# iharm3D: Vectorized General Relativistic Magnetohydrodynamics

Ben S. Prather111Corresponding author George N. Wong  Vedant Dhruv  Benjamin
R. Ryan  Joshua C. Dolence  Sean M. Ressler  Charles F. Gammie

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##  iharm3D Functionality and Purpose

iharm3D222https://github.com/AFD-Illinois/iharm3d is an open-source C code for
simulating black hole accretion systems in arbitrary stationary spacetimes
using ideal general-relativistic magnetohydrodynamics (GRMHD). It is an
implementation of the HARM (“High Accuracy Relativistic Magnetohydrodynamics”)
algorithm outlined in Gammie et al. (2003) with updates as outlined in
McKinney & Gammie (2004) and Noble et al. (2006). The code is most directly
derived from Ryan et al. (2015) but with radiative transfer portions removed.
HARM is a conservative finite-volume scheme for solving the equations of ideal
GRMHD, a hyperbolic system of partial differential equations, on a logically
Cartesian mesh in arbitrary coordinates.

## Statement of Need

Numerical simulations are crucial in modeling observations of active galactic
nuclei, such as the recent horizon-scale results from the Event Horizon
Telescope and GRAVITY collaborations. The computational simplicity of ideal
GRMHD enables the generation of long, high-resolution simulations and broad
parameter-exploration studies that can be compared to observations for
parameter inference.

Multiple codes already exist for solving the ideal GRMHD equations on regular
Eulerian meshes in 3D, including:

  * •

Athena++ (Stone et al. (2020), White et al. (2016))

  * •

BHAC (Porth et al. (2017))

  * •

Cosmos++ (Anninos et al. (2005), Fragile et al. (2012), Fragile et al. (2014))

  * •

ECHO (Londrillo & Zanna (2000), Londrillo & Zanna (2004))

  * •

H-AMR (Matthew Liska et al. (2019), M. Liska et al. (2020))

  * •

HARM-Noble (Noble et al. (2006), Noble et al. (2009), Noble et al. (2012),
Zilhão & Noble (2014), Bowen et al. (2018))

  * •

IllinoisGRMHD (Etienne et al. (2015))

  * •

KORAL (Sądowski et al. (2013), Sądowski et al. (2014))

  * •

GRHydro (Mösta et al. (2014))

  * •

Spritz (Cipolletta et al. (2020), Cipolletta et al. (2021))

As the length of this list illustrates, the field of GRMHD simulation is now
well established, and many codes now exist to serve different needs. These
codes can be distinguished by the trade-offs they make in prioritizing speed,
simplicity, and generality, with the latter encompassing, e.g., support for
dynamical spacetimes, adaptive mesh refinement, or higher-order integration
schemes.

In particular, iharm3D aims to be a simple and fast code capable of simulating
the original systems of interest when designing HARM, even at the cost of
features aimed at more general applicability. It provides a fast and scalable
update to HARM, but maintains the conventions and structure of the original
described in Gammie et al. (2003). The result is a code that is relatively
easy to understand and modify, yet capable of running simulations at state-of-
the-art scale.

## Implementation Notes

In MHD, uncorrected discretization errors inevitably lead to violations of the
no-monopoles condition $\nabla\cdot B=0$. As in the original HARM
implementation, iharm3D uses the “Flux-CT” scheme for cell-centered
constrained transport outlined in Tóth (2000).

iharm3D also retains numerical evaluation of all metric-dependent quantities,
allowing trivial modification of the coordinate system or background spacetime
so long as the line element is available in analytic form. This can be used as
a form of static mesh refinement, since the coordinates can be adapted to
place resolution in areas of interest (e.g., near the accretion disk
midplane).

In GRMHD, “conserved” variables (energy and momentum densities) are
complicated analytic functions of “primitive” variables (density, pressure,
and velocity). Conserved variables are stepped forward in time and then
inversion to primitives is done numerically. iharm3D uses the “$1D_{W}$”
scheme outlined in Noble et al. (2006).

As the equations of ideal GRMHD are rescalable, any consistent set of units
may be chosen to evolve them in the code. For numerical stability, when
simulating accretion systems we choose units in which $GM=c=1$ with $M$ the
mass of the central object, and scale the density of the initial conditions
such that the maximum value of $\rho$ is $1$.

To model a collisionless plasma, iharm3D implements an optional means of
tracking and partitioning dissipation into ions and electrons (Ressler et al.
(2015)). Currently the code implements five different heating models,
described in Howes (2010), Kawazura et al. (2019), Werner et al. (2018), Rowan
et al. (2017), and Sharma et al. (2007).

To avoid catastrophic failures caused by discretization error, especially in
low density regions, fluid variables are bounded at the end of each step.
Typically, the bounds in black hole accretion problems are enforced as
follows:

  * •

Density $\rho>10^{-6}k$, for $k\equiv\frac{1}{r^{2}(1+r/10)}$, with $r$ the
radial coordinate,

  * •

Internal energy density $u>10^{-8}k^{\gamma}$ where $\gamma\equiv$ adiabatic
index,

  * •

$\rho$ and $u$ are incremented until $\sigma\equiv\frac{2P_{b}}{\rho}<400$ and
$\beta\equiv\frac{P_{gas}}{P_{b}}>2.5\times 10^{-5}$ where
$P_{b}\equiv\frac{b^{2}}{2}$ is the magnetic pressure,

  * •

$\rho$ is incremented until $\frac{u}{\rho}<100$,

  * •

When evolving electron temperatures, $u$ is decremented until
$\frac{P_{gas}}{\rho^{\gamma}}<3$,

  * •

Velocity components are downscaled until Lorentz factor
$\Gamma\equiv\frac{1}{\sqrt{1-v^{2}}}<50$.

Global disk simulations inevitably invoke these bounds, most frequently those
on $\sigma$ and $\Gamma$.

## Tests

The convergence properties of HARM are well-studied in Gammie et al. (2003).
iharm3D implements most of the tests presented in that paper as integration
and regression tests. Figure 1 shows convergence results for linear modes and
for un-magnetized Bondi flow.

Figure 1: Results of convergence tests with iharm3D’s main branch, plotting L1
norm of the difference between the computed solution and the analytic or
stable result with increasing domain size. Wave solutions were performed on a
3D cubic grid $N$ zones to one side, the Bondi accretion problem was performed
on a logically Cartesian 2D square grid $N$ zones on one side.

iharm3D implements three additional tests which check that fluid evolution is
identical for different domain decompositions: one which initializes a new
fluid state, one which restarts from a checkpoint file, and one comparing the
initialized state to an equivalent checkpoint file.

## Scaling

Key iharm3D routines are written for effective compiler vectorization and
prioritize simple memory access patterns to make good use of high memory
bandwidth. Originally developed for Intel Knights Landing (KNL) chips on the
Stampede2 supercomputer at Texas Advanced Computing Center (TACC), iharm3D
also runs efficiently on TACC Frontera, which uses traditional Cascade Lake
(CLX) CPUs. Figure 2 presents scaling results for iharm3D on both Stampede2
and Frontera.

Figure 2: Strong and weak scaling performance of iharm3D. Performance is
measured in zones advanced by one cycle each second (Zone-Cycles per Second).
In the strong scaling test, an accretion torus problem of constant total size
$256^{3}$ was split among all nodes; in the weak scaling test, the total
problem size was varied to ensure that a mesh block of size $64^{3}$ was
allocated to each node.

## Research projects using iharm3D

iharm3D is one of several GRMHD codes used by the EHT Collaboration to produce
its library of fluid simulations. Images produced from this library were used
for validation tests in Event Horizon Telescope Collaboration et. al. (2019a)
and Event Horizon Telescope Collaboration et. al. (2021a) and for
interpretation of the M87 EHT results in total intensity (Event Horizon
Telescope Collaboration et. al. (2019b), Event Horizon Telescope Collaboration
et. al. (2019c)) and polarization (Event Horizon Telescope Collaboration et.
al. (2021b)).

Papers making use of the results of iharm3D simulations include Porth et al.
(2019), Johnson et al. (2020), Gold et al. (2020), Palumbo et al. (2020), Lin
et al. (2020), Ricarte et al. (2020), Wielgus et al. (2020), Tiede et al.
(2020), and Gelles et al. (2021).

## Acknowledgements

This work was supported by National Science Foundation grants AST 17-16327,
OISE 17-43747, AST 20-07936, AST 20-34306, and PHY 17-48958, by a Donald C.
and F. Shirley Jones Fellowship to G.N.W., by the Gordon and Betty Moore
Foundation through Grant GBMF7392, and by the US Department of Energy through
Los Alamos National Laboratory. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of the US Department of Energy (Contract No. 89233218CNA000001). This work has
been assigned a document release number LA-UR-21-23714.

This work used the Extreme Science and Engineering Discovery Environment
(XSEDE), which is supported by National Science Foundation grant number
ACI-1548562, specifically the XSEDE resource Stampede2 at the Texas Advanced
Computing Center (TACC) through allocation TG-AST170024. The authors
acknowledge the Texas Advanced Computing Center at The University of Texas at
Austin for providing HPC resources that have contributed to the research
results reported within this paper.

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# A Compactness Theorem for functions on Poisson point clouds

Marco Caroccia  Dipartimento di Matematica, Politecnico di Milano
<EMAIL_ADDRESS>
###### Abstract.

In this work we show a compactness Theorem for discrete functions on Poisson
point clouds. We consider sequences with equibounded non-local $p$-Dirichlet
energy: the novelty consists in the intermediate-interaction regime at which
the non-local energy is computed.

##  1\. Introduction

In this paper we prove a compactness property for sequences of discrete
functions defined on Poisson point clouds. We show (Theorem 2.3) that a
sequence with uniformly bounded $p$-Dirichlet non-local energy admits a
converging subsequence, in a suitable topology (see Definition 2.2). Non local
energies on discrete systems have been intensively treated, under the
variational point of view, in the last two decades, as a way of modelling
several phenomena: from fracture mechanics to image denoising and
crystallization. Typical discrete variational energies have their natural
definition on regular or stochastic lattices (see for instance [5], [6], [7],
[11], [13], [14], [15], [18], [21], [22], [24], [3], [4] and references
therein). In some situations we need to account in the model for a randomness
component. This leads towards a natural extension of classical models on
lattices to a more general point cloud framework. Among other interesting
features, variational energies on point clouds have been proposed also as a
tool to implement supervised or semi-supervised learning algorithms,
clustering problem, data denoising and image denoising (as a partial list of
literature on the topic we refer the reader to [19], [20], [26], [32], [33]
and references therein).

We here consider point clouds as realizations of a random variable $\eta$
called Poisson point process (we refer to Subsection 2.2 for a detailed
definitions). This formalism allows to account for point clouds that are
Poisson distributed, in the sense that

$\mathbb{P}(\\#(points\ in\
A)=m)=\mathrm{Po}(\gamma|A|,m)=e^{-\gamma|A|}\frac{(\gamma|A|)^{m}}{m!}$

where $\mathrm{Po}(t,m)$ is the probability mass function of the Poisson
distribution of parameter $t$ (we refer to Definition 2.2 for a more accurate
Definition of a Poisson point process). The parameter $\gamma$ is called the
intensity of the process. By introducing a parameter $\varepsilon$ we act on
the intensity of the process, obtaining a sequence of processes
$\eta_{\varepsilon}$ whose realizations converge (almost surely) to a
continuum set as $\varepsilon$ goes to $0$ (in the sense that the normalized
discrete measure induced by the process $\eta_{\varepsilon}$, weak*-converges
to the Lebesgue measure). For any realization $\omega$ of the process, and for
$\varepsilon>0$, we have a point cloud $\eta_{\varepsilon}(\omega)$. For
$s_{\varepsilon}>0$ the energy at scale $\varepsilon$ is then defined for
$u:\eta_{\varepsilon}(\omega)\rightarrow\mathbb{R}$ on
$A\subseteq\mathbb{R}^{d}$ as:

${\mathcal{F}}_{\varepsilon}^{\omega}(u;A):=\varepsilon^{d}\sum_{x\in\eta_{\varepsilon}(\omega)\cap
A}\frac{\varepsilon^{d}}{s_{\varepsilon}^{d}}\sum_{y\in\eta_{\varepsilon}(\omega)\cap
B_{s_{\varepsilon}}(x)}\left(\frac{|u(y)-u(x)|}{s_{\varepsilon}}\right)^{p}.$
(1.1)

Notice that the energy can be rephrased as

${\mathcal{F}}_{\varepsilon}^{\omega}(u;A):=\frac{\varepsilon^{2d}}{s_{\varepsilon}^{d}}\sum_{\begin{subarray}{c}x,y\in\eta_{\varepsilon}(\omega):\\\
|x-y|\leq
s_{\varepsilon}\end{subarray}}\left(\frac{|u(y)-u(x)|}{s_{\varepsilon}}\right)^{p}$

which express the fact that we are summing up the finite differences of $u$,
up to a distance of $s_{\varepsilon}$: the scale of averaging. The scale
$\\{s_{\varepsilon}\\}_{\varepsilon>0}$ is taken as a sequence
$s_{\varepsilon}\rightarrow 0$ that, here and in the sequel, will satisfy the
decay property:

$\liminf_{\varepsilon\rightarrow
0}\frac{s_{\varepsilon}}{\varepsilon(\log(\varepsilon^{-d}))^{1/d}}=\beta>0$
(1.2)

with $\beta<+\infty$.

The novelty of this work relies exactly in the assumption on the decay of
$s_{\varepsilon}$ in terms of $\varepsilon$, expressed by (1.2) with
$0<\beta<+\infty$. The typical approach for energies defined on point clouds
(as for instance in [20], [23], [34], [35]) makes use of the hypothesis
$\beta=+\infty$ in order to study the asymptotic behavior as
$\varepsilon\rightarrow 0$. In the long-range regime $\beta=+\infty$, the
topology of convergence is the strong one induced by the $TL^{1}$ distance
(see for instance [25] or [35]). In contrast to the case $\beta=+\infty$ we
can consider the case $\beta=0$. In the recent works [12], [17] the authors
deal with the short-range regime, where $s_{\varepsilon}\approx M\varepsilon$
or where nearest neighbors interactions are the only interactions involved. In
the short-range regime it is still possible, at least in the planar case, to
deduce the compactness of equibounded sequences (and thus the asymptotic
behavior of ${\mathcal{F}}_{\varepsilon}$) but in a much weaker topology than
the $TL^{1}$. In particular the analysis that seems to be missing in
literature, and that we aim to fill for what concerns a compactness property,
is when $0<\beta<+\infty$. As we state and prove in Theorem 2.3, when
$0<\beta<+\infty$, sequences with equibounded energy will be compact in the
topology yielding the convergence of Definition 2.2: a stronger topology than
the one accessible in the $\beta=0$ case but still weaker than the $TL^{p}$
topology (attainable in the $\beta=+\infty$ case). These differences in the
underlying topologies, arising in the asymptotic behavior analysis, are
related to the different behavior of the stochastic geometry of the point
cloud when looked at different scales.

The specialty of the comparison scale
$\sigma_{\varepsilon}:=\varepsilon\log(\varepsilon^{-d})^{1/d}$ can be naively
explained in this way. Given a stochastic Poisson point process
$\eta_{\varepsilon}$, and denoted by $\eta_{\varepsilon}(A)$ the number of
points of the point cloud that fall in $A\subset\mathbb{R}^{d}$, the expected
value of $\eta_{\varepsilon}(A)$ is (cf. Remark 2.1)

$\mathbb{E}[\eta_{\varepsilon}(A)]\approx\frac{|A|}{\varepsilon^{d}}.$

Thus, for a point $x\in\eta_{\varepsilon}$, the expected number of point
$y\in\eta_{\varepsilon}$ that will interact with $x$ at scale
$s_{\varepsilon}$ is approximately
$\nicefrac{{s_{\varepsilon}^{d}}}{{\varepsilon^{d}}}$
($\approx\eta_{\varepsilon}(B_{s_{\varepsilon}}(x))$). By invoking some well
known properties of Poisson distribution it is easy to see that the
probability that a point $x$ interacts, at a scale $s_{\varepsilon}$, with
either too much, or too few points is approximately

$p_{\varepsilon}:=\mathbb{P}\left(\eta_{\varepsilon}(B_{s_{\varepsilon}}(x))\geq
2\frac{s_{\varepsilon}^{d}}{\varepsilon^{d}}\ \ \ \text{or}\ \ \
\eta_{\varepsilon}(B_{s_{\varepsilon}}(x))\leq\frac{1}{2}\frac{s_{\varepsilon}^{d}}{\varepsilon^{d}}\right)\approx
e^{-C_{d}\frac{s_{\varepsilon}^{d}}{\varepsilon^{d}}}.$

Therefore depending on the behavior of $s_{\varepsilon}$ compared to
$\sigma_{\varepsilon}$ several phenomena can occur. If
$s_{\varepsilon}>>\sigma_{\varepsilon}$ then $p_{\varepsilon}$ tends to zero
fast enough to guarantee that, almost surely, each point $x$ will always
interact with approximately
$\nicefrac{{s_{\varepsilon}^{d}}}{{\varepsilon^{d}}}$ points, resulting in no
lack of information when approaching the continuum limit as
$\varepsilon\rightarrow 0$. This is precisely the approach that has been
implemented in [20], [25] to investigate energies on point clouds and that
allows to deduce convergence in $TL^{p}$. If $s_{\varepsilon}\approx
M\varepsilon$ then $p_{\varepsilon}\approx e^{-C_{d}M}$ and thus we expect to
see a fixed percentage of points that have the wrong number of interactions at
scale $M\varepsilon$ resulting in a loss of information. In particular, a
uniform bound on the energy ${\mathcal{F}}_{\varepsilon}$, might not contain
enough information to deduce convergence, up to a subsequence, in a strong
topology. Heuristically speaking, the geometry of the point cloud at this
scale, becomes closely related to the geometry of perforated domains: all the
points with few interactions act as holes in the domain. For the planar case,
in [12] and [17], a fine analysis of the geometry of the point clouds, and the
identification of regular subclusters, is required in order to describe the
asymptotic behavior of ${\mathcal{F}}_{\varepsilon}$. In the intermediate-
regime, given by (1.2), the probability
$p_{\varepsilon}\approx\varepsilon^{d\beta^{d}}$ vanishes as
$\varepsilon\rightarrow 0$, but not fast enough to guarantee that any point
$x$ will be well connected to its neighbors at scale $s_{\varepsilon}$ (as in
the case $\beta=+\infty$). However it is still fast enough to deduce that the
number of wrong points remains under control (see Proposition 3.7). This is
now similar to the study of energies on perforated domain, where we have some
controls on the size of the holes (see for instance [8], [16]). In this sense
we can deduce convergence, up to subsequences, in a stronger way than in the
case $\beta=0$ but in a weaker sense than in the case $\beta=+\infty$.

The paper is organized as follows. We briefly introduce the main ingredient of
our analysis and the main Theorem 2.3 in Section 2. We then proceed to the
study of the stochastic geometry of a general realization of the Poisson point
process in Section 3. Technically this analysis is implemented in this way: we
divide $\mathbb{R}^{d}$ in boxes of size $s_{\varepsilon}$ and we consider the
family of “bad boxes” that contains either too few or too many points. We show
an upper bound on the number of these boxes (Proposition 3.7) and on the
volume of each connected component of the union of the boxes (Lemma 3.6).
Finally, in Section 4, we present the proof of our main result 2.3. For
$u_{\varepsilon}:\eta_{\varepsilon}\cap Q\rightarrow\mathbb{R}$ we consider
the averaged function

$u^{I}_{\varepsilon}:=\frac{1}{\eta_{\varepsilon}(Q_{s_{\varepsilon}}(I))}\sum_{x\in\eta_{\varepsilon}\cap
Q_{s_{\varepsilon}}(I)}u(x)$

where $Q_{s_{\varepsilon}}(I)$ does not belong to the set of “bad boxes”. We
then have a function

$\hat{u}_{\varepsilon}(x):=\sum_{\begin{subarray}{c}Q_{s_{\varepsilon}}(I)\text{
not a}\\\ \text{``bad
box"}\end{subarray}}u_{\varepsilon}^{I}\mathbbmss{1}_{Q_{s_{\varepsilon}}(I)}(x).$

We use the results of Section 3, on the stochastic geometry of the process, to
build an extension operator that extends the function $\hat{u}_{\varepsilon}$
also on the “bad boxes” by keeping under control the finite differences (Lemma
4.6). We then use a compactness Theorem for functions on lattices to prove
compactness for the extended functions. Notice that this argument fails as
soon as we loose control on the size of the family of bad boxes, namely if
$\beta=0$.

### Acknowledgment

The author is grateful to Professor Andrea Braides for the many fruitful
discussions on the topics here contained. The author is grateful also to the
anonymous referee for his/her very precise report and for all the observations
that helped enrich the contribution here contained.

##  2\. Preliminaries and main result

###  2.1. General notation

Given $x=(x_{1},\ldots,x_{n})\in\mathbb{R}^{d}$, $r>0$ we denote by

$Q_{r}(x):=\left[x_{1}-\nicefrac{{r}}{{2}},x_{1}+\nicefrac{{r}}{{2}}\right]\times\ldots\times\left[x_{n}-\nicefrac{{r}}{{2}},x_{n}+\nicefrac{{r}}{{2}}\right]$

the closed cube centered at $x$ and with side length $r$. When $r=1$ and
$x_{0}=0$ we simply write $Q$. For a Borel set $E$, the notation $|E|$ stands
for the Lebesgue measure of $E$. For a discrete set $S:=\\{a,b,c,\ldots\\}$
the notation $\\#(S)$ stands for the cardinality of $S$. We will set, for
$A\subset\mathbb{R}^{d}$,

$\displaystyle\mathcal{I}_{s}(A)$ $\displaystyle:=\\{J\in s\mathbb{Z}^{d}\cap
A\\}.$

The notation $\mathbbmss{1}_{A}(x)$ stands for the characteristic function of
the set $A$. In all the estimates that follow, $C$ will be a constant
independent of the crucial quantities under analysis and that might change
from line to line.

###  2.2. Poisson point process

Given a probability space $(\Omega,\mathcal{F},\mathbb{P})$, a point process
is a random variable $\eta:\Omega\rightarrow\mathbf{N}_{s}(\mathbb{R}^{d})$,
where $\mathbf{N}_{s}(\mathbb{R}^{d})$ is the space of simple measures of
$\mathbb{R}^{d}$

$\mathbf{N}_{s}(\mathbb{R}^{d}):=\left\\{\left.\sum_{n\in\mathbb{N}}\delta_{x_{n}}\
\right|\ \\{x_{n}\\}_{n\in\mathbb{N}}\subset\mathbb{R}^{d}\ \text{any family
of distinct points}\ x_{n}\neq x_{k},\ k\neq n\right\\}.$

In particular, for any $\omega\in\Omega$, we have that $\eta(\omega)$ is a
measure concentrated on a countable family of points. A Poisson point
processes, or Poisson point cloud, of intensity $\gamma$ is a point process
$\eta:\Omega\rightarrow\mathbf{N}_{s}(\mathbb{R}^{d})$ such that

  * a)

for all $k\in\mathbb{N}$ it holds
$\mathbb{P}(\eta^{\omega}(A)=k):=e^{-\gamma|A|}\frac{(\gamma|A|)^{k}}{k!}$;

  * b)

for all $B_{1},\ldots,B_{k}$ essentially disjoint borel sets, the random
variable $X_{j}:\Omega\rightarrow\mathbb{N}$,
$X_{j}(\omega):=\eta(\omega)(B_{j})$ are independent.

For further details on Poisson point processes we refer the reader to [28].
Given $\eta$ a Poisson point process we define

$\eta_{\varepsilon}(\omega)(B):=\eta(\omega)(\varepsilon^{-1}B).$

Notice that $\eta_{\varepsilon}$ is just a Poisson point process with
intensity $\gamma\varepsilon^{-d}$. Also, for any realization,

$\operatorname*{spt}(\eta_{\varepsilon}(\omega))=\varepsilon\operatorname*{spt}(\eta(\omega)).$

We write, with a slight abuse of notation, $x\in\eta_{\varepsilon}(\omega)$,
meaning $x\in\operatorname*{spt}(\eta_{\varepsilon}(\omega))$.

######  Remark 2.1.

Observe that, for a generic Borel set $A\subset\mathbb{R}^{d}$,

$\displaystyle\mathbb{E}[\eta_{\varepsilon}(A)]$
$\displaystyle=\sum_{k=0}^{+\infty}k\mathbb{P}(\eta_{\varepsilon}(A)=k)=e^{-\gamma\varepsilon^{-d}|A|}\sum_{k=0}^{+\infty}k\frac{\left(\gamma\varepsilon^{-d}|A|\right)^{k}}{k!}$
$\displaystyle=\gamma\varepsilon^{-d}|A|e^{-\gamma\varepsilon^{-d}|A|}\sum_{k=1}^{+\infty}\frac{\left(\gamma\varepsilon^{-d}|A|\right)^{k-1}}{(k-1)!}=\gamma\varepsilon^{-d}|A|.$

###  2.3. $p$-Dirichlet energy

Given a Poisson point cloud
$\eta:\Omega\rightarrow\mathbf{N}_{s}(\mathbb{R}^{d})$ with intensity $\gamma$
and $\\{s_{\varepsilon}\\}_{\varepsilon>0}$ a sequence converging to zero, we
consider the $p$-Dirichlet energy on $A$ of
$u:\eta_{\varepsilon}(\omega)\rightarrow\mathbb{R}_{+}$ at scale
$s_{\varepsilon}$ to be

$\mathcal{F}^{\omega}_{\varepsilon}(u;A):=\varepsilon^{d}\sum_{x\in\eta_{\varepsilon}(\omega)\cap
A}|\mathrm{grad}_{s_{\varepsilon}}u(x)|^{p}$

where

$|\mathrm{grad}_{s_{\varepsilon}}u(x)|^{p}:=\frac{\varepsilon^{d}}{s_{\varepsilon}^{d}}\sum_{y\in\eta_{\varepsilon}(\omega)\cap
B_{s_{\varepsilon}}(x)}\left(\frac{|u(y)-u(x)|}{s_{\varepsilon}}\right)^{p}.$

In the sequel we omit to specify the dependence of ${\mathcal{F}}$ and
$\eta_{\varepsilon}$ on the realization when there is no ambiguity.

###  2.4. Main result

For $x\in\eta$ we consider the Voronoi cell to be

$V(x;\eta):=\\{y\in\mathbb{R}^{d}\ |\ |y-x|\leq|y-z|\ \text{for all
$z\in\eta\setminus\\{x\\}$}\\}.$

For a function $u_{\varepsilon}:\eta_{\varepsilon}\cap Q\rightarrow\mathbb{R}$
we set
$\hat{u}_{\varepsilon}:\mathcal{V}_{\varepsilon}(\eta_{\varepsilon})\cap
Q\rightarrow\mathbb{R}$

$\hat{u}_{\varepsilon}(x):=\sum_{y\in\eta_{\varepsilon}\cap
Q}u_{\varepsilon}(y)\mathbbmss{1}_{V(y;\eta_{\varepsilon})}(x).$

Set

$\eta^{\alpha}:=\left\\{x\in\eta\ :\ \mathrm{In}(V(x;\eta))>\alpha,\
\operatorname*{diam}(V(x;\eta)<\alpha^{-1}\right\\}$

being

$\mathrm{In}(E):=\sup_{r>0}\\{\text{there exists $B_{r}\subset E$}\\}.$

We refer the reader to Figure 2.1. The subcloud $\eta_{\alpha}$ has been
firstly introduced in [17], where the authors derive the continuum limit of an
interface-type energy on Poisson point clouds.

For $\alpha,\varepsilon>0$ we set also
$\eta^{\alpha}_{\varepsilon}:=(\eta_{\varepsilon})_{\alpha\varepsilon}$,
(observe that $\eta^{\alpha}_{\varepsilon}=\varepsilon\eta^{\alpha}$) and, for
a general subcloud of points $\vartheta\subset\eta_{\varepsilon}$, we define

$\mathcal{V}_{\varepsilon}(\vartheta):=\bigcup_{x\in\vartheta}V(x;\eta_{\varepsilon}).$

We are in the position to introduce the next Definition.

Figure 2.1. An illustration of a Poisson point cloud
$\eta_{\varepsilon}(\omega)$ on the left. On the right an illustration of
$\mathcal{V}_{\varepsilon}(\eta^{\alpha}_{\varepsilon})$ (generated by a sub-
cluster $\eta^{\alpha}_{\varepsilon}$) depicted in dark grey.

######  Definition 2.2 (Convergence).

We say that a sequence $u_{\varepsilon}\stackrel{{\scriptstyle
p}}{{\longrightarrow}}u$ in $A$ if for any $\alpha>0$ it holds

$\lim_{\varepsilon\rightarrow
0}\int_{\mathcal{V}_{\varepsilon}(\eta^{\alpha}_{\varepsilon})\cap
A}|\hat{u}_{\varepsilon}-u|^{p}\,\mathrm{d}x=0.$

With this notion of convergence we state our main result.

######  Theorem 2.3.

Let
$\\{u_{\varepsilon}:\eta_{\varepsilon}\rightarrow\mathbb{R}\\}_{\varepsilon>0}$.
Let $A\supset Q$ be any open set containing $Q$ and assume that
$s_{\varepsilon}$ is a sequence satisfying (1.2). Then for almost all
realization $\omega\in\Omega$ the following holds. Suppose that

$\sup_{\varepsilon>0}\left\\{{\mathcal{F}}_{\varepsilon}^{\omega}(u_{\varepsilon};A)+\sum_{x\in\eta_{\varepsilon}(\omega)\cap
Q}u_{\varepsilon}(x)^{p}\varepsilon^{d}\right\\}<+\infty.$ (2.1)

Then there exists a subsequence
$\\{u_{\varepsilon_{n}}\\}_{n\in\mathbb{N}}\subset\\{u_{\varepsilon}\\}_{\varepsilon>0}$
and a function $u\in W^{1,p}(Q)$ such that
$u_{\varepsilon_{n}}\stackrel{{\scriptstyle q}}{{\longrightarrow}}u$ in $Q$
for all $q<p$, in the sense of Definition 2.2.

######  Remark 2.4.

Notice that, since the energy is non local, to have compactness on $Q$ we need
some control also on the interactions between $x\in Q,y\in Q^{c}$ with
$|x-y|\leq s_{\varepsilon}$. This is why we require that the energy bound in
(2.1) to hold on a slightly bigger open set $A\supset Q$. As it will be clear
from the proof, Theorem 2.3 holds as well if we replace the cube $Q$ with a
generic open set.

##  3\. Stochastic geometry of the point cloud

For any realization $\omega$, given $s>0$ and $0<\kappa<1$ a parameter we set

$\displaystyle\mathcal{I}^{\kappa,\omega}_{s}(Q)$
$\displaystyle:=\left\\{J\in\mathcal{I}_{s}(Q)\ :\
\left|\eta_{\varepsilon}(\omega)(Q_{s}(J))-\gamma\frac{s^{d}}{\varepsilon^{d}}\right|\geq\kappa\gamma\frac{s^{d}}{\varepsilon^{d}}\right\\}$
(3.1) $\displaystyle E^{\kappa}_{s}(\omega)$
$\displaystyle:=\bigcup_{\mathcal{I}^{\kappa,\omega}_{s}(Q)}Q_{s}(J).$ (3.2)

The next Lemma is a standard concentration inequality for random variables
(see for instance [10, Corollary 2.11]). Again we will omit the dependence on
$\omega$ to lighten the notation.

######  Lemma 3.1 (Cramer-Chernoff concentration bounds).

If $X$ is a random variable with Poisson distribution then

$\mathbb{P}(|X-\mathbb{E}[X]|>t)\leq\exp\left(-\frac{2t^{2}}{(t+\mathbb{E}[X])}\right).$

We also employ the well-known Borel-Cantelli Lemma (see [29]).

######  Lemma 3.2 (Borel-Cantelli Lemma).

If $\\{E_{n}\subseteq\Omega\\}_{n\in\mathbb{N}}$ is a sequence of events such
that

$\sum_{n\in\mathbb{N}}\mathbb{P}(E_{n})<+\infty$

then

$\mathbb{P}\left(\limsup_{n\rightarrow+\infty}E_{n}\right)=\mathbb{P}\left(\bigcap_{n=1}^{+\infty}\bigcup_{k\geq
n}E_{k}\right)=0.$

######  Remark 3.3 (Chebishev inequality).

Given $X:\Omega\rightarrow\mathbb{R}$ a random variable, a useful way to
estimate the probability of the event $A:=\\{\omega\in\Omega\ :\ X(\omega)\geq
t\\}$ is the so-called Chebishev inequality

$\mathbb{P}(A)=\int_{A}\,\mathrm{d}\mathbb{P}(\omega)\leq\frac{1}{t}\int_{A}X(\omega)\,\mathrm{d}\mathbb{P}(\omega)\leq\frac{1}{t}\int_{\Omega}X(\omega)\,\mathrm{d}\mathbb{P}(\omega)=\frac{\mathbb{E}[X]}{t}.$

Let $\\{X_{J}\ :\ J\in\mathcal{I}_{s}(\mathbb{R}^{d})\\}$ be a countable
family of independent Bernoulli random variables taking value $X_{J}=1$ with
probability $p$. The probability $\mathbb{P}_{p}$ is defined

$\mathbb{P}_{p}(X_{J_{1}}=1,\ldots,X_{J_{k}}=1):=p^{k}$

and then extended to the $\sigma$-algebra generated by the cylinders

$A(J_{1},\ldots,J_{k})=\\{X_{J_{1}}=1,\ldots,X_{J_{k}}=1\\}.$

######  Definition 3.4.

We say that a subset of indexes
$\mathcal{J}\subset\mathcal{I}_{s}(\mathbb{R}^{d})$ is path connected if for
any $J,J^{\prime}$ there exists a family (called a path in $\mathcal{J}$),
$\\{J_{1},\ldots,J_{k}\\}$ with

  * a)

$J_{1}=J,J_{k}=J^{\prime}$;

  * b)

$J_{i}\in\mathcal{J}$ for all $i=1,\ldots k$;

  * c)

$|J_{i}-J_{i+1}|=s$ for all $i=1,\ldots,k-1$.

For $\mathcal{J}\subset\mathcal{I}_{s}(Q)$, $J\in\mathcal{J}$ the set

$U(J):=\\{J^{\prime}\in\mathcal{J}\ :\ \text{$J^{\prime}$ is connected to $J$
through a path in $\mathcal{J}$}\\}$

is called the connected component of
$\mathcal{J}\subset\mathcal{I}_{s}(\mathbb{R}^{d})$ containing $J$. Clearly if
$J^{\prime}\in U(J)$ then $U(J^{\prime})=U(J)$ (see Figure 4.1).

We will also employ the following Bernoulli bond percolation result. Lemma 3.5
summarizes a combination of standard results in Bernoulli bond percolation
theory (see [1], [2], [9], [27], [31]). We underline that the choice of the
origin $J=0$ in the following statement plays no role. In particular
(3.3),(3.4) remain valid for a generic $J$.

######  Lemma 3.5.

Let $\\{X_{J}\ :\ J\in\mathcal{I}_{s}(\mathbb{R}^{d})\\}$ be a countable
family of independent Bernoulli random variables taking value $X_{J}=1$ with
probability $p$. Then if $p<p_{c}(d)$, being $p_{c}$ a critical value
depending on the dimension only, the set
$\\{J\in\mathcal{I}_{s}(\mathbb{R}^{d}):X_{J}=0\\}$ is almost surely path
connected. Moreover, there exists $\psi_{p}$ depending on $p$ only such that,
setting $U(0)$ to be the (possibly empty) connected component of
$\\{J\in\mathcal{I}_{s}(\mathbb{R}^{d}):X_{J}=1\\}$ containing $J=0$, it holds

$\mathbb{P}_{p}(\\#(U(0))>N)\leq e^{-\psi_{p}N}\ \ \ \text{for all $N\geq
1$}.$ (3.3)

Finally for all $p<p^{\prime}$.

$\mathbb{P}_{p}(\\#(U(0))>N)\leq\mathbb{P}_{p^{\prime}}(\\#(U(0))>N).$ (3.4)

We now prove a statistical estimate which allows us to obtain information on
those regions of the point cloud having the wrong number of interactions.

######  Lemma 3.6.

Suppose that

$\limsup_{\varepsilon\rightarrow+\infty}\frac{\varepsilon}{s_{\varepsilon}}=0.$

Then there exists a subsequence $\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$ and a
constant $\Lambda$ such that almost surely the following hold. For all
$\kappa\in(0,1)$ we can find $n_{0}$ (depending on $\kappa$ and the
realization only) for which (setting for shortness
$s_{n}=s_{\varepsilon_{n}}$),
$\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{I}_{s_{n}}^{\kappa}(Q)$ is path
connected for any $n\geq n_{0}$ and it holds

$\sup\left\\{\\#(U)\ :\ \text{$U$ conn. comp. of
$\mathcal{I}_{s_{n}}^{\kappa}(Q)$}\right\\}\leq\Lambda\log(s_{n}^{-1})\ \
\text{for all $n\geq n_{0}$}.$ (3.5)

###### Proof.

For any $J\in\mathcal{I}_{s_{\varepsilon}}^{\kappa}(Q)$ consider $U(J)$ to be
the (possibly empty) connected component of
$\mathcal{I}_{s_{\varepsilon}}^{\kappa}(Q)$ containing $J$. Setting, for
$J\in\mathcal{I}_{s_{\varepsilon}}(Q)$,

$X_{J}^{\varepsilon}:=\left\\{\begin{array}[]{lr}1&\text{if
$J\in\mathcal{I}_{s_{\varepsilon}}^{\kappa}(Q)$}\\\
0&\text{otherwise}.\end{array}\right.$ (3.6)

we have, by invoking Lemma 3.1, that

$p_{\varepsilon}(\kappa)=\mathbb{P}(X_{J}^{\varepsilon}=1)\leq
e^{-\frac{2\kappa^{2}}{1+\kappa}\frac{s_{\varepsilon}^{d}}{\varepsilon^{d}}}.$

In particular, for any $\kappa>0$, $p_{\varepsilon}(\kappa)\rightarrow 0$.
Thus, for any $\kappa$ we can find $\varepsilon_{0}=\varepsilon_{0}(\kappa,d)$
such that $p_{\varepsilon}(\kappa)\leq p_{0}<p_{c}(d)$ and
$\mathcal{I}_{s_{\varepsilon}}(Q)\setminus\mathcal{I}_{s_{\varepsilon}}^{\kappa}(Q)$
is almost surely path connected for all $\varepsilon\leq\varepsilon_{0}$. In
particular

$\mathbb{P}_{p_{\varepsilon}(\kappa)}(\\#(U(0))\geq
N)\leq\mathbb{P}_{p_{0}}(\\#(U(0))\geq N)\leq e^{-\psi_{p_{0}}N}.$

for some $\psi_{p_{0}}$. Fix $\Lambda^{\prime}>d$ and consider the subsequence
$\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$ such that

$s_{\varepsilon_{n}}^{\Lambda^{\prime}-d}\leq\frac{1}{n^{2}}.$

Setting $s_{n}:=s_{\varepsilon_{n}}$, let us then consider the events

$A_{n}:=\left\\{\omega\in\Omega\ :\ \text{exists
$J\in\mathcal{I}_{s_{n}}(Q)$}\ \text{s.t.}\
\\#(U(J))\geq\frac{\Lambda^{\prime}}{\psi_{p_{0}}}\log(s_{n}^{-1})\right\\}.$

Then

$\displaystyle\mathbb{P}(A_{n})$
$\displaystyle\leq\mathbb{P}\left(\bigcup_{J\in\mathcal{I}_{s_{n}}(Q)}\left\\{\omega\in\Omega\
:\
\\#(U(J))\geq\frac{\Lambda^{\prime}}{\psi_{p_{0}}}\log(s_{n}^{-1})\right\\}\right)$
$\displaystyle\leq\sum_{J\in\mathcal{I}_{s_{n}}(Q)}\mathbb{P}\left(\left\\{\omega\in\Omega\
:\
\\#(U(J))\geq\frac{\Lambda^{\prime}}{\psi_{p_{0}}}\log(s_{n}^{-1})\right\\}\right)$
$\displaystyle=\sum_{J\in\mathcal{I}_{s_{n}}(Q)}\mathbb{P}_{p_{\varepsilon_{n}}(\kappa)}\left(\\#(U(J))\geq\frac{\Lambda^{\prime}}{\psi_{p_{0}}}\log(s_{n}^{-1})\right)$
$\displaystyle\leq s_{n}^{\Lambda^{\prime}}\\#(\mathcal{I}_{s_{n}}(Q))\leq
Cs_{n}^{\Lambda^{\prime}-d}\leq Cn^{-2}.$

By invoking Borel-Cantelli Lemma 3.2 we conclude that the probability that
$A_{n}$ occurs infinitely many time is $0$. Thus, naming
$\Lambda:=\frac{\Lambda^{\prime}}{\psi_{p_{0}}}$ we conclude. ∎

We will also make use of the following Proposition, providing bounds on the
cardinality of $\mathcal{I}_{s_{\varepsilon}}^{\kappa}(Q)$ (up to select a
subsequence $\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$).

######  Proposition 3.7.

Let $s_{\varepsilon}$ be a sequence satisfying

$\liminf_{\varepsilon\rightarrow
0}\frac{s_{\varepsilon}}{\varepsilon\log(\varepsilon^{-d})^{\nicefrac{{1}}{{d}}}}=\beta>0.$
(3.7)

Then the following hold:

  * a)

If $\beta>\frac{1}{\gamma^{1/d}}$, there exists $0<\kappa_{0}<1$ and a
subsequence $\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$ such that, setting
$s_{n}=s_{\varepsilon_{n}}$, for all $\kappa_{0}\geq\kappa<1$ and almost
surely we can find $n_{0}$ for which

$\\#(\mathcal{I}_{s_{{}_{n}}}^{\kappa}(Q))=0\ \ \ \text{for all $n\geq
n_{0}$};$

  * b)

If $0<\beta\leq\frac{1}{\gamma^{1/d}}$ there exist two constants
$\kappa_{0}>0$, $\varrho_{0}>0$ depending on $\beta,d,\gamma$ only and a
subsequence $\\{\varepsilon_{n}\\}_{n\in\mathbb{N}}$ such that, setting
$s_{n}:=s_{\varepsilon_{n}}$, for all $\kappa_{0}\leq\kappa<1$ and almost
surely we can find $n_{0}$ for which

$\\#(\mathcal{I}_{s_{n}}^{\kappa}(Q))\leq\frac{\varepsilon_{n}^{\varrho_{0}}}{s_{n}^{d}}\
\ \text{for all $n\geq n_{0}$.}$

###### Proof.

Set
$\sigma_{\varepsilon}:=\varepsilon\log(\varepsilon^{-d})^{\nicefrac{{1}}{{d}}}$.
We can select a subsequence such that, setting $\beta_{n}=\beta-\frac{1}{n}$,
$\sigma_{n}:=\sigma_{\varepsilon_{n}}$, we have

$s_{n}\geq\beta_{n}\sigma_{n}.$

Notice that (cf. Remark 2.1)

$\mathbb{E}[\eta_{\varepsilon}(Q_{s}(J))]=\gamma\frac{s^{d}}{\varepsilon^{d}}.$

In particular, Lemma 3.1 yields

$\displaystyle p_{n}$
$\displaystyle:=\mathbb{P}\left(\left|\eta_{\varepsilon_{n}}(Q_{s_{n}}(J))-\mathbb{E}[\eta_{\varepsilon_{n}}(Q_{s_{n}}(J))]\right|>\kappa\gamma\frac{s_{n}^{d}}{\varepsilon_{n}^{d}}\right)\leq\exp\left(-\frac{2\kappa^{2}\gamma}{(1+\kappa)}\frac{s_{n}^{d}}{\varepsilon_{n}^{d}}\right)$
$\displaystyle\leq\exp\left(-\frac{2\kappa^{2}\gamma}{(1+\kappa)}\log\varepsilon_{n}^{-d\beta_{n}^{d}}\right)=\varepsilon_{n}^{\beta_{n}^{d}d\gamma\frac{2\kappa^{2}}{(1+\kappa)}}.$

Set, for $J\in\mathcal{I}_{s_{n}}(Q)$

$X_{J}^{\kappa,n}:=\left\\{\begin{array}[]{lr}1&\text{if
$J\in\mathcal{I}_{s_{n}}^{\kappa}(Q)$}\\\
0&\text{otherwise}.\end{array}\right.$ (3.8)

Notice that each $X_{J}^{\kappa,n}$ is a Bernoulli random variable attaining
value $1$ with probability $p_{n}$ and $0$ with probability $1-p_{n}$ (Here we
are omitting the dependence of $p_{n}$ on $\kappa$ to lighten the notation).
Setting then

$T_{n}^{\kappa}:=\sum_{J\in\mathcal{I}_{s_{n}}(Q)}X_{J}^{\kappa,n}=\\#(\mathcal{I}_{s_{n}}^{\kappa}(Q))$

we have that

$\mathbb{E}[T_{n}^{\kappa}]=\\#(\mathcal{I}_{s_{n}}(Q))p_{n}\leq
C\frac{\varepsilon_{n}^{\beta_{n}^{d}d\gamma\frac{2\kappa^{2}}{(1+\kappa)}}}{s_{n}^{d}}$

for a constant $C$ depending on the dimension only (that in the sequel may
vary from line to line). Setting

$\varrho_{n}(\kappa):=\beta_{n}^{d}d\gamma\frac{2\kappa^{2}}{(1+\kappa)}$

the previous estimate writes

$\mathbb{E}[T_{n}^{\kappa}]\leq
C\frac{\varepsilon_{n}^{\varrho_{n}(\kappa)}}{s_{n}^{d}}.$

Proof of assertion a). If $\beta^{d}>\frac{1}{\gamma}$ then, for some $n_{0}$,
$\beta_{n}^{d}>\frac{1}{\gamma}$ for $n\geq n_{0}$. Then, for some
$0<\kappa_{0}<1$ we have $\varrho_{n}(\kappa_{0})>d$ whenever $n\geq n_{0}$.
Also recalling that, from the very definition of $\sigma_{\varepsilon}$, it
holds that
$\left(\frac{s_{n}}{\varepsilon_{n}}\right)^{d}\geq\frac{1}{d}\frac{\beta_{n}^{d}}{\log(\varepsilon_{n}^{-1})}$
we have

$\mathbb{E}[T_{n}^{\kappa_{0}}]\leq
C\frac{\varepsilon_{n}^{\varrho_{n}(\kappa_{0})-d}}{\log(\varepsilon_{n}^{-1})}.$

Consider the subsequence $\\{\varepsilon_{n_{j}}\\}_{j\in\mathbb{N}}$ such
that

$\varepsilon_{n_{j}}^{(\varrho_{j}(\kappa_{0})-d)/2}\leq\frac{1}{j^{2}}$

where we abbreviate $\varrho_{j}:=\varrho_{n_{j}}$ for the sake of clarity.
Define the events

$A_{j}(\kappa_{0}):=\left\\{\omega\in\Omega\ :\
T_{n_{j}}^{\kappa_{0}}(\omega)\geq\frac{\varepsilon_{n_{j}}^{(\varrho_{j}(\kappa_{0})-d)/2}}{\log(\varepsilon_{n_{j}}^{-1})}\right\\}.$

Then Chebishev inequality (cf. Remark 3.3) yields

$\displaystyle\mathbb{P}(A_{j}(\kappa_{0}))\leq\frac{\mathbb{E}[T_{n_{j}}^{\kappa_{0}}]\log(\varepsilon_{n_{j}}^{-1})}{\varepsilon_{n_{j}}^{(\varrho_{j}(\kappa_{0})-d)/2}}\leq
C\varepsilon_{n_{j}}^{(\varrho_{j}(\kappa_{0})-d)/2}\leq C\frac{1}{j^{2}}.$

In particular

$\sum_{j\in\mathbb{N}}\mathbb{P}(A_{j}(\kappa_{0}))<+\infty.$

Borel-Cantelli Lemma 3.2 now implies that

$\mathbb{P}\left(\bigcap_{j\in\mathbb{N}}\bigcup_{m\geq
j}A_{m}(\kappa_{0})\right)=0.$

Set $\Omega^{\prime}:=\Omega\setminus\bigcap_{j\in\mathbb{N}}\bigcup_{m\geq
j}A_{m}(\kappa_{0})$. We now observe that if $\omega\in\Omega^{\prime}$ then
there exists $j_{0}$ such that

$\\#(\mathcal{I}_{s_{n_{j}}}^{\kappa_{0}}(Q))<\frac{\varepsilon_{n_{j}}^{(\varrho_{j}(\kappa_{0})-d)/2}}{\log(\varepsilon_{n_{j}}^{-1})}\
\ \ \text{for all $j\geq j_{0}$}.$

Since $\varrho_{j}(\kappa_{0})-d>0$ we conclude that

$\\#(\mathcal{I}_{s_{n_{j}}}^{\kappa_{0}}(Q))=0\ \ \text{for all $j$ big
enough}.$

Since, for $\kappa>\kappa_{0}$ we have
$\mathcal{I}_{s_{n_{j}}}^{\kappa}(Q)\subset\mathcal{I}_{s_{n_{j}}}^{\kappa_{0}}(Q)$
we conclude the proof of assertion a).

Proof of assertion b). If $0<\beta^{d}\leq\frac{1}{\gamma}$ we have (for all
$n$ big enough)

$d>\varrho_{n}(\kappa)>\left(\frac{\beta}{2}\right)^{d}d\gamma\frac{2\kappa^{2}}{(1+\kappa)}=\varrho(\kappa)>0$

for all $\kappa\in(0,1)$ and then

$\mathbb{E}[T_{n}^{\kappa}]\leq
C\frac{\varepsilon_{n}^{\varrho(\kappa)}}{s_{n}^{d}}.$

Fix $\kappa_{0}\in(0,1)$ and consider the subsequence
$\\{\varepsilon_{n_{j}}\\}_{j\in\mathbb{N}}$ such that

$\varepsilon_{n_{j}}^{\varrho(\kappa_{0})/2}\leq\frac{1}{j^{2}}.$

We define the events

$A_{j}(\kappa_{0}):=\left\\{\omega\in\Omega\ :\
T_{n_{j}}^{\kappa_{0}}(\omega)\geq\frac{\varepsilon_{n_{j}}^{\varrho(\kappa_{0})/2}}{s_{j}^{d}}\right\\}.$

Then Chebishev inequality (cf. Remark 3.3) again yields

$\displaystyle\mathbb{P}(A_{j}(\kappa_{0}))\leq\frac{\mathbb{E}[T_{n_{j}}^{\kappa_{0}}]s_{j}^{d}}{\varepsilon_{n_{j}}^{\varrho(\kappa_{0})/2}}\leq
C\varepsilon_{n_{j}}^{\varrho(\kappa_{0})/2}\leq\frac{1}{j^{2}}.$

In particular

$\sum_{j\in\mathbb{N}}\mathbb{P}(A_{j}(\kappa_{0}))<+\infty.$

By invoking again Borel-Cantelli Lemma 3.2 we achieve

$\mathbb{P}\left(\bigcap_{j\in\mathbb{N}}\bigcup_{m\geq
j}A_{m}(\kappa_{0})\right)=0.$

Set
$\Omega^{\prime\prime}:=\Omega\setminus\bigcap_{j\in\mathbb{N}}\bigcup_{m\geq
j}A_{m}(\kappa_{0})$. We now observe that if $\omega\in\Omega^{\prime\prime}$
then there exists $j_{0}$ such that

$\\#(\mathcal{I}_{s_{n_{j}}}^{\kappa_{0},\omega}(Q))\leq\frac{\varepsilon_{n_{j}}^{\varrho(\kappa_{0})/2}}{s_{n}^{d}}\
\ \ \text{for all $j\geq j_{0}$}.$

Since, for $\kappa>\kappa_{0}$, we have
$\mathcal{I}_{s_{n_{j}}}^{\kappa}(Q)\subset\mathcal{I}_{s_{n_{j}}}^{\kappa_{0}}(Q)$
we conclude by setting $\varrho_{0}:=\varrho(\kappa_{0})/2$. ∎

##  4\. Proof of Compactness Theorem 2.3

Figure 4.1. On the left: The percolation properties of
$\mathcal{I}_{s_{n}}^{\kappa}(Q)$ stated in Lemma 3.6 and Proposition 3.7. On
the right: a depiction of a diagonal path $\mathbf{t}_{\partial U}$, a
connected component $U=U(J)$ of $\mathcal{I}_{s_{n}}^{\kappa}(Q)$ and the
boundary $\partial U(J)$.

Let us introduce some notation needed in the proof of the extension Lemma 4.6.
Given $U\subset\mathcal{J}$ a connected component of a subfamily of indexes
$\mathcal{J}\subset\mathcal{I}_{s_{\varepsilon}}(Q)$ we denote

${\color[rgb]{0,0,0}{\partial
U:=\left\\{J\in\mathcal{I}_{s_{\varepsilon}}(Q)\setminus\mathcal{J}\ |\
\exists\ J^{\prime}\in U\ :\partial
Q_{s_{\varepsilon}}(J^{\prime})\cap\partial
Q_{s_{\varepsilon}}(J)\neq\emptyset\right\\}.}}$

Notice that we allow also paths that can walk in diagonal, since our energy
allows us to control also the diagonal direction. In particular, for
$J\in\mathcal{I}_{s_{\varepsilon}}(Q)$ we define the set of neighbors of $J$
as

$N_{\varepsilon}(J):=\\{J^{\prime}\in(Q_{3s_{\varepsilon}}(J)\setminus\\{J\\})\cap\mathcal{I}_{s_{\varepsilon}}(Q)\\}.$

######  Definition 4.1 (diagonal paths).

We say that
$\mathbf{t}(J,J^{\prime})=\\{J_{1},\ldots,J_{k}\\}\subset\mathcal{I}_{s_{\varepsilon}}(Q)$
is a diagonal path connecting $J$ to $J^{\prime}$ if $J_{1}=J$,
$J_{k}=J^{\prime}$ and $J_{i+1}\in N_{\varepsilon}(J_{i})$ for all
$i=1,\ldots,k-1$.

Motivated by Lemma 3.6 and Proposition 3.7, we introduce the notion of
controlled subfamily of indexes
$\\{\mathcal{J}_{n}\subset\mathcal{I}_{s_{n}}(Q)\\}_{n\in\mathbb{N}}$.

######  Definition 4.2.

Let $\\{s_{n}\\}_{n\in\mathbb{N}}$ be a vanishing sequence. We say that
$\\{\mathcal{J}_{n}\subset\mathcal{I}_{s_{n}}(Q)\\}_{n\in\mathbb{N}}$ is a
controlled subfamily of indexes if there exists $n_{0}\in\mathbb{N}$,
$\varrho_{0}\in(0,1)$, $\Lambda\in\mathbb{R}_{+}$ universal constants such
that the following are in force for all $n\geq n_{0}$:

  * a)

$\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}$ is path connected in the
sense of Definition 3.4;

  * b)

there exists $0<\varrho_{0}<1$ such that $\\#(\mathcal{J}_{n})s_{n}^{d}\leq
s_{n}^{\varrho_{0}}$;

  * c)

for each connected component $U\subset\mathcal{J}_{n}$ it holds
$\\#(U)\leq\Lambda\log(s_{n}^{-1})$.

######  Remark 4.3.

Up to extract a subsequence, by combining Lemma 3.6 and Proposition 3.7 we can
ensure that (for a suitable $\kappa$) the family
$\mathcal{J}_{n}=\mathcal{I}_{s_{n}}^{\kappa}(Q)$ is a controlled subfamily of
index (we refer the reader to Figure 4.1).

######  Lemma 4.4.

Let $\\{\mathcal{J}_{n}\\}_{n\in\mathbb{N}}$ be a controlled subfamily of
indexes. There exists a universal constant $C$ such that for any
$U\subset\mathcal{J}_{n}$ connected component of $\mathcal{J}_{n}$, and for
any $J,J^{\prime}\in\partial U$, we can find a diagonal path
$\mathbf{t}_{\partial U}(J,J^{\prime})\subset\partial U$ such that

$\\#(\mathbf{t}_{\partial U}(J,J^{\prime}))\leq C\\#(\partial U).$ (4.1)

###### Proof.

Observe that the set

$E(U):=\bigcup_{J\in U}Q_{s_{n}}(J)$

is connected and simply connected due to properties a), b), c) of Definition
4.2. In particular the set $\partial E(U)$ is connected and thus connected by
arc. Fix $J,J^{\prime}\in\partial U$ and pick

$x_{J}\in\partial Q_{s_{n}}(J)\cap\partial E(U),\ \ \
y_{J^{\prime}}\in\partial Q_{s_{n}}(J^{\prime})\cap\partial E(U).$

Consider then an arc $\tau:[0,1]\rightarrow\partial E(U)$ such that
$\tau(0)=x_{J}$, $\tau(1)=y_{J}^{\prime}$. Since $\tau\in\partial E(U)$ we can
find $J_{1},\ldots,J_{k}\in\partial U$, $t_{1},\ldots,t_{k}\in(0,1)$ such that
$J_{1}=J$, $J_{k}=J^{\prime}$ and $\tau(t)\in Q_{s_{n}}(J_{i})$ for
$t\in(t_{i},t_{i+1})$. By eventually reducing loops and repetition, without
loss of generality we can assume that $\tau$ is a simple arc and do not passes
through each $J_{i}$ more than once. Now by defining $\mathbf{t}_{\partial
U}(J,J^{\prime}):=\\{J_{1},\ldots,J_{k}\\}$, by construction we have that
$\mathbf{t}_{\partial U}$ is a diagonal path. The bound (4.1) is also
immediate. ∎

######  Remark 4.5.

Notice that, if property a) of Definition 4.2 fails then Lemma 4.4 is, in
general, false.

######  Lemma 4.6 (An extension Lemma).

Let $\\{s_{n}\\}_{n\in\mathbb{N}}$ be a vanishing sequence. Let
$\\{\mathcal{J}_{n}\subset\mathcal{I}_{s_{n}}(Q)\\}_{n\in\mathbb{N}}$ be a
controlled subfamily of indexes in the sense of Definition 4.2. Then, for any
sequence of functions
$\\{u_{n}:\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}\rightarrow\mathbb{R}^{+}\\}_{n\in\mathbb{N}}$
such that

$\sup_{n\in\mathbb{N}}\left\\{\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}\\\
J^{\prime}\in
N_{\varepsilon_{n}}(J)\end{subarray}}|u_{n}(J)-u_{n}(J^{\prime})|^{p}s_{n}^{d-p}+\sum_{\begin{subarray}{c}J\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}\end{subarray}}|u_{n}(J)|^{p}s_{n}^{d}\right\\}<+\infty$
(4.2)

we can find a sequence of extended functions
$\\{Tu_{n}:\mathcal{I}_{s_{n}}(Q)\rightarrow\mathbb{R}^{+}\\}_{n\in\mathbb{N}}$
such that

$\displaystyle Tu_{n}$ $\displaystyle=u_{n}\ \ \ \text{on
$\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}$ }$

and for all $q<p$

$\sup_{n\in\mathbb{N}}\left\\{\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s_{n}}(Q)\\\
|J-J^{\prime}|=s_{n}\end{subarray}}|Tu_{n}(J)-Tu_{n}(J^{\prime})|^{q}s_{n}^{d-q}+\sum_{\begin{subarray}{c}J\in\mathcal{I}_{s_{n}}(Q)\end{subarray}}|Tu_{n}(J)|^{q}s_{n}^{d}\right\\}<+\infty.$
(4.3)

######  Remark 4.7.

Notice that we assume a bound (4.2) involving interaction also in diagonal
directions and we conclude a control on the energy of $Tu_{n}$ (4.3) in terms
of nearest neighbors. Bound (4.3) could, in principle, be improved to a bound
involving the full family of neighbors $N_{\varepsilon_{n}}(J)$ (consistently
with (4.2)) by carrying a careful analysis of the diagonal interactions of
$Tu_{n}$’s. However, as clarified by Lemma 4.8, bound (4.3) is already strong
enough to conclude compactness for the sequence $Tu_{n}$.

###### Proof.

We set, for $U$ a connected component of $\mathcal{J}_{n}$:

$\displaystyle(u_{n})_{U}$ $\displaystyle:=\frac{1}{\\#(\partial
U)}\sum_{J\in\partial U}u_{n}(J).$

We then set

$Tu_{n}(J)=\left\\{\begin{array}[]{ll}u_{n}(J)&\ \ \text{if
$J\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}$},\\\ (u_{n})_{U}&\
\text{if $J\in U$, for some $U$ connected component of
$\mathcal{J}_{n}$}.\end{array}\right.$ (4.4)

We now estimate the energy of $Tu_{n}$.

$\displaystyle\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s_{n}}(Q)\\\
|J-J^{\prime}|=s_{n}\end{subarray}}|Tu_{n}(J)-Tu_{n}(J^{\prime})|^{q}s_{n}^{d-q}\leq$
$\displaystyle
C\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}\\\
|J-J^{\prime}|=s_{n}\end{subarray}}|u_{n}(J)-u_{n}(J^{\prime})|^{q}s_{n}^{d-q}$
$\displaystyle+C\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\
\text{component}\end{subarray}}\sum_{\begin{subarray}{c}J\in\partial
U\end{subarray}}|u_{n}(J)-(u_{n})_{U}|^{q}s_{n}^{d-q}.$ (4.5)

Let us treat the second sum separately. Fix $U\subset\mathcal{J}_{n}$ a
connected component and observe, by considering the paths given by Lemma 4.4,
that

$\displaystyle\sum_{\begin{subarray}{c}J\in\partial U\end{subarray}}$
$\displaystyle|u_{n}(J)-(u_{n})_{U}|^{q}s_{n}^{d-q}\leq\frac{1}{\\#(\partial
U)}\sum_{\begin{subarray}{c}J,J^{\prime}\in\partial
U\end{subarray}}|u_{n}(J)-u_{n}(J^{\prime})|^{q}s_{n}^{d-q}$
$\displaystyle\leq\frac{1}{\\#(\partial
U)}\sum_{\begin{subarray}{c}J,J^{\prime}\in\partial
U\end{subarray}}(\\#(\mathbf{t}_{\partial
U}(J,J^{\prime}))^{q-1}\sum_{I\in\mathbf{t}_{\partial
U}(J,J^{\prime})}\sum_{\begin{subarray}{c}I^{\prime}\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}:\\\
I^{\prime}\in
N_{\varepsilon_{n}}(I)\end{subarray}}|u_{n}(I)-u_{n}(I^{\prime})|^{q}s_{n}^{d-q}$
$\displaystyle\leq\frac{C}{\\#(\partial
U)}\sum_{\begin{subarray}{c}J,J^{\prime}\in\partial
U\end{subarray}}(\\#(\mathbf{t}_{\partial
U}(J,J^{\prime}))^{q-1}\sum_{I\in\mathbf{t}_{\partial
U}(J,J^{\prime})}\left(\sum_{\begin{subarray}{c}I^{\prime}\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}:\\\
I^{\prime}\in
N_{\varepsilon_{n}}(I)\end{subarray}}|u_{n}(I)-u_{n}(I^{\prime})|^{p}s_{n}^{\frac{dp}{q}-p}\right)^{q/p}$
$\displaystyle\leq\frac{Cs_{n}^{\frac{d(p-q)}{p}}}{\\#(\partial
U)}\sum_{\begin{subarray}{c}J,J^{\prime}\in\partial
U\end{subarray}}(\\#(\mathbf{t}_{\partial
U}(J,J^{\prime}))^{q-1}\sum_{I\in\mathbf{t}_{\partial
U}(J,J^{\prime})}\left(\sum_{\begin{subarray}{c}I^{\prime}\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}:\\\
I^{\prime}\in
N_{\varepsilon_{n}}(I)\end{subarray}}|u_{n}(I)-u_{n}(I^{\prime})|^{p}s_{n}^{d-p}\right)^{q/p}$

where we applied Jensen’s inequality with the function $x\mapsto x^{p/q}$.
Call

$Eu_{n}(I):=\sum_{\begin{subarray}{c}I^{\prime}\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}:\\\
I^{\prime}\in
N_{\varepsilon_{n}}(I)\end{subarray}}|u_{n}(I)-u_{n}(I^{\prime})|^{p}s_{n}^{d-p}.$

Then, an application of Holder’s inequality yields

$\displaystyle\sum_{\begin{subarray}{c}J\in\partial U\end{subarray}}$
$\displaystyle|u_{n}(J)-(u_{n})_{U}|^{q}s_{n}^{d-q}\leq\frac{Cs_{n}^{\frac{d(p-q)}{p}}}{\\#(\partial
U)}\sum_{\begin{subarray}{c}J,J^{\prime}\in\partial
U\end{subarray}}(\\#(\mathbf{t}_{\partial
U}(J,J^{\prime}))^{q-1}\sum_{I\in\mathbf{t}_{\partial
U}(J,J^{\prime})}\left(Eu_{n}(I)\right)^{q/p}$ $\displaystyle\leq
C\frac{s_{n}^{\frac{d(p-q)}{p}}}{\\#(\partial
U)}\sum_{\begin{subarray}{c}J,J^{\prime}\in\partial
U\end{subarray}}(\\#(\mathbf{t}_{\partial
U}(J,J^{\prime}))^{q-1+\frac{p-q}{p}}\left(\sum_{I\in\mathbf{t}_{\partial
U}(J,J^{\prime})}Eu_{n}(I)\right)^{q/p}$ $\displaystyle\leq
Cs_{n}^{\frac{d(p-q)}{p}}\\#(\partial
U)^{q+\frac{p-q}{p}}\left(\sum_{I\in\partial U}Eu_{n}(I)\right)^{q/p}.$

In the last inequality we applied (4.1). In particular, by applying again
Holder inequality

$\displaystyle\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}$
$\displaystyle\sum_{\begin{subarray}{c}J\in\partial
U\end{subarray}}|u_{n}(J)-(u_{n})_{U}|^{q}s_{n}^{d-q}$ $\displaystyle\leq
s_{n}^{\frac{d(p-q)}{p}}\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)^{q+\frac{p-q}{p}}\left(\sum_{I\in\partial U}Eu_{n}(I)\right)^{q/p}$
$\displaystyle\leq
s_{n}^{\frac{d(p-q)}{p}}\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)^{\frac{pq}{p-q}+1}\right)^{\frac{p-q}{p}}\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\sum_{I\in\partial
U}Eu_{n}(I)\right)^{q/p}.$

By assumption (4.2) we have

$\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\ \text{connected}\\\
\text{component}\end{subarray}}\sum_{I\in\partial U}Eu_{n}(I)\right)^{q/p}<M.$

Moreover, by Properties b), c) of $\mathcal{J}_{n}$ (Definition 4.2) we can
infer (since $\\#(\partial U)\leq C\\#(U)$ for some universal constant $C$)

$\displaystyle
s_{n}^{\frac{d(p-q)}{p}}\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)^{\frac{pq}{p-q}+1}\right)^{\frac{p-q}{p}}$ $\displaystyle\leq
C\left(\log(s_{n}^{-1})^{\frac{pq}{p-q}}\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)s_{n}^{d}\right)^{\frac{p-q}{p}}$ $\displaystyle\leq
C\left(\log(s_{n}^{-1})^{\frac{pq}{p-q}}\\#(\mathcal{J}_{n})s_{n}^{d}\right)^{\frac{p-q}{p}}$
$\displaystyle\leq
C\left(\log(s_{n}^{-1})^{\frac{pq}{p-q}}s_{n}^{\varrho_{0}}\right)^{\frac{p-q}{p}}.$

This is enough to deduce that, for $q<p$,

$\lim_{n\rightarrow+\infty}\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\
\text{component}\end{subarray}}\sum_{\begin{subarray}{c}J\in\partial
U\end{subarray}}|u_{n}(J)-(u_{n})_{U}|^{q}s_{n}^{d-q}=0$ (4.6)

which, combined with (4.5), (2.1) is enough to conclude a uniform bound

$\sup_{n\in\mathbb{N}}\left\\{\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s_{n}}(Q)\\\
|J-J^{\prime}|=s_{n}\end{subarray}}|Tu_{n}(J)-Tu_{n}(J^{\prime})|^{q}s_{n}^{d-q}\right\\}<+\infty.$
(4.7)

The $L^{q}$ norm part can be estimated in a similar way as follows.

$\displaystyle\sum_{J\in\mathcal{I}_{s_{n}}(Q)}|Tu_{n}(J)|^{q}s_{n}^{d}$
$\displaystyle\leq\sum_{J\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}}|u_{n}(J)|^{q}s_{n}^{d}+\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\frac{\\#(U)}{\\#(\partial
U)}\sum_{J\in\partial U}|u_{n}(J)|^{q}s_{n}^{d}.$

Observe that, recalling the notation $E(U):=\bigcup_{J\in U}Q_{s_{n}}(J)$, we
have (for some constant $C$ depending on the dimension only)

$\begin{split}C^{-1}\\#(\partial U)s_{n}^{d-1}\leq P(&E(U))\leq C\\#(\partial
U)s_{n}^{d-1}\\\ C^{-1}\\#(\partial U)s_{n}^{d}\leq|&E(U)|\leq C\\#(\partial
U)s_{n}^{d}\end{split}$

(being $P(\cdot)$ the distributional perimeter). Thus, by the isoperimetric
inequality (see for instance [30])

$\frac{\\#(U)}{\\#(\partial U)}\leq C\frac{1}{s_{n}}\frac{|E(U)|}{P(E(U))}\leq
C\frac{1}{s_{n}}P(E(U))^{1/(d-1)}\leq C\\#(\partial U)^{1/(d-1)}$

where $C$ is a universal constant depending only on the dimension. Thus

$\displaystyle\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\frac{\\#(U)}{\\#(\partial
U)}\sum_{J\in\partial U}|u_{n}(J)|^{q}s_{n}^{d}$ $\displaystyle\leq
C\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\ \text{connected}\\\
\text{component}\end{subarray}}\\#(\partial U)^{1/(d-1)}\sum_{J\in\partial
U}|u_{n}(J)|^{q}s_{n}^{d}.$

Also

$\displaystyle\sum_{J\in\partial U}|u_{n}(J)|^{q}s_{n}^{d}$ $\displaystyle\leq
s_{n}^{d}\\#(\partial U)^{(p-q)/p}\left(\sum_{J\in\partial
U}|u_{n}(J)|^{p}\right)^{q/p}$ $\displaystyle\leq s_{n}^{d(p-q)/p}\\#(\partial
U)^{(p-q)/p}\left(\sum_{J\in\partial U}|u_{n}(J)|^{p}s_{n}^{d}\right)^{q/p}.$

Call

$\|u_{n}\|_{\ell^{p}(\partial U)}:=\left(\sum_{J\in\partial
U}|u_{n}(J)|^{p}s_{n}^{d}\right)^{1/p}$

hence

$\displaystyle\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}$
$\displaystyle\frac{\\#(U)}{\\#(\partial U)}\sum_{J\in\partial
U}|u_{n}(J)|^{q}s_{n}^{d}\leq
Cs_{n}^{d(p-q)/p}\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)^{\frac{1}{d-1}+\frac{p-q}{p}}\|u_{n}\|_{\ell^{p}(\partial U)}^{q}$
$\displaystyle\leq\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)^{\frac{p}{(p-q)(d-1)}+1}s_{n}^{d}\right)^{\frac{p-q}{p}}\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\
\text{component}\end{subarray}}\|u_{n}\|_{\ell^{p}(\partial
U)}^{p}\right)^{q/p}.$

By assumption

$\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\ \text{connected}\\\
\text{component}\end{subarray}}\|u_{n}\|_{\ell^{p}(\partial
U)}^{p}\right)^{q/p}<M.$

Moreover, by properties b), c) of $\mathcal{J}_{n}$ (Definition 4.2) using
again that $\\#\partial U\leq C\\#U$ we have

$\displaystyle\left(\sum_{\begin{subarray}{c}U\subset\mathcal{J}_{n}\\\
\text{connected}\\\ \text{component}\end{subarray}}\\#(\partial
U)^{\frac{p}{(p-q)(d-1)}+1}s_{n}^{d}\right)^{\frac{p-q}{p}}\leq
C\left(\log(s_{n}^{-1})^{\frac{p}{(p-q)(d-1)}}s_{n}^{\varrho_{0}}\right)^{\frac{p-q}{p}}.$

We conclude that

$\lim_{n\rightarrow+\infty}\sum_{J\in\mathcal{I}_{s_{n}}(Q)}|Tu_{n}(J)|^{q}s_{n}^{d}\leq
C\sum_{J\in\mathcal{I}_{s_{n}}(Q)\setminus\mathcal{J}_{n}}|u_{n}(J)|^{p}s_{n}^{d}.$
(4.8)

In particular, by combining (4.8), (4.2) and (4.7) we conclude (4.3). ∎

The last ingredient required to the proof of Theorem 2.3 is the following
Lemma 4.8, which comes as a consequence of [5, Theorem 3.1].

######  Lemma 4.8.

Let $\\{u_{s}:\mathcal{I}_{s}(Q)\rightarrow\mathbb{R}\\}_{s\in\mathbb{R}_{+}}$
be a sequence of function such that

$\sup_{s\in\mathbb{R}_{+}}\left\\{\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s}(Q):\\\
|J-J^{\prime}|=s\end{subarray}}|u_{s}(J)-u_{s}(J^{\prime})|^{q}s^{d-q}+\sum_{J\in\mathcal{I}_{s}(Q)}|u_{s}(J)|^{q}s^{d}\right\\}<+\infty.$

Then there exists a function $u\in W^{1,q}(Q)$ and a subsequence
$\\{s_{l}\\}_{l\in\mathbb{N}}$ such that the piecewise constant extension of
$u_{s_{l}}$ :

$\displaystyle\hat{u}_{s_{l}}(x):=\sum_{J\in\mathcal{I}_{s_{l}}(Q)}u_{s_{l}}(J)\mathbbmss{1}_{Q_{s_{l}}(J)}(x)$

converges to $u$ in $L^{q}(Q)$.

We are now ready to prove Theorem 2.3.

###### Proof of Theorem 2.3.

Define $s^{\prime}_{\varepsilon}:=\frac{s_{\varepsilon}}{4\sqrt{d}}$. Notice
that with this choice we have

$Q_{3s_{\varepsilon}^{\prime}}(J)\subseteq B_{s_{\varepsilon}}(x)\ \ \
\text{for all $x\in Q_{s_{\varepsilon^{\prime}}}(J)$}.$ (4.9)

Notice that $s_{\varepsilon}^{\prime}$ still satisfies (3.7). By invoking
Lemma 3.6 and Proposition 3.7 we can find $\kappa$ and extract a subsequence
(still denoted by $\varepsilon_{n},s^{\prime}_{n}$) for which
$\mathcal{J}_{n}:=\mathcal{I}^{\kappa}_{s^{\prime}_{n}}$ is a controlled
subfamily of indexes in the sense of Definition 4.2. Without loss of
generality we can also assume that

$\lim_{n\rightarrow+\infty}\frac{s_{n}^{\prime}}{\varepsilon_{n}\log\left(\varepsilon_{n}^{-d}\right)^{\nicefrac{{1}}{{d}}}}=\beta_{0}<+\infty.$

Define
$u_{n}:\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{J}_{n}\rightarrow\mathbb{R}_{+}$

$u_{n}(J):=\frac{1}{\eta_{\varepsilon_{n}}(Q_{s_{n}^{\prime}}(J))}\sum_{x\in\eta_{\varepsilon_{n}}\cap
Q_{s^{\prime}_{n}}(J)}u_{\varepsilon_{n}}(x)\ \ \ \text{for all
$J\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{J}_{n}$}.$

If $J,J^{\prime}\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{J}_{n}$,
$J^{\prime}\in N_{\varepsilon_{n}}(J)$ then we have

$\displaystyle|u_{n}(J)-u_{n}(J^{\prime})|^{p}(s_{n}^{\prime})^{d-p}\leq$
$\displaystyle
C\frac{\varepsilon_{n}^{2d}}{s_{n}^{2d}}\sum_{x\in\eta_{\varepsilon_{n}}\cap
Q_{s^{\prime}_{n}}(J)}\sum_{y\in\eta_{\varepsilon_{n}}\cap
Q_{3s^{\prime}_{n}}(J)}|u_{\varepsilon_{n}}(x)-u_{\varepsilon_{n}}(y)|^{p}s_{n}^{d-p}$
$\displaystyle\leq$ $\displaystyle
C\varepsilon_{n}^{d}\sum_{x\in\eta_{\varepsilon_{n}}\cap
Q_{s^{\prime}_{n}}(J)}|\mathrm{grad}_{s_{n}}u_{\varepsilon_{n}}(x)|^{p}$

where the prefactor $\frac{\varepsilon_{n}^{d}}{s_{n}^{d}}$ appears because of
double-counting and where we exploited (4.9). Henceforth

$\sum_{\begin{subarray}{c}J,J^{\prime}\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{J}_{n}:\\\
J^{\prime}\in
N_{\varepsilon_{n}}(J)\end{subarray}}|u_{n}(J)-u_{n}(J^{\prime})|^{p}(s_{n}^{\prime})^{d-p}<C\mathcal{F}_{\varepsilon_{n}}(u_{\varepsilon_{n}};A).$
(4.10)

In the same way

$\sum_{\begin{subarray}{c}J\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{J}_{n}\end{subarray}}|u_{n}(J)|^{p}(s_{n}^{\prime})^{d}\leq
C\sum_{x\in\eta_{\varepsilon_{n}}\cap
Q}|u_{\varepsilon_{n}}(x)|^{p}\varepsilon_{n}^{d}.$ (4.11)

By invoking the extension Lemma 4.6 we can find a
$Tu_{n}:\mathcal{I}_{s^{\prime}_{n}}(Q)\rightarrow\mathbb{R}_{+}$ which has
(thanks to Lemma 4.8 ) piecewise constant extension $\widehat{Tu_{n}}$
precompact in $L^{q}$ for each $q<p$. In particular, due to Sobolev embeddings
we conclude that $\widehat{Tu_{n}}\rightarrow u$ in $L^{p}$ where $u\in
W^{1,q}(Q)$ for all $q<p$. Notice that (recall the notation 3.1, 3.2) by weak
compactness we also infer that

$(\nabla\widehat{Tu_{n}})\mathbbmss{1}_{Q\setminus
E_{s_{n}^{\prime}}^{\kappa}}\stackrel{{\scriptstyle
L^{p}}}{{\rightharpoonup}}V,\ \ \text{for some $V\in L^{p}$}.$

Observe that

$|E_{s^{\prime}_{n}}^{\kappa}|=\\#(\mathcal{I}_{s_{n}^{\prime}}^{\kappa}(Q))(s^{\prime}_{n})^{d}$

If we are in situation a) of Proposition 3.7 (namely
$\beta_{0}>\gamma^{-\nicefrac{{1}}{{d}}}$) then
$|E^{\kappa}_{s_{n}^{\prime}}|=0$ for $n$ big enough. If instead case b) is in
force then

$|E_{s^{\prime}_{n}}^{\kappa}|=\\#(\mathcal{I}_{s^{\prime}_{n}}^{\kappa}(Q))(s^{\prime}_{n})^{d}\leq\varepsilon_{n}^{\varrho_{0}}\rightarrow
0.$

In both cases $|E_{s^{\prime}_{n}}^{\kappa}|\rightarrow 0$. Thus
$\mathbbmss{1}_{Q\setminus
E_{s_{n}^{\prime}}^{\kappa}}\rightarrow\mathbbmss{1}_{Q}$ strongly in $L^{q}$
and $(\nabla\widehat{Tu_{n}})\stackrel{{\scriptstyle
L^{q}}}{{\rightharpoonup}}\nabla u$ then

$(\nabla\widehat{Tu_{n}})\mathbbmss{1}_{Q\setminus
E_{s_{n}^{\prime}}^{\kappa}}\stackrel{{\scriptstyle
L^{q}}}{{\rightharpoonup}}\nabla u$

yielding that $\nabla u=V\in L^{p}$. In particular $u\in W^{1,p}(\Omega)$ and
$\widehat{Tu_{n}}\rightarrow u$ in $L^{p}(Q)$. Also

$\displaystyle\int_{\mathcal{V}(\eta^{\alpha}_{\varepsilon})\cap Q\setminus
E_{s^{\prime}_{n}}^{\kappa}}|\hat{u}_{\varepsilon_{n}}(x)-\widehat{Tu_{n}}(x)|^{p}$
$\displaystyle=\sum_{J\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{I}_{s^{\prime}_{n}}^{\kappa}(Q)}\int_{\mathcal{V}(\eta^{\alpha}_{\varepsilon})\cap
Q_{s^{\prime}_{n}}(J)}|\hat{u}_{\varepsilon_{n}}(x)-u_{n}(J)|^{p}\,\mathrm{d}x$
$\displaystyle\leq
C\varepsilon_{n}^{d}\sum_{J\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{I}_{s^{\prime}_{n}}^{\kappa}(Q)}\sum_{x\in\eta^{\alpha}_{\varepsilon}\cap
Q_{s^{\prime}_{n}}(J)}|u_{\varepsilon_{n}}(x)-u_{n}(J)|^{p}$
$\displaystyle\leq
C\frac{\varepsilon_{n}^{2d}}{s_{n}^{d}}\sum_{J\in\mathcal{I}_{s^{\prime}_{n}}(Q)\setminus\mathcal{I}_{s^{\prime}_{n}}^{\kappa}(Q)}\sum_{x,y\in\eta^{\alpha}_{\varepsilon}\cap
Q_{s^{\prime}_{n}}(J)}|u_{\varepsilon_{n}}(x)-u_{\varepsilon_{n}}(y)|^{p}$
$\displaystyle\leq
Cs_{n}^{p}{\mathcal{F}}_{\varepsilon_{n}}(u_{\varepsilon_{n}};Q)\rightarrow
0.$ (4.12)

Hence

$\displaystyle\int_{\mathcal{V}_{\varepsilon_{n}}(\eta^{\alpha}_{\varepsilon_{n}})\cap
E_{s^{\prime}_{n}}^{\kappa}}$
$\displaystyle|\hat{u}_{\varepsilon_{n}}(x)-\widehat{Tu_{n}}(x)|^{q}\,\mathrm{d}x$
$\displaystyle\leq\left(\int_{\mathcal{V}_{\varepsilon_{n}}(\eta^{\alpha}_{\varepsilon_{n}})\cap
E_{s^{\prime}_{n}}^{\kappa}}|\hat{u}_{\varepsilon_{n}}(x)-\widehat{Tu_{n}}(x)|^{p}\,\mathrm{d}x\right)^{\nicefrac{{q}}{{p}}}|E_{s^{\prime}_{n}}^{\kappa}|^{\frac{(p-q)}{p}}\rightarrow
0$ (4.13)

for all $q<p$. Observe that, due to the property (4.3) of the extended
function $Tu_{n}$, to the bound (2.1) and to the very definition of
$\eta_{\alpha}$ we can infer

$\left(\int_{\mathcal{V}_{\varepsilon_{n}}(\eta^{\alpha}_{\varepsilon_{n}})\cap
E_{s^{\prime}_{n}}^{\kappa}}|\hat{u}_{\varepsilon_{n}}(x)-\widehat{Tu_{n}}(x)|^{p}\,\mathrm{d}x\right)^{\nicefrac{{q}}{{p}}}<C$
(4.14)

By collecting (4.12), (4.13) and (4.14) we conclude that
$u_{\varepsilon_{n}}\stackrel{{\scriptstyle q}}{{\longrightarrow}}u\in
W^{1,p}(Q)$ in the sense of Definition 2.2 for all $q<p$. ∎

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