Asymptotic Behavior of the Dirichlet Energy on Poisson Point Clouds

We prove that quadratic pair interactions for functions defined on planar Poisson clouds and taking into account pairs of sites of distance up to a certain (large-enough) threshold can be almost surely approximated by the multiple of the Dirichlet energy by a deterministic constant. This is achieved by scaling the Poisson cloud and the corresponding energies and computing a compact discrete-to-continuum limit. In order to avoid the effect of exceptional regions of the Poisson cloud, with an accumulation of sites or with ‘disconnected’ sites, a suitable ‘coarse-grained’ notion of convergence of functions defined on scaled Poisson clouds must be given.


Introduction
The object of this paper is an analysis of the asymptotic behaviour of quadratic energies on Poisson random sets.Loosely speaking such sets are characterized by the property that the number of their points contained in a given set has a Poisson probability distribution, and that the random variables related to disjoint sets are independent.Even more loosely, in average the number of points contained in a set is proportional to the Lebesgue measure of the set.We denote by η such a random set.
In order to define some almost-sure properties of η we use a discrete-to-continuum approach that has been fruitfully used to derive continuum theories from microscopic interactions (see [2]).A simple interpretation of this method is as a finite-difference approximation.If η is a deterministic periodic locally finite discrete set in R d , then we can consider real-valued functions u : η → R and quadratic interaction potentials between points on η.The corresponding Dirichlet-type energy is x,y (u(x) − u(y)) 2 , ( where x, y indicates summation over nearest-neighbouring pairs (x, y) in η.We can then introduce a small parameter ε and scale both the environment and the energies accordingly; namely, considering u : ε η → R and x,y now summing over nearest-neighbouring pairs (x, y) in ε η.By letting ε → 0 we obtain a limit continuum energy, of the form ˆRd A∇u • ∇u dx, (1.3) where the matrix A carries information about the microstructure of the original set η.Note that in order to perform this passage to the limit we have to embed our energies in a common environment identifying functions on η with suitable interpolations.The limit is meant in the sense of Γconvergence, which implies that minimum problems for the limiting energies are approximations of the discrete ones, and can also be performed 'locally', by considering interactions only for x ∈ εη ∩ U for a fixed open set U .In order to define analogs of (1.1) and (1.2) for the realization of a Poisson cloud η we face a choice regarding what to consider as 'interacting sites', whether nearest neighbours in the sense of Voronoi cells or points 'close' in the sense of the ambient space.For the random set η these two choices are not equivalent since nearest-neighbouring points on η may be indeed arbitrarily distant in the ambient space, and conversely a very small distance between points of η does not ensure that they are nearest neighbours in η.We choose the second option, which also seems closer to applications; namely, we introduce an interaction radius λ > 0 and consider the energies defined for u : ε η ∩ Q → R, where Q is the unit coordinate cube centered in 0 (for ease of notation we treat only this case, which anyhow, up to scaling and localization, implies the result for any bounded Lipschitz open set in R d ).Note that, if a Γ-limit of such energies does exist then, thanks to the invariance properties by rotations of η, it must be a multiple of the Dirichlet integral (i.e., A in (1.3) is equal to a multiple of the identity matrix), which we expect to be almost surely deterministic.
The main issue in proving the convergence of F ε consists in providing a suitable notion of convergence for discrete functions u ε to a continuum parameter u, for which a compactness theorem can be proved under an assumption of boundedness of the energies.While for periodic η we can use piecewise-constant interpolations on Voronoi cells (or, equivalently, piecewise-affine interpolations on the related Delaunay triangulation), for a Poisson point process we cannot control the behaviour of such interpolation due to the presence of arbitrarily large and arbitrarily small Voronoi cells.Nevertheless, we can prove that the union of 'regular' Voronoi cells with suitably controlled dimensions form an infinite connected set in which we find 'paths' of cells such that also cells at distance λ are regular.This is done by exploiting a Bernoulli site-percolation argument.In the planar case d = 2 the complement of this set of Voronoi is composed of isolated sets with controlled dimensions, so that the ambient space can be thought of as a "perforated domain", in which we do not have a control of the discrete functions only in isolated 'holes' of controlled size.This allows to define a suitable convergence by choosing a subset G ε of εη composed of paths mentioned above whose union V ε (G ε ) has the geometry of a square grid (see Fig. 1.1)We thus use these grids to define a suitable convergence notion: given a sequence of function u ε : ε η → R we say that u is the L 2 -limit of u ε if ˆVε(Gε) namely, if the L 2 -distance between the piecewise-constant extensions ûε of u ε and u, restricted to the Voronoi cells of the grid G ε vanishes as ε → 0. Regular grids allow also to give a meaningful notion of boundary-value problems; in particular we can consider affine boundary condition as in the cell problems m(ξ; T Q) := inf (1.6) Using subadditive ergodic theorems we then can prove that, if ξ = 0, the constant Ξ given by Ξ := lim T →+∞ m(ξ; T Q) T 2 |ξ| 2 exists and is deterministic.Moreover, by the invariance properties of η it does not depend on ξ.This allows to state and prove the main result of the paper, which is the almost sure Γ-convergence in the planar case d = 2 of functionals (1.4) to with respect to the convergence in (1.5).Note that more in general we may consider energies of the form F a ε (u) = x,y∈Q∩εη with a positive and with compact support, recovering the case in (1.4) as a special case when a(ξ) is the characteristic function of the ball centered in 0 and radius λ.If a is radially symmetric the same limit result holds with obvious modifications in the statements.
The convergence theorem can be compared with various results in the literature.Our result is inspired by the recent paper [7], where perimeter energies on Poisson random sets are considered.In that context a simpler compactness result can be obtained with respect to the convergence is measure of sets, by using a covering lemma that ensures that the energy cannot concentrate on non-regular Voronoi cells.In the present context, this would correspond to an extension theorem for Sobolev functions from regular sets, which seems hard to obtain due to the random geometry of clusters of non-regular Voronoi cells.Furthermore, we can compare our approach to that in [4,3], where a notion of stochastic lattice η is given for which energies of the form (1.8) can be considered.Differently from Poisson random sets, stochastic lattices are more regular, in that all Voronoi cells have controlled dimension and hence are regular in the terminology above, a condition that seems a considerable restriction in terms of applications.The regularity of the lattice implies that functionals F a ε are coercive with respect to the L 2 convergence of piecewiseconstant interpolations on Voronoi cells.Conversely, in general the limits of functionals F a ε , which exist under ergodicity and stationarity assumptions, are not isotropic even if a is radially symmetric, except for specially constructed examples [16].More general random distributions of sites have been considered within problems in Machine Learning by D. Slepcev et al. [9,12,18] (see also the references therein).In their approach the convergence is given in terms of suitable interpolations of discrete functions using Optimal-Transport techniques.The presence of nonregular Voronoi cells is mitigated by considering kernels a ε with increasing support as ε → 0, which also allow to obtain isotropy in the limit (see also [8] for variational limits using a coarsegraining approach).Energies (1.8) have a continuum approximation in terms of a convolution double integral, for which random homogenization has been considered in [6] (see also [1]).We note that the existence of regular paths can be proved in any dimension d ≥ 2, but if d > 2 the geometry of regular grids can be thought as a set of "fibers" rather than a perforated domain.We believe that the same asymptotic result holds but with an even more complex argument.Finally, we mention, following a remark by D. Slepcev in a private communication, that our results may also have practical implications for the study of the graph Laplacian.Namely, one can show that if one drops all of the eigenmodes of the graph Laplacian on a low degree random geometric graph where the eigenvector has large L ∞ norm (in other words concentrates at few nodes) one still recovers the continuum spectrum.
We briefly outline the plan of the paper.In Section 2 we introduce the necessary definitions and notation for Poisson point clouds.This allows to give a definition of Dirichlet energy on a Poisson cloud η and to prove some asymptotic properties as the Poisson set is scaled by a small parameter η and correspondingly the energies F ε as in (1.4).We introduce the notation for Voronoi cells and define grids of paths of regular cells G ε,t depending on an additional parameter t > 0. This allows to define the convergence of piecewise-constant functions u ε on Voronoi cells to a continuum function u as the successive convergence of averages of u ε computed on G ε,t at the "mesoscopic scale" t to piecewise-constant functions u t defined on a square grid and then of such u t to a limit u as t → +∞ (Definition 2.7).This is proved to be equivalent to the L 2 convergence on grids as in (1.5), and actually independent of the choice of the family of grids.In Section 3 we state the main results of the paper.Theorem 3.1 states the pre-compactness the sense of the convergence above of sequences with equi-bounded Dirichlet energy; Theorem 3.3 is an almost-sure homogenization result characterizing the Γ-limit of Dirichlet energies as a deterministic quantity as in (1.7).Section 4 is devoted to the proof of the Compactness Theorem, based on the geometric properties of the grids that allow the use of Poincaré inequalities.In Section 5 we prove the Homogenization Theorem.The lower bound is obtained by using the Fonseca and Müller blow-up method, which is possible thanks to the use of cut-off functions that are locally constant close to non-regular Voronoi cells.The construction of recovery sequences is also made possible thanks to these 'regular' cut-off functions.Finally Section 6 (the Appendix) contains the proof of the existence of regular grids.

Notation and preliminaries
In this section we introduce the main ingredients required to perform our analysis.For the sake of simplicity, and since our analysis will take place in this context, we always consider the ambient space dimension to be two-dimensional, even though some of the definitions and results can be extended to general space dimension.

General notation. We let
2 denote the unit coordinate square centered in 0. We will also write Q r = rQ, Q r (x) = x + rQ.The same notation applies to B r (x), being B the unit ball of R 2 .For a Radon measure ν ∈ M + (R 2 ) the space L 2 (A; ν) is defined as the space of all measurable functions u : spt When ν = L 2 , the Lebesgue measure, we simply write L 2 (A).We denote by Bor(A) the collection of all Borel subsets of A. The notation |E| stands for the Lebesgue measure of E. For a set A and for t > 0 we define (A) t := {x ∈ R 2 | dist(x, A) ≤ t}.If no confusion arises, the notation {x i ∈ X i } i∈I is used for a family x i indexed by I, such that x i ∈ X i 2.2.Poisson point clouds.Some basic properties of the stationary stochastic point process called Poisson point cloud are here recalled.A complete treatment of this subject can be found in [10,17].In order to formally introduce this notion, we consider the family N s of simple measures; i.e., for all i, j ∈ I, and I a subset of N Here and in the sequel, δ x is the Dirac delta at x.For any Borel set E ∈ Bor(R 2 ) and k ∈ N we define the subset Definition 2.1.A Poisson point process on R 2 with intensity γ is a random element η on (N s , N ); that is, a map from a probability space (Ω, F, P) onto (N s , N ), such that 2) denoted by η(B) : (Ω, F, P) → N the random variable induced by η when fixing B, (namely η(B) := η(ω)(B) for ω ∈ Ω) then, for any B 1 , . . ., B m pairwise disjoint Borel sets we have that η(B 1 ), . . ., η(B m ) are independent.
Definition 2.1 implies (see [15,Proposition 8.3]) in particular that the Poisson point process on R 2 with intensity γ is stationary: if we define τ x η(A) := η(A + x), then τ x η is equal in distribution to η for any x ∈ R d .This implies in particular that P(η(R 2 ) < +∞) = 0 (see [15,Proposition 8.4]).In the sequel, whenever we speak of a Poisson point process we always mean a Poisson point process on R 2 with intensity γ.

Dirichlet energy on point clouds.
Let η be a Poisson point process.Without loss of generality, we fix the intensity to be γ = 1 and we carry out our analysis on the unit square Q.This is not a restriction since we may localize our energies on regular subsets of Q where the analysis applies unchanged, while we can deal with arbitrary bounded regular open sets by rescaling them to subsets of Q.
Let λ > 0 be a fixed parameter (the interaction radius) and for u ∈ L 2 (Q; η ε ), for a subset where we have set Remark 2.2.The definition of F ε takes into account the values of u in a λε-neighbourhood of A. As a consequence, the energy F ε is subadditive on essentially disjoint sets, namely Indeed, the energy of a function u on some open set A, takes into account also the contribution of all those points Note that F ε is not local, since we may not have F ε (u; A) = F ε (v; A) if u = v on A due to the interaction around boundary points.However, if u = v on A we can infer that We now give some estimates on the asymptotic behavior of F ε and η ε by means of a kind of (spatial) Mean Ergodic Theorem.We show that, almost surely, the average number of points in some open set A, lying at a distance less then λε can be bounded from above by the Lebesgue measure of A (the proof of Proposition 2.3 follow the lines of the proof of [15,Theorem 8.14]).
Proposition 2.3.There exists a constant C, depending on λ only, such that the following property holds almost surely: lim sup for any A ⊆ Q with Lipschitz boundary.
Proof.We consider a sequence {ε n } n∈N such that ε n → 0 and where C > 1 is a fixed universal constant.For an open set A we define the following objects and consider, for any i ∈ λZ 2 the random variable If we now invoke the law of large numbers (see for instance [15,Theorem B.11]), we have that for almost all ω ∈ Ω.Let now Since R 0 is a countable family of rectangles we have that P(Ω 0 ) = 1.Let now R := [p, p ]×[q, q ] ⊂ Q with p, p , q, q ∈ Q and define Since s n → 0 we immediately have µ n (R) → |R| λ 2 τ λ .In particular, having defined a realization and let {ε k = εk (ω)} k∈N be a sequence along which lim sup By taking the limit and exploiting that R ∈ R 0 , ω ∈ Ω , we achieve lim sup where C depends on λ only.In particular we have that lim sup For any open set A ⊂ Q and for any δ > 0 we can find a finite covering of disjoint rectangles Then, since µ ε is sub-additive on disjoint sets, we conclude that for a universal constant independent of λ, n, ω we obtain the claim for all ω ∈ Ω 0 , which has probability 1.
Corollary 2.4.Fix λ > 0. There exists a constant C depending on λ only such that almost surely it holds for any u ∈ C 1 (Q) and for any A ⊂⊂ Q with Lipschitz boundary. Then In particular by invoking Proposition 2.3, almost surely we have lim sup for a constant that depends on λ only.Since δ is arbitrary we get (2.4).Also Then Proposition 2.3 almost surely yields for a constant that depends on λ only.The arbitrariness of δ yields (2.3).

2.4.
Voronoi cells and paths.We now consider η : Ω → N s a Poisson point process.For a given realization ω we identify the Voronoi cell of x ∈ η as We say that x, y are nearest neighbors if C(x, η) shares a common edge with C(y, η).In this case we write x, y .Given x ∈ R 2 we define where, in case of multiple choices we consider the lexico-graphical order.We say that x im , x im+1 for m = 1, . . .M − 1.Moreover, we let (p) := #(p) denote the length of a path p.This notion induces a natural metric on η We say that a sub-cluster S ⊂ η is connected if for every x, y ∈ S there is a path p(x, y) := {x i1 , . . .x i M ∈ S} connecting x to y.

2.5.
Piecewise-constant extensions.Let q ⊆ η ε be a family of points in the point cloud.We define 2.6.Geometric structure of Poisson point processes.Now we state and prove some statistical properties of Poisson point processes that we find useful in the treatment of the Dirichlet energy on point clouds.For t ∈ R + and a Borel set A ⊂ R 2 we define When it is clear from the context that A = Q we sometimes write κ t in place of κ t (Q).For J ∈ I t (Q) we have J = t(i, j), i, j ∈ Z.Therefore, we set Q t J = Q t i,j ∈ Q t the square placed on the i-th row and j-th column.
For i = 1, . . ., k t , j = 1, . . ., k t we define the vertical and horizontal rectangles as 2.6.1.Selecting a sub-cluster: Percolation Theory.For a Poisson point process η we consider the sub-cluster where the in-radius of a set in(A) is defined as in(A) := sup{s > 0 | there exists B s (x) ⊂ A} Definition 2.5.Fix α and λ.Let ε, t > 0 be fixed.We say that a family of vertical and horizontal paths }} is a regular t-grid with Υ bounds for η ε , if the following properties are satisfied a) all the paths are in εη α (λ); b) for any m = 1, . . ., M ε,t , h i m connects the two opposite sides of R h i (t) of size t and is strictly contained in R h i (t); c) for any m = 1, . . ., M ε,t , v j m connects the two opposite sides of R v j (t) of size t and is strictly contained in R v j (t); ) neighboring points then |x − y| ≤ λε; For the family of (ε, t) regular grids with Υ bounds we use the notation G t (Υ; η ε ).
We invite the reader to confront Definition 2.5 with the situation depicted in Figure 2.1, which has the only purpose of being illustrative.For any t > 0 we consider the k 2 t ≈ 1/t 2 squares covering Q.Then we consider horizontal and vertical rectangles made by union of these squares and labeled suitably (from left to right for the vertical, and from bottom to top for the horizontal).We define a grid to be regular if inside each rectangles we can find a certain number of paths of points with properties (a)-(g) of Definition 2.5.These paths are further labeled inside each rectangles (from left to right in the vertical rectangles, and from bottom to top in the horizontal rectangles).In Figure 2.1 are depicted the sets V ε (v j m ) and V ε (h i m ) composed of the Voronoi cells of the points of the path.Theorem 2.6 ensures the existence of such grids with uniform bounds, by exploiting a Bernoulli site-percolation argument (see Section 6).
The following result ensures that we can find a universal constant Υ such that G t (Υ; η ε ) = ∅ for a suitable choice of the parameters ε, t, α, λ.It comes as a re-adaptation of a percolation result in [7] coupled with a technical geometric construction.We postpone the proof to Appendix 6.
Theorem 2.6.There exists α 0 , λ 0 with the following properties.Provided we can find a constant Υ > 0, depending on α only, and, for any t ∈ R + and for almost all realizations ω, a constant In order to lighten the notation we now choose α, λ satisfying (2.6) and we consider them to be fixed for the rest of the paper.We are covering Q with squares of size t and in horizontal, and vertical rectangles given by union of squares from the subdivision.We are considering the paths lying inside the rectangles and satisfying geometric properties (a)-(g).In particular, we find sometimes convenient to work on Q tkt ⊃ Q, which represent a slightly bigger square and can be divided in exactly k 2 t squares of size t.

2.7.
A notion of convergence for functions on Poisson point clouds.For a fixed t we introduce the set of simple function on Q as the space .
Fix now (ε, t) and consider a grid G ε,t ∈ G t (Υ; η ε ).For every i, j = 1, . . .k t we denote by (u) and we consider the operator and we simply write u ε → u, if there exists Υ ∈ R + such that, for any t ∈ R and for any sequence of grids Remark 2.8.We note that in order to have a meaningful notion of convergence, it is necessary to prove that the convergence is well defined; that is, it is independent of the choice of the grids.This will be a consequence of Lemma 4.3.Also note that this convergence implies (and, along sequences with equibounded energy, is in fact equivalent to) the L 2 convergence of the piecewise constant extension ûε restricted to the (Voronoi cells of the) grids (see Propositions 4.4 and 4.5) Remark 2.9 (Convergence up to subsequences).Note that with this notion of convergence, a sequence u ε converge to u up to subsequences if there exists {ε n } n∈N such that for any sequence of regular grids {{G εn,t } n∈N ∈ G t (Υ; η εn )} t∈R+ it holds

The main results
We have now introduced all the basic notation and we are thus ready to state our main results, which regard the asymptotic behavior of F ε .The first one is a compactness result.
where Q ⊂ A ⊂ Ω is any open set strictly containing Q, then there exists u ∈ W 1,2 (Q) such that u ε converge to u, up to subsequences, in the sense of Definition 2.7.
Remark 3.2.Note that in requiring to a sequence to have equibounded energy on Q we need to take into account Remark 2.2 and the fact that the energy is carried also on BQ.For this reason we state the compactness Theorem in terms of u ε ∈ L 2 (Ω; η ε ), with a uniform bound for the energy assumed on a set A, provided Q ⊂ A ⊂ Ω.In other words we are asking to the functions u ε to be defined on a slightly bigger open set than just Q.
With this compactness theorem in mind, that will be proved in Section 4, the following Γ convergence result is then meaningful.

Theorem 3.3 (Γ-convergence).
There exists a deterministic constant Ξ such that, for almost all realizations ω, the energy in the topology induced by the convergence of Definition 2.7.Namely (lim inf) if u ε → u in the sense of Definition 2.7 then The constant Ξ is identified by the relation where, for any open set A, m(•; A) denotes the cell problem m(ξ; A) := inf This theorem will be proved in Section 5.More precisely, in Section 5.1 we will prove the relation between Ξ and cell problem (3.2), in Section 5.3 the lower bound and finally in Section 5.4 the upper bound.

Proof of the compactness theorem
This section is entirely devoted to the proof of the compactness Theorem 3.1.We first need a few technical lemmas in order to guarantee that the convergence of Definition 2.7 is well defined and that any sequence with bounded energy is pre-compact.4.1.Preliminary lemmas.For u ∈ L 2 (Q; η ε ) we adopt the shorthand For any ε, t fixed we set Hor(i; G ε,t ) := {h i 1 , . . ., h i Mε,t } Ver(j; G ε,t ) := {v j 1 , . . ., v j Mε,t }.Next lemma ensures that we can choose a "skeleton" of G ε,t connecting two neighboring squares and carrying small energy (see Figure 4.1).This lemma is stated and proved for horizontal neighboring squares but it holds also for vertical neighboring squares with obvious changes in the proof.It will be used in the proof of Theorem 3.1 (in particular in the proof of Lemma 4.2).The "skeleton" obtained in Lemma 4.1.The blue part is carrying less energy than the grid (in red).In particular this allows to link each y ∈ Q t i,j to w ∈ Q t i,j+1 on particular paths that follows p h (i,j) .Again, for illustrative reason we are depicting the Voronoi cells of the point clouds.
(horizontal) neighboring squares there exists two vertical paths and a horizontal path such that, setting )), we have Proof.Since both horizontal and vertical paths are all disjoint we have that In particular there are three paths hi ∈ Hor(i; Note that, by the properties of the grid we have 1 Mε,t ≤ Υ ε t .Therefore, we conclude. We now provide a lemma that allows to estimate the difference of T Gε,t (u ε ), defined as in (2.8), between neighboring squares of the squares partition Q t .Once again, we limit ourselves to prove the statement for horizontal neighboring squares since the vertical case follows in the same way up to changing the notation accordingly.Lemma 4.2.There exists a constant C > 0 independent of ε and t such that for any G ε,t ∈ G t (Υ; η ε ) any u ∈ L 2 (Q; η ε ) and any pair Q t i,j , Q t i,j+1 of neighboring squares there holds Proof.Let hi , vj , vj+1 be the paths given by Lemma 4.1.Let p := p h i,j and let u(x).
We will make use of the above quantities (namely the average of the function u on the skeleton p) to estimate (u) Gε,t i,j .For any x ∈ p ∩ Q t i,j we now build a family of paths P i,j (x) which links x to all points y ∈ G ε,t ∩ Q t i,j .The construction will be such that we may control the number of times a path path passes through a point y ∈ G ε,t ∩ Q t i,j .We say that y ∈ G ε,t ∩ Q t i,j is a horizontal point if it belongs to some horizontal path h ∈ Hor(i; G ε,t ) and does not belongs to any v ∈ Ver(j; G ε,t ).We say instead that it is vertical if the converse happens.We say that it is nodal if it belongs to the intersection of horizontal and vertical paths.We briefly describe the construction: if y is horizontal, say it belongs to h i m the m-th horizontal path, then we consider the path starting from x, following p until we meet h i m and then following h i m until we reach y.If instead y is vertical and belongs to v j l we start from x, follow p until we meet v j l and then follow v j l to reach y.If it is nodal we follow any of the two possibilities.This family of paths, call it P i,j (x), have the following property, which is crucial in what follows.For any ) \ p belongs to not more than Ct ε paths t ∈ P i,j (x).Indeed, if y ∈ h i m then it belongs only to the paths with initial points in h i m , which do not exceed Ct/ε.Moreover, the length of each path does not exceed Ct/ε as well.
We now divide the rest of the proof in two steps.
Step one: Comparison between (u) p i,j and (u) Gε,t i,j .For the sake of shortness set ) Then, by using Jensen's inequality twice and property (g) of Definition 2.5 we have (u) By invoking property (d) we can further deduce (u) where the last inequality follows from the particular choice of p given by Lemma 4.1.Since the properties of the grid (property (d) of Theorem 2.6) also imply that The same exact computation shows also that (u) Step two: Comparison between (u) p i,j+1 and (u) p i,j .For x ∈ p ∩ Q t i,j let P j+1 (x) be the family of paths t on p which link each point of p ∩ Q t i,j+1 to x.By considering one path for each y ∈ p ∩ Q t i,j+1 we can build P j+1 (x) in a way that each t ∈ P j+1 (x) contains not more than Ct/ε) points.Moreover, each point z ∈ p is contained in at most Ct/ε paths.With the notation introduced above, we then compute

Conclusion. By means of Step one and
Step two, in particular by collecting (4.1), (4.2) and (4.3) and by means of a triangular inequality we conclude.4.2.Properties of the convergence for sequences with equibounded energy.We now state and prove some useful properties of the convergence in Definition 2.7.We start with the following lemma, which ensures that the limit of sequences with equibounded energy, when it exists, is unique and does not depend on the choice of the sequence of regular grids Proof.Fix Q t i,j ∈ Q t and let p, p be the union of paths given by Lemma 4.1 relative to Q t i,j (and one of its neighbors, say Q t i,j+1 without loss of generality) and to to the grid G ε,t , Ḡε,t respectively.Now by construction we have that p ∩ Q t i,j , p ∩ Q t i,j share at least two points.In particular, for any x ∈ p ∩ Q t i,j we can still build a family P(x) of paths that link each y ∈ p ∩ Q t i,j to x containing only points in p ∪ p ∩ Q t i,j .We can also ensure that each t ∈ P(x) contains not more than Ct/ε points and that any point in z ∈ p ∪ p ∩ Q t i,j is contained in at most Ct/ε paths in P(x).With the notation introduced in the proof of Lemma 4.2 we now compute where we have used the properties of p, p and of t ∈ P(x).Now, by recalling that (4.1),(4.2) are in force respectively on G ε,t , p and Ḡε,t , p by means of the same arguments used to prove Step one of Lemma 4.2, with a triangular inequality we obtain ).
If we sum up over all i, j and we observe that the energy of each square is counted at most a finite number of time (independent of t, n) we reach kt i,j=1 By means of this last relation we have In particular, if and we conclude.Now we proceed to state and prove Proposition 4.4 and Proposition 4.5, which will give us a useful characterization of the convergence in Definition 2.7.We will make use of the notion of piecewise-constant extension introduced in Section 2.5.for any sequence of regular grids Proof.By means of similar computations as the ones used in the proof of Lemmas 4.2 and 4.3, we can infer also that for any x ∈ G ε,t ∩ Q t i,j (adopting the same notation) for a constant independent of t, ε.In particular, by collecting (4.5) and (4.1) we have which, summed up over x ∈ G ε,t and i, j ∈ {1, . . ., k t }, and taking into accounting property (d) of Definition 2.5 In particular, if u ε → u, and {u t ∈ X t } t∈R+ denotes the sequence of intermediate functions with respect to G ε,t , then This implies that ˆVε(Gε,t)∩Q and then also (4.4)In particular, by relabeling the paths of G n as {h i m , m = 1, . . ., M n c n } ⊂ R h i (t) for each horizontal rectangle accordingly.We repeat the same argument for vertical paths and we obtain a grid Gn,t .Thanks to (4.9) we have that Gn,t ∈ G t (2Υ, η n ).
We now show that T Gn,t (u n ) → u t and u t → u in L 2 (Q).Let now J ∈ I t (Q) and consider Recalling the notation where where we have set u(y) dy.
We now concentrate on proving separately the estimates required.

and then
J ∈In(Qt(J)) We also have In particular, (4.12) and (4.13) yield Finally, we estimate Also, we have u(y) Hence, by taking the limit as n → +∞ we obtain T Gn,t (u n ) → u t in L 2 (Q) where Note that u t → u in L 2 (Q).In particular we exhibit a sequence of regular grids along which convergence in the sense of Definition 2.7 holds.Lemma 4.3 ensures then that it occurs on any sequence of regular grids, achieving thus the proof.
Remark 4.6.Note that Propositions 4.4 and 4.5 imply also that the stronger convergence on the piecewise-constant functions implies the convergence in the sense of Definition 2.7, provided the sequence has equibounded energy.Indeed if u ε is a sequence satisfying (2.5) and then, in particular it converges when restricted to any sequence of regular grids, allowing us to invoke Proposition 4.5 and thus to conclude that u ε → u in the sense of Definition 2.7.This has some useful consequence, as in the proof of the locality of the convergence (Lemma 4.7) and the fact that we can diagonalize in the L 2 convergence (Lemma 4.8).
Lemma 4.7.Let u ε → u in the sense of Definition 2.7 and satisfying (3.1).Let A ⊆ Q and suppose that u ε = w on A for some w ∈ C 1 (A) and for all ε > 0. Then u = w on A. Observe now that 1 Vε n (Gε n ,tn ) f weakly L 2 (A) and f (x) ≥ s 0 > 0 due to the good properties of the grids.In particular Being s 0 > 0 we conclude u = w on A.
If |J −J | = t we write J, J , meaning that the square Q t J = Q t i,j and Q t J = Q t i ,j are neighboring squares.
Lemma 4.9.Let {u t ∈ X t } t∈R+ be a sequence of function such that Then there exists a function u ∈ W 1,2 (Q) and a subsequence {t l } l∈N such that the piecewiseconstant interpolation of u t l converge to u in L 2 (Q).
Proof of Theorem 3.1.Let α, λ be fixed and ensuring the validity of Theorem 2.6 and choose {G ε,t ∈ G t (Υ; η ε )} ε,t to be a sequence of regular grids.Then, by invoking Lemma 4.2 and by summing up on i, j ∈ {1, . . ., k t }, we infer that where we have adopted the shorthand (u ε ) for J = (i, j).Moreover, by Jensen's inequality and the properties of the grid we have Fix now a sequence {t l } ∈N ∈ R + going to zero and observe that, for any J ∈ I t l (Q) we have Therefore, for such a fixed t l ∈ R + , we can find {ε m } m∈N such that (u εm ) Gε m ,t l J → u t l J .Up to a diagonal extraction argument we may find {ε m } m∈N such that This means that we can find a sequence of functions {u t l ∈ X t l } l∈N and a subsequence {ε m } m∈N such that If we now invoke Lemma 4.9, combined with estimates (4.17 and by invoking Proposition 4.5 this means that there is a subsequence of {u εm } m∈N converging to u in the sense of Definition 2.7.

Proof of the Γ-convergence Theorem 3.3
In this section we prove Theorem 3.3.We state and prove some preliminary results, subordinated to the identification of the constant Ξ and to the development of the technical machinery required to present the proof.

The cell problem. We recall the notation for the boundary-value problem m(ξ;
The first thing we need is the following lemma on the asymptotic behavior of the "cell problem" m(ξ; Q T ) when the boundary values are fixed on a cube Q T and T diverges.
Lemma 5.1.For any ξ ∈ R 2 it holds where Ξ is a constant independent of the realization.
Remark 5.2.In light of Lemma 5.1, the constant Ξ can be identified as The proof of Lemma 5.1 comes as a consequence of Proposition 5.3 below.
We denote by M (ϑ) a clockwise rotation of an angle ϑ around the origin.
as desired.
We now focus on proving Lemma 5.1.During the proof we find it convenient to explicit the dependence of F 1 and m δ on η and λ.In particular, if η is another Poisson point process, λ ∈ R we write and m(ξ; A, η , λ ) := inf We also refer, whenever needed, to f η (ξ, λ ) as the limit in T of m(ξ;A,η ,λ ) (which exists because the argument in Proposition 5.3 applies to a generic Poisson point process η ).
Step one: invariance by rotation.Observe that By dividing by T 2 and taking the limit, recalling that the limit exists (Proposition 5.3) , we get Since f η and f M (ϑ)η are independent of the realizations, as stated in Proposition 5.3, we conclude that f η (ξ, λ) = f M (ϑ)η (ξ, λ).
Step two: positive homogeneity of degree two.Fix r ∈ R + and observe that In particular, we also get that m(ξ; Q T , η, λ) = r 2 m(ξ; Q T /r , η/r, λ/r).By dividing by T 2 and taking the limit, using the existence result in Proposition 5.3, we get (setting T = T /r) Analogously we have . As a consequence we also have the equality By combining (5.5),(5.6)we thus get f η (rξ, λ) = r 2 f η (ξ, λ); that is, (5.2).

A boundary-value fixing argument.
We now concentrate on a key ingredient of these types of results; that is, the possibility of modifying boundary values.Before proceeding, we introduce the notation by referring to Figure 5.1.We fix δ > 0, N > 0 and we divide (∂Q 1 ) δ ∩ Q (depicted in soft grey on the left in Figure 5.1) in N sectors S i of size δ/N (see one of them in dark grey on the left).Given G εn,tn ∈ G tn (Υ; η n ) inside each sector we can find c i 1 , . . ., c i Kn disjoint "annuli" (one of them is depicted in red on the right) composed of portions of paths from the grid and all contained in S i .Still the good properties of the grid allows us to estimate K n ≈ δ/ε n .We consider this annuli labelled increasingly from the outer one, and we call Q(c i j ) the portion of the square bounded by the Voronoi cells of c i j and including them (on the right: the union of the region depicted in dark grey and all the red regions).Clearly Q(c i j ) ⊂ Q(c i j ) for j ≥ j.We use this geometry to build cut off functions h i that we will use to change the boundary data of a sequence of functions u n converging to u in the sense of Definition 2.7.
Proposition 5.4.Let U ⊂ Q, be an open set with Lipschitz boundary.Let {x n } n∈N be a sequence of points and pick {G εn,tn ∈ G tn (Υ; η n −ε n x n )} n∈N a sequence of grids and a sequence of functions for some u ∈ C 1 (U ).Suppose that the sequence also satisfies Then, for any δ > 0 there exists a sequence {v n } n∈N such that where C = C(α, λ) depends on α and λ only and P (U ) denotes the perimeter of U .
Proof.For the sake of simplicity we will prove the result only in the case U = Q since the general case results only in a heavier notation.So we assume U = Q and we fix δ > 0 and N > 0. Consider a regular grid, by joining the paths of the grid suitably, as in the proof of Theorem 2.6 (see Appendix) for any S i we can find (and eventually relabel) annuli c i 1 , . . ., c i Kn where δ Υε n ≤ K n ≤ Υδ ε n for a Υ uniform in n (see Figure 5.1).We moreover observe, due to the properties of the grid, that dist(c i j , c i j ) ≥ 3λε n for all i, j, i j ; ( dist(c i j , ∂S i ) ≥ 3λε n for all i, j. (5.9) Moreover, we have (5.10) Both properties (5.8) and (5.9) derive from property (e) of Definition 2.5 and from the fact that c i j is made of paths of G εn,tn .Property (5.10) is instead consequence of Property (f) of Definition 2.5.Let Q(c i j ) denote the portion of the square bounded by c i j and containing the Voronoi cells of points in c i j (we refer to Figures 5.1 and 5.2).For i = 1, . . ., N we set (5.11) Finally we adopt the shorthand ηn := ε n (η − x n ).Note that where the constant 2 arises since the interaction around ∂S i are counted twice when summed up over i.We can thus pick i = 1, . . ., N for which it holds Then, we set h n = h i n , c j = c i j and Set also G n := G εn,tn .We immediately conclude that giving v n → u.We also note that Thanks to the structure of v n we thus have where the second inequality exploits property (5.9) and the fact that v n agrees with u and u n on a slightly bigger sets than the two connected components of Q \ S i .The choice of i now allows to estimate where C, here and in the rest of the proof, stands for a constant depending on α, λ only and that may vary from line to line.We now exploit property (5.8): the annuli paths lie at a certain distance between each other and therefore we have that for some j and |x − y| ≤ λε n (5.12) In particular, we get where the last equality follows from property (5.10) of the annuli.For x ∈ ηn ∩ Q(c j ) c ∩ (c j ) λεn , let z x ∈ c j be such that |x − z x | ≤ λε n (see Figure 5.2).Then where the last inequality follow from property (5.10) and the fact that |x − z x | ≤ λε n .Thus, by summing up over j = 1, . . ., K n we obtain Then we conclude that where the constant C is independent of n, N, δ.We now use Proposition 2.3 (by observing that η − x n has the same distribution than η) and consider the limit in n and achieve A further limit in N yields (5.7).
The second part of the statement comes just from a diagonalization argument in n, t, ρ and by exploiting that x 0 is a Lebesgue point of ∇u.

5.3.
Proof of the lower bound.We follow the blow-up method by Fonseca and Müller [11] (see also [5] for its adaptation to homogenization).Let u ε → u in the sense of Definition 2.7.Without loss of generality we can assume that for L 2 -almost every x ∈ Q.This would imply that Since u n → u then, for any t ∈ R + and any grid G εn,t there exists a u t ∈ X t such that T Gε n ,t (u n ) → u t .With fixed δ, by invoking Proposition 5.5 we can find two subsequences ρ n , t n such that (5.14) and (5.15) holds.By relabeling εn := ε n /ρ n , tn = t n /ρ n , x n := x 0 /ε n and by invoking Proposition 5.4 we can find Moreover the following holds In particular, we have and finally In particular, we have εn and observing that . By considering the limit as δ → 0 we get dµ dL 2 (x 0 ) ≥ Ξ|∇u(x 0 )| 2 .as desired.
5.4.Proof of the upper bound.We prove the statement in several steps, in order to clarify the diagonalization process that we use.The strategy will be to approximate a generic function u ∈ W 1,2 (Q) with a sequence of functions {v k } k∈N which are piecewise affine on simplexes and then show how to recover the energy of each v k .Then we will exploit a diagonalization procedure (by means of Lemma 4.8).The recovery sequence for piecewise-affine maps is constructed in Step two below.The major technical point consists in handling the interaction at the common boundary between two simplexes S, S .To deal with this issue we build, for a generic simplex S and for an affine function v on S, an almost recovery sequence, which agrees with v in an internal neighborhood of ∂S.This is done in the Step one below.
Step one: We prove that for any triangle S ⊆ Q, for u(x) = ξ • x on S ⊆ Q and for any fixed δ > 0 there exists a sequence {u εn,δ ∈ L 2 (S; η n )} n∈N such that u n,δ → ξ • x in the sense of Definition 2.7 (5.16) (5.17) with C depending on S and |ξ| only.
The construction is illustrated in Fig. 5.3: we fix δ > 0, m ∈ N and we pick a grid of squares of size 1/m.For a square Q 1/m (J) that intersect in a non trivial way (∂S) δ ∩S we define u ε,m,δ = ξ •x on Q 1/m (J) ∩ S. If instead Q 1/m (J) is well contained in S we consider u ε,m,δ to agree with the quasi minimum v ε of the cell problem m on a the subsquare Q (1−δ)/m (J) and u ε,m,δ = ξ • x on Q 1/m (J)\Q (1−δ)/m (J).In this way u ε,m,δ = ξ •x on A δ m and this construction ensures convergence and the lim sup upper bound for the function u(x) = ξ • x on S. We now formalize this argument.Fix δ > 0, m ∈ N and introduce the sub-class of indexes and For J ∈ I 2,m (S) consider just In particular, setting we see that, for J ∈ I 1,m (S) we have and, by applying Corollary 2.4, S). Hence, we define In this way, by applying again Corollary 2.4 The existence of the limit of m(•; Q T ) given by Lemma 5.1 yields the existence of ε 0 such that, for all ε < ε 0 we have also Moreover sup Thanks to the compactness Theorem 3.1 we can thus conclude that v εn,m,δ → u m,δ in the sense of Definition 2.7.Observe also that v ε,m,δ (x) = ξ • x for all x ∈ A δ m , where This in particular implies that u m,δ = ξ • x on A δ m .Since u m,δ ∈ W 1,2 (S) and we also have that, up to sub-sequences u m,δ → ξ • x in L 2 (S).Hence, by applying Lemma 4.8 we can find m ε → +∞, with ε m ε → 0 and for which u ε,δ := v ε,mε,δ → ξ • x and also (5.18) and (5.17) hold.
Step two: we prove the existence of a recovery sequence for the piecewise-affine function with S a finite family of essentially disjoint triangles partitioning Q and v S (x) = v S (x) for x ∈ ∂S ∩ ∂S .Note that, being each v S affine it can be represented as v S (x) = ξ S • x + b S on S. Fix δ > 0 and for any S ∈ S let {u S n,δ ∈ L 2 (S; η n )} n∈N be the functions constructed in Step one and satisfying (5.18),(5.17),and (5.16) relatively to ξ S .Set We have that v ε,δ → v in the sense of Definition 2.7.Moreover, Then we just observe that Recall that, by construction we have that, for x ∈ S, v ε,δ (x) = v(x) on (S) λεn and, since it is continuous and piecewise affine, it is in particular a Lipschitz map.Then we have where in the last inequality we have used Proposition 2.3.Thus By now diagonalizing along δ, with the aid of Lemma 4.8, we find Step three: we prove the existence of a recovery sequence for a generic u ∈ W 1,2 (Q).We just observe that for any u ∈ W 1,2 (Q) we can find a sequence of piecewise-affine functions {v k } k∈N with the structure as in Step two such that v k → u in L 2 and ˆQ |∇v k | 2 dx → ˆQ |∇u| 2 dx.
The construction developed in Step two, for each k, and a further application of the diagonalizing procedure (Lemma 4.8) conclude the existence of the desired sequence.
6. Appendix 6.1.Existence of regular grids.We here focus on proving Theorem 2.6.Let Definition 6.1.Let {X j } j∈Z 2 be a sequence of i.i.d random variable such that X j = 1 with probability p 0 with probability 1 − p (6.1) We say that {j i } M i=1 is an open path for the realization ω if X ji (ω) = 1 for all i = 1, . . ., M and j i , j i+1 are neighboring squares.
We recall the following percolation property from [13].Theorem 6.2 (Property of Bernoulli site percolation).There exists a probability p cr such that the following holds.Let C be a compact set, for any δ > 0 there exists K δ such that for almost all ω ∈ Ω we can find T 0 (ω) > 0 for which any rectangles R v T,δ (x 0 ), R h T,δ (x 0 ) with T > T 0 and x 0 ∈ T C contains at least K δ T disjoint paths that connects the two opposite sides of R v T,δ (x 0 ), R h T,δ (x 0 ) respectively in the horizontal and in the vertical direction.
For the sake of brevity we introduce the following notation confined to this section.For R h T,δ (x 0 ), R v T,δ (x 0 ) we denote by h T 1 (x 0 ), . . ., h T M (x 0 ) and by v T 1 (x 0 ), . . ., v T M (x 0 ), the families of horizontal (and respectively vertical) disjoint paths connecting the two opposite sides of R h T,δ (x 0 ) (and R v T,δ (x 0 ) respectively).
Observe that the probability of the set of realizations satisfying properties (II) and (III) tends to 1 if α → 0 (we refer to the same argument as in the proof of [7,Lemma 4.1]).Moreover In particular, setting λ 2 for all j ∈ I(i) we have that P(A α,λ i ) = p λ (α) Λ 2 .
If α → +∞ we have p λ (α) → 1 − e −λ 2 .In particular, for any choice of Λ > 1, γ > 0 we can find α 0 (Λ), λ 0 (Λ) > 0 such that P(ξ α,λ i = 1) = p(α, λ) > 1 − γ for all α < α 0 , λ > λ 0 Thus for a suitably small α and big λ we can invoke Theorem 6.2 and find a K δ independent of the realization and T 0 = T 0 (ω, C) such that, for any R v T,δ (x 0 ), R h T,δ (x 0 ) (note that this is uniform as x 0 /T ∈ C) contains at least K δ T disjoint paths (connecting the two opposite sides of R v T,δ (x 0 ), R h T,δ (x 0 ) vertically and horizontally respectively) of neighboring squares from Q.We now define the Voronoi paths as follows.Consider, for instance {i j } N j=1 to be the first (from the bottom) horizontal path in R h T,δ (x 0 ) and let s 1 , . . ., s N be the segment joining the centers of neighboring squares (for instance (s m joins the centers of Q Λλ (i m ), Q Λλ (i m+1 ).We set h T 1 (x 0 ) := {x ∈ η : (C(x; η) ∩ s jm ) = ∅ for some m = 1, . . ., N }.We define all the other horizontal paths accordingly, as well as the vertical paths.We refer to h T 1 (x 0 ) without loss of generality in proving the properties.By arguing as in the proof of Lemma [7, Lemma 4.1] we can derive also that diam(C(x; η)) ≤ 1 α , I(C(x; η)) > α.
Moreover By choosing λ > max { 2 /α, λ 0 } we can guarantee additionally that In particular if x ∈ h T 1 (x 0 ) then x ∈ Q λ (m) ⊂⊂ Q Λλ (i j ) for some m ∈ J(i j ) intersecting s j .In particular, B λ (x) is contained in the union of the eight squares whose boundary intersects in a non trivial way the boundary of Q λ (m) and this union, call it O, is still contained in Q Λλ (i j ) and made by at most 8 of such squares.Therefore This implies that h T 1 (x 0 ) is a path on η α (λ) and we get property (a.1).Properties (b.1) and (c.1) are immediate from Bernoulli site percolation.Also property (e.1) is a consequence of the fact that any path is contained in a square around the segment joining the centers and thus the distance between two paths is at least 3λ.By the same principle, if Λ was chosen big enough from the very beginning (say bigger than 10) whenever x ∈ (h T 1 (x 0 )) 3λ then it belongs to a square of size λ contained in Q Λλ (i j ) and the same estimate applies on η(B λ (x)), achieving property (f.1).If instead x, y are neighboring points in a path, then |x − y| ≤ 2 α < λ yielding property (g.The notation adopted in the proof of Theorem 2.6.Here are depicted the sets V ε (v i j ), and V ε (h j ) for i = 1, 2 j = 1, . . ., M .
-We say that a point x lies on the right of v in R (and we write x £ R v) if any path in η ∩ R that links x to the left of R intersects v; Proof of Theorem 2.6.Let Υ, α 0 , λ 0 be given by Corollary 6.4.Fix α, λ satisfying (2.5), let t > 0.
Then for almost all realizations we can find ε 0 such that, if ε < ε 0 , the existence of paths in R h t, 1 /2 (x), R v t, 1 /2 (x) is guaranteed for x ∈ Q tkt , according to Corollary 6.4.We set the notation with the aid of Figure 6.1.Consider, for instance, the first square Q t 1,1 on the bottom-left corner corner.Consider the three rectangles depicted in Figure 6.1 with edges proportion of 1/2, namely the vertical one on the bottom left R v bl ⊂ Q t 1,1 , the horizontal one on the top R h top ⊂ Q t 1,1 and the vertical one on the top right R v tr intersecting R h top and the adjacent square Q t 2,1 .Each of them contains at least M ε,t ≥ Υt ε paths satisfying properties (a.2)-(g.2) of Corollary 6.4 relatively to their own rectangle.Now we label these paths from the , . . .v 2 M ∈ R v tr , with the shorthand M = M ε,t .Now the strategy is to join them and then refine the family in a way that all the properties (a)-(g) are ensured.
Consider a generic v We define the following path (see for instance Figure 6.2).
x R h M −j } Note that p j is a path which has the same starting point than v 1 j and the same ending point of v 2 j .Moreover with this definition we note that dist(p j , p j+1 ) > 3λε.
Indeed, suppose that for some (y j , y j+1 ) ∈ p j × p j+1 we have Then one of the following is necessarily in force (I) y j ∈ h M −j and y j+1 ∈ v 1 j+1 ;

Figure 1 . 1 .
Figure 1.1.A representation of a regular grid in Q with two 'paths' highlighted.

Figure 4 .
Figure 4.1.The "skeleton" obtained in Lemma 4.1.The blue part is carrying less energy than the grid (in red).In particular this allows to link each y ∈ Q t

Lemma 4 . 8 . 7 . 4 . 3 .
Let {u n,r ∈ L 2 (Q; η εn ), r > ε n > 0} be a sequence such that a) For any r > 0, u n,r → u r as n goes to +∞ in the sense of Definition 2.7;b) {u r } r>0 ⊂ W 1,2 (Q), u r → u as r goes to 0 in L 2 (Q) for some u ∈ W 1,2 (Q); c) sup r>εn>0 {F εn (u n,r ; A)} < +∞ on some open set A ⊃ Q.Then there is a sequence r n >> ε n such that r n → 0 and for which u n,rn → u in the sense of Definition 2.7.Proof.By invoking Proposition 4.4 we have (up to a subsequence) lim sup n→+∞ ˆVεn (Gn)∩Q |û n,r (x) − u r (x)| 2 dx = 0. Since also u r → u we have lim r→0 lim sup n→+∞ ˆVεn (Gn)∩Q |û n,r (x) − u(x)| 2 dx = 0. Hence, we can select the sought sequence r n to satisfy lim sup n→+∞ ˆVεn (Gn)∩Q |û n,rn (x) − u(x)| 2 dx = 0.Property (c) and Proposition 4.5 imply now that u n,rn → u in the sense of Definition 2.Proof of the compactness theorem.We are now ready to prove Theorem 3.1.We rely on the following Lemma 4.9, which comes as a consequence of[2, Theorem 3.1].Recall that ) and (4.18), we can find a subsequence of {t l } l∈N and a function u ∈ W 1,2 (Q) such that, with a slight abuse of notation, u t l → u in L 2 (Q) along the subsequence.But then, by Proposition 4.4 we have lim l→+∞ lim sup m→+∞ ˆVεm (Gε m,tl )∩Q |û εm (x) − u(x)| 2 dx = 0.

Proposition 5 . 3 .
For all ξ ∈ R 2 there exists almost surely the limitf (ξ) := lim T →+∞ m(ξ; Q T ) T d independent ofthe realization.Moreover, there exists a function g : R + → R + such that lim R→+∞ g(R) = +∞ and such that, for any sequence y T with |y T | ≤ T g(y T ) and any rotation M (ϑ) we have lim T →+∞ m((ξ; M (ϑ)Q T (y T )) T d = lim T →+∞ m(ξ; Q T ) T d = f (ξ).Proof.Note that m(ξ; A ∪ B) ≤ m(ξ; A) + m(ξ; B) whenever |A ∩ B| = 0.In particular, by arguing exactly as in the proof of [6, Lemma 5.1], invoking the uniform version of the sub-additive ergodic theorem, [14, Theorem 1], we can achieve the existence of g and f such that for any family of translations {y T } T ∈N satisfying |y T | ≤ T g(|y T |) it holds

Figure 5 . 1 .
Figure 5.1.In the proof of Proposition 5.4 we refer to the notation depicted here. .

Figure 5 . 2 .
Figure 5.2.The situation in the proof of Proposition 5.4.The difference h n (x)− h n (y) is different from 0 only on pairs (x, y) satisfying the relation (5.12).
holds for any sequence of regular grids{G ε,t ∈ G t (Υ; η ε )} ε,t>0 .Gn (u n )(x)| 2 ≤ Ct 2 n F εn (u n ; Q). ∈ Rand we provide a suitable rearrangement of G n in order to obtain a grid Gεn,t ∈ G t (Υ; η n ), provided n is large enough.We describe the construction.Inside each R h i (t) we can find at least c n = (t − 2tn) /tn horizontal rectangles R h Proposition 4.5.Let ε n , t n be two sequences.Set η n := η εn .Let {u n ∈ L 2 (Q; η n ))} n∈N be a sequence satisfying(2.5).If there exists a sequence of regular grids{G n ∈ G tn (Υ; η n )} n∈N such that lim n→+∞ ˆVεn(Gn)∩Q |û n (x) − u(x)| 2 dx = 0. (4.7)Then u n → u in the sense of Definition 2.7.i (t n ), . . .R h i +cn (t n ) strictly contained in R h i (t) and each containing M n disjoint horizontal paths h i 1 , . . ., h i Mn .Note that t 2 ≤ t n c n ≤ 2t (4.9) t>0 be a sequence of regular grids.Then, Proposition 4.4 gives ε (x) = w(x) on η ε ∩ A and since |x − y| ≤ λε for x ∈ G ε,t , y ∈ C(where C depends on α, λ and |A|.Here we have used Proposition 2.3.Then, for some subsequence {ε n , t n } n∈N and by means of a triangular inequality we have lim n→+∞ ˆVεn (Gε n,tn )∩A |u(x) − w(x)| 2 dx = 0.