Scaling limit of the odometer in divisible sandpiles

In a recent work Levine et al. (Ann Henri Poincaré 17:1677–1711, 2016. 10.1007/s00023-015-0433-x) prove that the odometer function of a divisible sandpile model on a finite graph can be expressed as a shifted discrete bilaplacian Gaussian field. For the discrete torus, they suggest the possibility that the scaling limit of the odometer may be related to the continuum bilaplacian field. In this work we show that in any dimension the rescaled odometer converges to the continuum bilaplacian field on the unit torus.


Introduction
The concept of self-organized criticality was introduced in Bak et al. [2] as a lattice model with a fairly elementary dynamics. Despite its simplicity, this model exhibits a very complex structure: the dynamics drives the system towards a stationary state which shares several properties of equilibrium systems at the critical point, e.g. power law decay of cluster sizes and of correlations of the height-variables. The model was generalised by Dhar [5] in the so-called Abelian sandpile model (ASP). Since then, the study of self-criticality has become popular in many fields of natural sciences, and we refer the reader to Járai [10] and Redig [20] for an overview on the subject. In particular, several modifications of the ASP were introduced such as non-Abelian models, ASP on different geometries, and continuum versions like the divisible sandpile treated in Levine and Peres [15,16]. We are interested in the latter one which is defined as follows. By a graph G = (V, E) we indicate a connected, locally finite and undirected graph with vertex set V and edge set E. By deg(x) we denote the number of neighbours of x ∈ V in E and we write "y ∼ V x" when (x, y) ∈ E. A divisible sandpile configuration on G is a function s : V → R, where s(x) indicates a mass of particles at site x. Note that here, unlike the ASP, s(x) is a real-valued (possibly negative) number. If a vertex x ∈ V satisfies s(x) > 1, it topples by keeping mass 1 for itself and distributing the excess s(x) − 1 uniformly among its neighbours. At each discrete time step, all unstable vertices topple simultaneously.
Given (σ (x)) x∈V i.i.d. standard Gaussians, we construct the divisible sandpile with weights (σ (x)) x∈V by defining its initial configuration as As in many models of statistical mechanics, one is interested in defining a notion of criticality here too. Let e (n) (x) denote the total mass distributed by x before time n to any of its neighbours. If e (n) (x) ↑ e V where e V : V → [0, +∞], then e V is called the odometer of s. We have the following dichotomy: either e V < +∞ for all x ∈ V (stabilization), or e V = +∞ for all x ∈ V (explosion). It was shown in Levine et al. [17] that if s(x) is assumed to be i.i.d. on an infinite graph which is vertex transitive, and if E[s(x)] > 1, s does not stabilize, while stabilization occurs for E[s(x)] < 1. In the critical case (E[s(x)] = 1) the situation is graph-dependent. For an infinite vertex transitive graph, with E[s(x)] = 1 and 0 < Var(s(x)) < +∞ then s almost surely does not stabilize.
For a finite connected graph, one can give quantitive estimates and representations for e V . It is shown in Levine et al. [17,Proposition 1.3] that the odometer corresponding to the density (1.1) on a finite graph V has distribution where η is a "bilaplacian" centered Gaussian field with covariance E[η(x)η(y)] = 1 deg(x)deg(y) w∈V g(x, w)g(w, y) setting g(x, y) = 1 |V | z∈V g z (x, y) (1.2) and g z (x, y) = E τ z −1 m=0 1 {S m =y} for S = (S m ) m≥0 a simple random walk on V starting at x and τ z := inf{m ≥ 0 : S m = z}. The field is called "bilaplacian" since a straightforward computation shows that where g denotes the graph Laplacian Hence the covariance is related to the Green's function of the discrete bilaplacian (or biharmonic) operator. The interplay between the odometer of the sandpile and the bilaplacian becomes more evident in the observation made by Levine  "We believe that if σ is identically distributed with zero mean and finite variance, then the odometer, after a suitable shift and rescaling, converges weakly as n → +∞ to the bilaplacian Gaussian field on R d ".
Note that, although they work with Gaussian weights in the proof of Proposition 1.3, their comment comprises also the case when σ has a more general distribution. Inspired by the above remark, we determine the scaling limit of the odometer in d ≥ 1 for general i.i.d. weights: we show that indeed it equals , the continuum bilaplacian, but on the unit torus T d (see Theorems 1 and 2). A heuristic for the toric limit is that the laplacian we consider is on Z d n , which can be seen as dilation of the discrete torus T d ∩(n −1 Z) d . We highlight that is not a random variable, but a random distribution living in an appropriate Sobolev space on T d . There are several ways in which one can represent such a field: a convenient one is to let be a collection of centered Gaussian random variables and 2 now is the continuum bilaplacian operator. We will give the analytical background to this definition in Sect. 2.2. As a by-product of our proof, we are able to determine the kernel of the continuum bilaplacian on the torus which, to the best of the authors' knowledge, is not explicitly stated in the literature.
Related work Scaling limits for sandpiles have already been investigated: in the ASP literature limits for stable configurations have been studied, for example, in Levine et al. [18] and Pegden and Smart [19]. Their works are concerned with the partial differential equation that characterizes the scaling limit of the ASP in Z 2 . They also provide an interesting explanation of the fractal structure which arises when a large number of chips are placed at the origin and allowed to topple. The properties of the odometer play an important role in their analysis. In the literature of divisible sandpiles models, the scaling limit of the odometer was determined for an α-stable divisible sandpile in Frómeta and Jara [6], who deal with a divisible sandpile for which mass is distributed not only to nearest-neighbor sites, but also to "far away" ones. Their limit is related to an obstacle problem for the truncated fractional Laplacian. In the subsequent work Cipriani et al. [4], the authors of the present paper extend the result to the case in which the assumption on the finite variance of the σ 's is relaxed, and obtain an alpha-stable generalised field in the scaling limit.
The discrete bilaplacian (also called membrane) model was introduced in Sakagawa [23] and Kurt [11,12] for the box of Z d with zero boundary conditions. In d ≥ 4 Sun and Wu [27] and Lawler et al. [13] construct a discrete model for the bilaplacian field by assigning random signs to each component of the uniform spanning forest of a graph and study its scaling limit. As far as the authors know, Levine et al. [17] is the first paper in which the discrete bilaplacian model has been considered with periodic boundary conditions.

Main results
Notation We start with some preliminary notations which are needed throughout the paper. Let T d be the d-dimensional torus, alternatively viewed as Moreover let B(z, ρ) a ball centered at z of radius ρ > 0 in the ∞ -metric. We will use throughout the notation z · w for the Euclidean scalar product between z, w ∈ R d . With · ∞ we mean the ∞ -norm, and with · the Euclidean norm. We will let C, c be positive constants which may change from line to line within the same equation. We define the Fourier transform of a function u ∈ L 1 (T d ) as u(y) := T d u(z) exp (−2πιy · z) d z for y ∈ Z d . We will use the symbol · to denote also Fourier transforms on Z d n and R d . We will say that a function We can now state our main theorem: we consider the piecewise interpolation of the odometer on small boxes of radius 1 2n and show convergence to the continuum bilaplacian field.  [12]).
We can now show the next Theorem, which generalises the previous one to the case in which the weights have an arbitrary distribution with mean zero and finite variance. We keep the proof separate from the Gaussian one, as the latter will allow us to obtain precise results on the kernel of the bilaplacian, and has also a different flavor. Moreover, the more general proof relies on estimates we obtain in the Gaussian case. With a slight abuse of notation, we will define a field n as in Theorem 1 also for weights which are not necessarily Gaussian (in the sequel, it will be clear from the context to which weights we are referring to). We now give an explicit description of the covariance structure of . Our motivation is also a comparison with the whole-space bilaplacian field already treated in the literature. More precisely, for d ≥ 5, Sun and Wu [27,Definition 3] define the bilaplacian field d on R d as the unique distribution on C ∞ c (R d ) * such that, for all Since we obtain a limiting field on T d , we think it is interesting to give a representation for the covariance kernel of the biharmonic operator in our setting. From now on, when we use the terminology "zero average" for a function u, we always mean T d u(x) d x = 0. (1.5) Remark 1 (Kernel of the biharmonic operator in lower dimensions) The convergence result of Theorem 2 allows us to determine the kernel in d ≤ 3 too. In fact, for such d interchanging sum and integrals is possible, so that we can write where we can define the kernel of the bilaplacian to be Outline of the articleThe necessary theoretical background is given in Sect. 2, together with an outline of the strategy of the proof of Theorem 1. Auxiliary results and estimates are provided in Sect. 3. The proof of Theorem 1 lies in Sect. 4, and of Theorem 2 in Sect. 5. Finally we conclude with the proof of Theorem 3 in Sect. 6.

Preliminaries
In this section we review the basics of the spectral theory of the Laplacian on the discrete torus from Levine et al. [17]. We also remind the fundamentals of abstract Wiener spaces which enable us to construct standard Gaussian random variables on a Sobolev space on T d . The presentation is inspired by Silvestri [25]. We also comment on the basic strategy of the proof of Theorem 1 and make some important remarks on the test functions we use for our calculations. We refer for the Fourier analytic details used in this article to Stein and Weiss [26] and for a survey on random distributions to Gel'fand and Vilenkin [7].

Fourier analysis on the torus
We now recall a few facts about the eigenvalues of the Laplacian from Levine et al. [17] for completeness. Consider the Hilbert space L 2 (Z d n ) of complex valued functions on the discrete torus endowed with the inner product The Pontryagin dual group of Z d n is identified again with Z d n . Let {ψ a : a ∈ Z d n } denote the characters of the group where ψ a (x) = exp(2πιx · a n ). The eigenvalues of the Laplacian g on discrete tori are given by Recalling (1.2), we use the shortcut g x (y) := g(y, x). Let g x denote the Fourier transform of g x . It follows that for all x ∈ Z d n (it can be seen in several ways, for example by translation invariance, that L is independent of x). Finally, we recall Levine et al. [17,Equation (20)]: for all

Gaussian variables on homogeneous Sobolev spaces on the torus
Since our conjectured scaling limit is a random distribution, we think it is important to keep the article self-contained and give a brief overview of analytic definitions needed to construct the limit in an appropriate functional space. Our presentation is based on Sheffield [ We remark that, in order to construct a measurable norm · B on H , it suffices to find a Hilbert-Schmidt operator T on H , and set · B := T · H .
Let us construct then an appropriate AWS. Choose a ∈ R. Let us define the operator (− ) a acting on L 2 (T d )-functions u with Fourier series ν∈Z d u(ν)e ν (·) as follows ((e ν ) ν∈Z d denotes the Fourier basis of L 2 (T d )): Let "∼" be the equivalence relation on C ∞ (T d ) which identifies two functions differing by a constant and let H a (T d ) be the Hilbert space completion of

Define the Hilbert space
We equip H a with the norm In fact, (− ) −a provides a Hilbert space isomorphism between H a and H a (T d ), which when needed we identify. For one shows that (− ) b−a is a Hilbert-Schmidt operator on H a (cf. also Silvestri [25,Proposition 5]). In our case, we will be setting a := −1. Therefore, by (2.3), for any The field associated to will be called and is the limiting field claimed in Theorem 1.
There is a perhaps more explicit description of which is based on Gaussian Hilbert spaces [9, Chapter 1]. The construction is taken from Janson [9, Example 1.25]. Let ( , A, P) be a probability space with A its Borel σ -algebra. Assume that on one can define a sequence of i.i.d. standard Gaussians (X m ) m∈N . Let further (X m ) m∈N be an orthonormal basis of H −1 (T d ). Then there is an isometric embedding The mapping can be extended by continuity to an isometry between H −1 and the corresponding closure of S. Taking := H − and P := μ − , this entails an alternative construction of : it is the unique Gaussian process indexed by H −1 such that

Strategy of the proof of Theorem 1
Firstly, we show that η can be decomposed into the sum of two independent fields, namely In particular, e n (·) admits the representation This decomposition is similar in spirit to the one in the proof of Levine et al. [17,Proposition 1.3], but we stress that the random fields we find are different. The proof of the above Proposition can be found in Sect. 3.1. As a consequence, to achieve Theorem 1 it will suffice to determine the scaling limit of the χ field, because test functions have zero average, and hence we can get rid of the minimum appearing in the odometer representation. We will therefore show (P2) From the above tightness result, there exists a subsequential scaling limit = lim k→+∞ n k for the convergence in law in the space H − . The proof is complete once we show this limit is unique: by Ledoux and Talagrand [14, Section 2.1], it suffices to prove that, for all mean-zero test functions u ∈ C ∞ (T d ), where the RHS is the characteristic function of . We will calculate the limit of the second moment of n , u directly in d ≤ 3 and through a mollifying procedure in d ≥ 4.
This will conclude the proof. Since the "finite dimensional" convergence is somewhat more interesting, we will defer the tightness proof to Sect. 4.2 and show (P2) in Sect. 4

.1.
A note on test functions By the above construction, the set of test functions we will consider is the set of smooth functions C ∞ (T d ) with zero mean. We need to stress at this juncture an important remark: C(T d ) does not correspond to the class of continuous functions on [− 1 2 , 1 2 ) d , but only to functions which remain continuous on R d when extended by periodicity. Similar comments apply to C ∞ (T d ) functions. See also Stein and Weiss [26, Section 1, Chapter VII] for further discussions. Therefore, when we consider u : R d → R which is periodic and belongs to C ∞ , we consider its restriction to [− 1 2 , 1 2 ) d while computing its integral on T d .

Auxiliary results
In this section we provide a proof of Proposition 4. The result helps us tackle the singularity arising from the zero eigenvalue of g and will also reduce the determination of the scaling limit to finding the scaling limit of (χ x ) x∈Z d n .

Proof of Proposition 4
Proof First, observe that, by Parseval's identity on the discrete torus, we can write the covariance of the Gaussian field (η(x)) x∈Z d First observe that using the description of g(x, y) in terms of the simple random walk One can notice that g x (0) is independent of x by translation invariance. Hence we get that the first term in the left-hand side of (3.1) is a constant equal to (2d) −2 n d L 2 having set L := n −2d As for the contribution from other sites, Define a centered Gaussian field (χ x ) x∈Z d n with covariance given by The field associated to H is well-defined and in fact H is positive definite. To see this, given a function c : where Y is a Gaussian random variable with mean zero and variance (2d) −2 n d L 2 independent of the field χ .
To conclude, note that the odometer function satisfies e n (x)

Proof of Theorem 1
We recall that it will suffice to prove the two properties (P1) and (P2) to achieve the Theorem. We first use to our advantage the fact that the test functions we consider have zero average, hence we can get rid of the minimum term which appears in the definition of the odometer. Let us recall the field in (1.3) We define a linear functional on C ∞ (T d ) by setting However using Proposition 4, and the fact that u has zero mean, one sees that By the theory of Gaussian Hilbert spaces of Sect. 2.2, n = n in distribution. Hence in the sequel we will, with a slight abuse of notation, consider n but denote it simply as n , since the law of the two fields is the same. We are now ready to begin with (P2).

Proof of (P2)
Overview of the proof We have just seen that We now replace the integral over the ball above by the value at its center and gather the remaining error term. More precisely we get Here the remainder R n (u) is defined by where using that the volume of B(z, 1 2n ) is n −d we have We observe that using the above decomposition one can split the variance of n , u as To deal with the convergence of the above terms we need two propositions. The first one shows that the first term yields the required limiting variance.

Proposition 5
In the notation of this Section, The second Proposition says the remainder term is small.

Proposition 6
In the notations of this Section, lim n→+∞ R n (u) = 0 in L 2 .
Then an application of the Cauchy-Schwarz inequality will allow us to deduce that lim n→+∞ E n , u 2 = u 2 −1 and the condition (P2) will be ensured. We give the proof of Proposition 5, which is the core of our argument, in Sect. 4.1.1 and of Proposition 6 in Sect. 4.1.2.

Proof of Proposition 5
Before we begin our proof we would like to prove a bound which would be crucial in estimating the eigenvalues of the Laplacian on the discrete torus. This lemma will be used later for other parts of the proof too.

Remark 2
The equation (4.5) is not enough to obtain sharp asymptotics for d i=1 n 2 sin 2 (πw i /n) as n → ∞. On the other hand, we will use it in the sequel while looking for a uniform lower bound for the same quantity for all w = 0.
We begin with the proof of Proposition 5. Let u : T d → R be a smooth function with zero mean. Define u n : Z d n → R as u n (z) := u( z n ). Note that To show the above expression converges it is enough to consider the convergence of This can be justified by showing that (4.6) can be bounded above and below appropriately by (4.7). Now observing that the lower bound of (4.3) immediately gives For the upper bound, using the bound in (4.3) we get Now we expand the square: the first term gives the correct upper bound as in (4.7) and the other two terms are negligible. In fact we show firstly that Using (4.8) and Parseval's identity we get we get that the second term converges to zero. Note that the same computation shows which again goes to zero as n → +∞. So this shows that we can from now on concentrate on showing the convergence of (4.7). We split now our proof, according to whether d ≤ 3 or d ≥ 4.
The case d ≤ 3 In the first case, the argument is more straightforward: we rewrite Since z∈T d n n −d u(z) exp(2πιz · w) is bounded above uniformly in n, and w∈Z d \{0} w −4 < +∞ in d < 4, we can apply the dominated converge theorem and obtain lim n→+∞ (4.7) = which concludes the proof of (P2) for d ≤ 3.

The case d ≥ 4
Here it is necessary to think of another strategy since w∈Z d w −4 is not finite. Let φ ∈ S(R d ), the Schwartz space, be a mollifier supported on It is a classical result [22,Theorem 7.22] that for δ = 0, 1, 2 . . . there exists A > 0 (depending on κ and δ) such that Now to show the convergence of (4.7) is equivalent to considering (4.10) Indeed, using the fact that Exploiting the fact that | exp(2πιx) − 1| 2 = 4 sin 2 (π x) and | sin(x)| ≤ |x| we obtain due to the fact that φ is supported on [− 1 2 , 1 2 ) d . Recalling u n (z) = u( z n ) and plugging the estimate (4.11) in (4.10) we get that Using w ≥ 1 we have Taking the limit κ → 0 in the previous expression we deduce the claim (4.10). Now we have to derive the limit of the following expression: Since φ κ has a fast decay at infinity, and we can apply the dominated convergence theorem to obtain The bound | φ κ (·)| ≤ 1 can be used to obtain a bound uniform in κ on the righthand side of the above expression: consequently we apply the dominated convergence letting κ → 0 to achieve This concludes the proof of Proposition 5.

Proof on the remainder: Proposition 6
We owe the reader now the last proofs on R n (see (4.1)). First we state the following Lemma 8 There exists a constant C > 0 such that sup z∈T d |K n (z)| ≤ Cn −1 .
Proof Using the mean value theorem as u ∈ C ∞ (T d ) we get that, for some c ∈ (0, 1), n .
We reprise now the proof on the limit of R n (u).

Proof of Proposition 6
We first compute E R n (u) 2 obtaining , thanks to Lemma 8 we have that the previous expression is equal to This shows immediately that R n (u) converges in L 2 to 0.
We are then done with the proof of (P2) on page 7.

Tightness: proof of (P1)
We proceed to prove tightness. Before that, we must introduce a fundamental result: Rellich's theorem.
is a compact linear operator. In particular for any radius R > 0, the closed ball Sketch of the proof The proof is readily adapted from the one in Roe [21,Theorem 5.8].
Let ω > 0 be arbitrarily small. Let B be the unit ball of H k 2 (T d ). We quotient then the space H k 2 (T d ) by the subspace Z := f : f (ν) = 0 for ν > N with N = N (ω) large enough so that f k 1 < ω for f ∈ B ∩ Z . The unitary ball in H k 2 /Z is then compact and thus can be covered by finitely many ω-balls, giving a finite 2ωcovering of balls for B in the H k 1 -norm as well. This shows the inclusion operator is compact. We take k 1 := − and k 2 := − 2 . By the definitions in Sect. 2.2, there is a Hilbert space isomorphism between H a (T d ) and H a (T d ). Applying the above observation, we get the result.

Proof of tightness
is a. s. finite, for fixed n, being a finite combination of Gaussian variables and their minimum.
s. By Rellich's theorem it will suffice to find, for all δ > 0, a R = R(δ) > 0 such that A consequence of Markov's inequality is that such an R(δ) can be found as long as we show that for some C > 0 Since n ∈ L 2 , it admits a Fourier series representation n (ϑ) = ν∈Z d n (ν)e ν (ϑ) with n (ν) = ( n , e ν ) L 2 (T d ) . Thus we can express Observe that This gives (4.14) Let us denote by F n, ν : T d n → R the function F n, ν (x) := B(x, 1 2n ) e ν (ϑ) d ϑ. Since e ν ∈ L 2 (T d ), the Cauchy-Schwarz inequality implies that F n,ν ∈ L 1 (T d ).
Assume we can prove Claim 10 There exists C > 0 such that (4.15) Using the above Claim and − < − d 2 , from (4.14) we get This concludes the proof, assuming Claim 10.
We are then left to show the claim we have made: Proof of Claim 10 First we use the bound (4.5) and the fact that Choose a mollifier φ κ as in the previous considerations (see below (6.1)). We rewrite the expression in the right-hand side of (4.16) accordingly as First we get a bound for the second term. Denote as G n, ν : Z d n → R the rescaled function G n, ν (z) := F n, ν ( z n ). Now we have 1 − φ κ (w) w 4 G n, ν (w) G n, ν (w) (4.11) ≤ Cκn 2d w∈Z d n G n, ν (w) 2 where in the last inequality we have used that w ≥ 1 and G n, ν (0) 2 ≥ 0. The description of G n, ν , the fact that |F n, ν (w)| ≤ n −d and Parseval give By means of (4.18) we get that We are back to bounding the first term in (4.17).
Using (4.9) we obtain a bound on the second term as Finally (4.9) tells us that where C possibly depends on κ and δ. Plugging in (4.15) the expressions (4.19), (4.20) and (4.21) we can draw the required conclusion.
This gives a proof of (P1) on page 7 and completes the proof of Theorem 1.

Proof of Theorem 2
Strategy of the proof We will argue as in Theorem 1 and need thus to show both (P1) and (P2). While (P2) will follow almost in the same way as in the Gaussian case, (P1) will require a different approach. Firstly, we will need to remove constants in defining e n so that we will end up working with a field depending only on linear combinations of (σ (x)) x∈Z d n . Secondly, we will show in Sect. 5.1 that, for σ bounded a. s., the convergence to the bilaplacian field is ensured via the moment method. Lastly, we will truncate the weights σ at a level R > 0 and show that the truncated field approximates the original one.
Reduction to a bounded field We first recall some facts from Levine et al. [17]. Note that odometer e n satisfies Observe that If we call by the mean-zero property of the test functions it follows that v n , u = w n , u . Therefore we shall reduce ourselves to study the convergence of the field w n . To determine its limit, we will first prove that all moments of w n converge to those of ; via characteristic functions, we will show that the limit is uniquely determined by moments.

Scaling limit with bounded weights
The goal of this Subsection is to determine the scaling limit for bounded weights, namely to prove Before showing this result, we must prove an auxiliary Lemma. It gives us a uniform estimate in n on the Fourier series of the mean of u in a small ball. Since u ∈ C ∞ (T d ), one can take derive under the integral sign and get that T n ∈ C ∞ (T d ), so z∈Z d T n (z) < +∞. Hence by the Fourier inversion theorem we have the following inversion formula to be valid for every y ∈ T d : First we split the sum above according to the norm of w and plug it in (5.3). Namely we get This finally gives that For the second summand of (5.4) observe that The parameter α will be chosen later so that the second summand is of lower order than the first. By (5.4) and (5.6) We use this estimate to get We can now start with the moment method, and we being with moment convergence.

Moment convergence
We now show that all moments converge to those of the required limiting distribution. This is explained in the following Proposition.

Proposition 13
Assume E [σ ] = 0, E σ 2 = 1 and that there exists K < +∞ such that |σ | ≤ K almost surely. Then for all m ≥ 1 and all u ∈ C ∞ (T d ) with zero average, the following limits hold: Proof We will first show that the m = 2 case satisfies the claim.
Case m = 2 We have the equality The independence of the weights gives With the same argument of the proof of Proposition 4 one has Now we break the above sum into the following 3 sums (recall K n (u) from (4.2)): A combination of Propositions 5 and 6 with the Cauchy-Schwarz inequality shows that the first term converges to u 2 −1 in the limit n → +∞ and the other two go to zero.
Having concluded the case m = 2, we would like to see what the higher moments look like. Let us take for example m = 3, in which case 858 A. Cipriani et al.
More generally, let us call P(n) the set of partitions of {1, . . . , n} and as P 2 (n) ⊂ P(n) the set of pair partitions. We denote as a generic block of a partition P and as | | its cardinality (for example, = {1, 2, 3} is a block of cardinality 3 of P = {{1, 2, 3}, {4}} ∈ P(4)). Observe that For a fixed P, let us consider in the product over ∈ P any term corresponding to a block with | | = 1: this will give no contribution because σ is centered. Consider instead ∈ P with := | | > 2. We see that is finite with probability one, since σ is bounded. One can then go along the lines of the proof of (P1) in Sect. 4.2 and get to (4.14) which will become, in our new setting, From this point onwards, the computations of the proof of (P1) can be repeated in a one-to-one fashion.

Truncation method
At the moment we are able to determine the scaling limit when the weights are bounded almost surely. To lift this condition to zero mean and finite variance only, we begin by defining a truncated field and show it will determine the scaling limit of the global field. Fix an arbitrarily large (but finite) constant R > 0. Set Clearly w n (·) = w <R n (·) + w ≥R n (·). To prove our result, we will use for all τ > 0, and X n, u ⇒ n Z u ⇒ u X , where " ⇒ x indicates convergence in law as x → +∞, then X n ⇒ n X.
(S2) For a constant v R > 0, we have w <R n ⇒ n √ v R ⇒ R in the topology of As a consequence we will obtain that w n converges to in law in the topology of H − .

Proof of (S1)
We notice that by definition, for every realization of (σ (x)) x∈Z d n . Since, for every τ > 0, it will suffice to show that the numerator on the right-hand side goes to zero to show (S1 where F n, ν (x) was defined as B(x, 1 2n ) e ν (ϑ) d ϑ. We have at hand (4.15), which we can use to upper-bound the previous expression by C 16π 4 E σ (w) 2 for some C > 0. The sum over ν is finite as long as > d/4, and E σ (w) 2 is going to zero as R → +∞ (note that σ has finite variance). We will show that the second term obtained by inserting the second summand of (5.13) in (5.12) is zero to complete the proof of (S1). In fact we obtain We consider the second line in the previous expression to deduce that it equals