $x \in O$
is finite and Hölder continuous in $x$.  More generally, any $q$ of the form,

\[
q(x) = \int_{O} G_{O}(x, y) f (y) d \mu_h(y) \qquad \mbox{for some $f \in L^{\infty}(\overline O)$}
\]

is finite and Hölder continuous in $\overline{O}$.

Finiteness follows immediately from Lemma <ref>.
Hölder continuity uses the embedding of Campanato spaces into Hölder spaces together with Lemma <ref>.
See, for example, Section 16.2 (or the remark after Proposition 13.5) in <cit.>.

The bounds also lead to continuity of the LBM heat kernel using the main result of <cit.>.
Continuity of the LBM heat kernel (for other versions of the GFF) was previously established by <cit.>	and <cit.>.

Let $K$ be a square in $\C$ and for $x \in \C$ let $\{ \mathcal{B}_t^x\}_{t > 0}$ denote $\gamma$-LBM with respect to the field $h$ started from $x$ with Neumann (reflecting) boundary conditions on $K$. The heat kernel $p^K_t$ of reflected LBM in $K$ is the
function $p^K_t(x,y) : (0,\infty) \times K \times K \to [0,\infty)$ such that

\begin{equation} \label{eq:defining-property-of-kernel}
\P[ \mathcal{B}_t^x \in dy | h] = p^K_t(x,y) dy.
\end{equation}

Let $K$ be a square in $\C$. Almost surely, the heat kernel $p^K_t(x,y)$ associated to $\gamma$-LBM with Neumann boundary conditions on $K$ exists, is finite, jointly continuous, and strictly positive for all $(t,x,y) \in (0,\infty) \times K \times K$.

This is <cit.>   with input given by Lemma <ref>.
Strictly speaking, <cit.>   concerns the transition density of reflecting $\gamma$-LBM in the unit square. A scaling argument shows that <cit.> applies
to the transition density of reflecting $\gamma$-LBM in any fixed square.

§ CONSTRUCTION OF HARMONIC BALLS VIA HELE-SHAW FLOW

In this section we construct domains which we will later show are $\gamma$-LQG harmonic balls.
Specifically, we construct a family of sets $\{{\cluster{}{t}}\}_{t > 0}$ via an obstacle problem involving the Green's function for the ball.
This family describes the (restricted) flow of a viscous fluid injected onto an LQG surface.
For the precise relationship between the obstacle problem and this flow, Hele-Shaw flow, see, for example, Chapter 3 of <cit.>. Our construction is a standard one, see, , <cit.> and originates from the work of Sakai <cit.>.

In subsequent sections we show the existence of $T > 0$ so that $\{{\cluster{}{t}}\}_{0 < t < T}$ are a family of harmonic balls satisfying the conditions in Theorem
<ref>. We then use scale invariance and compatibility to extend this construction to all $t > 0$.

§.§ Definition of the obstacle problem

We construct candidate harmonic balls via a technique similar to the Perron method involving the measure $\mu_h$ and the Green's function for the ball.
For each $t > 0$ and ball $B_r$ the set of supersolutions

\begin{equation} \label{eq:super-solutions}
\mclS{B_r}{t}  = \{ w \in C(\overline{B_r}) :  \Delta w \leq \mu_h \mbox{ in $B_r$}  \mbox{ and }  w \geq  -t G_{B_r}(0, \cdot) \mbox{ in $\overline{B_r}$}\},
\end{equation}

where $C(\overline{B_r})$ denotes the set of continuous functions on the closed ball.
The least supersolution is defined as the pointwise infimum of all functions in $\mclS{B_r}{t}$,

\begin{equation} \label{eq:least-super-solution}
\lss{B_r}{t} = \inf\{ w \in \mclS{B_r}{t} \}
\end{equation}

and the cluster as

\begin{equation} \label{eq:non-co-set}
\cluster{B_r}{t} = \{ x \in B_r : \lss{B_r}{t}(x) > -t G_{B_r}(0, x) \}.
\end{equation}

We also consider the odometer

\begin{equation} \label{eq:limit-odometer}
\odometer{B_r}{t} = \lss{B_r}{t}  + t G_{B_r}(0, \cdot).
\end{equation}

Note that $\mclS{B_r}{t}$ is non-empty as it contains the zero function — thus $\lss{B_r}{t}$ always exists.
This equation in (<ref>) is known as an obstacle problem with obstacle given by the Green's function.
When $B_r$ is the unit ball (which for most of the paper will be the case), the distinguishing superscripts are omitted.

We think of the above obstacle problem as modeling the flow of liquid on a rough surface.
A mass $t$ of fluid is injected into the origin and its growth is dictated by the infinitesimal capacity of the surface, namely, the measure $\mu_h$. The cluster ${\cluster{}{t}}$ represents the settled fluid and $\odometer{}{t}$ captures the `work' needed to spread the fluid.

From this physical picture, one expects that if $t$ is larger than $\mu_h(B_r)$, then ${\cluster{}{t}}$ should fill the entire ball.
Moreover, if $\mu_h$ is regular enough, then for $t$ small the clusters should be strictly contained in
$B_r$. Further, clusters with closures which do not intersect their boundaries should be compatible with clusters on smaller domains.  We provide rigorous statements of these heuristics below.

§.§ Basic properties of the obstacle problem

We assert existence and basic regularity of solutions to the obstacle problem. These results are standard but for completeness
are proved in Appendix <ref>.
Each of the statements below made for $B_1$ also hold for each fixed $B_r$ for any $r > 0$.

We first note that the least supersolution is indeed a supersolution.

On an event of probability 1, for all $t > 0$, $\lss{}{t}$ is finite, continuous, and an element of $\mclS{}{t}$.

The next lemma is a consequence of being the least supersolution.

On an event of probability 1, for all $t > 0$,  the cluster ${\cluster{}{t}}$ is open and connected and

\[
\Delta \lss{}{t} = \mu_h \vert_{{{\cluster{}{t}}}} + \nu \vert_{\partial {\cluster{}{t}}} \quad \mbox{on $B_1$},
\]

where $\nu$ is a Radon measure which is absolutely continuous with respect to $\mu_h$ and satisfies $0 \leq \nu \leq \mu_h$.
In particular,

\[
\Delta \odometer{}{t} = -t \delta_0 + \mu_h \vert_{{{\cluster{}{t}}}} + \nu \vert_{\partial {\cluster{}{t}}} \quad \mbox{on $B_1$}.
\]

We will eventually show that a.s. $\mu_h(\partial {\cluster{}{t}}) = 0$ for each $t > 0$, which implies that $\nu = 0$. However, for the time being we need to allow for the possibility that there is some mass on $\partial {\cluster{}{t}}$.

We also have monotonicity of the clusters in $t$.

On an event of probability 1, for all $t_1 \leq t_2$, $\cluster{}{t_1} \subseteq \cluster{}{t_2}$.

Clusters also have a conservation of mass property.

On an event of probability 1, for all $t > 0$, $\mu_h({\cluster{}{t}}) \leq t$ and $\odometer{}{t} = 0$ on $\partial B_1$.  Moreover, if $\overline{{{\cluster{}{t}}}} \subset B_1$ and $\mu(\partial {\cluster{}{t}}) = 0$, then $\mu_h({\cluster{}{t}}) = t$.

We conclude with a compatibility result for ${\cluster{}{t}}$ across different domains.

The following holds for each $R > 0$ on an event of probability 1.
For all $s_1 \leq R$, if, for some $s_2 \in [s_1, R]$, we have $\overline{\cluster{B_{s_2}}{t}} \subset B_{s_1}$,
then  $\cluster{B_{s}}{t} = \cluster{B_{s_2}}{t}$ for all $s \in [s_1, R]$.

§ BASIC PROPERTIES OF CLUSTERS

In this section we note some basic properties of the clusters $\{{\cluster{}{t}}\}_{t > 0}$  and
odometers $\{\odometer{}{t}\}_{t > 0}$. As in Section <ref>, these results are fairly standard, , <cit.>, but (short) proofs are included for completeness.

§.§ Lower bound

We first show that each ${\cluster{}{t}}$ contains a Euclidean ball of sufficiently small radius and eventually the family
coincides with the unit ball.

On an event of probability 1, for each $t > 0$ there exists a random $\epsilon = \epsilon(t) > 0$ so that

\begin{equation}
B_{\epsilon} \subset {\cluster{}{t}}.
\end{equation}

Moreover, for each $\delta \in (0,1)$, there exists a random $t(\delta)>0$ such that for all $t \geq t(\delta)$,

\begin{equation}
B_{1-\delta} \subset  {\cluster{}{t}}
\end{equation}

and there exists a random $t^+ > 0$ so that for all $t \geq t^+$

\begin{equation}
\overline{{{\cluster{}{t}}}} \cap \partial B_1 \neq \emptyset.
\end{equation}

Our proof uses the fact that the logarithm function blows up near the origin together with
the finiteness of the expected exit time of Liouville Brownian Motion from a ball.

Let $q_1$ denote the expected exit time of Liouville Brownian Motion from the unit ball,

\[
\begin{cases}
\Delta q_1 = -\mu_h \quad &\mbox{in $B_1$} \\
q_1 = 0 \quad &\mbox{on $\partial B_1$}.
\end{cases}
\]

Let $t > 0$ be given.
As $\Delta \lss{}{t} \leq \mu_h$ (Lemma <ref>), $\lss{}{t} + q_{1}$ is superharmonic in $B_1$. Hence, as $\lss{}{t} + q_1 \geq 0$ on $\partial B_1$, we have $\lss{}{t} \geq -q_{1} $
in $\overline{B_1}$. Since

\[
\lim_{\epsilon \to 0} \sup_{x \in B_{\epsilon}} t \log |x| \to -\infty,
\]

we have that

\begin{equation} \label{eq:lower-bound-green}
\lss{}{t}(x) \geq  -q_{1}(x) > t \log |x|, \quad \forall x \in B_{\epsilon}, \quad \forall \epsilon > 0 \mbox{ sufficiently small}.
\end{equation}

Indeed, by Proposition <ref>, $q_{1}$ is finite. By the definition (<ref>) of ${\cluster{}{t}}$ and the fact that $G_{B_1}(0,x) = -\log|x|$, this shows that $B_{\epsilon} \subset {\cluster{}{t}}$. 	Similarly, for each $\delta \in (0,1)$, for all $t > t(\delta)$, (<ref>)
is satisfied for all $x \in B_{1-\delta}$. 	The last assertion follows by choosing $t^+ = \mu_h(B_1)$
and using Lemma <ref>.

§.§ Hölder continuity of the odometer

We observe that $\lss{}{t}$ is Hölder for a deterministic exponent depending only on $\gamma$.

There exists a deterministic exponent $\alpha = \alpha(\gamma)$ so that on an event of probability 1, there
exists a constant $C > 0$

\[
|\lss{}{t}(x) - \lss{}{t}(y)| \leq C |x-y|^{\alpha}
\]

for all $t$ and $x,y \in B_1$.

By Lemma <ref>, $\Delta \lss{}{t} = \mu_h |_{{{\cluster{}{t}}}} + \nu |_{\partial {\cluster{}{t}}}$
and $\nu$ is absolutely continuous with respect to $\mu_h$. Thus, the claim follows by Proposition  <ref>.

§.§ Non-degenerate clusters are subharmonic balls

We prove that clusters which do not intersect the boundary of $B_1$ are harmonic balls.
In fact, we observe a stronger property — each ${\cluster{}{t}}$ strictly contained in $B_1$
is a subharmonic ball. That is, subharmonic functions satisfy the sub-mean-value property
on such ${\cluster{}{t}}$.

Specifically, an open set $\Lambda$ is a subharmonic ball with respect to a Radon measure $\mu$ if

\begin{equation} \label{eq:subharmonic-ball}
\mu(\Lambda) f(0)  \leq \int_{\Lambda} f(z) \mu(d z)
\end{equation}

for all functions $f: \overline{O} \to \R$ of the form

\begin{equation} \label{eq:sub-harmonic-potential}
f(x) = \int_{O} G_{O}(x,y) d \nu(y) + q(x)
\end{equation}

where $O$ is an open set containing a neighborhood of the closure of $\Lambda$,
$\nu$ is a signed Radon measure, $\nu |_{\Lambda} \leq 0$, with compact support in $O$,
and $q:\overline{O} \to \R$ is a harmonic function on $O$.

We note that every subharmonic ball is a harmonic ball in the sense described just above Theorem <ref>. Indeed, the set of harmonic functions in the definition of a harmonic ball (as described above Theorem <ref>) is the same as the set of functions $f$ of the form (<ref>) with $\nu|_\Lambda=0$. Since this set of functions is closed under replacing $f$ with $-f$, the inequality (<ref>) gives both the sub-mean-value property and the super-mean-value property for functions in this set.

On an event of probability 1, for all $t > 0$, if  $\overline{{{\cluster{}{t}}}} \subset B_1$ and $\mu_h(\partial {\cluster{}{t}}) = 0$,
then ${\cluster{}{t}}$ is a subharmonic ball.

We do not know a priori that the hypotheses $\overline{{{\cluster{}{t}}}} \subset B_1$ and $\mu_h(\partial{\cluster{}{t}}) = 0$ are satisfied for any value of $t>0$. We will prove that these hypotheses are satisfied, at least when $t$ is small, in Sections <ref> and <ref>, respectively.
By the Riesz decomposition theorem, see, , Section 4 in <cit.>, functions of the form (<ref>)
include functions which are subharmonic in a neighborhood of $\overline{\Lambda}$.

Let $t > 0$, an open set $O \supset \overline{{{\cluster{}{t}}}}$, and $f, q, \nu$ as in (<ref>) be given. By Lemma <ref>, Lemma <ref>, and Proposition <ref> and our assumption that $\mu_h(\partial {\cluster{}{t}}) = 0$, ${\cluster{}{t}}$ is open and

\begin{equation} \label{eq:v-characterization}
\begin{cases}
\odometer{}{t} = 0  &\mbox{on $\partial B_1$} \\
\Delta \odometer{}{t} = -t \delta_0 + \mu|_{{{\cluster{}{t}}}} &\mbox{on $B_1$}.
\end{cases}
\end{equation}

and $\mu_h({\cluster{}{t}}) = t$. As $\overline{{{\cluster{}{t}}}} \subset O$, we can find a smooth domain ${\cluster{}{t}}'$ with ${\cluster{}{t}} \subset \Lambda'_t \subset O$
so that

\begin{align*}
0 &= \int_{{{\cluster{}{t}}}} \Delta q(z) \odometer{}{t}(z) dz \\
&= \int_{{\cluster{}{t}}'} \Delta q(z) \odometer{}{t}(z) dz  \qquad \mbox{(since $\odometer{}{t} = 0$ on $B_1 \backslash {\cluster{}{t}}$)} \\
&= \int_{{\cluster{}{t}}'} q(z) \Delta \odometer{}{t}(z)dz  \qquad \mbox{(integration by parts)} \\
&= -t q(0) + \int_{{{\cluster{}{t}}}} q(z) \mu_h(dz) \qquad \mbox{(by \eqref{eq:v-characterization}) } .
\end{align*}

\begin{align*}
&t (f-q)(0) - \int_{{{\cluster{}{t}}}}(f-q)(y) d \mu_h(y)  \\
&=\int_{{{\cluster{}{t}}}} \int_{{{\cluster{}{t}}}} (G_{O}(0,z) - G_{O}(y,z)) d \nu(z) d \mu_h(y)  \qquad \mbox{(definition of $f$)} \\
&= \int_{{}{t}} \int_{{{\cluster{}{t}}}} (G_{O}(0,z) - G_{O}(y,z)) d \mu_h(y) d \nu(z) \qquad \mbox{(by Fubini)} \\
&= \int_{{{\cluster{}{t}}}} \odometer{}{t}(z) d \nu(z)  \qquad \mbox{(by \eqref{eq:v-characterization}) } \\
&\leq 0 \qquad \mbox{(since $\odometer{}{t} \geq 0$ and $\nu \leq 0$)}.
\end{align*}

We conclude by combining the above two expressions.

§ NON-DEGENERACY OF THE FLOW

In this section, we set up the proof of our main result Theorem <ref>
by dividing it into several intermediate results which will be proven in Sections <ref> through <ref>. We then show how these intermediate results imply the claim.
In the last subsection we observe that the clusters which we construct are locally determined in the sense of Proposition <ref>.

§.§ Properties of the restricted flow

We first show in Section <ref> that clusters do not immediately exit the unit ball.

On an event of probability 1, there exists $T = T(\gamma, h) > 0$ so that for all $0 < t < T$

\begin{equation}
\overline{{{\cluster{}{t}}}} \subset B_1.
\end{equation}

We show in the second part of Section <ref> that the family is continuous.

On an event of probability 1, the clusters decrease to the origin,

\begin{equation} \label{eq:contains-origin}
\bigcap_{t > 0} \overline{{{\cluster{}{t}}}} = \{0\}
\end{equation}

and continuously increase in $t$:  for each $t > 0$, for all $\epsilon > 0$ sufficiently small, there exists $\delta > 0$ so that for all
$t' \in [t, t + \delta]$,

\begin{equation} \label{eq:continuity}
\overline{{\cluster{}{t'}}} \subset {\cluster{}{t}} + B_{\epsilon}.
\end{equation}

In Section <ref>, we show that each cluster has zero boundary area measure.

On an event of probability 1,

\begin{equation}
\mu_h (\partial {\cluster{}{t}}) = 0
\end{equation}

for all $t > 0$ such that $\overline{{{\cluster{}{t}}}} \subset B_1$.

In order to ensure exact uniqueness, the family of harmonic balls appearing in our final theorem differ from the above clusters
via a set of $\mu_h$-measure zero,

\begin{equation} \label{eq:regular-cluster}
{\regcluster{}{t}} := \mathrm{int}(\overline{{{\cluster{}{t}}}}), \quad \forall t >0.
\end{equation}

Indeed, by definition, ${\cluster{}{t}} \subset \mathrm{int}(\overline{{{\cluster{}{t}}}})$ and as $\mu_h(\partial {\cluster{}{t}}) = 0$, $\mu_h(\mathrm{int}(\overline{{{\cluster{}{t}}}}) \backslash {\cluster{}{t}}) = 0$. This shows that ${\regcluster{}{t}}$ is a subharmonic ball and Propositions <ref> through <ref>
hold with ${\regcluster{}{t}}$ in place of ${\cluster{}{t}}$. Thus, we may combine Proposition <ref>, Lemma <ref>, and Propositions <ref> through <ref> into the following statement.

On an event of probability 1, there exists a family of clusters $\{{\regcluster{}{t}}\}_{0 < t < T}$ strictly contained in $\overline{B_1}$
satisfying the conditions of Theorem <ref> for $0 < t<T$ and $\overline{{\regcluster{}{t}}} \cap \partial B_1 \neq \emptyset$.

In the next two subsections we use the compatibility property Lemma <ref> together with a certain scale invariance of clusters
to extend the construction in Proposition <ref> to the entire plane.

On an event of probability 1, there exists a family of clusters $\{{\cluster{}{t}}\}_{t > 0}$
satisfying the conditions of Theorem <ref>.
Moreover, each cluster is a subharmonic ball related to the clusters of (<ref>) in the following way:
if ${\regcluster{}{t}} \Subset B_s$, then ${\regcluster{}{t}} = \mathrm{int}(\overline{\cluster{B_r}{t}})$ for all $r \geq s$.

In Section <ref>, we prove that the family given by Theorem <ref> is the unique such family,
completing the proof of Theorem <ref>.

Let $\{{\regcluster{}{t}}\}_{t > 0}$ be given by Theorem <ref>.
On an event of probability 1, if $\{A_t\}_{t > 0}$ is a family
of harmonic balls satisfying the assumptions in Theorem <ref>,
then $A_t={\regcluster{}{t}}$ for all $t> 0$.

§.§ Scale invariance

We now give the relevant scale invariance property which we then use to prove Theorem <ref>.
Specifically, we show that the law of a cluster stopped upon exiting a ball of arbitrary radius coincides with the law of a rescaled cluster which is stopped upon exiting the unit ball.

For each $r > 0$, the laws of $\{ \cluster{B_1}{t}\}_{t > 0}$ and $\{ r^{-1} \cluster{B_r}{A_r t}\}_{t > 0}$ coincide, where
$A_r := e^{\gamma (Q \log r + h_r(0))}$.

Let $t > 0$ be given.
Consider the continuous function $\tilde{w}_t \in C(\overline{B_1})$ defined by

\begin{equation} \label{eq:definition-of-tildewt}
\tilde{w}_{t} := A_r^{-1} \lss{B_r}{A_r t}(r\cdot) \qquad \mbox{on $B_1$}
\end{equation}

and the fields

\begin{equation}
\tilde{h} = h(r\cdot) + Q \log r, \quad \mbox{where} \quad Q = \frac{2}{\gamma} + \frac{\gamma}{2}.
\end{equation}

\begin{equation}
h' = h(r\cdot) - h_r(0)
\end{equation}

where $h_r(0)$ is the average of $h$ on the circle of radius $r$ around 0. By (<ref>), we have $h' \overset{d}{=} h$.

We claim that

\begin{equation}
\tilde{w}_{t} = \tilde{u}_t := \inf\{ w \in \mathcal{S}^{h'}_t\}
\end{equation}

where $\tilde{u}_t$ is the pointwise infimum over the family

\begin{equation} \label{eq:super-solution-other-h}
\mathcal{S}^{h'}_t  = \{ w \in C(\overline{B_1}) :  \Delta w \leq \mu_{h'} \mbox{ in $B_1$}  \mbox{ and }  w \geq  -t G_{B_1}(0, \cdot) \mbox{ in $\overline{B_1}$}\},
\end{equation}

defined in the same manner as $\mclS{}{}$ but with $h'$ instead of $h$. Write $\mathcal{S}^h_{t}(B_r)$ when $B_1$ in (<ref>)
is replaced by $B_r$ and $h'$ by $h$.

We first show that $\tilde{w}_t \in \mathcal{S}^{h'}_t$. By the fact that $\Delta \lss{}{t} \leq \mu_h$ (Lemma <ref>), the LQG coordinate change formula, and Weyl scaling (Fact <ref>),

\begin{equation}
\Delta \tilde{w}_{t} \leq A_r^{-1} \mu_h(r\cdot) = A_r^{-1} \mu_{\tilde h} = \mu_{h'}  \quad  \mbox{on $B_1$}.
\end{equation}

Also, since $\lss{B_r}{A_r t} \in \mathcal{S}^h_{A_r t}(B_r)$ and $G_{B_r}(0,  r x) = G_{B_1}(0, x)$,

\begin{equation}
\tilde{w}_{t}(x) \geq -t A^{-1}_{r} A_r G_{B_r}(0, r x) = - t G_{B_1}(0, x) \quad  \mbox{on $B_1$}.
\end{equation}

Hence, $\tilde{w}_t \in \mathcal{S}^{h'}_t$.
Similarly, $A_r \tilde{u}_t(\cdot/r) \in \mathcal{S}^h_{A_r t}(B_r)$, which shows
$\tilde{u}_t = \tilde{w}_t$.
Indeed, $A_r \tilde{u}_t(\cdot/r) \leq \lss{B_r}{A_r t}(\cdot)$ implies $\tilde{u}_t(\cdot/r) \leq A_r^{-1} \lss{B_r}{A_r t}(\cdot) = \tilde{w}_{t}(\cdot/r)$ by (<ref>).
Hence, as $h'$ has the same law as $h$,

\begin{equation}
\cluster{B_1}{t} \overset{d}{=} \{ x \in B_1 : \tilde{u}_t(x) > -t G_{B_1}(0, x) \} = r^{-1} \cluster{B_r}{A_r t};
\end{equation}

the last equality uses $\tilde{u}_t = \tilde{w}_t$.

§.§ Proof of Theorem <ref> assuming Proposition <ref>

By combining Lemma <ref> together with Proposition <ref> and a union bound,
on an event of probability 1, for each $r \in \N$, there exists $T^{(r)} > 0$ so that the family of sets

\begin{equation}
\{\cluster{B_r}{t}\}_{0 < t < A_r T^{(r)}}
\end{equation}

is compactly embedded in $\overline{B_r}$, and $\mathrm{int}(\overline{\cluster{B_r}{t}})$ satisfies the properties of Theorem <ref> for $t < A_r T^{(r)}$,
and $\overline{\cluster{B_r}{A_r T^{(r)}}} \cap \partial B_r \neq \emptyset$.

\begin{equation}
{\cluster{}{t}}  = \begin{cases}
\cluster{B_1}{t} \qquad &\mbox{for $t < A_{1} T^{(1)}$} \\
\cluster{B_r}{t} \qquad &\mbox{if $A_{r-1} T^{(r-1)} \leq t < A_{r} T^{(r)}$ for some $r \in \N$}.
\end{cases}
\end{equation}

and ${\regcluster{}{t}} = \mathrm{int}(\overline{\cluster{B_r}{t}})$.
By the compatibility property, Lemma <ref>, the times $A_r T^{(r)}$ are increasing
and hence the construction is well-defined.
Compatibility also implies that the family $\{{\regcluster{}{t}}\}_{t > 0}$ satisfies the properties of Theorem <ref>.

It remains to show that for each $t$ there exists an $r$ so that $A_{r-1} T^{(r-1)} \leq t < A_{r} T^{(r)}$.
That is, we must show that

\begin{equation} \label{eq:strictly-increasing-times}
A_{r} T^{(r)} \to \infty \qquad \mbox{with probability 1}.
\end{equation}

Indeed, if this were the case, this would give us a complete family $\{\regcluster{}{}\}_{t > 0}$ satisfying the properties of Theorem <ref>, and we have uniqueness of such a family by Proposition <ref>.

First note that by Lemma <ref>, $T^{(r)} {\buildrel d \over =} T^{(1)}$ for each $r$.
Since $T^{(1)}$ is strictly positive, for each $p \in (0,1)$, there exists
$c_p > 0$ so that

\[
\P[T^{(r)} > c_p] \geq p
\]

for all $r \in \N$. In particular,

\begin{equation} \label{eq:positive-io}
\P[\cap_{m=1}^{\infty} \cup_{r=m}^{\infty} T^{(r)} > c_p] \geq p.
\end{equation}

We claim that also

\begin{equation} \label{eq:geometric-bm}
\lim_{r \to \infty} A_r = \infty.
\end{equation}

Indeed, the process,

\[
t \to h_{e^{t}}(0)
\]

has a continuous modification which is a standard two-sided Brownian motion <cit.>.

\[
t \to A_{e^{t}} = e^{\gamma (Q t + h_{e^{t}}(0))}
\]

is a geometric Brownian motion with percentage drift $\gamma Q + \gamma^2/2$
and percentage volatility $\gamma$ — this implies (<ref>). Combining (<ref>) with (<ref>) and using that $T^{(r)}$ is increasing in $r$ shows that with probability at least $p$,

\[
A_r T^{(r)} \to \infty.
\]

Since this holds for any $p \in (0,1)$, we have (<ref>),
completing the proof.

§.§ Harmonic balls are local

In this subsection we prove that the harmonic balls given by Theorem <ref> are local; that is,
we prove Proposition <ref>.

Before doing so, we note that we have constructed clusters and stated Lemma <ref> for clusters restricted to domains which are balls.
However, the definition of $\cluster{B_r}{t}$ and the proof of Lemma <ref> extend essentially verbatim to the case when $B_r$ is replaced by any bounded open set containing the origin.

Let $U$ be a deterministic bounded open set containing the origin.
For all $t > 0$, we have $\overline{{\regcluster{}{t}}} = \overline{\cluster{U}{t}}$ if either ${\regcluster{}{t}} \Subset U$ or $\cluster{U}{t} \Subset U$.

This is immediate from the proof of Lemma <ref> together with Theorem <ref>.

Let the deterministic open set $U$ and cluster ${\regcluster{}{t}}$ be given. As $\mathbb \P[\overline{{\regcluster{}{t}}} \subset U]  = 0$ if $0\notin U$
we suppose $0 \in U$. As we may approximate $U$ by an increasing sequence of bounded open sets, we further suppose $U$ is bounded.

The cluster $\cluster{U}{t}$ depends only on $\mu_h |_{U}$ and hence, by locality (Fact <ref>), only on $h |_{U}$.
Therefore, it suffices to observe from Lemma <ref> that

\begin{equation} \label{eq:cluster-coincides}
\overline{{\regcluster{}{t}}} \subset U  \iff  \overline{\cluster{U}{t}} \subset U.
\end{equation}

§ HARNACK-TYPE ESTIMATE

Recall the notation for Euclidean annuli from (<ref>). The main goal of this section is to prove the following Harnack-type estimate for clusters:

\begin{equation} \label{eq:harnack-type}
\mu_h({\cluster{}{t}} \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0)) \implies {\cluster{}{t}} \cap B_{\rho/2}(x_0) = \emptyset
\end{equation}

for all $t > 0$ where $\overline{B_{\rho}}(x_0) \subset B_1 \backslash \{0\}$ and $\alpha \in (0,1)$ is some fixed, small constant. See Figure <ref> for a visualization of this condition.

An example of the Harnack-type property. The cluster ${\cluster{}{t}}$ is in gray with a solid boundary
and an annulus $\A_{\rho/2,\rho}(x_0)$ for which $\overline E_\rho(x_0)$ occurs is displayed in light-gray with a dashed-line boundary. Proposition <ref> asserts that if $\mu_h( {\cluster{}{t}} \cap \A_{\rho/2,\rho}(x_0) )$ is small, then ${\cluster{}{t}} \cap B_{\rho/2}(x_0) = \emptyset$.

Due to the variable nature of the Liouville measure, we cannot show this
holds for every annulus but rather for `most' annuli. 	Specifically, we show the following.

There exists $\alpha = \alpha(\gamma) > 0$ and events
$\overline E_\rho(x_0)$ for $\rho \in (0,1)$ and $x_0 \in \C$
with the following properties: if $\overline E_\rho(x_0)$ occurs, $\overline{B_{\rho}}(x_0) \subset B_1 \backslash \{0\}$, and $\mu_h({\cluster{}{t}} \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0))$, then ${\cluster{}{t}} \cap B_{\rho/2}(x_0) = \emptyset$.

Moreover,  there is a universal constant $c > 0$ so that
with polynomially high probability as $\epsilon \to 0$ for each $x_0 \in (B_{1 + \epsilon} \backslash B_{10 \sqrt{\epsilon}}) \cap \frac{\epsilon}{100} \Z^2$ there are at least $c \log \epsilon^{-1/2}$ radii $\rho \in [\epsilon, \epsilon^{1/2}] \cap
\{8^{-n}\}_{n \in \N}$ for which $\overline E_{\rho}(x_0)$ occurs.

In fact, we prove the following stronger statement.

Assume we are in the setting of Proposition <ref> and let $\rho \in (0,1)$ and $x_0 \in \C$. If $\overline E_\rho(x_0)$ occurs, $\overline{B_{\rho}}(x_0) \subset B_1 \backslash \{0\}$, and $\widetilde\Lambda_t$ is a connected component of ${\cluster{}{t}} \cap B_{\rho}(x_0)$
then the following occurs.  If $\mu_h(\widetilde \Lambda_t \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0))$, then $\widetilde \Lambda_t \cap \overline{B_{\rho/2}(x_0)} = \emptyset$.

We note that Proposition <ref> implies the first part of Proposition <ref>. Indeed, if $\mu_h({\cluster{}{t}} \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0))$, then each connected component $\widetilde\Lambda_t$ as in Proposition <ref> satisfies $\mu_h(\widetilde \Lambda_t \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0))$. So, if $\overline E_\rho(x_0)$ occurs, then Proposition <ref> implies that none of these connected components intersect $\overline B_{\rho/2}(x_0)$.

Proposition <ref> is sufficient for most of our applications, but Proposition <ref> is needed in Section <ref> to show that the boundaries
of the complementary connected components of ${\cluster{}{t}}$ are curves.

The proof of the Harnack-type estimate is inspired by the `no thin tentacles' argument of Jerison-Levine-Sheffield <cit.>
and the IDLA bound of <cit.>.
The idea is as follows. If $A$ is an annulus, then with high probability for every set $Y\subset A$ such that $\mu_h(Y)$ is much smaller than $\mu_h(A)$, it is unlikely for a Brownian motion to cross between the inner and outer boundaries of $A$ without exiting $Y$ (Lemma <ref>). Hence, if $\mu_h({\cluster{}{t}} \cap A) / \mu_h(A)$ is small, then a Brownian motion is unlikely to cross between the inner and outer boundaries of $A$ before exiting ${\cluster{}{t}}$. Since the cluster ${\cluster{}{t}}$ is `grown according to harmonic measure', one can show that if $A'$ is a smaller annulus which is disconnected from 0 by $A$, then $\mu_h({\cluster{}{t}} \cap A') / \mu_h(A')$ is even smaller than $\mu_h({\cluster{}{t}} \cap A) / \mu_h(A)$. See Section <ref> for precise statements to this effect. Iterating this across several nested annuli (with decreasing aspect ratios) allows us to prevent ${\cluster{}{t}}$ from intersecting an appropriate Euclidean ball.

We make the above argument precise via a combination of potential theory and LQG arguments.
We simultaneously study the odometer and the mass of the cluster. Specifically, we study the decay of the odometer and mass in disjoint shells of geometrically decreasing radii. We show that whenever an annulus is `very good' (as defined in Section <ref>) and the mass is small in that annulus, then the odometer has to decrease by a geometric amount from one shell to the next (Lemma <ref>). If the odometer is small, then the mass is also small (Lemma <ref>). This together with the prevalence of `very good' annuli established below forces the odometer (and mass) to go to zero.

We start by introducing notions of `good' and `very good' annuli
in Section <ref>. We show in Sections <ref>
and <ref> that there are many `good' and `very good' annuli.
We then establish harmonic comparison lemmas which allow us to compare the size of the odometer and the LQG mass of the cluster
in Section <ref>. In Section <ref>
we use these harmonic comparison lemmas to show that very good annuli satisfy the Harnack-type estimate. This result
combined with the prevalence of very good annuli leads to the proof of Proposition <ref>.

§.§ Good and very good annuli

In this subsection we define what it means for an annulus to be good and very good.
Roughly, for a good annulus it is difficult for Brownian motion to stay within a set
of relatively small LQG measure until it exits the annulus. Very good annuli
contain lots of good annuli and satisfy additional regularity properties.
We will later see that very good annuli satisfy (<ref>).

§.§.§ Good annuli

We start with defining good annuli. For $z \in \C$, $r > 0$ and parameters $a,b \in (0,1)$,
let $E_r(z) = E_r(z; a,b)$ be the event that the following is true. For each Borel set $Y \subset \A_{3r, 5r}(z)$
such that $\mu_h(Y) \leq a \mu_h(\A_{3 r, 5r}(z))$ we have

\begin{equation} \label{eq:good-annulus-harmonic}
\sup_{u \in \partial B_{4 r}(z)} P\left[ \mbox{$\mathcal{B}^u$ exits $\A_{3 r, 5 r}(z)$ before exiting $Y$}  \, |\, h  \right] \leq b ,
\end{equation}

where $\mathcal B^u$ denotes standard planar Brownian motion started from $u$.
We note that $E_r(z) \in \sigma(h |_{\A_{3r,5r}(z)})$.
The annuli $\A_{3 r, 5r}(z)$ for which $E_r(z)$ holds are good.

§.§.§ Alternative measures of the LQG size of an annulus

It will be convenient to go back and forth between Liouville measure and two other notions of size when using
(<ref>).  To that end, let $\beta^-$ be the growth lower bound exponent appearing in Lemma <ref> and define

\begin{equation} \label{eq:alternate-size}
\Mrho{\rho}{x_0} = \inf_{z \in \A_{\rho/2, \rho}(x_0)} \inf_{r \in (0, \rho/4)} \frac{ \mu_h(B_r(z))}{ (r/\rho)^{\beta^-}} .
\end{equation}

By Lemma <ref>, a.s. $0 < \Mrho{\rho}{x_0} < \infty$ for each $\rho \in (0,1)$ and each $x_0 \in \C$.
We also define

\begin{equation} \label{eq:alternate-size-2}
\SGrho{\rho}{x_0} = \sup_{x \in \A_{\rho/4, 2 \rho}(x_0)} \left( \int_{\A_{\rho/4, 2 \rho}(x_0)} G_{B_{2 \rho}(x_0)}(x, y) d \mu_h(y) \right),
\end{equation}

where $G_{A}: \overline{A} \times \overline{A} \to \overline \R$ is the Green's function for the set $A$.
By Proposition <ref>, a.s. $\SGrho{\rho}{x_0}$ is finite and positive for each $\rho\in (0,1)$ and each $x_0 \in \C$.
For later use, we also define a version of  $\SGrho{\rho}{x_0}$ with a variable aspect ratio,

\begin{equation} \label{eq:alternate-size-2-full}
\SGrhofull{\rho}{x_0}{s_1}{s_2} = \sup_{x \in \A_{s_1 \rho , s_2 \rho}(x_0)} \left( \int_{\A_{s_1 \rho, s_2 \rho}(x_0)} G_{B_{s_2 \rho}(x_0)}(x, y) d \mu_h(y) \right).
\end{equation}

§.§.§ Very good annuli

For $x_0 \in \C, \rho > 0$, and parameters $N_0 \geq 1,  a \in (0,1),b \in(0,1), C^{\pm}_1 > 0, C^{\pm}_2 > 0, C_3 > 0, $ let $\overline E_\rho(x_0) = \overline E_\rho(x_0; N_0, a, b,C^{\pm}_1, C^{\pm}_2, C_3)$ be the event that the following are true:

* $C^-_1  \leq \frac{\Mrho{\rho}{x_0}}{\mu_h(A_{\rho/4, 2 \rho}(x_0))} \leq C^+_1$.

* $C^-_2  \leq \frac{\SGrho{\rho}{x_0}}{\mu_h(A_{\rho/4, 2 \rho}(x_0))} \leq C^+_2$.

* For each $\epsilon \in \{2^{-n}\}_{n \geq N_0}$
and each $z \in \frac{\epsilon \rho}{100} \Z^2 \cap \A_{\rho/2, \rho}(x_0)$,
there are at least $C_3 \log_7 \epsilon^{-1/2}$ radii $r \in [\epsilon \rho, \epsilon^{1/2} \rho] \cap \{\rho 7^{-k}\}_{k \geq 1}$
for which $E_r(z; a, b)$ occurs.

The annuli for which $\overline E_{\rho}(x_0)$ occur are very good.
Our goal in the next two sections is to show that with high probability there are many very good annuli surrounding each point in $B_1$.

§.§ There are many good annuli

We start by showing that good annuli are prevalent, that is we prove the following.

Fix $b \in (0,1)$. There exists $a = a(b,\gamma) > 0$ and a universal constant $c > 0$ such that the following holds
with polynomially high probability as $\epsilon \to 0$.
For each $z \in (B_{1 + \epsilon} \backslash B_{10 \sqrt{\epsilon}}) \cap \frac{\epsilon}{100} \Z^2$ there are at least $c \log_7 \epsilon^{-1/2}$ radii $r \in [\epsilon, \epsilon^{1/2}] \cap
\{7^{-n}\}_{n \in \N}$ for which $E_r(z)$ occurs, where $E_r(z) = E_r(z; a, b)$ is as in (<ref>).

We note the similarity between Lemma <ref> and condition <ref> in the definition of very good annuli.
To prove Lemma <ref> we will first show that for each $z \in \C$ and $r > 0$, the event $E_r(z)$ occurs
with high probability provided $a$ is chosen to be sufficiently small depending on $b$ (Lemma <ref>).
We will then use the near-independence of the GFF across disjoint concentric annuli (Lemma <ref>)
to show that for each fixed $z \in \C$, it holds with very high probability when $\epsilon$ is small
that there are many radii $r \in [\epsilon, \epsilon^{1/2}] \cap \{7^{-n}\}_{n \in \N}$ for which $E_r(z)$ occurs.
Finally, we will take a union bound over all $z \in B_{1 + \epsilon} \cap \frac{\epsilon}{100} \Z^2$.

Let $b \in (0,1)$ and $p \in (0,1)$. There exists $a = a(b,p,\gamma) \in (0,1)$ such that the event $E_r(z) = E_r(z;a,b)$ of (<ref>) satisfies

\[
P\left[ E_r(z) \right] \geq p ,\quad \forall r > 0,\quad \forall z \mbox{ such that $\dist(\A_{3 r, 5 r}(z), \{0\}) \geq r/100$}.
\]

We first show that it suffices to prove the lemma for $h^{\C}$, , the GFF without a log-singularity ($\boldsymbol{\alpha}_0 = 0$ in (<ref>)).
We then prove the lemma for the case $\boldsymbol{\alpha}_0 = 0$.

Step 1: Reduction to $\boldsymbol{\alpha}_0 = 0$.

Recall that $h = h^\C -\boldsymbol{\alpha}_0 \log |\cdot|$, where $h^\C$ is a whole-plane GFF.
We first explain why it is sufficient to prove the lemma with $h^\C$ in place of $h$.
Suppose the statement of the lemma holds for $h^{\C}$ with $a' \in (0,1)$ in place of $a$.

Fix $r > 0$ and then  $z$ such that the annulus $\A_{3 r, 5 r}(z)$ lies at Euclidean distance at least $r/100$ from the origin.
By Weyl scaling (Fact <ref>), there are constants $C_1,C_2>  0$ depending only on $\gamma$ such that

\begin{equation}
\mu_h(Y) \geq C_1 |z|^{-\boldsymbol{\alpha}_0 \gamma} \mu_{h^{\C}}(Y), \quad \forall Y \subset \A_{3 r, 5 r}(z)
\end{equation}

\begin{equation}
\mu_h(Y) \leq C_2 |z|^{-\boldsymbol{\alpha}_0 \gamma} \mu_{h^{\C}}(Y), \quad \forall Y \subset \A_{3 r, 5 r}(z) .
\end{equation}

Thus, for all $Y \subset \A_{3 r, 5 r}(z)$

\[
\mu_{h^\C}(Y) \leq a' \mu_{h^\C}(\A_{3 r, 5 r}(z)) \implies  \mu_h(Y) \leq \frac{C_2}{C_1} \times a' \mu_{h}(\A_{3 r, 5 r}(z))
\]

and hence $\P[E_r(z)] \geq p$ for $a := \frac{C_2}{C_1} \times a'$.

Step 2: Case when $\boldsymbol{\alpha}_0 = 0$.

For the rest of the proof we assume that $\boldsymbol{\alpha}_0 = 0$. The law of $h = h^\C$ is both scale and translation invariant modulo additive constant. By the Weyl scaling property of the measure $\mu_h$ (Fact <ref>), the event $E_r(z)$ is a.s. determined by $h$ viewed modulo additive constant. From this and the LQG coordinate change formula for $\mu_h$, we infer that the probability of $E_r(z)$ does not depend on $r$ or $z$. Hence, it suffices to find $a  \in (0,1)$ as in the lemma statement such that $\P[E_1(0)] \geq p$.

To this end, for $u \in \C$ let $\mathcal{B}^u$ denote $\gamma$-Liouville Brownian motion with respect to the field $h$, started from $u$ with
reflecting boundary conditions in a square $K$ centered at the origin with side length 100.

By <cit.>, the conditional law of $\mathcal B^u$ stopped when exiting $\A_{3, 5}(0)$ depends continuously on $u$.
Although the proof in <cit.> is for the whole-plane massive GFF, as explained in  <cit.>, Proposition 2.19 in  <cit.>
extends to the massless GFF in a finite domain. Moreover, ordinary LBM and reflected LBM coincide until the first exit from $\A_{3,5}(0)$.
Hence, by the compactness of the circle $\partial B_4(0)$, we may therefore find a random $T = T(h) > 0$ such that a.s.
\begin{equation} \label{eq:annulus-hit-time}
\sup_{u \in \partial B_4(0)}  P\left[ \text{$\mathcal B^u$ exits $\A_{3,5}(0)$ before time $T$} \,|\, h \right] \leq  \frac{b}{2} .
\end{equation}

For $t>0$, let $p^K_t(u,\cdot)$ be the time $t$ Liouville heat kernel for $\mathcal B^u$, so that $p^K_t(u,\cdot)\, d\mu_h$ is the law of $\mathcal B^u_t$.
By Proposition <ref> a.s. $p^K_t(u,v)$ is a continuous function of $(t,u,v)$ and  $p^K_t(u,v) > 0$ for all $t > 0$ and all $u,v\in K$.
Again using the compactness of $\partial B_4(0)$, we infer that with $T$ as in (<ref>), there exists a random $C = C(h)  > 0 $ such that a.s.
\begin{equation}
\label{eq:heat-kernel-max}
\sup_{u\in\partial B_4(0)} \sup_{v\in B_5(0)} p_{T}(u,v) \leq C  .
\end{equation}

From (<ref>), we get that for each Borel set $Y \subset  \A_{3,5}(0)$,

\begin{equation}
\label{eq:heat-kernel-int}
\sup_{u\in \partial B_4(0)} P\left[ \mathcal B^u_{T} \in Y \,|\, h \right] \leq C \mu_h(Y) .
\end{equation}

Hence, if $\mu_h(Y) \leq [ C \mu_h(\A_{3,5}(0) ) ]^{-1} (b/2)  \times \mu_h(\A_{3,5}(0) )$, then $P\left[ \mathcal B^u_{T} \in Y \,|\, h \right] \leq b/2$ for each $u\in \partial B_4(0)$. Combining this with (<ref>) shows that for every such Borel set $Y$,

\begin{equation}
\sup_{u \in \partial B_4(0)}  P\left[ \text{$\mathcal B^u$ exits $\A_{3,5}(0)$ before exiting $Y$} \,|\, h \right] \leq b .
\end{equation}

That is, a.s. $E_1(0)$ occurs with $a$ replaced by the random variable $[ C \mu_h(\A_{3,5}(0) ) ]^{-1} (b/2)$. This random variable is a.s. positive, so we can find a deterministic $a \in (0,1)$ such that

\begin{equation}
P\left[ [ C \mu_h(\A_{3,5}(0) ) ]^{-1} (b/2) \geq a \right] \geq p .
\end{equation}

Hence, for this choice of $a$ we have $\P[E_1(0)] \geq p$, as required.

The following lemma is a consequence of the fact that the restrictions of the GFF to disjoint concentric annuli, viewed modulo additive constant, are nearly independent. See <cit.> for a slightly more general statement.

Fix $0 < s_1<s_2 < 1$. Let $\{r_k\}_{k\in\N}$ be a decreasing sequence of positive numbers such that $r_{k+1} / r_k \leq s_1$ for each $k\in\N$ and let $\{E_{r_k} \}_{k\in\N}$ be events such that $E_{r_k}$ is measurable with respect to $h |_{\A_{s_1 r_k , s_2 r_k}(0)  } $, viewed modulo additive constant, for each $k\in\N$.
For $K\in\N$, let $N(K)$ be the number of $k\in [1,K] \cap \Z$ for which $E_{r_k}$ occurs.
For each $\alpha > 0$ and each $\beta \in (0,1)$, there exists $p = p(\alpha,\beta,s_1,s_2) \in (0,1)$ and $C = C(\alpha,\beta,s_1,s_2) > 0$ (independent of the particular choice of $\{r_k\}$ and $\{E_{r_k}\}$) such that if

\begin{equation} \label{eqn-annulus-iterate-prob}
P\left[ E_{r_k}  \right] \geq p , \quad \forall k\in\N  ,
\end{equation}

\begin{equation} \label{eqn-annulus-iterate}
P\left[ N(K)  < \beta K\right] \leq C e^{-\alpha K} ,\quad\forall K \in \N.
\end{equation}

We now prove the desired claim.

The event $E_r(z)$ depends only on the measure $\mu_h|_{\A_{3 r,5r}(z)}$. Moreover, multiplying this measure by a constant does not change whether $E_r(z)$ occurs.
Therefore, $E_r(z)$ is a.s. determined by $h|_{\A_{3 r,5r}(z)}$ viewed modulo additive constant.

We now apply Lemma <ref> with $K = \lfloor \log_7  \epsilon^{-1/2} \rfloor$, the radii $r_1,\dots,r_K \in [ \epsilon, \epsilon^{1/2}] \cap \{7^{-n}\}_{n\in\N}$, the events $E_{r_k} = E_{r_k}(z)$, and appropriate universal constant choices of $\alpha$ and $\beta$. We find that there exist universal constants $p\in (0,1)$ and $c>0$ such that if $\P[E_r(z)] \geq p$
for each $r > 0$ and each $z \in \C \backslash B_{10 r}$,  then for all $z \in \C \backslash B_{10 \sqrt{\epsilon}}$,

\begin{equation} \label{eqn-annulus-hit-iterate}
P\left[ \text{$E_r(z)$ occurs for at least $c\log\epsilon^{-1/2}$ values of $r \in [ \epsilon, \epsilon^{1/2}] \cap \{7^{-n}\}_{n\in\N}$} \right] \geq 1 - O_\epsilon(\epsilon^3)
\end{equation}

with a universal implicit constant in the $O_\epsilon(\cdot)$.

By Lemma <ref>, there exists $a = a(b,\gamma) > 0$ such that for this choice of $a$, one has $\P[E_r(z)] \geq p$ for each $r > 0$ and each $z \in \C \backslash B_{10 r}$. Therefore, the estimate (<ref>) holds for this choice of $a$. We now conclude by means of a union bound over all $z\in  (B_{1 + \epsilon} \backslash B_{10 \sqrt{\epsilon}}) \cap \frac{\epsilon}{100} \Z^2$.

§.§ There are many very good annuli

In this section we prove that very good annuli are prevalent, following the same strategy as the last section.

Fix $b \in (0,1)$. There exists $a = a(b,\gamma) > 0$, universal constants $c, C_3 > 0$, $C^{\pm}_1, C^{\pm}_2 > 0$ depending on $a, \gamma$, and $N_1 = N_1(b,\gamma) \geq 1$ such that for all $N_0  \geq N_1 $ the following holds with polynomially high probability as $\epsilon \to 0$. For each $x_0 \in (B_{1 + \epsilon} \backslash B_{10 \sqrt{\epsilon}}) \cap \frac{\epsilon}{100} \Z^2$ there are at least $c \log_8 \epsilon^{-1/2}$ radii $\rho \in [\epsilon, \epsilon^{1/2}] \cap
\{8^{-n}\}_{n \in \N}$ for which $\overline E_\rho(x_0)$ occurs, where $\overline E_\rho(x_0) = \overline E_\rho(x_0; N_0, a, b,C^{\pm}_1, C^{\pm}_2, C_3)$ is as in <ref>, 	<ref>, 	<ref>.

We start by showing the event $\overline E_\rho(x_0)$ occurs with high probability
with $a,b$ chosen as in Lemma <ref>,  $N_0$ large, and $C^{\pm}_1$,$C^{\pm}_2$
chosen appropriately.

Let $b \in (0,1)$ and $p \in (0,1)$.
There exists $a = a(b,p,\gamma) \in (0,1)$,  $C^{\pm}_1, C^{\pm}_2$ depending on $a, \gamma, p$
and a universal constant $C_3>0$ such that the event $\overline E_\rho(x_0)$ of <ref> <ref> <ref> satisfies

\[
P\left[ \overline E_\rho(x_0) \right] \geq p ,\quad \forall \rho > 0,\quad \forall x_0 \mbox{ such that $\dist(\A_{\rho/4, 2 \rho}(x_0), \{0\}) \geq \rho/100$}
\]

for all $N_0 \geq N_1(b,p,\gamma) \geq 1$ sufficiently large.

Recall that $h = h^\C -\boldsymbol{\alpha}_0 \log |\cdot|$. The proof is similar to that of Lemma <ref>. We show that
we can reduce to the case $\boldsymbol{\alpha}_0 = 0$ and then give a proof in that case.

Step 1: Reduction to $\boldsymbol{\alpha}_0 = 0$.

Suppose the statement of the lemma holds for $h^{\C}$ with constants $\underline{C}^{\pm}_1$ in place of $C^{\pm}_1$.  Write $M^{h}_{\rho}(x_0)$ to indicate the dependence in the definition of $\Mrho{\rho}{x_0}$
on the GFF.

Fix $\rho > 0$ and then $x_0$ such that the annulus $\A_{\rho/4, 2\rho}(x_0)$ lies at Euclidean distance at least $\rho/100$ from the origin.
By Weyl scaling (Fact <ref>), there are constants $A_1,A_2>  0$ depending only on $\gamma$ such that

\begin{equation}
\frac{M^{h}_{\rho}(x_0)}{\mu_h(A_{\rho/4, 2 \rho}(x_0))} \geq A_1  \frac{M^{h^\C}_{\rho}(x_0)}{\mu_{h^\C}(A_{\rho/4, 2 \rho}(x_0))}
\end{equation}

\begin{equation}
\frac{M^{h}_{\rho}(x_0)}{\mu_h(A_{\rho/4, 2 \rho}(x_0))} \leq A_2  \frac{M^{h^\C}_{\rho}(x_0)}{\mu_{h^\C}(A_{\rho/4, 2 \rho}(x_0))}.
\end{equation}

\[
\underline{C}^{-}_1 \leq  \frac{M^{h^\C}_{\rho}(x_0)}{\mu_{h^\C}(A_{\rho/4, 2 \rho}(x_0))} \leq \underline{C}^{+}_1 \implies A_1 \underline{C}^{-}_1 \leq \frac{M^{h}_{\rho}(x_0)}{\mu_h(A_{\rho/4, 2 \rho}(x_0))} \leq A_2 \underline{C}^{+}_1
\]

and hence <ref> occurs for $C^+_1 := A_2 \underline{C}^{+}_1$ and $C^-_1 := A_1 \underline{C}^{-}_1$ if it occurs under $h^{\C}$.
The argument for <ref> is nearly identical. Step 1 of the proof of Lemma <ref> also shows that we can
reduce to the case $\boldsymbol{\alpha}_0 = 0$ for <ref>.

Step 2: Case when $\boldsymbol{\alpha}_0 = 0$.

As in Step 2 of the proof of Lemma <ref> it suffices to find $a \in (0,1)$
and  $C^{\pm}_1, C^{\pm}_2, C_3 > 0$ such that $\P[\overline E_1(0)] \geq p$.

Note that by Lemma <ref>, $\Mrho{1}{0}$ is a strictly positive, finite random variable.
Since $\mu_h(\A_{1/4, 2}(0))$ and $\SGrho{1}{0}$ are also strictly positive and finite random variables,
there exists $C^{\pm}_1, C^{\pm}_2 > 0$ so that

\begin{equation}
P\left[C^-_1  \leq \frac{\Mrho{1}{0}}{\mu_h(A_{1/4, 2}(0))} \leq C^+_1\right] \geq p_1
\quad \text{and} \quad
P\left[C^-_2  \leq \frac{\SGrho{1}{0}}{\mu_h(A_{1/4, 2}(0))} \leq C^+_2\right] \geq p_2.
\end{equation}

Also, by Lemma <ref>, for $N_0$ sufficiently large, with probability at least $p_3$, the event in <ref> occurs with $\rho = 1$ and $C_3$ an appropriate universal constant.

By adjusting our choices of parameters so that $(1-p_1) +  (1-p_2) + (1-p_3) \leq 1-p$, we may conclude via a union bound.

Given Lemma <ref>, the  argument is identical to the proof of Lemma <ref>.

§.§ Harmonic comparison

In this section we prove lemmas which let us compare the size of the odometer to the LQG mass of the cluster.
Our first lemma allows us to show that the LQG mass of the cluster is small in annuli where the odometer is small.

Fix $0 < s_1 < s_2 < s_3 < s_4$.
Let $\A_{s_1 r, s_4 r}(z) \subset B_1$ be an annulus not containing the origin
and let $\hat \Lambda_t$ be a union of connected components of $\A_{s_1 r, s_4 r}(z) \cap {\cluster{}{t}}$.
There exists a constant $C$, depending only on $s_{1},\ldots,s_4$,
so that for all such annuli

\[
\mu_h(\A_{s_2 r, s_3 r}(z) \cap \hat \Lambda_t) \leq C \sup_{x \in \hat \Lambda_t} \odometer{}{t}(x).
\]

First note that there is a positive constant $c_1$ so that the annulus $\A_{s_2, s_3}$ can be covered by $c_1^{-1}$
balls of radius $c_2 := \min(s_2-s_1, s_3 -s_2, s_4-s_3)/4$ centered at points in $\A_{s_2, s_3}$. Therefore, by scaling,
this implies the annulus $\A_{s_2 r, s_3 r}(z)$ can be covered by $c_1^{-1}$ balls of radius $c_2 r$
centered at points in $\A_{s_2 r, s_3 r}(z)$. By the pigeonhole principle, there is at least one such ball $B_{2 c_2 r}(x) \Subset \A_{s_1 r, s_4 r}(z)$

\begin{equation} \label{eq:ball-lower-bound-mass}
\mu_h(B_{c_2r}(x) \cap \hat \Lambda_t) \geq c_1 \mu_h(\A_{ s_2 r,  s_3 r}(z) \cap \hat \Lambda_t).
\end{equation}

Write $\odometer{}{t} |_{\hat \Lambda_t}$ for $\odometer{}{t}(\cdot) 1\{ \cdot \in\hat \Lambda_t\}$.
As we will explain just below, one can deduce from Lemma <ref> that

\begin{equation} \label{eq:laplacian-of-indicator}
\Delta( \odometer{}{t} |_{\hat \Lambda_t} ) =
\begin{cases}
\mu_h &\mbox{ on $\hat \Lambda_t$ } \\
0 &\mbox{on $\A_{s_1 r, s_4 r}(z) \backslash \overline{\hat \Lambda_t}$} \\
\geq 0 &\mbox{ on $\partial \hat \Lambda_t \cap \A_{s_1 r, s_4 r}(z)$}.
\end{cases}
\end{equation}

Indeed, $\hat \Lambda_t$ is a union of connected components of ${\cluster{}{t}} \cap \A_{s_1 r, s_4 r}(z)$,
an open set. The odometer, $\odometer{}{t}$ is non-negative and continuous on $B_1$ and $\odometer{}{t} = 0$ on $\partial \hat \Lambda_t \cap \A_{s_1 r, s_4 r}(z)$. Therefore $\odometer{}{t} |_{\hat \Lambda_t}$ is continuous on $\hat \Lambda_t \cap \A_{s_1 r, s_4 r}(z)$
and $\odometer{}{t} |_{\hat \Lambda_t}$  satisfies the sub-mean-value property on $\partial \hat \Lambda_t \cap \A_{s_1 r, s_4 r}(z)$.
As $\odometer{}{t} |_{\hat \Lambda_t}$ coincides with  $\odometer{}{t}$ on $\hat \Lambda_t$, this shows (<ref>) by Lemma <ref>.

Let $G_A$ denote the Green's function for the domain $A$ with zero boundary conditions
and let

\[
u(\cdot) = \int_{B_{2 c_2 r}(x) \cap \hat \Lambda_t} G_{B_{2 c_2 r}(x)}(\cdot,y) \mu_h(dy).
\]

Observe that $\Delta (u+\odometer{}{t} |_{\hat \Lambda_t} ) \geq 0$ on $B_{2 c_2 r}(x)$: indeed,
by the definition of $u$ we have $\Delta u = -\mu_h$ on $\hat \Lambda_t \cap B_{2 c_2 r}(x)$ and $\Delta u = 0$ elsewhere
which, together with (<ref>), shows $u+\odometer{}{t} |_{\hat \Lambda_t} $ is subharmonic on $B_{2 c_2 r}(x)$.

Hence, by the maximum principle, on $B_{2 c_2 r}(x)$,

\[
u+\odometer{}{t} |_{\hat \Lambda_t}  \leq \sup_{\partial B_{2 c_2  r}(x)} (u + \odometer{}{t} |_{\hat \Lambda_t} ) = \sup_{\partial B_{ 2 c_2 r }(x)} \odometer{}{t} |_{\hat \Lambda_t}  \leq \sup_{\A_{s_1 r, s_4 r}(z)} \odometer{}{t} |_{\hat \Lambda_t}  = \sup_{\hat \Lambda_t} \odometer{}{t}
\]

as $u = 0$ on $\partial B_{2 c_2 r}(x)$ and $B_{2 c_2 r}(x) \Subset \A_{s_1 r, s_4 r}(z)$. Thus, as $\odometer{}{t} |_{\hat \Lambda_t}  \geq 0$,

\begin{equation} \label{eq:mass-ineq-upper-bound}
0 \leq \left(\sup_{\hat \Lambda_t} \odometer{}{t} \right) - u ,\quad \text{on $B_{2 c_2 r}(x)$} .
\end{equation}

We now estimate $u$ at the center of $B_{2 c_2 r}(x)$. By the definition of $u$ and then the scale invariance of the Green's function for a ball,

\begin{align*}
u(x) &= \int_{B_{2 c_2  r}(x) \cap \hat \Lambda_t} G_{B_{2 c_2 r}(x)}(x,y) \mu_h(dy) \\
&\geq \int_{B_{c_2 r}(x) \cap \hat \Lambda_t} G_{B_{1}(0)}(0, (2 c_2 r)^{-1}(y-x)) \mu_h(dy) \\
&\geq C \mu_h(\A_{s_2 r, s_3 r}(z) \cap \hat \Lambda_t) \qquad \mbox{(by \eqref{eq:ball-lower-bound-mass})}
\end{align*}

where $C := c_1 \inf_{y \in B_{1/2}(0)} G_{B_{1}(0)}(0, y) > 0$ is independent of $r$.  We conclude the proof by combining this lower bound for $u(x)$ with (<ref>).

We next show that if it is difficult for Brownian motion to get through a domain without
exiting ${\cluster{}{t}}$, then the odometer must be small within the domain.

Let $A$ denote an open set in $B_1$ not containing the origin.
For all $x \in A \cap {\cluster{}{t}}$

\begin{equation}
\odometer{}{t}(x) \leq \left( \sup_{\partial A \cap \overline \Lambda^x_t} \odometer{}{t} \right)  \P[\mbox{$\mathcal{B}^x$ exits  $A$ before hitting  $A \backslash \Lambda^x_t$} | h],
\end{equation}

where $\mathcal{B}^x$ denotes an independent Brownian motion started at $x$ and $\Lambda^x_t$
is the connected component of $A \cap {\cluster{}{t}}$ containing $x$.

Write $f_A(z) =\P[\mbox{$\mathcal{B}^z$ exits  $A$ before hitting  $A \backslash {\cluster{}{t}}$} | h]$. Observe that

\begin{equation} \label{eq:expression-for-fA}
\begin{cases}
\Delta f_A = 0 \quad &\mbox{in $A \cap {\cluster{}{t}}$} \\
f_A = 1 \quad &\mbox{on $\partial A \cap {\cluster{}{t}}$} \\
f_A = 0 \quad &\mbox{on $A \cap \partial {\cluster{}{t}}$}
\end{cases}
\end{equation}

and by Lemma <ref>,

\begin{equation}\label{eq:expression-for-odometer}
\begin{cases}
\Delta \odometer{}{t} \geq 0 \quad &\mbox{in $A \cap {\cluster{}{t}}$} \\
\odometer{}{t} \leq \sup_{\partial A \cap {\cluster{}{t}}} \odometer{}{t} \quad &\mbox{on $\partial A \cap {\cluster{}{t}}$} \\
\odometer{}{t} = 0 \quad &\mbox{on $A \cap \partial {\cluster{}{t}}$}.
\end{cases}
\end{equation}

Now, fix $x \in A \cap {\cluster{}{t}}$ and consider $\Lambda^x_t$, the connected component of $A \cap {\cluster{}{t}}$ containing $x$.
As $A \cap \Lambda^x_t$ is a connected component of $A \cap {\cluster{}{t}}$, (<ref>) and (<ref>)
show that

\[
\odometer{}{t}(\cdot) - \left(\sup_{\partial A \cap \overline \Lambda^x_t} \odometer{}{t}\right) f_A(\cdot)
\]

is subharmonic in $A \cap \Lambda^x_t$, equal to 0 on $A \cap \partial \Lambda^x_t$, and less than or
equal to 0 on $\partial A \cap \overline \Lambda^x_t$.
Hence, by the maximum principle,

\[
\odometer{}{t}(z) \leq  \left(\sup_{\partial A \cap \overline \Lambda^x_t} \odometer{}{t}\right) f_A(z) , \quad \mbox{for $z \in A \cap \Lambda^x_t$} .
\]

\[
f_A(z) = \P[\mbox{$\mathcal{B}^z$ exits  $A$ before hitting  $A \backslash \Lambda^x_t$} | h ] \quad \mbox{for $z \in A \cap \Lambda^x_t$}
\]

since there is no path in $A$ from $z$ to any point of $\Lambda_t\setminus \Lambda_t^x$.
The previous two sentences imply

\[
\odometer{}{t}(z) \leq  \left(\sup_{\partial A \cap \overline \Lambda^x_t} \odometer{}{t}\right) \P[\mbox{$\mathcal{B}^z$ exits  $A$ before hitting  $A \backslash \Lambda^x_t$} |h ]  \quad  \mbox{for $z \in A \cap \Lambda^x_t$}
\]

completing the proof.

We next provide a weak upper bound on the growth of the odometer around its zeros.

Fix $0 < s_1  < s_2 < s_3 < s_4$.
There exists a constant $c > 0$, depending only on $s_1, \ldots, s_4$,  so that, with $\SGrhofull{\rho}{x_0}{s_1}{s_4}$ as in (<ref>),

\[
\sup_{x \in \A_{s_2 \rho,s_3 \rho}(x_0)} \odometer{}{t}(x) \leq c  \SGrhofull{\rho}{x_0}{s_1}{s_4}
\]

for all $\A_{s_1 \rho,s_4 \rho}(x_0) \subset B_1 \backslash \{0\}$ such that ${{{\cluster{}{t}}}}^c \cap  \A_{s_2 \rho,s_3 \rho}(x_0) \neq \emptyset$.

Fix an annulus $\A_{s_1 \rho,s_4 \rho}(x_0)$ and let $\lambda = \Delta \odometer{}{t}$.
Consider the positive function

\[
q(z) = \int_{\A_{s_1 \rho,s_4 \rho}(x_0)} G_{B_{s_4 \rho}(x_0)}(z,y) d \lambda(y).
\]

for $z \in \A_{s_1 \rho,s_4 \rho}(x_0)$.
Since $\lambda \leq \mu_h$ (Lemma <ref>),

\begin{equation} \label{eq:q-inequality}
\sup_{z \in \A_{s_1 \rho,s_4 \rho}} q(z) \leq \SGrhofull{\rho}{x_0}{s_1}{s_4}.
\end{equation}

Hence the statement of the lemma will follow once we bound $\odometer{}{t}$ by $q$. We do this via Harnack's inequality for positive harmonic functions.

Our choice of $q$ ensures it is positive and

\[
\Delta q = -\lambda \qquad \mbox{ on $\A_{s_1 \rho,s_4 \rho}(x_0)$}.
\]

This implies that the function $g: \A_{s_1 \rho,s_4 \rho}(x_0) \to \R$ defined by

\[
g := \odometer{}{t} + q
\]

is harmonic and non-negative in $\A_{s_1 \rho,s_4 \rho}(x_0)$.
Fix $s_2', s_3'$ so that

\[
0 < s_1 < s_2' < s_2 < s_3 < s_3' < s_4.
\]

By Harnack's inequality for positive harmonic functions,

\begin{equation} \label{eq:g-harnack-step}
\sup_{z \in \A_{s_2' \rho,s_3' \rho}} g(z) \leq c \inf_{y \in \A_{s_2 \rho,s_3 \rho}(x_0)} g(y)
\end{equation}

for a constant $c$ (depending only on the ratio of the domains on the left and right of (<ref>)).
Note that the assumption ${{{\cluster{}{t}}}}^c \cap  \A_{s_2 \rho,s_3 \rho}(x_0) \neq \emptyset$ implies the existence of $z_0 \in \A_{s_2 \rho,s_3 \rho}(x_0)$ with $ \odometer{}{t}(z_0) = 0$.

\begin{align*}
\sup_{z \in \A_{s_2' \rho,s_3' \rho}(x_0)} \odometer{}{t}(z) &\leq \sup_{z \in \A_{s_2' \rho,s_3' \rho}(x_0)} g(z)  \qquad \mbox{ (since $q \geq 0$)} \\
&\leq c \inf_{y \in \A_{s_2 \rho,s_3 \rho}(x_0)} g(y) \qquad \mbox{(by \eqref{eq:g-harnack-step})} \\
&\leq c g(z_0)   \qquad \mbox{ (since $z_0 \in \A_{s_2 \rho,s_3 \rho}(x_0)$)} \\
&= c q(z_0) \qquad \mbox{(since $\odometer{}{t}(z_0) = 0$)} \\
&\leq c \sup_{z \in \A_{s_2 \rho,s_3 \rho}(x_0)} q(z),
\end{align*}

which together with (<ref>) completes the proof.

§.§ Very good annuli satisfy the Harnack-type estimate

In this section we prove that there are choices of parameters so that every annulus $\A_{\rho/2, \rho}(x_0)$ with $\overline{B}_{\rho}(x_0) \subset B_1 \backslash \{0\}$  which is very good also satisfies the Harnack-type property.

There exists a universal constant $C_3 > 0$ and $b = b(\gamma) \in (0,1)$ so that the following is true.
For every choice of $C_1^\pm , C_2^\pm  > 0$ and $a \in (0,1)$ there exists

* $N_1 \geq 200$ depending only on $a,\gamma,C_1^-, C_2^+$;

* for each $N_0 \geq N_1$, a parameter $\alpha \in (0,1)$ depending on $a, N_0, \gamma$, and $C^-_1$;

with the following property. If $\rho \in (0,1)$ and $x_0 \in B_1$ are such that $\overline{B}_{\rho}(x_0) \subset B_1 \backslash \{0\}$,  the event $\overline E_\rho(x_0)$ of <ref>,
<ref>, and <ref> occurs, and $\widetilde\Lambda_t$ is a connected component of ${\cluster{}{t}} \cap B_{\rho}(x_0)$
for which  $\mu_h(\widetilde \Lambda_t \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0))$, then $\widetilde\Lambda_t \cap \overline{B_{\rho/2}(x_0)} = \emptyset$.

Illustration of the choice of constants and dependencies in the proof of Proposition <ref>.
A line between two constants indicates
that the downwards constant is chosen in a way which depends directly on the upwards constant. Where the constant is chosen is written directly next to it. Note that $b = b(\gamma)$. Otherwise, dependence on $\gamma \in (0,2)$ and some universal constants is not indicated.
$N_0$ can be any number larger than $N_1$. This is done so that there is flexibility later (the proof of Proposition <ref>) to choose the initial scale $N_0$ to be large.

This leads to the Harnack-type inequality.

Combine Lemma <ref> and Lemma <ref>.
See Figure <ref> for an illustration of how the constants are chosen.

Visual explanation of the proof of Lemma <ref>.
We display some of the shells in the shell decomposition, the grid of points in each shell surrounded by good annuli, and some of the good annuli $\A_{r_i, 7 r_i}(z)$ surrounding one of the grid points $z_i$. The shells are displayed in gray, the good annuli are in purple, and the grid of points are black dots. Aspect ratios of annuli are not shown to scale.

The proof of Lemma <ref> is purely deterministic. We now outline the proof — also see Figure <ref>.
As mentioned previously, we will use <ref> and <ref>  to switch between $\mu_h$, $\SGrho{\rho}{\cdot}$, and $\Mrho{\rho}{\cdot}$ when convenient.

* Set up the iteration by decomposing $\A_{\rho/2, \rho}(x_0)$ into a disjoint, sparse collection of shells $\{S_j\}_{j \geq 0}$
contained in $\{\A_{\rho/2, d_j}(x_0)\}_{j \geq 0}$ for some infinite geometric sequence $d_j \downarrow d_{\infty} > \rho/2$ (Lemma <ref>).

* Show that if $\mu_h(\A_{\rho/2, d_j}(x_0) \cap {\cluster{}{t}})$ is very small, then the supremum of the odometer
decreases by a multiplicative factor from $\A_{\rho/2, d_j}(x_0)$ to $\A_{\rho/2, d_{j+1}}(x_0)$ (Lemma <ref>).

* As  $\mu_h(\A_{d_j, \rho}(x_0) \cap {\cluster{}{t}})$ is small, we may use <ref>
to cover $S_j$ by a dense grid of points
surrounded by a large number $N$ of concentric good annuli.

* For each good annulus, apply harmonic comparison, Lemma <ref>, together
with (<ref>) to see that the supremum of the odometer
decreases by a factor of $b$ from one concentric annulus to the next.

* Since the concentric annuli cover $S_j$ and $\odometer{}{t}$ is subharmonic, iterating shows that
the odometer decreases by a factor of $b^N$.

* Show that if $\sup_{\A_{\rho/2, d_{j+1}}(x_0)} \odometer{}{t}$ is small, then $\mu_h(\A_{3r, 5r}(z) \cap {\cluster{}{t}})$ is small for each annulus $\A_{r, 7r}(z) \subset \A_{\rho/2, d_{j+1}}(x_0)$ (using Lemma <ref>).

* Start with a weak initial bound on the odometer, Lemma <ref>
and the initial assumption that $\mu_h(\A_{\rho/2, \rho}(x_0) \cap {\cluster{}{t}}) \leq \alpha \mu_h(\A_{\rho/2,\rho}(x_0))$ and iterate the previous two steps to see that $\lim \left( \sup_{\A_{\rho/2, d_j}(x_0)}  \odometer{}{t} \right) \to 0$.

We start with the shell decomposition.

Fix $\rho \in (0,1)$ and $x_0 \in B_1$. Define the collection of shells

\begin{equation}
S_j =  \partial B_{d_j}(x_0) + B_{l_j}  \qquad \mbox{for $j \geq 0$}
\end{equation}

\begin{equation}
l_j = \rho \times 2^{-j/4} \times 2^{-50} \qquad \mbox{for $j \geq 0$}
\end{equation}

\begin{equation}
\begin{aligned}
d_0 &= \frac34 \rho  \\
d_j &= d_{j-1} - 32   l_{j-1} \qquad \mbox{for $j \geq 1$}.
\end{aligned}
\end{equation}

For each $N_0 \geq 200$, the shells satisfy the following properties.

* The union of fattened shells is contained in an annulus:

\begin{equation} \label{eq:contained-in-annulus}
\bigcup_{j \geq 0} (S_j + B_{8 l_j}) \Subset \A_{\rho/2, \rho}(x_0).
\end{equation}

* Shells are sufficiently far apart:

\begin{equation} \label{eq:sparse-shells}
(S_j + B_{8 l_j}) \cap \left( \cup_{j' \neq j} S_{j'} \right) = \emptyset \qquad \mbox{for $j \geq 0$}.
\end{equation}

* Balls centered at points of a shell are contained in an annulus: for each $j \geq 0$, let

\begin{equation} \label{eq:def-epsilon-j}
\epsilon_j := 2^{-(N_0 + j)} \quad \text{and} \quad d^+_j := d_j + 8 l_j .
\end{equation}

Then for each $z\in S_j$,

\begin{equation} \label{eq:annulus-contained-in-ball}
B_{8 \epsilon^{1/2}_j \rho}(z) \subset \A_{\rho/2, d^+_j}(x_0) \subset \A_{\rho/2, \rho}(x_0).
\end{equation}

The first two properties (<ref>) and (<ref>) are immediate from the definitions of $d_j$ and $l_j$.
The second inclusion in (<ref>) follows from unpacking the definitions:

\[
d^+_j = d_j + 8 l_j \leq d_0 + 8 l_0 =  (3/4 + 8 \times 2^{-100}) \rho < \rho.
\]

We now check check the first inclusion in (<ref>). First note

\begin{equation}
d_j - l_j \leq |z - x_0| \leq d_j + l_j, \quad \forall z \in S_j
\end{equation}

\begin{equation} \label{eq:shell-ball-contained}
d_j - l_j - 8 \epsilon^{1/2}_j \rho \leq |z'-x_0| \leq d_j + l_j + 8 \epsilon^{1/2}_j \rho, \quad \forall z' \in B_{8 \epsilon^{1/2}_j \rho}(z), z \in S_j.
\end{equation}

We claim that the first inclusion in (<ref>) follows from the following inequality
which we verify below,

\begin{equation} \label{eq:desired-inequality-shell-decomp}
l_j + 8 \epsilon^{1/2}_j \rho \leq 4 l_j.
\end{equation}

Indeed, if (<ref>) holds, then by (<ref>)
and the definitions of $d^+_j$ and $S_j$,

\[
B_{8 \epsilon^{1/2}_j \rho}(z) \subset S_j + B_{4 l_j} \subset \A_{\rho/2, d^+_j}, \quad \forall z \in S_j.
\]

It remains to check (<ref>):

\begin{align*}
l_j + 8 \epsilon^{1/2}_j \rho &= \rho \times 2^{-j/4} \times 2^{-50} + 8 \times  2^{-N_0/2-j/2} \times \rho \\
&\leq \rho \times 2^{-j/4} \left(  2^{-50} + 8 \times 2^{-N_0/2}  \right) \qquad \mbox{(since $2^{-j/4} \geq 2^{-j/2}$ for $j \geq 0$)} \\
&\leq 2 \times \rho \times 2^{-j/4} \times 2^{-50} \qquad \mbox{(since $N_0 \geq 200$)} \\
&= 2 l_j.
\end{align*}

We next show that an upper bound for the amount of mass in $\A_{\rho/2, d^+_j}(x_0)$ implies an upper bound for the odometer in $\A_{\rho/2, d^+_{j+1}}(x_0)$. This will be a key input in the induction argument in the proof of Lemma <ref>.

Fix an annulus $\A_{\rho/2,\rho}(x_0)$ for which <ref> occurs
with parameters $a,b \in (0,1)$, $N_0 \geq 200$, and universal constant $C_3 > 0$. Let the shell decomposition $\{S_j, d^+_j\}_{j \geq 0}$ be given by Lemma <ref> and let $\widetilde \Lambda_t$ be a connected component of ${\cluster{}{t}} \cap B_{\rho}(x_0)$.
For $j\geq 0$, let

\begin{equation} \label{eq:k-j-parameter}
K_j = a \times \Mrho{\rho}{x_0} \times \epsilon^{\beta^-}_j \qquad \mbox{($\epsilon_j$ from \eqref{eq:def-epsilon-j} and $\beta^-$ from Lemma \ref{lemma:volume-growth})} .
\end{equation}

Then for each $j\geq 0$,

\begin{align} \label{eq:low-mass-small-odometer}
& \sup_{\substack{r > 0 , z \in B_1 : \\ \A_{r, 7r}(z) \subset \A_{\rho/2, d^+_j}(x_0)}}  \mu_h(\widetilde \Lambda_t \cap \A_{3r, 5r}(z)) \leq K_j \notag \\
&\qquad\qquad \implies
\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{j+1}(x_0)}} \odometer{}{t}  \leq  b^{C' \times C_3 \times (N_0 + j)}  \left( \sup_{\tilde \Lambda_t \cap \A_{\rho/2, d^+_j}(x_0)} \odometer{}{t} \right)
\end{align}

where $C' > 0$ is a universal constant.

Let $j \geq 0$ be given and let $\widetilde \Lambda_t$ be a connected component of ${\cluster{}{t}} \cap B_{\rho}(x_0)$. We first check that the assumption

\[
\sup_{\substack{r > 0 , z \in B_1 : \\ \A_{r, 7r}(z) \subset \A_{\rho/2, d^+_j}(x_0)}}    \mu_h(\widetilde \Lambda_t \cap \A_{3 r, 5r}(z)) \leq K_j
\]

allows us to use (<ref>) on sufficiently many annuli covering $S_j$.

In particular, there exists a finite set of points $Z\subset  S_j \cap \frac{\epsilon_j \rho}{100} \Z^2$ so that $\cup_{z\in Z} B_{\rho \epsilon_j}(z)$
covers $S_j$. Fix one such $z \in Z$,

\[
N = \lfloor C_3 \log \epsilon_j^{-1/2} \rfloor,
\]

and let $r_1, \ldots, r_N \in [\epsilon_j \rho, \epsilon_j^{1/2} \rho] \cap \{\rho 7^{-n}\}_{n \in \N}$ be distinct radii $r_1 > r_2 > \cdots > r_N$ for which the event $E_r(z)$ occurs, as provided by <ref>.

Step 1: Small mass in annuli.

For each $r \in \{ r_1, \ldots, r_N\}$,

\begin{align*}
\mu_h(\widetilde \Lambda_t \cap \A_{3r,5r }(z))
&\leq K_j \qquad \mbox{($\A_{r, 7r}(z) \subset \A_{\rho/2, d_j^+}(x_0)$ by \eqref{eq:annulus-contained-in-ball} and since $r  \leq \epsilon_j^{1/2} \rho$)} \\
&= a \times \Mrho{\rho}{x_0} \times \epsilon^{\beta^-}_j  \qquad \mbox{(definition of $K_j$)}\\
&\leq a \times \frac{\mu_h(B_{\epsilon_j \rho }(z + 4 r e_1))}{\epsilon_j^{\beta^-}} \times  \epsilon_j^{\beta^-} \qquad \mbox{(definition of $\Mrho{\rho}{x_0}$ and \eqref{eq:annulus-contained-in-ball})}  \\
&= a \times \mu_h(B_{\epsilon_j \rho}(z + 4 r e_1)) \\
&\leq a \times \mu_h(\A_{3 r, 5 r}(z)) \qquad \mbox{(since $r \geq r_N \geq \epsilon_j \rho$)}.
\end{align*}

Hence, we may use the estimate on the exit probability  (<ref>) on each such annulus with $Y = \widetilde \Lambda_t \cap \A_{3r,5r }(z)$.
In fact, we may use it with $Y$ set to be any connected component of  $\widetilde \Lambda_t \cap \A_{3r,5r }(z)$.

Step 2: Small mass in good annulus implies small odometer.

First note that as $\A_{3 r_1, 5 r_1}(z) \Subset \A_{\rho/2, d^+_j}(x_0)$,

\begin{equation} \label{eq:outer-odom-sup}
\sup_{\partial \A_{3 r_1, 5 r_1}(z) \cap \widetilde \Lambda_t} \odometer{}{t} \leq \sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_j}(x_0)} \odometer{}{t}.
\end{equation}

For each $x \in \widetilde \Lambda_t \cap \partial B_{4 r_1}(z)$, let $\mathcal{B}^x$ denote an independent Brownian motion started at $x$ and $\Lambda^x_t$
the connected component of $\A_{3r_1, 5r_1}(z) \cap {\cluster{}{t}}$ containing $x$.
Note that for each such $x$, we have $\Lambda_t^x \subset \widetilde \Lambda_t$ (and hence $\mu_h(\Lambda_t^x) \leq \mu_h(\tilde \Lambda_t))$.
We use this to see that for each $x \in \widetilde \Lambda_t \cap \partial B_{4 r_1}(z)$,

\begin{align*}
&\odometer{}{t}(x) \\
&\leq \left( \sup_{\partial \A_{3r_1, 5r_1}(z) \cap \overline \Lambda^x_t} \odometer{}{t} \right)  \P[\mbox{$\mathcal{B}^x$ exits  $\A_{3r_1 , 5r_1}(z)$ before  $\Lambda^x_t$} | h]
\quad \mbox{(Lemma \ref{lemma:harmonic-odometer-comparison} with $A = \A_{3r_1, 5r_1}(z)$)} \\
&\leq b \left( \sup_{\partial \A_{3r_1, 5r_1}(z) \cap \overline \Lambda^x_t} \odometer{}{t} \right)  \quad \mbox{(\eqref{eq:good-annulus-harmonic} with $Y = \Lambda^x_t$)} \\
&\leq b  \left( \sup_{  \A_{3 r_1, 5 r_1}(z) \cap  \widetilde \Lambda_t}   \odometer{}{t} \right) \quad \mbox{($\Lambda_t^x \subset \widetilde \Lambda_t$)}.
\end{align*}

As $\widetilde \Lambda_t$ is a connected component
of ${\cluster{}{t}} \cap \A_{\rho/2, \rho}(x_0)$,  $\widetilde \Lambda_t \cap B_{4 r_1}(z)$ is a union
of connected components of ${\cluster{}{t}} \cap B_{4 r_1}(z)$.
The same argument in the proof of Lemma <ref> shows that $\odometer{}{t} 1\{ \cdot \in \widetilde \Lambda_t\}$
is subharmonic in $B_{\rho}(x_0)$.

This together with the maximum principle and the most recent indented inequality shows that

\[
\sup_{\widetilde \Lambda_t \cap B_{4 r_1}(z)} \odometer{}{t} \leq 	\sup_{\widetilde \Lambda_t \cap \partial B_{4 r_1}(z)} \odometer{}{t} \leq  b \left( \sup_{ \A_{3 r_1, 5 r_1}(z) \cap \widetilde \Lambda_t} \odometer{}{t} \right).
\]

Since the next annulus, $\A_{r_2, 7 r_2}(z) \subset B_{4 r_1}(z)$ ($r_2 \leq r_1/7$ by construction), this implies that

\[
\sup_{  \A_{r_2, 7 r_2}(z) \cap \widetilde \Lambda_t } \odometer{}{t} \leq  b  \left( \sup_{  \A_{3 r_1, 5 r_1}(z) \cap \widetilde \Lambda_t} \odometer{}{t} \right) .
\]

Step 3: Iterate.

We have shown in Step 1 that each annulus $\A_{r, 7 r}(z)$ for $r \in \{r_1, \ldots, r_N\}$ satisfies the conditions required to use (<ref>), so we may iterate Step 2 $(N-1)$ times, then apply (<ref>), to get

\begin{equation} \label{eq:first-iteration}
\sup_{\widetilde \Lambda_t \cap \A_{r_N, 7 r_N}(z)} \odometer{}{t}
\leq b^{N-1}  \left( \sup_{  \A_{3 r_1, 5 r_1}(z)  \cap \widetilde \Lambda_t} \odometer{}{t} \right)
\leq b^{N-1} \left( \sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_j}(x_0)} \odometer{}{t} \right).
\end{equation}

Since the estimate (<ref>) holds for all $z\in Z$, $r_N \geq \rho \epsilon_j$, and $\cup_{z\in Z} B_{\rho \epsilon_j}(z ) \supseteq S_j$, the maximum principle applied in $B_{r_N}(z)$ for each $z\in Z$ gives

\begin{equation} \label{eq:odom-sup-S}
\sup_{\widetilde \Lambda_t \cap S_j} \odometer{}{t}  \leq b^{N-1} \left( \sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_j}(x_0)} \odometer{}{t}  \right).
\end{equation}

As previously mentioned, $v_t 1\{ \cdot \in \widetilde \Lambda_t\}$ is subharmonic in $B_{\rho}(x_0)$.
Therefore, as $S_j$ disconnects $\partial B_{d^+_{j+1}}(x_0)$ from $\partial B_{\rho}(x_0)$,

\[
\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{j+1}}(x_0)} \odometer{}{t} \leq \sup_{\widetilde \Lambda_t \cap S_j} \odometer{}{t}.
\]

This combined with (<ref>) gives

\begin{equation} \label{eq:odom-sup-end}
\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{j+1}}(x_0)} \odometer{}{t} \leq b^{N-1} \left( \sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_j}(x_0)} \odometer{}{t}  \right) .
\end{equation}

We now recall that $N = \lfloor C_3 \log \epsilon_j^{-1/2} \rfloor$ and $\epsilon_j = 2^{-(N_0 + j)}$ (<ref>). Hence the lemma statement follows from (<ref>).

Schematic outline of the proof of Lemma <ref>

We conclude with the proof of the desired claim.

Let $C^{\pm}_1 > 0, C^{\pm}_2 >0$, corresponding to <ref> and <ref>,
and $a \in (0,1)$, $C_3 > 0$, corresponding to <ref>, be given.  Fix $b = b(\gamma) \in (0,1)$, $N_0 \geq N_1(a,C^-_1, C^+_2) \geq 200$, and then $\alpha = \alpha(a, N_0, \gamma, C^-_1) \in (0,1)$ which will be specified
in (<ref>), (<ref>), and (<ref>) respectively.

Fix $\rho \in (0,1)$ and $x_0 \in B_1$ such that $\overline E_\rho(x_0)$ occurs and let $\widetilde \Lambda_t$ be a connected component of ${\cluster{}{t}} \cap \A_{\rho/2, \rho}(x_0)$ with $\overline{\widetilde \Lambda_t} \cap \partial B_{\rho}(x_0) \neq \emptyset$. We assume that

\begin{equation} \label{eq:ind-assumption}
\mu_h(\widetilde{\Lambda}_t \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0))
\end{equation}

and we seek to show that $\widetilde{\Lambda}_t \cap \partial B_{\rho/2}(x_0) = \emptyset$.

Start by decomposing the annulus via the shell decomposition  $\{S_j, d^+_j\}_{j \geq 0}$ given by Lemma <ref>.
We will iteratively apply Lemma <ref> and Lemma <ref>.
Suppose $r>0$ and $z \in B_1$ such that $\A_{r,7r}(z) \subset \A_{\rho/2, d^+_{j+1}}(x_0)$.
As $\widetilde \Lambda_t$ is a connected component of $\A_{\rho/2,\rho}(x_0)$ and $\A_{r,7r}(z) \Subset \A_{\rho/2,\rho}(x_0)$, we have that $\A_{r, 7r}(z) \cap \widetilde \Lambda_t$ is a union of connected components of ${\cluster{}{t}} \cap \A_{r, 7r}(z)$. Hence, we may use the following consequence of Lemma <ref>:

\begin{align} \label{eq:small-odometer-implies-low-mass}
&\mu_h(\A_{3r, 5r}(z) \cap \widetilde \Lambda_t) \leq C_4 \sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{j+1}}(x_0)} \odometer{}{t} \notag\\
&\qquad\qquad \forall j \geq 0 \mbox{ and all $r>0$ and $z\in B_1$ such that } \A_{r, 7r}(z) \subset \A_{\rho/2, d^+_{j+1}}(x_0),
\end{align}

where $C_4 > 0$ is a universal constant.

Step 1: Choose $\alpha$ and set up iteration.

We set up the iteration of Lemma <ref> and Lemma <ref>.
Start by choosing

\begin{equation} \label{eq:choice-of-alpha}
\alpha := a \times 2^{-N_0 \beta^-} \times \frac{1}{C^-_1 \vee 1}
\end{equation}

so that our assumption (<ref>) implies

\begin{equation} \label{eq:step-4-initial-assumption}
\mu_h(\widetilde \Lambda_t \cap \A_{\rho/2, \rho}(x_0)) \leq a \times 2^{-N_0 \beta^-} \times \frac{1}{C^-_1 \vee 1}  \times \mu_h(\A_{\rho/2, \rho}(x_0)).
\end{equation}

Let $K_j$ be as in Lemma <ref>. We will show

\begin{equation} \label{eq:step-4-induct-mass}
\sup_{\substack{r > 0 , z \in B_1 : \\ \A_{r, 7r}(z) \subset \A_{\rho/2, d^+_j}(x_0)}}    \mu_h(\widetilde \Lambda_t \cap \A_{3 r, 5r}(z)) \leq K_j, \quad \forall j \geq 0
\end{equation}

\begin{equation} \label{eq:step-4-induct-odometer}
\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{j}(x_0)}} \odometer{}{t} \leq \frac{1}{C_4 \vee 1} K_j, \quad \forall j \geq 1 \quad
\end{equation}

where the universal constant $C_4$ is from (<ref>).

Once we show this, then we may take $j \to \infty$ in (<ref>) and use that $\lim_{j\to \infty} K_j = 0$ to get that $\odometer{}{t} $ is zero on $\widetilde\Lambda_t \cap \partial B_{d_\infty^+}(x_0)$, where $ d_\infty^+:= \lim_{j\to\infty} d_j^+ \in (\rho/2,\rho)$.
Since $\odometer{}{t} 1\{ \cdot \in {\widetilde \Lambda_t}\}$ is subharmonic in $B_{\rho}(x_0)$, this implies that

\begin{equation} \label{eq:zero-odometer-in-inner-ball}
\sup_{\widetilde \Lambda_t \cap \overline B_{\rho/2}(x_0)} \odometer{}{t} = 0,
\end{equation}

implying the desired statement by the definition of ${\cluster{}{t}}$.

Hence, it remains to prove (<ref>) and (<ref>).
Our strategy is to induct on $j \geq 0$ and show the following chain of implications:

\[
\{ \mbox{\eqref{eq:step-4-induct-mass} for $j$ and \eqref{eq:step-4-induct-odometer} for $(j+1)$} \} \implies 	\{ \mbox{\eqref{eq:step-4-induct-mass} for $(j+1)$ and \eqref{eq:step-4-induct-odometer} for $(j+2)$} \}.
\]

We start with the base case.

Step 2: Base case $j = 0$.

By <ref> and the inequality (<ref>)

\begin{equation} \label{eq:step-4-initial-mass-bound}
\mu_h(\widetilde \Lambda_t \cap \A_{\rho/2, \rho}(x_0)) \leq a \times \Mrho{\rho}{x_0} \times 2^{-N_0 \beta^-}  = K_0,
\end{equation}

which is (<ref>) for $j = 0$.
Since $\alpha \in (0,1)$, by (<ref>),

\[
\A_{\rho/2, \rho}(x_0) \cap \widetilde{\Lambda}_t^c \neq \emptyset.
\]

Therefore, by Lemma

\begin{equation} \label{eq:step-4-odometer-upper-bound}
\sup_{\A_{\rho/2, \rho}(x_0)} \odometer{}{t} \leq c \times \SGrho{\rho}{x_0} 	\leq c \times C^+_2 \times \frac{1}{C^-_1 \vee 1} \times \Mrho{\rho}{x_0}
\end{equation}

with the latter inequality following from <ref> and <ref>. The inequality (<ref>) allows us to use Lemma <ref>
to see that

\begin{equation} \label{eq:first-step-initial-bound}
\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{1}}(x_0)} \odometer{}{t} \leq b^{C' \times C_3 \times N_0} \left(	\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{0}}(x_0)} \odometer{}{t} \right).
\end{equation}

Now, pick

\begin{equation} \label{eq:choice-of-b}
b := 2^{-\frac{2 \beta^-}{(C' \times C_3) \vee 1}}  .
\end{equation}

We emphasize that $b$ depends only on $\gamma$ (through $\beta^-$) since $C_3$ is a universal constant.
We also choose $N_1$ sufficiently large so that

\begin{equation} \label{eq:choice-of-N1}
2^{-\beta^- N_1} \leq a \times 2^{-\beta^-} \times \frac{\frac{1}{C_4 \vee 1}}{(c \times C^+_2 \times \frac{1}{C^-_1 \vee 1}) \vee 1}.
\end{equation}

Fix some $N_0 \geq N_1$. With these choices of $b$ and $N_0$, we have

\begin{equation} \label{eq:choice-of-b-and-m0}
b^{C' \times C_3 \times N_0} \leq
2^{-N_0 \times 2 \beta^-} \leq
\frac{ \frac{1}{C_4 \vee 1} \times a 2^{-\beta^-(N_0+1)}}{(c \times C^+_2 \times \frac{1}{C^-_1 \vee 1}) \vee 1}.
\end{equation}

Hence, by (<ref>) followed by (<ref>) and (<ref>),

\begin{equation} \label{eq:base-case-odometer-size}
\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, d^+_{1}(x_0)}} \odometer{}{t} \leq b^{C' \times C_3 \times N_0} \left(	\sup_{\widetilde \Lambda_t \cap \A_{\rho/2, \rho}(x_0)} \odometer{}{t} \right) \leq \frac{1}{C_4 \vee 1} a 2^{-\beta^-(N_0+1)} \Mrho{\rho}{x_0}	= \frac{1}{C_4 \vee 1} K_1,
\end{equation}

which is (<ref>) for $j = 1$.

Step 3: Inductive step, $j \to (j+1)$.

If (<ref>) holds for $(j+1)$, then
by (<ref>), we have (<ref>) for $(j+1)$.
It remains to show that

\[
\{\mbox{\eqref{eq:step-4-induct-mass} for $(j+1)$ and \eqref{eq:step-4-induct-odometer} for $(j+1)$} \} \implies 	\{\mbox{\eqref{eq:step-4-induct-odometer} for $(j+2)$} \}.
\]

This is similar to the argument of the base case however we will not need the full strength of the inductive step (unlike the base case). In particular we will use the very crude bound $b^{C' \times C_3 \times (N_0 + j)} \leq 2^{-\beta^-}$.

By (<ref>) for $(j+1)$
we may use Lemma <ref>. Hence,

\begin{align*}
\sup_{	\A_{\rho/2, d^+_{j+2}}(x_0)} \odometer{}{t}
&\leq b^{C' \times C_3 \times (N_0 + j + 1)} \sup_{	\A_{\rho/2, d^+_{j+1}}(x_0)} \odometer{}{t} \quad \mbox{(by \eqref{eq:low-mass-small-odometer})} \\
&\leq b^{C' \times C_3 \times (N_0 + j)} \frac{1}{C_4 \vee 1} K_{j+1} \quad \mbox{(by \eqref{eq:step-4-induct-odometer} for $(j+1)$)}  \\
&\leq 2^{-\beta^- } \times \frac{1}{C_4 \vee 1} K_{j+1} \qquad \mbox{(by \eqref{eq:choice-of-b-and-m0})} \\
&= \frac{1}{C_4 \vee 1} K_{j+2} \quad \mbox{(by \eqref{eq:k-j-parameter})}
\end{align*}

which is (<ref>) for $(j+2)$, completing the proof.

§ UPPER BOUND AND CONTINUITY

In this section we prove, using the Harnack-type estimate Proposition <ref>, that clusters do not immediately exit the unit ball and are in fact Hölder-continuous in the parameter $t$.
We start with the upper bound.

For each $r \in (0,1)$, on an event which occurs with polynomially high probability as $T \to 0$,

\[
\overline{{{\cluster{}{t}}}} \subset B_r ,\quad \forall t \leq T.
\]

Fix $r \in (0,1)$ and let $\overline E_{\rho}(x_0)$ and $\alpha \in (0,1)$ be the event and parameter from Proposition <ref>.
By Proposition <ref>, it holds with polynomially high probability as $\epsilon \to 0$ that for each $x\in (B_{1 + \epsilon} \backslash B_{10 \sqrt{\epsilon}}) \cap \frac{\epsilon}{100} \Z^2$, there exists $\rho_x \in [\epsilon,\epsilon^{1/2}]$ such that $\overline E_{\rho_x}(x)$ occurs. Henceforth assume that this is the case for some $\epsilon \in (0, 2^{-20} r^2)$.

Let $\mathbf{X} \subset  (B_{1 + \epsilon} \backslash B_{10 \sqrt{\epsilon}}) \cap \frac{\epsilon}{100} \Z^2$ be a set such that

\begin{equation}
\bigcup_{x \in \mathbf{X}} \A_{\rho_x/2, \rho_x}(x) \Subset B_{1} \backslash B_{r/4}
\end{equation}

\begin{equation} \label{eq:cover-inner-ball}
\partial B_{r/2} \subset \bigcup_{x \in \mathbf{X}} B_{\rho_x/2}(x).
\end{equation}

This is possible since for each $x$, we have $\epsilon \leq \rho_x$ and $\rho_x^{1/2} \leq r/100$ — see Figure <ref>.

By Lemma <ref>, it holds with polynomially high probability as $\epsilon \to 0$ that

\begin{equation} \label{eq:upper-bound-annuli}
\mu_h(\A_{\rho_x/2,\rho_x}(x )) \geq \rho_x^{2 \beta^-}, \quad \forall x \in \mathbf{X}, \quad \mbox{$\beta^-$ from Lemma \ref{lemma:volume-growth}}.
\end{equation}

Henceforth assume that $\epsilon \in  (0, 2^{-20} r^2)$ is such that (<ref>) holds.

Now choose $T \leq \alpha \epsilon^{2 \beta^-}$. Then for each $x \in \mathbf{X}$ and each $t \leq T$,

\begin{align*}
\mu_h({\cluster{}{t}} \cap \A_{\rho_x/2,\rho_x}(x)) &\leq \mu_h({\cluster{}{t}}) \quad \mbox{(monotonicity, Lemma \ref{lemma:monotonicity})} \\
&\leq  \alpha \epsilon^{2 \beta^-} \qquad \mbox{(choice of $T$ and Lemma \ref{lemma:conservation-of-mass})} \\
&\leq \alpha \rho_x^{2 \beta^-} \qquad \mbox{($\rho_x \geq \epsilon$)} \\
&\leq \alpha \mu_h(\A_{\rho_x/2,\rho_x}(x )) \qquad \mbox{(by \eqref{eq:upper-bound-annuli})}.
\end{align*}

Thus, we may apply  Proposition <ref> and (<ref>) to see that

\begin{equation}
\sup_{\partial B_{r/2}} \odometer{}{t} = 0, \quad \forall t \leq T.
\end{equation}

Since $\sup_{\partial B_1} \odometer{}{t} = 0$ by Lemma <ref> and $\odometer{}{t}$ is subharmonic away from the origin, this implies

\begin{equation} \label{eq:upper-bound-odometer}
\sup_{\A_{r/2,1}} \odometer{}{t} = 0, \quad \forall t \leq T.
\end{equation}

By the definition (<ref>) of ${\cluster{}{t}}$, (<ref>) implies that ${\cluster{}{t}} \subset B_{r/2}$.
We conclude by recalling that both the condition in the first paragraph and (<ref>) hold with polynomially high probability as $\epsilon\to 0$.

Covering of $\partial B_{r/2}$ by very good annuli as in the proof of Proposition <ref>. The cluster ${\cluster{}{t}}$ is displayed in gray with a
solid black border and the good annuli are light gray with a dotted black border.

We next prove continuity of the clusters, using a similar argument as in the proof of Proposition <ref>.

The following occurs on an event of probability 1.
For each $t > 0$ such that ${\cluster{}{t}} \Subset B_1$, for all $\epsilon > 0$ sufficiently small, depending on $t$,

\[
{\cluster{}{t + \epsilon}} \Subset {\cluster{}{t}} + B_{\delta} ,\quad \text{for $\delta = C \epsilon^{1/(4\beta^-)}$}
\]

where $\beta^- > (2+\gamma)^2/2$ is from Lemma <ref> and  $C = C(\beta^-) >0$ is a deterministic constant.

Let $\overline E_{\rho}(x_0)$ and $\alpha \in (0,1)$ be the event and parameter from Proposition <ref>.
By Proposition <ref>, it holds with polynomially high probability as $\delta \to 0$ that for each $x\in (B_{1 + \delta} \backslash B_{10 \sqrt{\delta}}) \cap \frac{\delta}{100} \Z^2$, there exists $\rho_x = \rho_x(\delta) \in [\delta,\delta^{1/2}]$ such that $\overline E_{\rho_x}(x)$ occurs.
By Lemma <ref>, it also holds with polynomially high probability as $\delta \to 0$ that

\begin{equation} \label{eq:volume-growth-lower-bound}
\mu_h\left( \A_{\rho_x/2, \rho_x}(x ) \right)  \geq \rho_x^{2 \beta^-}, \quad \forall x \in (B_{1 + \delta } \backslash B_{10 \sqrt{\delta }})  \cap \frac{\delta }{100} \Z^2 .
\end{equation}

By the Borel-Cantelli lemma, a.s. there exists $M_0$ sufficiently large such that the preceding two conditions hold for each $\delta \in \{2^{-n}\}_{n \geq M_0}$.

Let $t > 0$ and $r \in (0,1)$ such that ${\cluster{}{t}} \Subset B_r$. By Proposition <ref>, there exists $\epsilon_0 > 0$ so that $B_{\epsilon_0} \subset {\cluster{}{t}}$.
Hence, since ${\cluster{}{t}} \Subset B_1$, by taking $M_0$ possibly larger (depending on $\epsilon_0$), we can arrange that

\begin{equation} \label{eq:boundary-inside}
({\cluster{}{t}} + B_{20 \sqrt{\delta }}) \backslash ({\cluster{}{t}} + B_{\sqrt{\delta }}) \Subset  B_1 \backslash B_{10 \sqrt{\delta }}, \quad \forall \delta  \in \{2^{-n}\}_{n \geq M_0} .
\end{equation}

By (<ref>) and the fact that $\rho_x \in [\delta , \delta^{1/2}]$, we obtain that for each $\delta  \in \{2^{-n}\}_{n \geq M_0}$ there exists $\mathbf{X} = \mathbf{X}(\delta) \subset (B_{1 + \delta } \backslash B_{10 \sqrt{\delta }})  \cap \frac{\delta }{100} \Z^2$ for which

\begin{equation} \label{eq:cover-continuous}
({\cluster{}{t}} + B_{4 \sqrt{\delta }}) \backslash ({\cluster{}{t}} + B_{3 \sqrt{\delta }}) \Subset \bigcup_{x \in \mathbf{X}} B_{\rho_x/2}(x ) \Subset B_{1- 10 \sqrt{\delta}}
\end{equation}

\begin{equation} \label{eq:cover-in-complement}
\bigcup_{x \in \mathbf{X}} B_{\rho_x}(x) \Subset {{{\cluster{}{t}}}}^c
\end{equation}

for each $\delta  \in \{2^{-n}\}_{n \geq M_0}$.

As ${\cluster{}{t}} \Subset B_1$, by Lemma <ref> we have $\mu_h({\cluster{}{t}}) = t$.
Thus, by monotonicity, Lemma <ref>,

\[
\mu_h({\cluster{}{t + \epsilon}} \backslash {\cluster{}{t}}) \leq \epsilon, \quad \forall \epsilon > 0.
\]

Therefore, by (<ref>), whenever $\delta  \in \{2^{-n}\}_{n \geq M_0}$ we have

\begin{equation} \label{eq:expand-cluster-mass}
\mu_h(\A_{\rho_x/2,\rho_x}(x ) \cap {\cluster{}{t+\epsilon}}) \leq \epsilon, \quad \forall x \in \mathbf{X}, \quad \forall \epsilon > 0.
\end{equation}

Now set

\[
\epsilon = \epsilon(\delta) := \alpha \delta^{2 \beta^-},
\]

(where $\alpha$ is as in Proposition <ref>). Then for each $\delta  \in \{2^{-n}\}_{n \geq M_0}$ and each $x \in \mathbf{X}$, we have

\begin{align*}
\mu_h(\A_{\rho_x/2,\rho_x}(x) \cap {\cluster{}{t+\epsilon }}) &\leq \alpha \delta^{2 \beta^-} \quad \text{(by \eqref{eq:expand-cluster-mass})} \\
&\leq \alpha \rho_x^{2 \beta^-} \quad \mbox{($\rho_x \geq \delta $)} \\
&\leq \alpha \mu_h(\A_{\rho_x/2,\rho_x}(x)) \quad \mbox{(by \eqref{eq:volume-growth-lower-bound})}.
\end{align*}

Therefore, as the event $\overline E_{\rho_x }(x )$ occurs for each $x \in \mathbf{X}$, by Proposition <ref>,

\[
B_{\rho_x/2}(x ) \cap {\cluster{}{t+\epsilon }} = \emptyset, \quad \forall x \in \mathbf{X}, \quad \forall \delta  \in \{2^{-n}\}_{n \geq M_0} .
\]

This implies by (<ref>) that

\begin{equation} \label{eq:zero-on-shell}
\sup_{({\cluster{}{t}} + B_{4 \sqrt{\delta}}) \backslash ({\cluster{}{t}} + B_{3 \sqrt{\delta}})} \odometer{}{t+\epsilon } = 0, \quad \forall \delta  \in \{2^{-n}\}_{n \geq M_0}.
\end{equation}

Since $\sup_{\partial B_1} \odometer{}{t+\epsilon } = 0$ by Lemma <ref> and $\odometer{}{t+\epsilon }$ is subharmonic away from the origin, (<ref>) implies

\begin{equation}
\sup_{B_1 \backslash ({\cluster{}{t}} + B_{3 \sqrt{\delta }})} \odometer{}{t+\epsilon } = 0, \quad \forall \delta  \in \{2^{-n}\}_{n \geq M_0} .
\end{equation}

By the definition (<ref>) of $\cluster{}{t+\epsilon }$, this implies that

\begin{equation}
{\cluster{}{t+\epsilon }} \Subset {\cluster{}{t}} + B_{4\sqrt{\delta}}, \quad \forall \delta  \in \{2^{-n}\}_{n \geq M_0} .
\end{equation}

Recalling that $\epsilon = \alpha \delta^{2 \beta^-}$, this concludes the proof.

For completeness, we indicate how the above two results imply Proposition <ref>.

The fact that the clusters continuously increase in $t$, (<ref>), is implied by Proposition <ref>.
We show here that Proposition <ref> implies that the clusters decrease to the origin, (<ref>).

By Proposition <ref>, for each $t > 0$, there exists $\epsilon(t) > 0$ so that $B_{\epsilon(t)} \subset {\cluster{}{t}}$.
By Proposition <ref> and the Borel-Cantelli lemma, applied to a sequence of dyadic radii, $r_n := \{2^{-n}\}_{n \geq 2}$,
a.s. there exists a sequence of positive times $\{t_n\}_{n \geq 2}$ so that

\[
\overline{{\cluster{}{t_n}}} \Subset B_{r_n}, \quad \forall n \geq 2.
\]

The desired claim follows by combining the previous two sentences with monotonicity of the clusters in $t$, Lemma <ref>.

§ BOUNDARY HAS MEASURE ZERO

We show that the boundary of a cluster compactly embedded in the unit ball has LQG-measure zero.

On an event of probability 1,

\[
\mu_h(\partial {\cluster{}{t}}) = 0 \qquad \mbox{for all ${\cluster{}{t}} \Subset B_1$}.
\]

Our strategy for doing so is to use the Lebesgue density theorem together with some of the intermediate results from Section <ref>.

The Lebesgue density theorem for general Radon measures on $\C$ <cit.> shows that a.s. for every Borel set $X$, the set of $\mu_h$-density points of $X$ has full $\mu_h$-mass, i.e.,

\begin{equation} \label{eq:density-0}
\lim_{r \to 0} \frac{\mu_h(B_r(z) \cap X)}{\mu_h(B_r(z))} = 1,\quad \text{for $\mu_h$-a.e.\ $z\in X$}.
\end{equation}

We want to deduce Proposition <ref> from the density theorem for $\mu_h$ in Euclidean balls (<ref>)
together with the Harnack-type estimate from Section <ref>.
However, the results of Section <ref> are in terms of the LQG mass of the intersection of a cluster with an annulus. In order to compare the $\mu_h$-masses of balls and annuli, we require a doubling property for the $\mu_h$-masses of Euclidean balls, , an up-to-constants comparison of the $\mu_h$-masses of $B_{2^{-n}}(z)$ and $B_{2^{-n-1}}(z)$ with the constant independent of $n$ and $z$. That is, we require an event of the form,

\begin{equation} \label{eq:annulus-ball-comparison}
G_{n}(z) = G_n(z; m) := \left\{\mu_{h}(\A_{2^{-n-1}, 2^{-n}}(z)) \geq m \mu_{h}(B_{2^{-n}}(z)) \right\}
\end{equation}

for $z \in \C$ and $m > 0$.
Due to the randomness of $\mu_h$, such an event does not hold uniformly over all choices of $n$ and $z$. Instead, we will show that for $\mu_h$-a.e. $z \in \C$, one has this estimate
and a Harnack-type property for `most' large values of $n$. For convenience, write

\begin{equation} \label{eq:very-good-annuli-event}
\overline E_{n}(z) := \overline E_{2^{-n}}(z)
\end{equation}

for $z \in \C$ and $n \in \N$ where $\overline E_{\rho}(z)$ is the very good event from Section <ref>.

Let $\zeta > 0$ and $b \in (0,1)$. There exists $m > 0$ and parameters corresponding to the event $\overline E_{\rho}(z) = \overline E_{\rho}(z; N_0, a, b, C_1^{\pm}, C_2^{\pm}, C_3)$ (from Section <ref>) such that a.s. for $\mu_h$-a.e. $z\in  B_1$, it holds for each large enough $N\in \N$ (depending on $z$) that

\begin{equation} \label{eq:mu_h-doubling}
\#\left\{ n \in [N+1,2N] \cap \Z : \mbox{$\overline E_{n}(z)$ and $G_{n}(z)$ occur} \right\}  \geq (1-\zeta) N.
\end{equation}

Recall that $h^{\C} = h + \boldsymbol{\alpha}_0 \log |\cdot|$ is the whole plane GFF as defined in (<ref>). It is a standard fact from LQG theory that if $U\subset  \C$ is open and $Z$ is sampled from $\mu_{h^\C}|_U$, normalized to be a probability measure, then near $Z$ the field $h$ locally looks like $\tilde h - \gamma \log |\cdot-Z|$, where $\tilde h$ is a GFF sampled independently from $Z$ (see, e.g., <cit.>). Hence, Lemma <ref> will turn out to be a consequence  of the following statement for a GFF with a logarithmic singularity at 0.

Let $\zeta > 0$ and $b \in (0,1)$. There exists $m > 0$ and parameters corresponding to the event $\overline E_{\rho}(0) = \overline E_{\rho}(0; N_0, a, b, C_1^{\pm}, C_2^{\pm}, C_3)$ (from Section <ref>) such that a.s. for each large enough $N \in \N$, the condition (<ref>) holds.

By the scale invariance of the law of $h$, viewed modulo additive constant (<ref>), and the LQG coordinate change formula for $\mu_{h}$ (Fact <ref>), the law of

\[
\mu_{h}(\A_{2^{-n-1}, 2^{-n}}(0))/\mu_{h}(B_{2^{-n}}(0))
\]

does not depend on $n$.
Furthermore, this random variable is a.s. finite and strictly larger than 0. Hence,
$   \P[G_n(0; m)]$ does not depend on $n$ and we can find $m = m(\alpha,\zeta,\gamma) > 0$ such that

\begin{equation} \label{eq:doubling-prob}
\P[G_n  ] \geq 1 - \zeta /8 , \quad\forall n\in\N ,\quad \text{where $G_n := G_n(0; m)$} .
\end{equation}

By  Lemma <ref> there is a choice of parameters so that

\begin{equation} \label{eq:harnack-prob}
\P[\overline E_n] \geq 1 - \zeta /8 , \quad \forall n \geq 0 ,\quad \text{where $\overline E_n := \overline E_n(0)$} .
\end{equation}

Hence, by a union bound

\begin{equation} \label{eq:doubling-and-harnack-prob}
q := \P[\overline E_n \cap G_n] \geq 1- \zeta/4.
\end{equation}

By the scale invariance of the law of $h$ modulo additive constant and the fact that the occurrence of the event $G_n \cap \overline E_n$ does not depend on the choice of additive constant for $h$, the sequence of random variables

\[
\{1_{G_n \cap \overline E_n}\}_{n \geq 0}
\]

is stationary.
Hence, by the Birkhoff ergodic theorem,

\[
\frac{1}{N} \sum_{n=1}^N  1_{G_n \cap \overline E_n}
\]

converges a.s. and in $L^1$ to a (possibly random) limit. The limiting random variable is measurable with respect to the $\sigma$-algebra $\bigcap_{\epsilon > 0} \sigma(h|_{B_\epsilon(0)})$, which is trivial (see <cit.> for a proof of the analogous tail triviality statement for a free-boundary GFF; the proof for a whole-plane GFF is similar).
Therefore, the limiting random variable is a.s. constant, and hence is a.s. equal to the number $q$ from (<ref>). Consequently, a.s.

\begin{equation}
\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N  1_{G_n \cap \overline E_n} = q.
\end{equation}

\begin{equation}
\lim_{N \to \infty} \frac{1}{N} \sum_{n=N+1}^{2N}  1_{G_n \cap \overline E_n}
= \lim_{N \to\infty} \frac{1}{N} \sum_{n=1}^{2N}  1_{G_n \cap \overline E_n} - \lim_{N \to\infty} \frac{1}{N} \sum_{n=1}^{N}  1_{G_n \cap \overline E_n}
= 2q - q
\geq 1-\zeta/2 .
\end{equation}

By the definition of $G_n$ and $\overline E_n$, this implies the lemma statement.

Recall that $h = h^{\C} - \boldsymbol{\alpha}_0 \log|\cdot|$, where $h^{\C}$ is a whole-plane GFF normalized so that $h^{\C}_1(0) = 0$.
By Weyl scaling, we have $\mu_{h^{\C}} = |\cdot|^{\boldsymbol{\alpha}_0 \gamma} \mu_h$.
Conditional on $h$, let $Z$ be sampled from $|\cdot|^{\boldsymbol{\alpha}_0 \gamma} \mu_{h}|_{B_1}$, normalized to be a probability measure.
By <cit.>, the law of the pair $(h, Z)$ is mutually absolutely continuous with respect to the law of the pair $(\tilde h , \tilde Z)$, where $\tilde Z$ is sampled from Lebesgue measure in $B_1$ independently from $h$ and $\tilde h = h  - \gamma\log|\cdot-\tilde Z|  + \gamma \log \max\{|\cdot|,1\}$, with $h$ and $\tilde h$ viewed as distributions modulo additive constant.

From the definitions of $G_n(\tilde Z)$ and $\overline E_n(\tilde Z)$ and the locality property of $\mu_{\tilde h}$ (Fact <ref>), we have

\begin{equation} \label{eq:doubling-locality}
G_n(\tilde Z) \cap \overline E_n(\tilde Z) \in \sigma\left( \tilde Z  , h |_{B_{2^{-n+1}}(\tilde Z)} \right).
\end{equation}

Almost surely, $\tilde Z \not= 0$. If $r < |\tilde Z|$, then the restriction of $\tilde h$ to $B_r(\tilde Z)$ is equal to the restriction of a whole-plane GFF to $B_r(\tilde Z)$ plus $- \gamma\log|\cdot-\tilde Z|$ plus the function $-\boldsymbol{\alpha}_0 \log|\cdot|  + \gamma \log\max\{|\cdot|,1\}$, which is smooth on $B_r(\tilde Z)$. By standard absolute continuity results for the GFF (see, e.g., <cit.>), the conditional law of $\tilde h|_{B_r(\tilde Z)}$ given $\tilde Z$ is absolutely continuous with respect  to the law of the corresponding restriction of a whole-plane GFF plus $-\gamma\log|\cdot-\tilde Z|$.
From this, (<ref>), Lemma <ref> (with $\boldsymbol{\alpha}_0 =\gamma$), and the translation invariance of the law of the whole-plane GFF, viewed modulo additive constant, we get that if the parameters for $\overline E_n(\cdot)$ and  $G_{n}(\cdot)$ are chosen as in Lemma <ref>, then a.s. for each large enough $N\in\N$,

\begin{equation} \label{eq:mu_h-doubling-reweight}
\#\left\{ n \in [N+1,2N] \cap \Z  : \mbox{$\overline E_{n}(\tilde Z)$ and $G_{n}(\tilde Z)$ occur with $\tilde h$ in place of $h$} \right\}  \geq (1-\zeta) N.
\end{equation}

By absolute continuity, the same is also true with $(h,Z)$ in place of $(\tilde h , \tilde Z)$. Since $Z$ is sampled from $|\cdot|^{\boldsymbol{\alpha}_0 \gamma} \mu_{h}|_{B_1}$, we get that a.s. the lemma statement holds for $\mu_h$-a.e. $z\in B_1$.

We conclude with a proof of the desired claim.

By Lemma <ref>
and Lemma <ref>
we can choose parameters $b$, $N_0$, $a,C^{\pm}_1, C^{\pm}_2, C_3, c, m$
so that for some fixed $\alpha \in (0,1)$, it holds for each $x_0 \in B_1 \setminus \{0\}$ and each $\rho \in (0,|x_0|)$ that

\begin{equation} \label{eq:good-annulus-implies-harnack}
\mbox{$\overline E_{\rho}(x_0)$ occurs} \implies  \{ 		\mu_h({\cluster{}{t}} \cap \A_{\rho/2, \rho}(x_0)) \leq \alpha \mu_h(\A_{\rho/2, \rho}(x_0)) \implies {\cluster{}{t}} \cap B_{\rho/2}(x_0) = \emptyset \}
\end{equation}

and the implication of Lemma <ref>
holds with $\zeta = 1/2$.

We will now show that $\mu_h(\partial{\cluster{}{t}}) = 0$ for every $t>0$ such that ${\cluster{}{t}} \Subset B_1$. By (<ref>), a.s. for every $t>0$ we have

\begin{equation} \label{eq:density-0-apply0}
\lim_{\rho \to 0} \frac{\mu_h(B_{\rho}(z) \cap \partial {\cluster{}{t}})}{\mu_h(B_{\rho}(z))} = 1, \quad \text{for $\mu_h$-a.e.\ $z\in \partial {\cluster{}{t}}$} .
\end{equation}

Since ${\cluster{}{t}}$ is open, we have ${\cluster{}{t}} \cap \partial {\cluster{}{t}} = \emptyset$, so (<ref>) implies that

\begin{equation} \label{eq:density-0-apply}
\lim_{\rho \to 0} \frac{\mu_h(B_{\rho}(z) \cap {\cluster{}{t}})}{\mu_h(B_{\rho}(z))} = 0 , \quad \text{for $\mu_h$-a.e.\ $z\in \partial {\cluster{}{t}}$} .
\end{equation}

In particular, for $\mu_h$-a.e. $z\in \partial {\cluster{}{t}}$ it holds for each large enough $n\in\N$ that

\begin{equation} \label{eq:density-theorem-apply}
\mu_h(B_{2^{-n}}(z) \cap {\cluster{}{t}}) \leq  \alpha \times m \times \mu_h(B_{2^{-n}}(z))
\end{equation}

where $\alpha$ is as in Lemma <ref> and $m$ is as in the definition (<ref>) of $G_n(z; m)$.

By Lemma <ref>, it is a.s. the case that for $\mu_h$-a.e $z\in B_1$, there are arbitrarily large values of $n$ such that $\overline E_{n}(z)$ and $G_{n}(z)$ occur.
Hence, a.s. for each $t>0$, it holds for $\mu_h$-a.e. $z\in\partial {\cluster{}{t}}$ that there are arbitrarily large values of $n$ such that $\overline E_n(z) \cap G_n(z)$ occurs and

\begin{align*}
\mu_h(\A_{2^{-n-1}, 2^{-n}}(z) \cap {\cluster{}{t}}) &\leq \mu_h(B_{2^{-n}}(z) \cap {\cluster{}{t}})   \\
&\leq \alpha \times m \times \mu_h(B_{2^{-n}}(z)) \qquad \mbox{(by \eqref{eq:density-theorem-apply})} \\
&\leq \alpha \mu_h(\A_{2^{-n-1}, 2^{-n}}(z) \cap {\cluster{}{t}}) \qquad \mbox{(since $G_n(z)$ holds)}.
\end{align*}

Since $\overline E_{n}(z)$ holds, this implies by (<ref>) that

\[
{\cluster{}{t}} \cap B_{2^{-n-1}}(z) = \emptyset,
\]