Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes
Arianna Giunti Richard Höfer Juan J. L. Velázquez




This paper deals with the homogenization of the Poisson equation in a bounded domain of , d>2, which is perforated by a random number of small spherical holes 
with random radii and positions. We show that for a class of stationary short-range correlated measures for the centres and radii of the holes, we recover in the homogenized 
limit an averaged analogue of the “strange term” obtained by Cioranescu and Murat in the periodic case [D. Cioranescu and F. Murat, Un term étrange venu d'ailleurs (1986)]. We stress that we only require that the random radii have finite (d-2)-moment, which is the minimal assumption in order to ensure that the average of the capacity of the balls is finite. Under this assumption, there are holes which overlap with probability one. However, we show that homogenization occurs and that the clustering holes do not have any effect in the resulting homogenized equation.



Exact description of the boundary theory of the Kitaev Toric Code with open boundary conditions
    Stefan Kehrein
    December 30, 2023
===============================================================================================














§ INTRODUCTION



In this paper, we study the homogenization of the Poisson equation 

    -Δ u_ = f         in D^
      u_ = 0         on ∂ D^,

where the domain D^ is obtained by removing from a bounded set D ⊂, d>2, the union of properly rescaled spherical holes: Given a collection of points Φ= { z_i }_i∈⊂ and associated radii {ρ_i}_z_i ∈Φ⊂_+, we define

    D^:= D \⋃_z_i ∈Φ∩1/D B_ρ_i(  z_i),

where 1/D:= { 	x ∈   x ∈ D }.



In this paper, we assume that D^ is a random set. More precisely, we assume that the collection Φ of the centres is generated according to a stationary point process on  and that the radii {ρ_i}_z_i ∈Φ are unbounded random variables with short-range correlations. We show that, -almost surely, when ↓ 0^+, the solutions of (<ref>) weakly converge in H^1_0(D) to the solution of

    (-Δ  + C_0)  = f         in D
      = 0         on ∂ D.

Here, the constant C_0> 0 may be expressed in terms of an averaged density of capacity generated by the holes. We thus recover in the limit the analogue of 
the well-known “strange term” obtained by Cioranescu and Murat in the case of deterministic and periodic holes <cit.>. In this latter case,
which is equivalent to taking Φ = and ρ_i ≡ r for r >0, C_0 equals the capacity of a ball of radius r. More precisely,

    C_0= (d-2) ℋ^d-1( S^d-1) r^d-2,

where S^d-1 is the (d-1)-dimensional unit sphere in .



The main contribution of this paper is to show that the homogenization of u_ε to  of (<ref>) takes place under minimal assumptions
on the decay
of the density functions for the radii {ρ_i}_z_i ∈Φ of the holes. Namely, we only assume that
the average capacity of each hole is finite. More precisely, we will
show that the homogenization result holds assuming that the
configurations of the holes in ℝ^d, d>2, are assigned to a class of
probability measures for which the expectation of each radius ρ_i satisfies
only

    ⟨ρ^d-2_i⟩ <∞.



 In view of (<ref>), this is the minimal assumption under which one can expect C_0 in (<ref>) to be finite 
and therefore the limit problem (<ref>) to be meaningful.  On the other hand, to assume only
(<ref>) on the hole distributions poses some difficulties due to
the fact that the presence of balls in (<ref>) with large radii could allow the onset of clusters having large capacity. These
clusters might prevent the convergence of u_ε to the solutions
of the homogenized problem . In fact, it is known that the onset of
large clusters take place with probability one in systems of spherical holes
filling a small volume fraction of the space , if the radii
are distributed according to probability distributions with sufficiently fat tails (cf.
<cit.>). However, in <cit.> the holes are not rescaled 
as in (<ref>). In this paper, we prove that with the rescaling of the balls as in
(<ref>), the assumption (<ref>) is sufficient to ensure that  no percolating-like
structures appear in the limit ε↓ 0^+.




Stochastic homogenization problems for (<ref>) have been considered before in
the literature. The earliest results were the ones in <cit.>
and <cit.>. In <cit.>, the evolution version of (<ref>)
(i.e. the linear heat equation) is considered, and it is shown that the corresponding
solutions u_ converge to the analogous evolution version of  assuming that
ε^-d spherical holes are distributed independently according to some density function V ∈ C_0^∞(
ℝ^d). All the holes are assumed to have constant and fixed radius r_j=ε^d/d-2
. 

In <cit.>, the authors consider spherical hole configurations constituted of
ε^-3  balls in a bounded domain D⊂ℝ^3,
selected according to some classes of probability measures in which the
balls cannot overlap due to the presence of a hard-sphere potential. These
probability measures allow also to have short range correlations between two holes. As in our paper, also in <cit.>, the balls have random size ρ_i,
where the random variables ρ_i are assumed to satisfy

    ⟨ρ_i^3+β⟩ <∞ with β
    >0.

Under these assumptions, which we further discuss below, it is proved in <cit.> that the
solutions of (<ref>) converge to the solutions of (<ref>).


Stochastic homogenization for equations related to (<ref>) and domains D^ perforated as in (<ref>) has been considered also
more recently in <cit.>. In
<cit.>, the authors consider the
homogenization limit for the obstacle problem associated to a Dirichlet
functional in D ⊂ℝ^d, d≥2, in which the solutions must satisfy u_ε≥ 0 in the collection of small compact sets D \ D^.
Differently from our setting, the compact sets constituting D \ D^ are centred on a periodic lattice, but they can have random shapes which are uniformly bounded by M, for a fixed constant M > 0. 
Assuming an ergodicity condition of the probability measure on the shapes of the sets, the authors of <cit.> prove that the minimizers u_ converge to the solution  of the
semi-linear equation -Δ +α( )_-=f in D.

In <cit.>, the main focus is to study the stochastic homogenization of elliptic equations which not only include
(<ref>) but also singular elliptic operators like the p-Laplacian
operator. The probability measures considered in these works
allow to have hole configurations having random shapes which are uniformly bounded
by . Moreover, a stringent condition is assumed on the probability measure for the positions of the holes to ensure that the minimal distance between the holes is of order
ε with probability one.



In all the papers listed above with the exceptions of <cit.>, it is assumed
that the size of each hole is of order ε^d/d-2 with
probability one. We emphasize that the main technical difficulty in our paper
is due to the fact that under the sole assumption (<ref>), namely for distributions of the size of holes having fat tails
decreasing slowly enough, there exist, with probability one, domains D^ with the form (<ref>) punctured by clusters of two or more overlapping holes. These
clusters do not occur (with probability tending to one as ε↓ 0^+) under the assumption (<ref>) which is made in <cit.>.



In order to prove the homogenization results mentioned above there are
different methods in the literature. The first one, which was introduced by
Cioranescu and Murat in <cit.>, is related to the energy method of Tartar
<cit.>. It is based on the construction of some oscillating test
functions w_ε. A related approach has been used in the analysis
of several deterministic and stochastic homogenization problems (cf.
<cit.>) and this
is also the approach that will be used in this paper. 

A second approach is based on the construction of suitable projection operators in
Hilbert spaces which are defined using the geometry of the perforated domains.
This approach was introduced by Marchenko and Khruslov (cf. <cit.> and the references therein). Related to this approach is
the so-called method of reflections which yields
 a formal series representation for the Dirichlet problem (<ref>) that is frequently used in the physics literature. In
 <cit.>, the authors gave a
precise meaning to some of these formal series and used this method to prove homogenization results.

A third approach, used for instance in <cit.>, employs the probabilistic interpretation of the solutions u_ of 
(<ref>)  (and its evolution analogue), in terms of the properties of the
Brownian motion. In particular, the solutions of (<ref>) as well as the term C_0 arising in the limit equation can be obtained in terms of
expectations of functions of the survival time of a Brownian walker among
obstacles. 

Finally, we also mention that for problems related to (<ref>), a different approach has been introduced in <cit.>. In this series of papers, the main goal is to study a dynamical version of (<ref>), where the holes evolve according to the function u_ itself. In this case, the main challenge is thus to obtain estimates for the solution in the space L^∞ instead of the Sobolev space H^1. The starting point used in <cit.> is an ansatz for the structure of the solution of (<ref>) which gives rise to an explicit expression for an approximate solution of  (<ref>). 
The difference between this approximate solution and the solution of (<ref>)  is
then estimated using the maximum principle. 

Stochastic homogenization results have been obtained using this
approach in <cit.>, and in the case of solutions of (<ref>) in unbounded domains, they rely on the study of screening
properties <cit.>. Concerning the introduction and the study of such screening phenomena for interacting particles, we also refer to 
<cit.> and <cit.>.



In the problems of stochastic homogenization, two different types of
convergence results are obtained. One approach consists in introducing a
probability measure ℙ on the space of hole configurations
Ω (positions and shapes) in ℝ^d. The Dirichlet
problem (<ref>) is then solved for each fixed realization ω∈Ω in
a bounded domain which is obtained by means of (<ref>). It is then proved
that u_ε converges for ℙ-a.s. as
ε↓ 0^+ to the solutions of (<ref>). This is the type of
results obtained in <cit.>, and also in
this paper. 

The second approach consists in creating  configurations
containing ε^-d holes in a bounded domain according to a
family of probability measures ℙ_ε defined on a space
of configurations Ω_ε.  The homogenization results is thus expressed in terms of convergence in probability, namely that for any
δ>0, lim_ε↓ 0^+ℙ_ε( {‖ u_ε-u‖ >δ})=0, where ‖·‖ is a suitable norm. The results obtained in <cit.>, <cit.> and  <cit.> are of this type.



 §.§ Main ideas and organisation of the paper

As already mentioned in the previous discussion, in this paper, we focus on probability measures where the radii of the balls in (<ref>) satisfy merely the minimal condition (<ref>) on their moments,
and the centres of the balls are distributed according to a stationary point process on the whole space. We allow that both the centres and the radii have short-range correlations. This class of measures
includes the cases of balls having independent and identically distributed radii and centres either periodic or distributed according to an homogeneous Poisson point
process (cf. settings (a) and (b) in the next section). We also give some explicit examples of short-range correlated measures which are constructed starting from clustering 
or repulsive point processes for the centres of the holes (cf. setting (c) in the next section). 

In order to prove the main homogenization result for these measures, we adapt the argument of <cit.> to translate the conditions on the geometry of
the holes of D^ into properties of the associated oscillating test function w_.
These functions account for the presence of the holes in the domain D^ by correcting any admissible test function ϕ∈ C^∞_0(D) for (<ref>)
into  an admissible test function w_ϕ∈ H^1_0( D^) for (<ref>). The main breakthrough of <cit.> 
is the formulation of sufficient conditions on w_ which allow to treat the error terms generated by the presence of w_ in the weak formulation for 
(<ref>). In the limit ↓ 0^+, these errors are the ones giving rise to the additional term C_0 in (<ref>).
In the case of periodic balls in ^d, the authors in <cit.> explicitly construct the test functions w_ and obtain (<ref>) with the value for C_0 given by (<ref>); from this construction, the link between the term C_0 and the density of capacity generated by the holes becomes apparent and motivates the necessary
choice in (<ref>) of the length-scales  for the radii and  for the distance between the centres.

The main challenge of this paper is that with the sole assumption (<ref>) on the radii, we need to deal in (almost) all the configurations with the presence
of large radii. In spite of the scaling of (<ref>), the associated big balls may overlap and potentially break down the construction of the functions
w_ε. The main idea of our proof is to show that, even though with probability one there are regions where the balls overlap,
the moment assumption on the radii is sufficient to ensure that almost surely these regions have a capacity which vanishes in the limit ↓ 0^+.
This yields that the contribution of the functions w_ to the new term in the limit equation is restricted to the region of the domain D^ where the balls 
are small and well-separated.



The structure of this paper is the following: In the next section, we give a precise definition of the processes generating the holes in 
(<ref>) and introduce some examples which are included in our setting; we then state the main homogenization result
(Theorem <ref>). 
Section <ref> contains the proof of the theorems provided that the oscillating test functions exist, while Section <ref> is devoted to 
the crucial arguments for the construction of such oscillating test functions. Section <ref> provides some probabilistic results for marked point processes on
which the previous section relies and which make the arguments of this paper totally self-contained. To this purpose, we also include an appendix with a proof of a Strong
Law of Large Numbers which is tailored to the processes that we consider. In the sake of what we think is a more comfortable reading,
we do not prove our main result directly for a general probability measure, but we first give the argument in the case of holes with periodic centres and independent and identically distributed radii. By relying on the abstract results of Section <ref>, we then show how to adapt this proof to a general measure with short-range correlations. This, we believe, gives a more intuitive structure to the arguments of this paper.




§ SETTING AND MAIN RESULT


Let D ⊂^d, d >2, be an open and bounded set that it is star-shaped with respect to the origin[This assumption ensures that the sets in the family {1/ D }_ > 0 (see definition after formula (<ref>)) are nested. This is not a necessary condition for our results to hold, but it avoids some technicalities in our proof and it keeps our arguments and our notation leaner.]. For ε > 0, we denote by D^ε⊂ D the domain obtained by 
removing from D the closure of a set of “small” holes H^ε⊂^d of the form:

    H^ε := ⋃_z_j ∈Φ∩1/D B_ε^d/d-2ρ_j (ε z_j),

where 1/D := { x ∈  x ∈ D }, the set Φ⊂^d is a random collection of (countably many) points and the radii
{ρ_i}_z_i ∈Φ⊂^+ are random variables. The set H^ may thus be thought as being generated  by a marked point process
(Φ, ℛ) on ×_+, where  Φ is a point process on  for the centres of the balls, and the marks
ℛ={ρ_i}_z_i ∈Φ⊂_+ are the radii associated to each centre. We refer to <cit.> 
for a rigorous definition of marked point processes as a class of random measures on ×_+. We remark indeed that there is a one-to-one correspondence between
representing each realisation of the process as a collection of points and radii { (z_i, ρ_i) }_i∈⊂×_+ as we do in this paper, and as the
atomic measure μ:=  ∑_i∈δ_(z_i, r_i ) on ×_+. We also note that both the previous representations are invariant under permutation of the indices
i ∈ and thus that there is no preferred ordering of the centres of the balls generating H^.



We denote by (Ω, ℱ, ) the probability space associated to the process (Φ, ℛ) and, for every ω∈Ω, we write H^( ω) for the set defined in (<ref>) with 
(Φ, ℛ)(ω).  Throughout this paper, we assume that (Φ, ℛ) satisfies the the following properties:

 
 
 
  * The process Φ is stationary: For every x∈ we have τ_x  ∘Φℒ=Φ, where for each
 { z_i}_i∈⊂ the translations are defined as
 
    τ_x({z_i }_i∈)= {z_i + x }_i∈.

 
 
  * There exists λ< +∞ such that for any unitary cube Q ⊂
 
    ⟨#(Φ∩ Q)^2 ⟩^1/2≤λ ,

where #(S) ∈∪∞ denotes the cardinality of a set S and ⟨·⟩ is the integration over Ω with respect
 to the measure . Note that, by stationarity of Φ, the left-hand side of (<ref>) does not depend on the position of Q.
 
 
  * The point process Φ satisfies a strong mixing condition: For any bounded Borel set A ⊂^d, let ℱ(A) be the smallest σ-algebra with respect to which the random variables N(B)(ω):= #(Φ∩ B) are measurable for every Borel set B ⊂ A. Then, there exist C_1 < +∞ and γ > d such that for every A⊂ as above, every x ∈^d with |x| > diam(A) and every ξ_1, ξ_2 measurable with respect to ℱ(A) and ℱ(τ_xA), respectively, we have
   
    | ⟨ξ_1 ξ_2 ⟩ - ⟨ξ_1⟩⟨ξ_2 ⟩  | ≤C_1/1+(|x|- diam(A))^γ⟨ξ_1^2 ⟩^⟨ξ_2^2 ⟩^.


 
  * The marginal _ℛ of the marks with respect to the process Φ has 1- and 2- correlation functions 
  
    f_1((z, ρ))    = h(ρ),
    
    f_2(z_i, ρ_i , z_j, ρ_j)    = h( ρ_i) h(ρ_j) + g(|z_i - z_j|, ρ_i, ρ_j)      ∀ i≠ j

with 

    ∫ρ^d-2 h(ρ) d ρ < +∞     | g(r , ρ_1, ρ_2) | ≤c/(1+r^γ)(1 + ρ_1^p)(1 + ρ_2^p)

for p > d-1,  γ > d and c ∈_+.

The previous assumptions imply that the (d-2)-moment of the radii of the balls is finite and that, conditioned to the positions of the centres, the radii for two balls with centres in z_1, z_2 have correlations which vanish when the distance |z_1-z_2| → +∞. These correlations are short-range in the sense that the function g above is integrable in the variable r:=|z_1- z_2|.
 



Throughout this paper, we denote D^ε(ω):= D\ H^(ω) with H^(ω) as in (<ref>), and we identify any v ∈ H^1_0(D^(ω)) with the function ṽ∈ H^1_0(D) obtained by extending v as v ≡ 0 in H^(ω). Then we have: 


Let the holes in (<ref>) be generated by a marked point process (Φ, ℛ). Let Φ satisfy (<ref>) and (<ref>), and
let the marginal _ℛ satisfy (<ref>) and (<ref>). For f ∈ H^-1(D) 
 and ε > 0,
let u_ε= u_ε( ω, · ) ∈ H^1_0( D^ε(ω) ) solve (<ref>). Then, there exist a constant C_0> 0 and ∈ H^1_0(D) solving 
(<ref>) such that for -almost every ω∈Ω

    u_ε(ω, ·) ⇀    in H^1_0(D),   for ε↓ 0^+.




Moreover, we have that the constant C_0 in (<ref>) is defined as

    C_0= (d-2) σ_d ⟨  N(Q)  ⟩⟨ ρ^d-2 ⟩,

where σ_d = ℋ^d-1(S^d-1) and N(Q) is the number of centres falling into any fixed unitary cube Q.
 





 §.§ Some examples of processes generating the holes H^.


Among the processes which satisfy the conditions required in the previous theorem, we mention the following three examples: For the first two examples, it is immediate that  (<ref>), (<ref>), (<ref>) and (<ref>) are satisfied. Also in the case of the examples given in (c), the previous conditions are satisfied and this follows by easy calculations that we postpone to the appendix.   




  (a) The set H^ is a collection of balls with periodic centres and i.i.d. radii.
 
Here, the centres have deterministic positions Φ= and the marks ℛ for the radii are a family of independent and identically distributed random variables which satisfy (<ref>). 
In this case, we have that the constant C_0 in (<ref>) reads C_0= (d-2) σ_d ⟨ ρ^d-2 ⟩. 




  (b) The set H^ is a collection of balls with centres generated by a Poisson point process and i.i.d. radii. 
The process Φ is an homogeneous Poisson point process, and, conditioned to Φ, the marks ℛ are as in case (a). In this case, we have that C_0= (d-2) σ_d λ⟨ ρ^d-2 ⟩ with λ> 0 being the intensity of the Poisson point process Φ.



  (c) The balls of H^ have correlated radii and centres generated by a clustering or repulsive point process. The process Φ is an attractive or a repulsive point processes with short-range correlations, respectively:





  (c.1)  Neymann-Scott cluster process on  (see, e.g. <cit.>): Let (Φ_1, { r_i }_i ∈Φ_1) be a marked point process where Φ_1 is a homogeneous Poisson point process and the marks are independent and uniformly distributed on (0, R_c), with 0< R_c < +∞. For λ_2 ∈ L^∞() and x ∈, let Φ^x_2 be the heterogeneous Poisson point process having intensity λ_2( · - x). Then, we define

    Φ := ⋃_z_i ∈Φ_1Φ^z_i_2 ∩ B_r_i(z_i).





  (c.2)  Strauss process Φ on   with parameters α > 0, β∈ [0,1] and interaction distance r_c > 0 <cit.>,
<cit.>. 
For each bounded Borel set B ⊂, we define 

    (  #(Φ∩ B)= n ) = Z_B^-1α^n/n!∫_B ×⋯× Bβ^R({x_1, ⋯ , x_n }) dx_1  ⋯  dx_n ,

with 

    R({x_1, ⋯ , x_n }):= 1/2∑_i,j  i ≠ j^n _[0, r_c](|x_i - x_j|),

and 

    Z_B =  ∑_n=0^+∞α^n/n!∫_B ×⋯× Bβ^R({x_1, ⋯ , x_n }) dx_1  ⋯  dx_n.

This probability measure is well-defined in the repulsive case β∈ [0,1), while it requires further assumptions in the attractive case β≥ 1. For β =0, we remark that Φ is the hard-core process with radius r_c and intensity α (see also  <cit.> and <cit.>).
We remark that this process is the same as the macrocanonical Gibbs ensemble of Statistical Physics at temperature T=1 for the pair-interaction potential

    ψ(r) = 
    
    -logβ  if  r ≤ r_c
    
    0   if r > r_c

and the chemical potential μ= logα.
  



For each one of the previous point processes Φ, we let the marginal _ℛ be any probability measure satisfying (<ref>)
and (<ref>).




§ PROOF OF THEOREM <REF>

As already discussed in the introduction, our strategy is to adapt the method of <cit.> and to show that, in spite of the unboundedness of the radii of the holes in H^, we may almost surely construct a sequence of suitable oscillating test function. The crucial result of our paper is indeed the following:


 Let  H^ε= H^ε(ω) be as in Theorem <ref>. Then, for -almost every ω∈Ω, there exists a sequence { w_ε( ω, ·) }_ε > 0⊂ H^1(D)  which satisfies
  
 
  (H1) For every ε > 0,  w_ε(ω, ·) = 0 in H^(ω);
 
  (H2) w_ε( ω, ·) ⇀ 1 in H^1(D) for ↓ 0^+;
 
  (H3) For every sequence v_ε⇀ v in H^1_0(D) such that v_ε∈ H^1_0( D^ε) it holds that
  
    ( -Δ w_ε(ω, ·), v_ε )_H^-1, H^1_0→ C_0 ⟨ρ^d-2⟩∫_D v

for ↓ 0^+ and where C_0 defined as in Theorem <ref>.
 


By relying on the previous lemma, the proof of Theorem <ref> follows exactly as in <cit.>:

Let ω∈Ω be fixed, and let it belong to the full-probability set Ω' ⊂Ω made of configurations for which, according to  Lemma <ref>, the functions  { w_}_ > 0:= {w_(ω, ·)}_>0 exist and satisfy hypothesis (H1), (H2) and (H3). 



Since u_∈ H^1_0(D^), we may test equation (<ref>) with u_ itself and get by the standard energy estimate

    u__H^1≤  C f_H^-1,
 
with a constant C that depends only on the domain D.
By weak-compactness of H^1_0(D), we infer that, up to a subsequence which may depend on ω,

    u_⇀    in H^1_0(D) for ↓ 0^+.

We show that ∈ H^1_0(D) is the solution of (<ref>); by uniqueness, this extends the weak convergence of the solutions u_ to the continuum limit ↓ 0 and concludes the proof of the theorem.

To prove that  solves (<ref>), let us fix any function ϕ∈ C^∞_0(D). Since (H1) yields w_ϕ∈ H^1_0(D^), we can test the equation (<ref>) with w_ϕ and obtain

    ∫∇( w_ϕ) ·∇ u_ = (f, w_ϕ)_H^-1, H^1_0.

By (H2), the right-hand side above converges to

    (f, w_ϕ)_H^-1, H^1_0→ (f, ϕ)_H^-1, H^1_0.

We now use the product-rule and an integration by parts to rewrite the left-hand side in (<ref>) as 

    ∫∇( w_ϕ) ·∇ u_   = ∫ϕ∇ w_·∇ u_ + ∫ w_∇ϕ·∇ u_
        = ( -Δ w_, ϕ u_ )_H^-1, H^1_0 - ∫ u_∇ w_·∇ϕ + ∫ w_∇ϕ·∇ u_.

Since by (H2) of Lemma <ref> and (<ref>), both u_ and w_ converge strongly in L^2_loc(D), the last two terms on the right-hand side above converge to ∫∇ϕ·∇. Furthermore, by (<ref>) and the assumption on ϕ, we apply hypothesis (H3) of Lemma <ref> to the first term on the right-hand side above and conclude that

    ∫∇( w_ϕ) ·∇ u_→ C_0 ∫ϕ + ∫∇ϕ·∇.

This, together with  (<ref>), (<ref>) and the arbitrariness of ϕ∈ C^∞_0(D), yields that  weakly solves (<ref>). The proof of Theorem <ref> is complete. 



§ EXISTENCE OF THE OSCILLATING TEST FUNCTIONS (PROOF OF LEMMA <REF>)

As already mentioned in Subsection <ref>, we proceed to prove Lemma <ref> in two steps: We first give an argument in the simplest case of random holes
H^ having periodic centres and i.i.d. radii (cf. example (a) of Section <ref>). In that case, the crucial role played by assumption (<ref>) on the random geometry of the set H^ becomes clear.
We then generalize this argument to an arbitrary process (Φ, ℛ) that satisfies the assumptions of Theorem <ref>. We observe that, as it becomes apparent in the proofs of this section, the full-probability set of realizations for which the statement of Lemma <ref> holds true is selected by countable repeated applications of Strong Laws of Large Numbers-type of results. The final set Ω' ⊂Ω in which we prove the existence of the oscillating test functions is thus a countable intersection of full-probability sets and remains of full probability. 





Before giving the proof of Lemma <ref>, we fix the following notation: For any two open sets A ⊂ B⊂^d, we define

    (A, B) := inf{∫ |∇ v|^2   :   v ∈ C^∞_0(B),   v ≥_A}.




For a point process Φ on ^d and any bounded set E ⊂^d, we define the random variables

    Φ(E)   := Φ∩ E,       Φ^(E):= Φ∩(1/E ), 
    
      N(E)    := # (Φ (E)),        N^(E) := # (Φ^(E)).

For δ > 0, we denote by  Φ_δ a thinning for the process Φ obtained as 

    Φ_δ(ω):= { x ∈Φ(ω)   min_y ∈Φ(ω),  y ≠ x | x- y| ≥δ},

i.e. the points of Φ(ω) whose minimal distance from the other points is at least δ. Given the process Φ_δ, we set Φ_δ(E), Φ_δ^(E), N_δ(E) and N_δ^(E) for the analogues for  Φ_δ of the random variables defined in (<ref>).



For a fixed M>0, we define the truncated marks

    ℛ^M:= {ρ_j,M}_z_j ∈Φ,     ρ_j,M := ρ_j ∧ M.


Furthermore, throughout the proofs, we write 

    a ≲ b

whenever a ≤ C b for a constant C=C(d) depending only on the dimension d. 


Finally, we remark that, under the assumptions of the process (Φ, ℛ) in Theorem <ref>,
the process (Φ, {ρ^d-2}_z_i∈) satisfies the assumptions of Section <ref>, and we therefore may apply
all the results stated in that section.



 §.§ Case (a): Periodic centres

In this setting, the holes H^ε are generated by Φ= and a collection of i.i.d. random variables {ρ_i }_i ∈ satisfying (<ref>).
It is immediate to check that the marked process 





Since the centres of the holes are periodically distributed, the only challenge in the construction of the functions w_ of Lemma <ref> is due to the random variables {ρ_i}_i∈ which 
might generate very large holes under the mere condition (<ref>). In fact, in <cit.> the construction of w_ relies on the assumption that each hole B_( z_i), z_i ∈, is strictly contained in the concentric cube of size ; this allows to explicitly construct w^ by locally solving a PDE on each of these cubes. In our case, the sole assumption (<ref>) does not exclude that there are big holes which overlap and where the previous construction breaks down. The main auxiliary result on which Lemma <ref> for the periodic case (a) relies is the following Lemma <ref> on the asymptotic geometry of the set H^. Roughly speaking, this lemma ensures that H^ may be almost surely partitioned into two subsets, a “good” and a “bad” set of holes which we denote by H^_g and H^_b, respectively. The set H^_g contains most of the holes of H^ and is made of small balls where the construction of w_ may be carried out similarly to <cit.>. The remaining holes, some of which overlap with full probability, are all included in H^_b.  This set is well separated from H^_g and small with respect to the macroscopic size of the domain D: We may indeed enclose  H^_b into a set D^_b ⊂ D which is still separated from H^_g and such that the harmonic capacity of H^_b with respect to this “safety layer” D^_b vanishes in the limit ↓ 0^+. This allows us to implicitly define w^ in D^_b as the capacitary function of H^_b in D^_b; this choice ensures that the H^1-norm of w_ on D^_b converges to zero. Hence, in the verification of (H2) and (H3) of Lemma <ref>, we only need to focus on the construction of w_ on D\ D^_b.

 Let δ∈ (0,2/d-2) be fixed. Then, there exists _0= _0(δ) > 0 such that for -almost every ω∈Ω and for all ≤_0 there exist H^ε_g(ω), H^ε_b(ω), D^_b(ω) ⊂^d such that

    H^ε(ω) = H^ε_g(ω)    ∪ H^ε_b(ω),       H^ε_b(ω) ⊂ D^_b(ω), 
       ( H^ε_g(ω),  D^_b(ω) )≥ε/2,

where

    lim_ε↓ 0^+( H^ε_b(ω),D_b^(ω)) = 0.

Moreover, H^ε_g(ω) may be written as the following union of  disjoint balls centred in n^(ω) ⊂^d ∩1/εD:

    H^ε_g(ω): =    ⋃_ z_j ∈ n^ε B_ε^d/d-2ρ_j( ε z_j ), 
    ε^d/d-2ρ_j ≤ε^1+ δ< /2,      lim_↓ 0^+^d#(n^ε) = |D|.






The partition of the set H^(ω) in the statement of the lemma clearly depends on the realization ω∈Ω; in the sake of a leaner notation, though,  in the rest of the proof we omit the argument ω and write H^, H^_b, H^_g instead of H^(ω), H^_b(ω), H^_g(ω). For each z_i ∈^d, we denote by Q^_i the cube of length  centered at z_i. 



We begin by constructing the set H^_b and its “safety layer” D_b^ε. We first  include in H^_b the particles which are large compared to the size of the cubes Q_i^: For δ as in the statement of the lemma,we consider the subset of  given by

    J_^b := {z_i ∈^d ∩1/ D ^d/d-2ρ_j ≥^1+δ},

and the corresponding union of balls

    H̃^_b := ⋃_ z_j ∈ J^_b B_2^d/d-2ρ_j( z_j).

We now extend J^_b by including the centres of the balls for which, independently from the size of their radius, the corresponding cell Q^_i intersects H̃^_b: We define

    Ĩ^_b := { z_i ∈^d    Q^_i ∩H̃^_b ≠∅}⊃ J^_b,    I^_b := Ĩ^_b ∩1/ D.

We finally set

    H^_b := ⋃_z_j ∈ I^_b B_^d/d-2ρ_j( z_j),     D_b^	 := ⋃_z_j ∈Ĩ^_b Q^_j.

By (<ref>) it is immediate that H^_b ⊂ D_b^. To show (<ref>) we first argue that provided  ≤_0(δ), with  _0(δ) such that 2_0^1+δ≤_0, for every  z_j ∈ I^_b it holds

    B_2^d/d-2( z_j)  ⊂ D_b^.

Indeed, since by definition H̃^_b ⊂ D_b^, if z_j ∈ J^_b, then (<ref>) follows immediately. If, otherwise,
z_j ∈ I_b^∖ J^_b, then the assumption  ≤_0 implies B_2^d/d-2( z_j) ⊂ Q_j^⊂ D^_b. 
By the subadditivity of the capacity (see definition (<ref>)) we estimate

    Cap(( H^ε_b(ω),D_b^(ω)) 
    	   (<ref>)≤∑_j ∈ I^_bCap( B_^d/d-2ρ_j(x_j),D_b^(ω)) 
       (<ref>)≤∑_j ∈ I^_bCap( B_ρ_j(x_j),B_2 ^d/d-2ρ_j(x_j)) 
         ≲∑_j ∈ I^_b^d ρ_j^d-2.

To conclude (<ref>), it remains to show that the right-hand side above vanishes almost surely in the limit ↓ 0^+. This follows
from Lemma <ref> for the process (, {ρ^d-2_i }_z_i ∈) provided

    lim_↓ 0^+^d # (I^_b) = 0 .




To show (<ref>), we first bound by (<ref>) 

    ^d # (I^_b) ≤^d # (J^_b)  + ∑_z_i ∈ I_b^\ J^_b |Q_i^|.

We note that by (<ref>) and (<ref>), there exists a constant c=c(d) such that, provided ≤_0(d)
(with _0(d) possibly smaller than the one above), for any cube Q^_i with z_i ∈ I^_b, there exists z_j ∈ J^_b such that

    Q^_i ⊂ B_2cρ_j( z_j).

Since all the cubes Q^_i are (essentially) disjoint, we use the previous inclusion and the definition of H̃^_b to bound

    ^d # (I^_b)≲^d # (J^_b)  + |H̃^_b |    ≲^d # (J^_b)  + ∑_z_j ∈ J^_b(ρ_j)^d
       ≲^d # (J^_b)  +(max_z_j ∈1/D ∩^dρ_j )^2^d∑_z_j ∈ J^_bρ_j^d-2.

By Lemma <ref>, we have almost surely 

    lim sup_↓ 0^+max_z_j ∈1/D ∩^dρ_j 
    ≤lim_↓ 0^+( ^d ∑_z_j ∈1/D ∩^dρ_j^d-2)^1/d-2
    = ⟨ρ^d-2⟩^1/d-2,

and thus estimate for  small enough (this time depending on ω)

    ^d # (I^_b) ≲^d # (J^_b)  + ⟨ρ^d-2⟩^2/d-2∑_z_j ∈ J^_b(ρ_j)^d-2.

The first term on right-hand side above tends to zero thanks to

    ^d #( J^_b) = ^d  ∑_z_j ∈1/D ∩^d_ρ_j ≥^1+δ≤^2-δ(d-2)^d ∑_1/D ∩^dρ_j^d-2

and the choice δ< 2/d-2 together with the right-hand side of (<ref>). By this estimate and Lemma <ref>, also the second term on the right-hand side of (<ref>) vanishes almost surely in the limit ↓ 0^+. We thus established (<ref>) and therefore also (<ref>).



We now define H^ε_g:= H^ε\ H^ε_b, which allows to write H^ε_g as in (<ref>)
with n_ = (∩1/D) \  I^_b.  The first property in (<ref>) is immediately implied by J_b^⊂ I^_b and (<ref>). The second property in (<ref>) follows from (<ref>).



 It remains to prove the last inequality in (<ref>): By the definition of H^_g itself,
 if B_ρ_j( z_j) ⊂ H^_g, then ρ_j ≤^1+δ. We choose ≤_0(δ) as in (<ref>), such that B_ρ_j( z_j) ⊂ Q^_j
 and

    /2≤dist( B_ρ_j( z_j) , ∂ Q^_j) 
    (<ref>)≤dist(  B_ρ_j( z_j) , D^ε_b).

This concludes the proof of Lemma <ref>.





 Let us fix δ and _0(δ) as  in  the statement of Lemma <ref>. Then, we know that we may fix -almost any event  ω∈Ω such that we find H_b^ε(ω), H^ε_g(ω) and D^_b(ω) as in Lemma <ref>. Also in this proof, to keep the notation leaner,  we omit the argument ω in the oscillating test functions and in the set of holes and write, for instance, w_ε, H^ instead of w_ε(ω, ·) and H^(ω).



 Step 1. We begin by a reduction argument: We claim that we may separately treat the two regions D_b^ and D \ D_b^, 
which contain H_b^  and H_g^ respectively,  and give an explicit construction for w^ only in the set D \ D_b^ . We indeed claim that, for -almost every ω∈Ω we may set w^ = w^_1 ∧ w^_2 with
 w_1, w_2 ∈ H^1(D) and such that

    w^_1    ≡ 1   in  D ∖ D_b^,    w^_1    = 0   in H^_b,
    
    0    ≤ w_2^≤ 1,     w^_2    ≡ 1   in D^_b,    w^_2    = 0    in H^_g,

with, in addition,

    w^_1 → 1      in H^1(D).

If true, this decomposition for w^ yields that (H1) is satisfied and, by (<ref>), that (H2) needs to be argued only for  the sequence {w^_2 }_ > 0. Finally,  since ∇ w^_1 and ∇ w^_2 have disjoint support, for any sequence { v_}_>0⊂ H^1(D) as in (H3) we have that 

    ( -Δ w^, v_ )_H^-1, H^1_0 = ∫∇ w^_1 ·∇ v_ + ∫∇ w^_2 ·∇ v_,

and, by (<ref>), that the first term on the right-hand side vanishes in the limit ↓ 0^+. Since by an integration by parts, the second term on the right-hand side may be rewritten as ( -Δ w^_2, v_ )_H^-1, H^1_0, we deduce that with the previous decomposition we may verify (H3) only for the measures {-Δ w^_2}_ >0.



Step 2. Construction of w^_1 and w^ε_2. We begin with w^_1: Thanks to  (<ref>) of Lemma <ref> for H^_b, H^_g and D^_b, together with (<ref>) and (<ref>), for every ≤_0 there exists a function w̃^_1 ∈ H^1_0( D_b^), such that w̃^_1 = 1 in H^_b, which satisfies

    ∫_D_b^ |∇w̃^ _1|^2 ≤ 2  Cap( H^_b, D^_b).

If we now set  w^ε_1= 1 - w̃^_1, and trivially extend w_1^ by 1 outside D^_b, we immediately have that (<ref>) for w^_1 is satisfied. In addition, thanks to (<ref>) and our choice of w̃^_1, also (<ref>) follows. 




We now turn to the construction of w^_2: By the properties of H^_g, H^_b and D^_b of Lemma <ref>, the set D\ D^_b contains only the holes of H^_g, which are all disjoint balls, each strictly contained in the concentric cube Q^_i of size .
We define w^_2 ≡ 1 on D_b^, and explicitly construct w^_2 on D \ D^_b as done in <cit.>:
For each z_i ∈ n^, with n^ defined in the statement of Lemma <ref>, we write T_i^ = B_^d/d-2ρ_i ( z_i) and B_i =  B_/2( z_i) and define

    w^ε_2 = 1 - ∑_z_i ∈ n^ w^ε, i_2,

with each w^ε, i_2 solving

    -Δ w^ε, i_2 = 0   in  B_i ∖ T_i
    
    	w^ε, i_2 = 1   in  T_i
    
    	w^ε, i_2 = 0   in  D ∖ B_i.




Since by Lemma <ref> all the balls B_i are disjoint and contained in D \ D^_b, definitions (<ref>) and (<ref>) yield that w^_2 satisfies (<ref>) of Step 1. We thus constructed w^_1, w^_2 satisfying (<ref>) and (<ref>) of Step 1. We conclude this step by remarking that definition (<ref>) also implies that 

    w^ε,i_2 = 1 - {(T^_i,B^_i) },

and that each w^,i_2 may be written explicitly as

    w^ε, i_2(x)    = |x-   z_i|^-(d-2)- (/2)^-(d-2)/^-dρ_i^-(d-2) -(/2)^-(d-2)  in  B_i ∖ T_i 
    
    	w^ε, i_2    = 1   in  T_i
    
    	w^ε, i_2    = 0   in  D ∖ B_i.




Step 3.  Equipped with w^_1 and w^_2 constructed above, we show that w^ = w^_1 ∧ w_2^ satisfies properties (H1)-(H3).
As already discussed in Step 1, it suffices to prove that { w^_2}_> 0 satisfies (H2) and (H3). 


We begin with (H2): By (<ref>), (<ref>) and (<ref>) of Lemma
<ref>, a direct calculation leads to

    ∇ w_2^^2_L^2(D)   ≲^d∑_z_i ∈ n^ρ_i^d-2≤^d∑_z_i ∈∩1/Dρ_i^d-2.

By Lemma <ref> applied to the right hand side, we infer that, almost surely,

    lim sup_↓ 0^+∇ w^_2 ^2_L^2(D)≤ C.

In addition, since 1 - w^_2 = 0 in \( ⋃_z_i ∈ n^B_i ), and the balls {B_i }_z_i ∈ n^ are essentially disjoint, by Poincaré's inequality we obtain also

    1- w^_2 _L^2(D)^2 ≤∑_z_i ∈ n^ 1- w^_2 _L^2(B_i)^2 ≲^2 ∑_z_i ∈ n^∇ w^_2 _L^2(B_i)^2.

This, together with (<ref>) and (<ref>), yields that almost surely w^_2  1 in H^1(D) when ↓ 0^+. We thus established  (H2).



To prove (H3) for w^_2, we first use  (<ref>) and (<ref>) to decompose

    -Δ w^_2 = ∑_i=1^n^ (μ^,i- γ^,i),    μ^,i= -∂_ν w^,i_2  δ_∂ B_i,  γ^,i=-∂_ν w^,i_2  δ_∂ T_i,

with ν denoting the outer normal and δ_∂ B^i and δ_∂ T^i being the (d-1)-dimensional Hausdorff measure restricted to ∂ B_i
and ∂ T_i, respectively. We start by remarking (see (H5)' of <cit.>) that, since in (H3) the functions v_ are always assumed to be vanishing
on each T_i, we only need to focus on the convergence (H3) for the sequence of measures 

    μ^:=  -∑_i ∈ n^∂_ν w^,i_2  δ_∂ B_i : = ∑_i ∈ n^μ^,i.

More precisely, we claim that for every v_ϵ⇀ v in H^1_0(D) such that v_∈ H^1_0(D^), it holds

    ( μ^ , v_)_H^-1, H^1_0(D)→ C_0 ∫_D v,

where C_0:=(d-2)σ_d ⟨ ρ^d-2 ⟩ corresponds to the definition (<ref>) for the case Φ= under consideration.



We begin by arguing that it suffices to prove (<ref>) above for any truncated process (, ℛ^M), with M ∈ and ℛ^M defined in (<ref>). From now on, we use the lower index M to distinguish
the objects constructed with the truncated marks ℛ^M and the ones coming from ℛ. For instance, we denote by w^_2,M, μ^_M the analogues
of w^ and μ^ introduced above. Note that, for ℛ^M, the constant in (<ref>) reads 
C_0,M= (d-2)σ_d ⟨ ρ_M^d-2 ⟩.


For any M∈, since  |C_0 - C_0,M| ≲⟨ ρ^d-2_ρ≥ M ⟩,
 we bound by the triangular inequality, an integration by parts and Cauchy-Schwarz inequality  

    |( - Δ w^ , v_)_H^-1, H^1_0(D) - C_0 ∫_D v|
       ≲∇ (w^_M - w^)_L^2(D)∇ v__L^2(D)
        + |( μ^ϵ_M , v_)_H^-1, H^1_0(D) - C_0,M∫_D v | +⟨ ρ^d-2_ρ≥ M ⟩v_L^1,

By an argument similar to the one in
(<ref>), we estimate

    lim sup_↓ 0^+∇ (w^_M - w^)_L^2(D)≲⟨ ρ^d-2_ρ≥ M ⟩,

so that by letting ↓ 0^+ in (<ref>), this  and the boundedness of the sequence 
v_ in H^1 yield

    lim sup_↓ 0^+   |( - Δ w^ , v_)_H^-1, H^1_0(D) - C_0 ∫_D v| 
       ≲lim sup_↓ 0^+|( μ^ϵ_M , v_)_H^-1, H^1_0(D) - C_0,M∫_D v | + ⟨ ρ^d-2_ρ≥ M ⟩ (v_L^1 + 1).

Hence, provided that (<ref>) holds for μ^_M and any fixed M∈, we may then send M ↑ +∞ and establish (H3) by assumption (<ref>).



To argue that almost surely and for every M ∈ the convergence in (<ref>) holds for μ^_M, we follow <cit.>.
First, by the definition of w_i,M^, we compute

    μ^_M = ∑_z_i ∈ n_∩1/D2^d-1(d-2) (ρ_i,M)^d-2/1- 2^d-2^2 (ρ_i,M)^d-2δ_∂ B_i.

Since ρ_i,M≤ M, to obtain (<ref>) it suffices to prove

    μ̃^_M := ∑_z_i ∈ n_ 2^d-1 (d-2) ρ_i,M^d-2δ_ B_i→ C_0,M  strongly in  W^-1,∞(D).

 To show this, we fix M∈ and split the convergence (<ref>) into the two following steps: If we define 

    η^_M := ∑_z_i ∈^d ∩1/D 2^d (d-2) d ρ_M,i^d-2_B_i,

then  we argue that

    μ̃^_M- η^_M → 0   strongly in  W^-1,∞(D),

and

    η^_M → C_0,M  strongly in  W^-1,∞(D).



To show (<ref>), we consider the auxiliary problems

    -Δ q^_i,M   = 2^d (d-2)d ρ_i,M^d-2  in  B_i^
    ∂ q^_i,M/∂ν   = 2^d-1 (d-2) ρ_i,M^d-2  on ∂ B_i^,

which are in particular satisfied by the functions 

    q^_i,M(x) = 2^d-1 (d-2) ρ_i,M^d-2(|x-z_i|^2 - (/2)^2).

As q^_i,M = 0 on  ∂ B_i^, we may extend q^_i,M by zero outside of B_i^ and estimate

    ∇ q^_i,M_L^∞(B_i) = 2^d-1 (d-2)  ρ_i,M^d-2≲  M^d-2.

Using Poincaré's inequality, and since the balls B_i^ are disjoint, we infer that

    q^_M := ∑_z_i ∈^d ∩1/D q^_i,M→ 0   in  W^1,∞(^d).

We observe that by (<ref>)

    η^_M - μ̃^_M  =
    	 - Δ q^_M + ∑_z_i ∈ (^d ∩1/D) \ n_ 2^d (d-2) d ρ_M,i^d-2_B_i =: - Δ q^_M + R_^M.

We have by (<ref>) that - Δ q^_M → 0 in W^-1,∞(^d). On the other hand, for the term R_^M above, we have that by Lemma <ref> and (<ref>)

    lim_↓ 0^+R^_L^1≲lim_↓ 0^+^d ∑_z_i ∈ (^d ∩1/D) \ n_ρ_M,i^d-2 = 0,

almost surely. 
Since R^ is bounded in L^∞, we also have that R^*⇀ 0 in L^∞(D) and R^→ 0 in W^-1,∞(D). This yields (<ref>). 





In order to show (<ref>), we first remark that it suffices to argue that

    η^_M *⇀  C_0,M  in  L^∞(D).

Since  the family of functions {η^_M}_> 0 is uniformly bounded in L^∞(D), we identify the w^*-limit by testing η^_M with any function  ζ∈ C^1_0(D): Indeed, by Lemma <ref> applied to (, {ρ^d-2_i,M}_i∈) in the domain B=D we infer almost surely that

    ( η^_M , ζ)_H^-1, H^1_0(D)→ C_0,M∫_D ζ.

This establishes (<ref>)  and thus concludes the proof for (H3) and for the whole lemma.






 §.§ Proof of Lemma <ref> in the general case

Let (Φ, ℛ) be a marked point process satisfying the assumptions
of Theorem <ref>.
In contrast with the previous subsection, when the centres of the holes are distributed according to a general point process Φ, there is not a deterministic
positive lower bound for the minimal distance between the points of Φ. This requires some technical changes in the arguments
of Subsection <ref> but still allows us to obtain a statement on the asymptotic geometry of H^ similar to  Lemma <ref> and to prove Lemma <ref>.


 There exist an _0=_0(d) and a family of random variables {r_}_>0⊂_+ 
such that for -almost every ω∈Ω

    lim_↓ 0^+    r_(ω) = 0,

and for any ≤_0 there exist H^ε_g(ω), H^ε_b(ω), D^_b(ω) ⊂^d  such that 

    H^ε(ω) = H^ε_g(ω) ∪ H^ε_b(ω),        H^ε_b(ω) ⊂ D^_b(ω),
    (H^_g(ω), D^_b(ω))    ≥ r_(ω) /2,

where

    lim_ε↓ 0^+    ( H^ε_b(ω),D_b^(ω)) = 0.

Moreover, H^ε_g(ω) may be written as the following union of disjoint balls centred in 
n^ε(ω) ⊂Φ(1/εD):

    H^ε_g(ω)   : = ⋃_ z_j ∈ n^ε B_ε^d/d-2ρ_j( ε z_j ), 
    min_z_i≠ z_j ∈ n^ |z_i - z_j|≥ 2 r_,      ε^d/d-2ρ_j    ≤ r_(ω) /2,      lim_↓ 0^+^d#(n^ε) = ⟨ N(Q) ⟩ |D|.


Furthermore, if for δ > 0 the process Φ_δ is defined as in (<ref>),  then

    lim_↓ 0^+^d #( {  z_i ∈Φ^_2 δ(D)(ω)   ( z_i , D^_b ) ≤δ})= 0.





We remark that the lower bounds in (<ref>) and (<ref>) differ from the ones of Lemma <ref> by the factor r_. 
This implies, by (<ref>), that the minimal distance between the balls of the “good” set H^_g is only o() for 
↓ 0^+ and not of order  as required in the construction of the functions {w_}_>0 carried out
in the previous subsection. We overcome this technical issue by comparing w_ again with
the oscillating test functions w^_M obtained by approximating H^_g by a simpler set H^,M_g.
Here, H^,M_g is obtained not only by truncating the radii at size M as in the previous subsection, but also by considering in H^,M_g only the balls whose
centres (in n^⊂Φ^(D)) satisfy (<ref>) and (<ref>) with M^-1 instead of r_. As in the previous
subsection, we show that the sets H^,M are a good approximation of H^, in the sense that the associated functions w^_M and w^ are close in H^1. 
This follows from the fact that the centres removed from H^_g, which are either too close to each other or to the “safety layer” D^_b, are few and may be taken
care of by studying the properties of the thinnings Φ_δ of a process Φ defined in (<ref>). In fact, the last limit (<ref>)
of the previous lemma states that the main error in considering the approximate holes H^,M_g is given by the points which are too close to each other.



 
As in the previous subsection, we suppress the argument ω∈Ω for the random sets involved in the argument below. Let us fix an α∈ (0, 2/d-2)  and let us define

    r_ := (max_z_j ∈Φ^(D)ρ_j)^1/d∨^α/4.

With this choice, we prove that the decomposition of H^ required by the lemma holds true. We begin by showing that with this definition r_ satisfies (<ref>): Let

    F^ := { z_j ∈Φ^(D)   |  ρ_j ≥}.

If F^ = ∅, then r_≤^1/d∨^α/4. If otherwise, we estimate

    ^d max_z_j ∈Φ^(D)ρ_j^d-2 =  ^d max_z_j ∈ F^ρ_j^d-2≤^d ∑_z_j ∈ F^ρ_j^d-2.

To get (<ref>), it suffices to show that almost surely the right hand side above tends to zero in the limit ↓ 0^+. By Lemma <ref>, this holds provided 

    ^d #(F^) → 0      ↓ 0^+.

We show this by bounding

    ^d #(F^) ≲^2 ^d∑_z_j ∈Φ^(D)ρ_j^d-2,

and using Lemma <ref>. We thus established (<ref>).



Equipped with (<ref>), we now set η_= r_ and begin by constructing the sets H^_b and D^_b. As in Lemma <ref>, we denote by I^_b the set of points in Φ^(D) which generate the sets H^_b and D^_b. We start by requiring that I^_b contains the points in Φ^(D) whose associated radii are “too big”, namely
the set

    J^_b= { z_j ∈Φ^(D) ρ_j ≥η_/2}.

Similarly to the periodic case, we set

    H̃^_b := ⋃_z_j ∈  J^_b B_2 ρ_j( z_j).

We now include in I^_b also the points in Φ^(D) \ J^_b which, in spite of having radii below the threshold set in definition (<ref>), are too close to each other. We indeed define

    K^_b:= Φ^(D) \(Φ^_2 r_(D) ∪ J^_b).

 Finally, we include into I^_b also the set of points which are not in J^_b ∪ K^_b, but which might are close to H̃^_b: We denote them by

    Ĩ^_b := { z_j ∈Φ^(D) \(J^_b ∪ K^_b) H̃^_b  ∩ B_η_( z_j) ≠∅}.

We thus set

    I^_b    = Ĩ^_b ∪ J^_b ∪  K^_b,
    
    H^_b:=  ⋃_z_j ∈ I^_b B_ρ_j( z_j),    H^_g   := H^∖ H^_b,      D^_b:= ⋃_z_j ∈ I^_b  B_2ρ_j( z_j).





It remains to show that, with H^_b, H^_g and D^_b defined as in (<ref>), properties  (<ref>), (<ref>), (<ref>) and (<ref>) are satisfied. We start with (<ref>). As in the proof of Lemma <ref>, the definition of D^_b allows us to estimate by the sub-additivity of the capacity

    Cap(H^_b, D^_b) ≲^d∑_z_j ∈ I^_bρ_j^d-2.

Thanks to Lemma <ref>, we conclude that, almost surely, when ↓ 0^+, the right-hand side above vanishes provided that almost surely

    lim_↓ 0^+^d #(I^_b) = 0.

We show this by using definition (<ref>) and proving that each of the sets which constitute I^_b satisfies the limit above. We begin with J^_b: Definitions (<ref>) and (<ref>) yield

    ^d #(J^_b) ≲^2 r_^-(d-2)^d∑_z_j ∈Φ^(D)ρ_j^d-2≤^2 - α(d-2)^d∑_z_j ∈Φ^(D)ρ_j^d-2.

By Lemma <ref> and the assumption α < 2/d-2, the right-hand side almost surely vanishes in the limit ↓ 0^+. We thus established

    lim_↓ 0^+^d #(J^_b)=0,

i.e. limit (<ref>) for J^_b. We now turn to K^_b: Let {δ_k }_k ∈ be any sequence such that δ_k ↓ 0^+. Since r_ satisfies (<ref>), we estimate for any δ_k

    lim sup_↓ 0^+^d #(K^_b)(<ref>)=lim sup_↓ 0^+^d (N^(D) - N^_r_(D)) 
    	 (<ref>)≤lim sup_↓ 0^+^d (N^(D) - N^_δ_k(D)).

 We now apply Lemma <ref> to Φ and each Φ_δ_k to deduce that almost surely and for every δ_k

    lim sup_↓ 0^+^d #(K^_b) ≤⟨  N(Q) - N_δ_k(Q) ⟩ |D|,

where Q is a unit cube. By sending δ_k ↓ 0^+, Lemma <ref> yields 

    lim_↓ 0^+^d #(K^_b) = 0.




To conclude the proof of (<ref>), it remains to show that almost surely also

    ^d #(Ĩ^_b) → 0      ↓ 0^+.

By definitions (<ref>), (<ref>) and (<ref>), for each z_i ∈Φ^(D)\ (J^_b ∪ K^_b), we have 

    min_z_j ∈Φ^(D) \{ z_i}| z_j - z_i| ≥ 2η_,     ρ_i < η_/2.

On the one hand, by the first inequality above, the balls { B_η_(  z_i ) }_z_i ∈Ĩ^_b are all disjoint and satisfy

    ^d #(Ĩ^_b)    ≲^d ∑_z_i ∈Ĩ^_bη_^-d |B_η_( z_i)| =  r_^-d∑_z_i ∈Ĩ^_b |B_η_( z_i)| .

In addition, the inequalities (<ref>), definitions (<ref>), (<ref>) and  (<ref>) also imply that  |⋃_z_i ∈Ĩ^_b B_η_(  z_i )| ≲ |H̃^_b|. By wrapping up the previous two inequalities, we bound

    ^d #(Ĩ^_b)    ≲ r_^-d |H̃^_b| (<ref>)≲  r_^-d∑_z_j ∈ J_b^(ρ_j)^d 
       ≲ r_^-d(max_z_j∈ J^_bρ_j)^2 ∑_z_j ∈  J^_b(ρ_j)^d-2.

By definition (<ref>), the inequality above reduces to

    ^d #( Ĩ^_b) ≲^d ∑_j ∈  J^_b(ρ_j)^d-2.

Thanks to (<ref>), we apply Lemma <ref> and deduce (<ref>). This, together with (<ref>) and (<ref>), yields (<ref>) and concludes the proof of (<ref>).



To show (<ref>), we recall the definitions of D^_b, H^_b and H^_g in (<ref>) and set n^:= Φ^(D)\ I^_b. Since all z_i ∈ n^ satisfy (<ref>) and thus also

    dist( B_ρ_i( z_i) , ∂ B_η_( z_i)) ≥η_/2,

 by definition (<ref>) of D_b^, it suffices to show  that for all z_i ∈Φ^(D)\ I^_b and all z_j ∈ I^_b we have

    B_η_( z_i) ∩  B_2 ρ_j( z_j) = ∅.

For all z_j ∈ J_b^⊂ I^_b, this identity holds by (<ref>) and the definition of n^. If z_j ∈ I_b^\ J_b^,
then we know that 2 ρ_j ≤η_ and, by (<ref>) for z_i, we obtain (<ref>) also in this case. This establishes  (<ref>) and also (<ref>).



Finally, the properties (<ref>) of the set H^_g are a consequence of (<ref>), definition (<ref>) and (<ref>) of  Lemma <ref>. 



 To show (<ref>), we resort to the definition of D^_b to estimate
  
    {  z_i ∈Φ^_2δ(D)(ω)     ( z_i , D^_b ) ≤δ}
       ⊂ I_b^   ∪{  z_i ∈ n^(ω)  ( z_i , ⋃_z_j ∈ J^_b B_2 ρ_j( z_j)  ) ≤δ}
       ∪{  z_i ∈ n^(ω) ∩Φ^_2δ(D)(ω) ( z_i , ⋃_z_j ∈Ĩ^_b ∪ K^_b B_2 ρ_j( z_j)  ) ≤δ}
       := I_b^∪ E^∪ C^ .

We already know ^d # (I_b^) → 0.
Next, we argue that 

    ^d #( E^ ) → 0.

This follows by an argument similar to the one for (<ref>): Then, we may choose _0=_0(d) such that for all ≤_0, property (<ref>) yields η_=  r_≤δ. By definition of J_b^ in (<ref>) and of E^ above, we infer that for such ≤_0, for all z_j ∈ E^ there exists z_i ∈ J_b^ such that

    B_η_( z_j) ⊂ B_2 δ + 2 ρ_i( z_i) ⊂ B_6 δ r_^-1ρ_i( z_i),

where in the second inequality we use that r_^-1δ≥  1.  We note that by (<ref>) the balls B_η_( z_j) with z_j ∈ n^ are all disjoint. Hence,

    ^d #( E^ )= r_^-dη_^d #( E^ )   (<ref>)≲ r_^-d | ⋃_z_i ∈ J^_b B_6 δ r_^-1ρ_i( z_i) | 
       ≲δ^d r_^-2d∑_z_i ∈ J^_b(ρ_i )^d 
    (<ref>)≤δ^d ^d ∑_z_i ∈ J^_bρ_i^d-2 .

By Lemma <ref>  and (<ref>), almost surely the right hand side tends to zero in the limit ↓ 0^+.



We conclude the argument for  (<ref>) by showing that the set C^ is empty when  is small: In fact, by construction, if z_i ∈ n_ satisfies

    (  z_i , ⋃_z_j ∈Ĩ^_b ∪ K^_b B_2 ρ_j( z_j)  ) ≤δ,

then there exists a z_j ∈Ĩ^_b ∪ K^_b such that for ≤_0

    |z_i - z_j| ≤(  z_i, B_2 ρ_j( z_j) ) + η_≤ 2δ.

 This yields C^⊂Φ^(D) \Φ_2 δ(D) and thus that it is empty since by definition we also have C^⊂Φ^_2δ(D). 
The proof of Lemma <ref> is complete.




 We split the proof of the lemma in the same steps as in the proof for the case of periodic centres (case (a)).
Some of these steps may be proven exactly as in the previous subsection by relying on Lemma <ref> instead of Lemma <ref>. We thus focus below only on the parts of the proof which differ from Subsection <ref>.



Step 1. Since by Lemma <ref>, the sets H^_g and D^_b are disjoint, the splitting w^= w^_1 ∧ w^_2 with
w^_1, w^_2 solving (<ref>), (<ref>) and (<ref>) remains unchanged from the case of periodic centres.

 Step 2. Again by Lemma <ref>, we construct the sequence {w^_1 }_> 0 satisfying (<ref>) and (<ref>) as
in Subsection <ref>. We thus only need to focus on the construction of the functions { w^_2}_>0, which we set equal to 1 on D^_b. 
For each z_j ∈ n^, with n^ being  the set of centers of the particles in H^_g (see Lemma <ref>), we denote
the random variables 

    d^_j := min{dist(  z_j, D^_b), 1/2min_ i ≠ j |z_i - z_j|, } .

We remark that, in contrast with case (a) where we had by Lemma <ref> that d^_j ≥/2, here Lemma <ref> only implies that, for  small, d^_j ≥ r_ with r_ satisfying (<ref>).
By defining for each z_j ∈ n^ the sets 

    T_j^ = B_^d/d-2ρ_j ( z_j)     B_j =  B_d^_j( z_j),

we consider the function w_2^,j solving (<ref>) in B^j \ T^j and hence defined as

    w^ε, j_2(x) =  |x-  z_j|^-(d-2) - (d^_j)^-(d-2)/^-dρ_i^-(d-2) - (d^_j)^-(d-2)  in  B_j ∖ T_j 
    
    		 	 1   in  T_j
    
    	 0   in  D ∖ B_j.


Note, that by definition of d^_j, (<ref>), the functions ∇ w^,j have disjoint support.
Moreover, for  sufficiently small,

    d^_j ≥  2 ρ_j.

Indeed, by Lemma <ref>,

    2 ρ_j ≤ r_≤min{1/2min_ i ≠ j |z_i - z_j|, },

and

    2 ρ_j ≤ρ_j +  r_/2≤ρ_j + dist( T_j, D^_b)
    	= dist(  z_j, D^_b).


We thus set 

    w^_2 = 1 - ∑_z_j ∈  n^ w^,j_2,

which immediately satisfies condition (<ref>). Therefore, as discussed in Step 1, the function w^= w^_1 ∧ w^_2 satisfies (H1) and it suffices
to prove (H2)-(H3) only for w^_2.



Step 3. We begin by showing that w^_2 satisfies (H2): By the triangular inequality 
and definitions (<ref>) and (<ref>),  we estimate 

    ∇ w^_2 ^2_2 = ∑_z_i ∈  n^∇ w^,i_2^2_L^2(B_i)   ≲∑_z_i ∈ n^^d ρ_i^d-2/1- (ρ_i/d_)^d-2
       (<ref>)≲∑_i ∈ n^^dρ_i^d-2n^⊂Φ^(D)≲∑_i ∈Φ^ (D)^dρ_i^d-2 .

By Lemma <ref>, the right-hand side above is almost surely bounded in the limit ↓ 0^+. This, together with Poincaré's inequality for
1- w^_2 in D, yields that almost surely, up to a subsequence, we have w^_2 ⇀ w in H^1(D) when ↓ 0^+. 



We claim that w≡ 1. To this purpose, it is useful to consider  the following “truncated” processes  (n^_M, {ρ_j,M}_j∈ n^) which
we construct in the following way: For any M ∈, we set

    n^_M:= { z_i ∈  n^ d_j^≥/M},     ρ_j,M = ρ_j ∧ M.

In addition, let

    H^,M_g := ⋃_z_j ∈ n^_M B_ρ_j,M( z_j),    D^,M:= D \ ( H^,M_g ∪ H^_b),

and let w_2^,M be the function constructed as in (<ref>) and (<ref>) for the set H^,M_g.



By the same argument above for w^_2, almost surely and up to a subsequence ,it holds w_2^,M⇀ w^M for every M ∈. Moreover, since 1 - w^,M_2=0 on \( ⋃_z_i ∈ n^_M B_i ) and the balls B_i are pairwise disjoint and have radii in [M^-1, ] e may argue as in Subsection <ref> and infer that almost surely, and for every M∈, w^,M_2  1 in H^1(D), and therefore also strongly in L^2(D).  This implies by the triangular inequality 

    lim sup_↓ 0^+ w^_2 - 1 _2^2 ≤lim sup_M ↑∞lim sup_↓ 0^+ w^_2 - w_2^,M_2^2 .

Condition (H2) holds for w^_2 provided that the limit on the right-hand side above vanishes. By Poincaré's inequality in D, it suffices to prove that 

    lim_M↑∞lim sup_↓ 0^+∇ ( w^_2 - w_2^,M) _2^2 = 0.

To show this, we argue as follows: By construction

    w^_2 = w_2^,M    in B_i ⊂ D\ D^_b, whenever ρ_i ≤ M and d_i ≥ M^-1,
       w_2^,M≡ 1       in B_i, whenever d_i ≤ M^-1.

This implies that the L^2-norm on the left-hand side above reduces to

    ∇( w^ - w_2^,M) _2^2    = ∑_z_i ∈ n^∇( w_2^,i - w_2^,M,i) _2^2 _ρ_i ≥ M_ d_i ≥ M^-1
           
      +   ∑_z_i ∈ n^∇ w_2^_2^2 _d_i ≤ M^-1.

Similarly to (<ref>), we use the explicit formulation for w^_2, w^,M_2 to control the first term on the right-hand side above by

    ∑_z_i ∈ n^   ∇( w_2^,i - w_2^,M,i) _2^2 _ρ_i ≥ M_ d_i ≥ M^-1≲∑_z_i ∈ n^^d ρ_i^d-2_ρ_i ≥ M.

By Lemma <ref> applied to the process Φ with marks {ρ_i^d-2_ρ_i ≥ M}_z_i ∈Φ, and the assumption (<ref>), we
obtain that almost surely

    lim_M↑ +∞lim sup_↓ 0^+∑_z_i ∈ n^   ∇( w_2^,i - w_2^,M,i) _2^2 _ρ_i ≥ M_ d_i ≥ M^-1
    =  0.

By using again the same estimate as in (<ref>) for w^_2, we have that

    ∑_z_i ∈ n^∇ w_2^,i_2^2 _d_i ≤ M^-1
       ≲∑_z_i ∈ n^^d ρ_i^d-2_d_i ≤ M^-1.

By Definition of  d_i in (<ref>), we have that if d_i ≤ M^-1, then either 
z_i ∈Φ^(D) \Φ_2 M^-1^(D) or 

    z_i ∈  I^_M := {  z_i ∈ n_∩Φ^_2 M^-1(D)   ( z_i , D^_b ) ≤/M}.
 

Hence, from  (<ref>) we obtain

    lim sup_↓ 0+∑_z_i ∈ n^∇ w_2^,i_2^2 _d_i ≤ M^-1   ≤lim sup_↓ 0+^d∑_ z_i ∈Φ^(D) \Φ^2 M^-1, (D)ρ_i^d-2 +  lim sup_↓ 0^+^d∑_ z_i ∈ I^_Mρ_i^d-2

On the one hand, by (<ref>) and Lemma <ref>, the second term on the right-hand side vanishes. On the other hand, Lemma <ref> for both Φ and
 Φ^2M^-1 imply that

    lim sup_↓ 0+∑_z_i ∈ n^∇ w_2^,i_2^2 _d_i ≤ M^-1≤⟨  N(Q) - N_2 M^-1(Q) ⟩⟨ρ^d-2⟩,

where Q is a unit cube. Finally, the right-hand side in the above estimate converges to zero in the limit M →∞ again by  Lemma <ref>.
If we now wrap up the previous estimate with (<ref>) and (<ref>), we conclude  (<ref>). 
We thus established (H1) and (H2) for the sequence w^_2 (and thus also fo w^).





It remains to prove (H3). 
 We consider again the truncated sequences {w^,M_2}_>0 above and start by arguing that it is enough to show that, for every M ∈ fixed, condition (H3) is
 satisfied by w^,M_2, namely

    ( -Δ w^,M_2, v_ε)_H^-1, H^1_0→ C_0,M∫_D v,

where C_0,M:= (d-2) σ_d ⟨ N_2M^-1(Q) ⟩⟨ ρ_M^d-2 ⟩. In fact, Cauchy-Schwarz's
inequality yields 

    |( -Δ(w^_2 -  w^,M_2), v_ε)_H^-1, H^1_0| 
       ≤( ∫ |∇ v_|^2)^1/2( ∫ |∇ ( w^_2 - w^,M_2)|^2)^1/2,

and, as the family { v^}_>0 is uniformly bounded in H^1(D), by (<ref>) we get

    lim_M→∞lim sup_↓ 0^+ |( -Δ(w^_2 -  w^,M_2), v_ε )_H^-1, H^1_0| =0.

Since C_0,M→ C_0 when M ↑ +∞ by (<ref>) of Lemma <ref>, the above limit and the triangular inequality yield that to prove (H3) it suffices to show  (<ref>).





 The proof of (<ref>) follows the same lines of  Step 3. in Subsection <ref> (in particular, (<ref>) for w^,M_2), so we just point out the differences: By arguing as in that case, it suffices to prove that

    η_M^ := ∑_z_i ∈ n^_M	d(d-2)ρ_i,M^d-2^d/d_i^d_B_i   *⇀ C_0,M  in L^∞(D)

The factor ^d d_i^-d in the above expression is due to the fact that the balls B_i have now radii d_i instead of .
Hence, by including this factor, we have 

    ^d d_i^-d_B_i_L^1 = _B_( z_i)_L^1 = ^d

as in the periodic case.

Since

    η_M^_L^∞≲  M^d(d-2),

to show (<ref>) it suffices to test η_M^ with functions ζ∈ C^1_0(D).
We observe that if we define

    η̃_M^ := ∑_z_i ∈Φ^_2M^-1  (D) d(d-2)ρ_i,M^d-2^d/d_i^d_B_i ,

then as in Step 3 of Subsection <ref>, we use Lemma <ref> for Φ_2M^-1 and Lemma <ref> applied to (Φ_2M^-1, {ρ_i,M^d-2})
to infer that, almost surely and for all ζ∈ C_0^1(D),

    ∫_D  η̃_M^ζ→ C_0,M∫_D ζ.

We conclude (<ref>) by arguing that, almost surely and for all
ζ∈ C^1_0(D), we have

    |∫_D (η_M^ - η̃_M^) ζ| → 0.

Indeed,

    |∫_D (η_M^ - η̃_M^) ζ| 
       ≲  M^d-2∑_z_i ∈Φ^_2M^-1(D) \  n^_M∫_B_( z_i)  |ζ| 
       ≲ M^d-2ζ_∞^d  #( { z_i ∈Φ^_2M^-1 (D)  d_i ≤/M}).
 
By definition of d_i and (<ref>), the right-hand side tends to zero almost surely in the limit → 0.
This concludes the proof of (<ref>) and thus establishes (H3) for the sequence { w^_2 }_>0. 
The proof of Lemma <ref> is complete.



§ AUXILIARY RESULTS

In this section, we prove some auxiliary results on which  the proofs of Section <ref> rely. 
We give these results for a general marked point process (Φ , ) with Φ satisfying (<ref>) and (<ref>) and with the marks
:={X_i }_z_i∈Φ satisfying (<ref>) with 

    ⟨ X ⟩ = ∫_0^+∞ x h(x) dx < +∞

and with the function g being bounded as in (<ref>) (with ρ substituted by x and with p > 2).  

 
Let Q a unitary cube and let (Φ, ) be a marked point process as introduced above.
Then, for every bounded set B ⊂ which is star-shaped with respect to the origin, we have 
 
    lim_↓ 0^+^d N^(B)= ⟨ N(Q) ⟩  |B|     almost surely,

and

    lim_↓ 0^+^d∑_ z_i ∈Φ^(B) X_i =  ⟨ N(Q) ⟩⟨ X ⟩ |B|     almost surely.

 
 Furthermore, for every δ < 0 the process Φ_δ obtained from Φ as in (<ref>) satisfies the analogues of (<ref>), (<ref>) and

    lim_δ↓ 0^+⟨ N_δ(A) ⟩ = ⟨ N(A) ⟩

for every bounded set A ⊂.
 



  
	In the same setting of Lemma <ref>, let { I_}_ > 0 be a family of collections of points such that I_⊂Φ^( B) and
       
    lim_↓ 0^+^d# I_ = 0    almost surely.

       Then,
	
    lim_↓ 0^+^d∑_z_i ∈ I_ X_i → 0   almost surely.




In the same setting of Lemma <ref>, let us assume that in addition the marks satisfy ⟨ X^2 ⟩ < +∞. 
For z_i ∈Φ and >0, let r_i, > 0, and assume there exists a constant C>0  such that for all z_i ∈Φ and >0

    r_i,≤ C .

Then, almost surely, we have

    lim_↓ 0^+ ∑_z_i ∈Φ^(B) X_i ^d/r_i,^d∫_B_r_i,( z_i)ζ(x)   dx = σ_d/d⟨ N(Q) ⟩⟨  X  ⟩∫_Bζ(x)   dx,

for every ζ∈ C_0^1(B). 
 
 
 

  
 
 Without loss of generality we assume that B= Q^R, with Q^R the cube of size R centred at the origin and Q^R/=1/B. Moreover, we denote by { Q_i }_i∈ the partition of  made of (essentially) disjoint unit cubes centred in the points of the lattice = {x_i }_i∈.
 
 The limit (<ref>) is an easy consequence of a Strong Law of Large Numbers for correlated random variables (see Lemma <ref> in the Appendix): For all μ > 0 and all  small enough

    ^d∑_ z_i ∈Φ^( Q^R) X_i ≤^d∑_x_j ∈∩ Q^R+μ/ Z_i,

where  Z_j:= ∑_ z_i  ∈Φ( Q_j) X_i are identically distributed random variables by stationarity of (Φ, ).  Moreover, they have finite average 

    ⟨∑_ z_i  ∈Φ( Q) X_i ⟩ = ⟨ N(Q) ⟩⟨ X ⟩ < +∞

and satisfy for every i, j ∈ with i ≠ j

    |⟨ Z_i Z_j ⟩ - ⟨ Z ⟩^2|    (<ref>)=|⟨∑_ z_k  ∈Φ( Q_i)  z_l ∈Φ(Q_j) X_k X_l  ⟩  - ⟨ N(Q) ⟩^2 ⟨ X ⟩^2| 
       (<ref>)≤| ⟨ X ⟩^2 ⟨  N(Q_i) N(Q_j) ⟩ -  ⟨ N(Q) ⟩^2 ⟨ X ⟩^2| + C/|x_i - x_j|^γ⟨  N(Q_i) N(Q_j) ⟩,

where the constant C depends on the constants in (<ref>).
We now appeal to condition (<ref>): By the stationarity assumption on Φ, we have that for any i,j ∈

    ⟨ N(Q_i ) N(Q_j) ⟩ = ⟨ N( Q_i-j) N (Q) ⟩,

so that (<ref>) applied to the random variables N( Q_i-j) and N (Q) yields

    |⟨ N(Q_i) N( Q_j) ⟩ - ⟨ N(Q) ⟩^2 | ≲⟨ N(Q )^2 ⟩/| x_i - x_j|^γ.

We thus insert this bound into (<ref>) and get

    |⟨ Z_i Z_j ⟩ - ⟨ Z ⟩^2| ≤C ⟨ N(Q )^2 ⟩/|x_i - x_j|^γ.
 
Hence, condition (<ref>) is satisfied with constant C ⟨ N(Q )^2 ⟩  < +∞, where C depends on the constants in (<ref>) and (<ref>). 
We apply Lemma <ref> to the sequence {Z_i }_i∈ in  (<ref>) and conclude 

    lim sup_↓ 0^+^d∑_ z_i ∈Φ^( Q^ R ) X_i ≤⟨ N(Q) ⟩⟨ X ⟩ |Q^R+μ|.

Arguing analogously for the limit inferior and taking the limit μ→ 0 yields (<ref>). 
Limit (<ref>) follows exactly as (<ref>) by substituting the marks  X_i with 1.



 For δ > 0 be fixed. We show that Φ_δ satisfies the analogues of (<ref>) and (<ref>) together with (<ref>). Since by definition (<ref>) we have that N^δ(B) ≤ N(B), the limit in (<ref>)  follows from the Dominated Convergence Theorem. 
To show (<ref>) and (<ref>) we may argue exactly as above for the original process Φ and apply Lemma <ref> to the random variables Z^δ_i := ∑_ z_i  ∈Φ^δ( Q_j) X_i. Since for each x_i∈ we have 0≤ Z^δ_i ≤ Z_i, the only condition that remains to be shown for  the collection { Z^δ_i}_x_i∈ is (<ref>). By arguing as in (<ref>), we use again (<ref>) to reduce ourselves to show (<ref>) for the random variables {N^δ(Q_i)}_i ∈. To do so, for any x∈ we define
	
    d_x:=  min_y ∈Φ(ω),  y ≠ x | x- y|,

	so that
	
    N^δ( Q) = ∑_z_i ∈Φ∩ Q_d_x > δ(z_i),    N^δ(Q_i)=  ∑_z_i ∈τ_-x_iΦ∩ Q_d_x > δ(z_i).

	Since _d_x  > δ = _N( B_δ(x) \{x } = 0), each N^δ(Q_i) are measurable random variables with respect to ℱ( B_δ(Q_i)) defined in (<ref>), with 
	
    B_δ(Q_i):= { x ∈  dist(x, Q_i) ≤δ}.
 
We thus apply (<ref>) as above and conclude that, with a constant depending on δ, condition  (<ref>) is satisfied by the sequence N^δ(Q_i). This yields (<ref>) for Φ^δ and, by the same argument, also (<ref>). 
 
 

 
 
	Let M ∈. We define for every z_i∈Φ the truncated marks  which satisfy assumption (<ref>) and (<ref>) thanks to the corresponding assumptions for the original marks {X_i}_i∈. Since
	
    ⟨ Y_i ⟩≤⟨ X ⟩ < +∞,

	 we apply Lemma <ref> to the point process Φ with the truncated marks { Y_i}_z_i∈Φ to infer that almost surely
	 
    ^d ∑_z_i ∈Φ^(B) Y_i →⟨ X _[M,∞)(X) ⟩.

	 This yields
	
    lim sup_↓ 0^+^d ∑_z_i ∈ I_ X_i 
    		   ≤lim sup_↓ 0^+^d ∑_z_i ∈ I_ X_i _[0,M)(X_i) +  ⟨ X _[M,∞)(X) ⟩
       ≤  Mlim sup_↓ 0^+^d # I_ + ⟨ X _[M,∞)(X) ⟩(<ref>)=⟨ X _[M,∞)(X) ⟩.

	Since ⟨ X ⟩ < +∞, we may take the limit M →∞ and conclude the proof.




First, we argue that it suffices to prove the case r_i, = for all z_i ∈Φ and >0.
Indeed, for ζ∈ C^1_0(B) we use a change of coordinates to get almost surely,

    lim sup_↓ 0^+∑_z_i ∈Φ^ (B)|^d/r_i,^d∫_B_r_i,( z_i)ζ(x)  x - ∫_B_( z_i)ζ(x)  x| 
    	≤lim sup_↓ 0^+ C ∇ζ_L^∞^d N^(B) = 0

since ^d N^(B) is bounded by Lemma <ref>.
 


Without loss of generality we therefore assume r_i, = and |B| = 1. 



Next we observe that it suffices to argue that the assertion holds for any fixed ζ∈ W^1,∞_0(B). Indeed, once we have shown this,
the statement follows because there exists a countable subset of W^1,∞_0(B) which is dense in C^1_0(B).



We fix ζ∈  W^1,∞_0(B) and we begin by rewriting the term in the limit as 

    ∑_z_i ∈Φ^ (B) X_i ∫_B_( z_i )ζ( x)   dx    = ∑_z_i ∈Φ^ (B) (X_i - ⟨ X ⟩ )∫_B_( z_i )ζ( x)   dx  
          +  ⟨ X ⟩∑_z_i ∈Φ^ (B)∫_B_( z_i )ζ( x)   dx,
 
so that

    |   ∑_z_i ∈Φ^ (B) X_i ∫_B_( z_i )ζ( x)   dx - σ_d/d⟨ N(Q) ⟩⟨ X⟩∫_B ζ|
           ≤|∑_z_i ∈Φ^ (B) (X_i - ⟨ X ⟩ )∫_B_( z_i )ζ( x)   dx|  +⟨ X ⟩| ∑_z_i ∈Φ^ (B)∫_B_( z_i )ζ( x)   dx - ⟨ N(Q) ⟩∫_B ζ|.
 
Let { Q_i}_i∈ be a partition of  into (essentially) disjoint unitary cubes and let { y_i }_i∈ be the collection of their centres. We claim that if

    T_(ζ)    := ∫_B ζ ,         T̃_(ζ) := ^d ∑_Q_i ∩1/B ≠∅ζ( y_i),
    
       R_(ζ)    := ∑_z∈Φ^( B)∫_B_(  z)ζ(x)   dx,        R̃_(ζ) := ^d σ_d/d∑_Q_i ∩1/B ≠∅ N(Q_i)ζ( y_i),

then

    lim_↓ 0^+ | T^(ζ) - T̃^(ζ) |= 0,    lim_↓ 0^+ | R^(ζ) - R̃^(ζ) | = 0      almost surely.

The first limit is a standard Riemann sum; for the second limit we argue in a similar way: Since ζ∈  W^1,∞_0(B), we have that 
 
    |R_(ζ) - R̃_(ζ)|    = |∑_Q_i ∩1/B ≠∅( ∑_z_j ∈Φ(Q_i)∫_B_( z_j)ζ - ^d σ_d/d N(Q) ζ( y_i) ) |
        = |∑_Q_i ∩1/B ≠∅∑_z_j ∈Φ(Q_i)∫_B_( z_j) (ζ(x) -  ζ( y_i) ) |≤
    		 2∇ζ_∞ ^d+1  N^(B).

We now apply (<ref>) of Lemma <ref> to infer (<ref>). This, together with (<ref>) and the triangular inequality implies that almost surely

    lim sup_↓ 0^+|∑_z_i ∈Φ^ (B) X_i ∫_B_( z_i )ζ( x)   dx -σ_d/d⟨ N(Q) ⟩⟨ X⟩∫_B ζ|
           ≤lim sup_↓ 0^+|∑_z_i ∈Φ^ (B) (X_i - ⟨ X ⟩ )∫_B_( z_i )ζ( x)   dx|
             + lim sup_↓ 0^+| ^d ⟨ X ⟩σ_d/d ∑_Q_i ∩1/B ≠∅ζ( y_i)  (N(Q_i)- ⟨ N(Q) ⟩) |.
 
It remains to show that also the previous two terms on the right-hand side above vanish almost surely. Since to do this we follow an argument very similar to the one of Lemma <ref>, we only give the details of the parts in which the proof differs.  For >0 let

    a_i,:=  ∫_B_( z_i)ζ(x)   dx ,      X̃_i := X_i - ⟨ X_i ⟩

and

    S_:= ∑_z_i ∈Φ^(B) a_i, X_i,     S̃_:= ∑_z_i ∈Φ^(B) a_i,X̃_i.

 We start by proving that the the first term on the right-hand side of (<ref>), i.e. S̃_ above, vanishes in the limit; we may argue analogously that also second term on the right-hand side of (<ref>)  vanishes.

As in the proof of Lemma <ref>, we may use Chebyshev's inequality to estimate for each δ >0 
 
    ( S̃_ > δ ) ≤δ^-2⟨S̃_^2 ⟩

and rewrite 
 
    ⟨S̃_^2 ⟩   = ⟨∑_z_i, z_k ∈Φ( 1/B) a_i, a_k,X̃_i X̃_k ⟩ = ∑_Q_j ∩1/B ≠∅ Q_i ∩1/B ≠∅⟨(∑_z_l ∈Φ( Q_j) a_j,X̃_j  ) ( ∑_z_k ∈Φ( Q_i )  a_k,X̃_k ) ⟩.

If we now set Y_i := ∑_z_l ∈Φ( Q_j) a_j,X̃_j, since all |a_,i|≤ζ_L^∞^d, we argue as for the random variables { Z_i}_i∈ in the proof of Lemma <ref> and infer that 
 
    ⟨S̃_^2 ⟩   ≤∑_Q_i ∩1/B ≠∅ζ_∞^2 ^2d⟨ N(Q)^2 ⟩(X) +  C ^2dζ_∞^2 ∑_Q_j ∩1/B ≠∅ Q_i ∩ Q_j = ∅⟨ N(Q)^2 ⟩⟨ X ⟩^2/|x_i - x_j|^γ
       γ > d≲^dζ_∞^2 ⟨ N(Q)^2 ⟩⟨ X^2 ⟩.

Therefore, if we plug this into (<ref>) and apply Borel-Cantelli's lemma to the subsequence _n = 1/n with n∈, we get that

    lim_n ↑ +∞S̃__n =0    almost surely.

We appeal to an estimate similar to this one also for the second term on the right hand side of (<ref>)  (this time using the assumption (<ref>)) and conclude from (<ref>) that for the sequence {_n }_n∈ we have almost surely that

    lim_n ↑ +∞∑_z_i ∈Φ (1/_n B) X_i ∫_B__n(_n z_i  )ζ( x)   dx =   σ_d/d⟨ N( Q) ⟩⟨  X ⟩∫_Bζ(x)   dx.


 To extend (<ref>) to any sequence _j ↓ 0 and for the same full-probability set, we argue again similarly to Lemma <ref>. We first fix the following notation: 
 For 0 <  < 1, we define 
 
    : = ( ⌊1/⌋ + 1 )^-1 ,      :=  ( ⌊1/⌋)^-1.

 Note that ^-1, ^-1∈ and ≤≤.
 By writing ζ = ζ_+ + ζ_- and using linearity, we observe that it suffices to consider non-negative functions ζ and thus reduce ourselves to the case a_i,≥ 0.
 
 For any _j ↓ 0^+ we may use the triangle inequality and the assumptions on the sign of the weights and the X_i's to bound 

    S__j≤   S__j + ∑_i= 1^N^_j(B) |a_i ,_j - a_i, _j | X_i  
    ≤  S__j + max_i=1, ⋯, N^_j(B)  |a_i ,_j - a_i, _j |  ∑_i= 1^N^_j(B)X_i .

We now claim that the weights are uniformly continuous in the second index, uniformly in the first index: More precisely we have that almost surely
 
    lim_↓ 0^+max_i ≤#N^(B) |a_i,  - a_i, |/^d = lim_↓ 0^+max_i ≤#N^(B) |a_i,  - a_i, |/^d=0.

We first argue that, if this is true, the proof of the lemma is concluded:  From (<ref>) we indeed obtain

    S__j≤ S__j  + max_i=1, ⋯, N^_j(B)  |a_i ,_j - a_i, _j |/_j^d _j^d ∑_i= 1^N^_j(B)X_i.

Limit (<ref>), Lemma <ref> and the second limit in (<ref>) yield

    lim sup__j ↓ 0 S__j≤σ_d/d⟨ N( Q) ⟩ ⟨ X ⟩ ∫ζ.

We may argue similarly as in (<ref>) for the bound from below and get that

    lim inf__j ↓ 0 S__j≥σ_d/d⟨ N(Q ) ⟩ ⟨ X ⟩∫ζ .

This yields the claim of Lemma <ref>.



It thus remains to establish (<ref>): Since ζ  W^1,∞_0(B), for any choice of z_i ∈ B and _1 ≤_2 we estimate

    |a_i, _1 - a_i, _2| = ∫_B__1( 0) | ζ( x + _1 z_i) - ζ( x+ _2 z_i ) |   dx + ∫_B__2(0) \ B__1(0)ζ( x +  z_i)   dx 
    ≤∇ζ_∞ |_2- _1|  |z_i | _1^d + ζ_∞( (_2/_1)^d - 1 ) _1^d.
 
Since N^_2 (B)≤ N^_1(B) and thus i ≤ N^_2(B), we have that | z_i | ≤_2^-1 and

    |a_i, _1 - a_i, _2|  ≤ζ_W^1,∞( (1- _1/_2) + ( (_2/_1)^d - 1 ) )_1^d.

Therefore, for the choice _1=, _2= this yields

    |a_i,  - a_i, |  ≤ζ_W^1,∞ (  + (1/1-)^d -1 ) ^d .

and hence also the first limit in (<ref>) by (<ref>) of Lemma <ref>. The second limit may be argued in a similar way. 


 

§ APPENDIX



 §.§ Strong Law of Large Numbers for sums of random variables with correlations.  
 This result is an easy adaptation to our setting, and to our needs, of the standard argument for the Strong Law of Large Numbers.

 
Let {x_i}_i ∈ =, and let { X_i}_i ∈ be identically distributed random variables with X_i ≥ 0 and ⟨ X ⟩ <+∞. 
Let us assume that for every i,j ∈ with i ≠ j

    |⟨ X_i X_j ⟩ - ⟨ X ⟩^2 | < C/|x_i-x_j|^γ     γ > d.

Then for every bounded Borel set B ⊂ which is star-shaped with respect to the origin, we have

    lim_↓ 0^+^d∑_x_i ∈∩1/B X_i = ⟨ X ⟩ |B|     almost surely.


 
The proof of this lemma is an easy adaptation of the standard argument for independent and identically distributed random variables: In particular, we adapt to the case of correlated variables the argument
of <cit.>. 

Without loss of generality, we assume that B= Q, where Q is the unitary cube centred at the origin and Q^1/= 1/Q. 
Moreover, we may assume that |x_i| is monotone in i ∈. Thus, there exists a constant c = c(d) such that for all i ∈

    |x_i| ≥ c i^1/d.




The first step is to reduce the study of (<ref>) to the sum of the truncated random variables Y_i := X_i _X_i ≤ i.
Indeed, 

    ∑_i=1^∞(X_i > i) ≤∫_0^∞(X > t)  t = ⟨ X ⟩ < ∞.

Thus, by Borel-Cantelli theorem applied to the events E_i:= { X_i > i } we have that almost surely in (<ref>) we may substitute the variables X_i with their truncated versions Y_i. Clearly, also the sequence { Y_i}_i∈ satisfies (<ref>).



We define Ỹ_i := Y_i - ⟨ Y_i ⟩. The next step of the proof is the following estimate:

    ∑_i=1^∞⟨Ỹ_i^2 ⟩/i^2≤ 4 ⟨ X ⟩ < ∞.

We estimate

    ⟨Ỹ_i^2 ⟩≤⟨ Y_i^2 ⟩∫_0^∞ 2 y  ( Y_i > y)  y ≤∫_0^i 2 y  ( X > y)  y.

Using the monotone convergence theorem, this yields

    ∑_i=1^∞⟨Ỹ_i^2 ⟩/i^2≤∑_i=1^∞1/i^2∫_0^∞_(0,i)(y) 2 y  ( X > y)  y
    	≤∫_0^∞∑_i > y^∞1/i^2 2 y  ( X > y)  y.

Since ∫_0^∞ ( X > y)  y = ⟨ X ⟩, to prove (<ref>) it suffices to show

    y ∑_i > y1/i^2≤ 2.

If y ≥ 1, then

    y ∑_i > y1/i^2   = y ∑_i= ⌊ y ⌋+ 1^∞1/i^2≤ y ∫_⌊ y ⌋^∞1/t^2 t = y/⌊ y ⌋≤ 2.

If 0 < y < 1,

    y ∑_i > y1/i^2≤ 1 + ∑_i=2^∞1/i^2≤ 1 +  ∫_1^∞1/t^2 t = 2.

This concludes the proof of (<ref>).



Next, we define 

    S_:= ∑_i ∈∩ Q^1/ Y_i,     S̃_:= ∑_i ∈∩ Q^1/Ỹ_i.

Then, for every δ >0, we estimate by Chebyshev's inequality

    (^d S̃_ > δ ) ≤^2d⟨S̃_^2⟩/δ^2 = δ^-2^2d⟨∑_j,i ∈∩ Q^1/Ỹ_i Ỹ_j ⟩ .

By definition of Y_i and assumption (<ref>)  the last term is bounded by

    (^d S̃_ > δ ) ≤δ^-2^2d∑_i ∈∩ Q^1/⟨Ỹ_i^2 ⟩ + δ^-2^2d∑_j, i ∈∩ Q^1/ i ≠ jC/ |i - j|^γ.

We now restrict ourselves to consider the sequence _k:=α^k, k∈ and α∈ (0, 1) and use the previous inequality to estimate

    ∑_k=1^+∞(_k^d S̃__k > δ ) ≤δ^-2∑_k=1^+∞_k^2d∑_i ∈∩ Q^ 1 /_k⟨Ỹ_i^2 ⟩ + δ^-2∑_k=1^+∞_k^2d∑_j, i ∈∩ Q^ 1/_k i ≠ jC/ |z_i - z_j|^γ.

For the second term on the right-hand side above, thanks to assumption γ > d, we have

    ∑_k=1^+∞_k^2d∑_j≠ i ∈∩ Q^1/_kC/ |z_i - z_j|^γ≤∑_k=1^+∞_k^d < +∞.

To estimate the first term on the right-hand side in (<ref>), we can interchange the order of the sums since all terms are nonnegative.
Thus,

    ∑_k=1^+∞_k^2d∑_i ∈∩ Q^1/_k⟨Ỹ_i^2 ⟩   = ∑_i ∈⟨Ỹ_i^2 ⟩∑_k=1^+∞_k^2d_x_i ∈ Q^1/_k(<ref>)≤∑_i ∈⟨Ỹ_i^2 ⟩∑_k _k^-d≤ C i_k^2d
       ≲∑_i ∈⟨Ỹ_i^2 ⟩1/i^2d1/1-α^2d(<ref>)≲⟨ X ⟩ < ∞

Therefore, for every δ >0  we have that ∑_k=1^+∞(_k^d S̃__k > δ ) < +∞ and by Borel-Cantelli's lemma and the Dominated Convergence theorem we get

    S̃__k→ 0     almost surely.

Since lim_i →∞⟨ Y_i ⟩ = ⟨ X ⟩, this implies also

    S__k→⟨ X ⟩    almost surely.




To pass to the continuum limit ↓ 0^+ for the same full-probability set, we argue as in <cit.> by monotonicity.
Indeed, for _k+1≤≤_k, we have 

    ∩ Q^1/_k⊂∩ Q^1/⊂∩ Q^1/_k+1

Hence, since Y_i ≥ 0, it holds

    #(Q^1/_k+1)/#(Q^1/_k) S__k≤ S_≤#(Q^1/_k+1)/#(Q^1/_k) S__k+1 .

By the choice _k=α^k, we obtain 

    #(Q^1/_k+1)/#(Q^1/_k)→α^-d.

We now combine (<ref>) and (<ref>) to infer

    lim inf_→ 0 S_≥α^d ⟨ X ⟩, 
         lim sup_→ 0 S_≤α^-d⟨ X ⟩.

Since α∈ (0,1) is arbitrary, we conclude the proof by sending α→ 1^- in the inequalities above.






 §.§ Conditions of Theorem <ref> for the processes defined in case (c) of Section <ref>.

By construction, the processes are stationary. Moreover, the marginal _ℛ satisfies   (<ref>), (<ref>). Therefore, it suffices to prove that the point process Φ as defined in either (c.1) or (c.2) satisfies (<ref>) and (<ref>).



We begin with case (c.1): For a bounded set D ⊂, r >0 and a point x_i∈, we define N_1(D):= #Φ_1(D) and N_r^i(D)= #Φ_r^x_i∩ (B_r_i(x_i) ∩ D). For R > 0, let

    B_R(D):= { x ∈  ( x, D) ≤ R}.

Then, by (<ref>), we  estimate 

    ⟨ N(Q)^2 ⟩   =   ⟨( ∑_z_i ∈Φ_1(  )  N_r_i^i( B_r_i(z_i) ∩ Q) )^2  ⟩≤⟨ N_1(  B_R_c(Q )) ∑_z_i ∈Φ_1(  B_R_c(Q) ) ( N_r_i^i( B_r_i(z_i)) )^2 ⟩
       =  ⟨ N_1( B_R_c (Q))^2 ⟩⟨ N^0_R_c(B_R_c(0))^2 ⟩≤λ_1^2 λ_2^2_∞ R_c^2d |B_R_c(Q) |^2.
 
After taking the square-root of the above inequality, we conclude (<ref>).



Condition (<ref>) is an easy consequence of the fact that the process under consideration has finite range of dependence R_c, namely that if dist(A, B) > R_c,
then the random variables N(A) and N(B) are independent. Thus, condition (<ref>) is satisfied for any γ> 0 and with a constant depending on R_c. 



We now turn to case (c.2). In this case, property (<ref>) is an immediate consequence of the choice β < 1 and the fact that, if Φ̃ is the Poisson point process on  with intensity α, then for every m∈ and bounded set B⊂

    ⟨ N(B)^m ⟩≤⟨ ( #Φ̃(B))^m ⟩ .

Furthermore, as in the previous case, the process considered in (c.2) has finite range of dependence given by r_c and thus satisfies (<ref>) for any γ > 0.



§ ACKNOWLEDGEMENTS

The authors acknowledge support through the CRC 1060 (The Mathematics of Emergent Effects) that is funded through the German Science Foundation (DFG), and the Hausdorff Center for Mathematics (HCM) at the University of Bonn.




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