Quenched invariance principle for random walks on Delaunay triangulations

We consider simple random walks on Delaunay triangulations generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on the point processes, we show that the random walk satisfies an almost sure (or quenched) invariance principle. This invariance principle holds for point processes which have clustering or repulsiveness properties including Poisson point processes, Mat{\'e}rn cluster and Mat{\'e}rn hardcore processes. The method relies on the decomposition of the process into a martingale part and a corrector which is proved to be negligible at the diffusive scale.


Introduction
Let us first describe the model. Given an infinite locally finite subset ξ of R d , the Voronoi tessellation associated with ξ is the collection of the Voronoi cells: The point x is called the nucleus or the seed of the cell. The Delaunay triangulation DT(ξ) of ξ is the dual graph of its Voronoi tiling. It has ξ as vertex set and there is an edge between x and y in DT(ξ) if Vor ξ (x) and Vor ξ (y) share a (d − 1)-dimensional face. Another useful characterization of DT(ξ) is the following: a simplex ∆ is a cell of DT(ξ) iff its circumscribed sphere has no point of ξ in its interior. Recall that this is a well defined triangulation when ξ is in general position. In the sequel, we denote by N (resp. N 0 ) the set of infinite locally finite subsets of R d (resp. the set of infinite locally finite subsets of R d containing 0).
Given a realization ξ of a suitable point process with law P, we consider the variable speed nearest-neighbor random walk (X t ) t≥0 on the Delaunay triangulation of ξ, that is the Markov process with generator: where c ξ x,y is the indicator function of 'y ∼ x in DT(ξ)'. We denote by P ξ x the law of the walk starting from x ∈ ξ and by E ξ x the corresponding expectation. We study for almost every realization of the point process the behavior of the random walk at the diffusive scale and we prove the following theorem.
Theorem 1. Assume that ξ is distributed according to a simple, stationary, isotropic point process with law P a.s. in general position, satisfying assumptions (V), (SD), (Er) and (PM) (see Subsection 1.1).
For P−a.e. ξ, for all x ∈ ξ, under P ξ x , the rescaled process (X ε t ) t≥0 = (εX ε −2 t ) t≥0 converges in law as ε tends to 0 to a Brownian motion with covariance matrix σ 2 I where σ 2 = σ 2 VSRW is positive and does not depend on ξ.
Note that the assumptions of Theorem 1 are satisfied by Poisson point processes, Matérn hardcore processes and Matérn cluster processes. The same result holds for the discrete-time nearest-neighbor random walk (X n ) n∈N with diffusion coefficient σ 2 DTRW related with σ 2 VSRW by the formula: σ 2 VSRW = E 0 deg DT(ξ 0 ) (0) σ 2 DTRW , where E 0 denotes the expectation w.r.t. the Palm measure P 0 associated with the (stationary) point process with law P.
The main idea for proving such results is to show that the random walk behaves like a martingale up to a correction which is negligible at the diffusive scale. Actually, well-known arguments (see e.g. [CFP13,§3.3.1], [BP07, p. 1340[BP07, p. -1341 or [BB07, §6.1 and §6.2]) show that the last claim follows from Theorem 2.
Theorem 2. Under the assumptions of Theorem 1, there exists a so-called corrector χ : N 0 × R d −→ R d such that for: (1) ϕ(ξ 0 , x) := x − χ(ξ 0 , x) is harmonic at 0 for P 0 − a.e. ξ 0 , i.e.: x∈ξ 0 c ξ 0 0,x ϕ(ξ 0 , x) < ∞ and L ξ 0 ϕ(ξ 0 , 0) = 0 for P 0 − a.e. ξ 0 ; (2) χ is a.s. sublinear: The arguments to deduce Theorem 1 from Theorem 2 are rather standard. We have chosen not to do develop the arguments which can be found in the references cited above and we only indicate the main lines of the proof in Section 10.2. Various methods to prove quenched invariance principle for random walks among random conductances on Z d or related models were developed during the last ten years (see [SS04, BB07, BP07, MP07, BZ08, Mat08, Bis11, GZ12, ZZ13, BD14]). Theorem 2 is proved by adapting the approach developed in [BP07]. Actually, we first prove the sublinearity of the corrector restricted to a suitable subgraph of the Delaunay triangulation and extend it by harmonicity. Let us note that this method was also successfully used in the context of random walks on complete graphs generated by point processes with jump probability which is a decreasing function of the distance between points in [CFP13].
Recurrence and transience results for random walks on Delaunay triangulations generated by point processes were obtained in [Rou13] and an annealed invariance principle was proved in [Rou14]. In [FGG12], the existence of an harmonic corrector was recently established by a different (constructive) method. Nevertheless, the authors of this paper obtained the sublinearity of the corrector only in dimension 2. In order to extend the quenched invariance principle in higher dimensions, they suggested to prove full heat-kernel bounds similar to the one obtained by Barlow in [Bar04] in the setting of random walks on supercritical percolations clusters. This approach would require much more sophisticated arguments and a better control of the regularity of the full graph than the one used in the present paper. It is worth noting that, as in [Rou13,Rou14], the arguments given in the sequel can be used to obtain quenched invariance principles for random walks on other graphs constructed according to the geometry of a realization of a point process.
1.1. Conditions on the point process. In this section, we list the assumptions on the point process needed to obtain the quenched invariance principle.
We assume that ξ is distributed according to a simple and stationary point process with law P. In the sequel, we will denote by E the expectation with respect to P. We suppose that P is isotropic and almost surely in general position (see [Zes08]): there are no d + 1 points (resp. d + 2 points) in a (d − 1)-dimensional affine subspace (resp. in a sphere). We also assume that the point process satisfies: (V) there exists a positive constant c 1 such that for L large enough: In order to prove the sublinearity of the corrector, we need to restrict the study to a subgraph of the Delaunay triangulation of ξ which has good regularity properties. To this end, we will define a notion of 'good boxes' that in particular allows us to bound the maximal degree of vertices in an infinite subgraph of the Delaunay triangulation of ξ. Precise definitions and assumptions are given below.
1.1.1. Good boxes, good points and the stochastic domination assumption. For s ∈ N * , let us divide R d into boxes of side K := 3 √ d s: Each box B z is then subdivided into smaller sub-boxes b z for the critical probability for site percolation in Z d , the stochastic domination hypothesis is the following. (SD) For any p site c (Z d ) < p < 1, if α and s 0 are well chosen, for any s ≥ s 0 the process of good boxes X : If (SD) is satisfied, we can find a coupling P K,p of the processes X and Y such that B z is good when Y z = 1. With a slight abuse of notation, we will omit the superscript and denote by P the probability measure P K,p on N := N × {0, 1} Z d whose marginal distributions are respectively the law of the point process ξ and the law of the independent site percolation process Y with parameter p being fixed large enough. The precise value of p is not stated explicitely but we assume that it is large enough to ensure that all the percolation results we need are satisfied. A generic element of N is denoted by ξ = (ξ, (y z ) z∈Z d ).
Let us denote by G (L) (resp. G ∞ ) the largest ( · 1 -)connected component of Y contained in [−L, L] d ∩Z d (resp. the a.s. unique infinite component of Y). We then define G (L) = G (L) ( ξ) (resp. G ∞ = G ∞ ( ξ)) as the set of points of ξ whose Voronoi cell intersects a K−box with index in G (L) (resp. G ∞ ). The points of G ∞ are called good points. Let us note that G (L) and G ∞ are connected in the Delaunay triangulation of ξ.
We claim that good points have their degrees uniformly bounded. More precisely: Lemma 3. There exists D = D(d, K, α) such that for every x ∈ ξ with Voronoi cell intersecting an α-good box: Remark 4. As it appears in the following proof, D is also an upper bound for the maximal number of Voronoi cells which intersect a good box. It will be used to bound these two quantities throughout the paper.
Proof. Let x ∈ ξ be such that its Voronoi cell intersects an α-good box B z . By definition, each box B z with z − z ∞ ≤ 1 is α-nice. Since any sub-box of side s that belongs to a nice box contains at least one point of ξ, the points in nice boxes are whithin a distance at most √ ds from the nucleus of their Voronoi cell. In particular, x is whithin a distance at most √ ds from B z . Similarly, the nuclei of the Voronoi cells which share a face with Vor ξ (x) are in 1.1.2. The ergodicity assumption. As in [CFP13], we have an ergodicity assumption but we make more explicit the use of the coupling. We will use it to adapt the method developed in [BB07,§4] for proving the sublinearity of the corrector along coordinate directions.
(Er) For each (K, p) such that (SD) holds and each e, P = P K,e is ergodic with respect to the transformation: Note that τ is invertible and P is invariant with respect to τ due to the stationarity of the point process.
1.1.3. Polynomial moments. We also make the following assumption: (PM) the number of points in a unit cube admits a polynomial moment of order 2 under P and deg DT(ξ 0 ) (0) and max x∼0 in DT(ξ 0 ) x admit respectively a moment of order 2 and 4 under the Palm distribution P 0 associated with the point process.
1.1.4. The case of point processes with a finite range of dependence. We verify that, when the point process has a finite range of dependence, assumption (Er) is always satisfied, and assumptions (SD) and (PM) are implied by assumptions (V), (D), (V') and (EM) which are described below. Let us check that (Er) is satisfied if the point process has a finite range of dependence. Let A be a measurable set such that τ A = A. Fix ε > 0 and B with P(A∆B) ≤ ε which depends only on ξ restricted to a compact subset of R d and on finitely many y z s. It is thus possible to find n such that B and τ n B are independent. Then, P[B ∩ τ n B] = P[B] 2 by invariance of P w.r.t. τ . Hence, Since ε is arbitrary, this implies that P[A] is 0 or 1.
Since deciding if a box is good depends only on the behavior of the point process ξ in a neighborhood of the box, the process of good boxes X is a Bernoulli process on Z d with a finite range of dependence when the point process has itself a finite range of dependence. In this case, if P[X z = 1] is as close to 1 as we wish for s and α large enough, [LSS97, Theorem 0.0] ensures that (SD) holds. One can easily bound P[X z = 1] from below when the point process satisfies (V) and (D) there exist positive constants c 2 , c 3 such that for L large enough: In Lemma 31, we prove that deg DT(ξ 0 ) (0) and max x∼0 in DT(ξ 0 ) x admit exponential moments when the point process has a finite range of dependence and satisfies: Let us finally note that these assumptions are in particular satisfied by homogeneous Poisson point processes, Matérn cluster processes and type I or II Matérn hardcore processes. Indeed, these point processes have finite range of dependence and it is quite classical to check assumptions (V), (D), (V') and (EM) for these processes (see [Rou14,Appendix]).
1.2. Outline of the paper. As announced at the beginning of the introduction, the crux is to prove Theorem 2 and we follow the approach of [BP07]. The main steps of the proof are stated explicitly in Theorem 2.4 of that paper. We prove the existence of the corrector in Section 2. Next, we verify that the corrector grows at most polynomially in Section 3 and at most linearly in each coordinate direction in Section 4. The sublinearity on average is treated in Section 5 while diffusive bounds for a related random walk are proved in Sections 7 and 8. In Section 9, we prove the a.s. sublinearity of the corrector in the set of 'good points'. The proofs of Theorems 1 and 2 are finally completed in Section 10.

Construction of the corrector and harmonic deformation
Let us define the measure µ on N 0 × R d by: where c ξ 0 0,x = 1 0∼x in DT(ξ 0 ) . This measure has total mass E 0 [deg DT(ξ 0 ) (0)] which is finite thanks to assumption (PM). We denote by (·, ·) µ the scalar product in L 2 (µ).
2.1. Weyl decomposition of L 2 (µ). As in [MP07], [BB07] or [CFP13], we work with the orthogonal decomposition of L 2 (µ) in the subspaces of square integrable potential and solenoidal fields. This decomposition is quite standard (see e.g. [LP14, Chap. 9]) and generally called Weyl decomposition. Let us denote by (τ x ) x∈R d the group of translations in R d which acts naturally on N 0 as follows: τ x ξ 0 = y∈ξ 0 δ y−x .
Definition 5. For ψ : N 0 → R, the gradient field ∇ψ : Note that gradients of measurable bounded functions on N 0 are elements of L 2 (µ) thanks to assumption (PM).
Definition 6. The space L 2 pot (µ) of potential fields is defined as the closure of the subspace of gradients of measurable bounded functions on N 0 . Its orthogonal complement is the set of solenoidal (or divergence-free) fields and is denoted by L 2 sol (µ). Let us recall some additional definitions: (2) shift-covariant if u(ξ 0 , x) = u(ξ 0 , y) + u(τ y ξ 0 , x − y), ∀ξ 0 ∈ N 0 , ∀x, y ∈ ξ 0 ; (3) curl-free if it satisfies the following co-cycle relation: for any ξ 0 ∈ N 0 , any n ∈ N * , and any collection of points x 0 , . . . , x n ∈ ξ 0 with x 0 = x n , one has A function of L 2 (µ) is called antisymmetric (resp. shift-covariant, curl-free) if it is antisymmetric (resp. shift-covariant, curl-free) for P 0 -a.a. ξ 0 . In each case, it admits a representative wich satisfies the corresponding property everywhere. By taking x = y = 0 in the definition, one can see that any shift-covariant function u must satisfy u(ξ 0 , 0) = 0 for any ξ 0 ∈ N 0 . Next proposition lists simple but useful links between definitions above: Proposition 8. Let u ∈ L 2 (µ).
(2) If u is curl-free, then it is also antisymmetric and shift-covariant.

Proof.
(1) Gradients fields are clearly curl-free. The general case is obtained by a standard approximation argument.
We can now define the divergence of an integrable field and derive an integration by parts formula.
Triangle inequality clearly implies that divergences of L 1 (µ) functions are in L 1 (P 0 ). Moreover, if u ∈ L 1 (µ) is a positive function, we have the equality: We derive the following integration by parts formula: Lemma 10. Let ψ be a bounded measurable function on N 0 and let u ∈ L 2 (µ) be an antisymmetric field. It holds that: Proof. Observe that: c ξ 0 0,x = c τxξ 0 0,−x . Due to the antisymmetry of u, one has: By Neveu exchange formula (see [SW08,Theorem 3.4.5] in the special case where X = Y ), one has for any integrable function f on N 0 × R d : The conclusion is then obtained by applying the identity above with: This lemma implies the immediate following corollary.

2.2.
Construction of the corrector. In [FGG12], the authors relied on an Harness-type process to obtain the existence harmonic deformations of Delaunay triangulations which corresponds to the existence of the corrector. Here, we recall how the decomposition of L 2 (µ) allows us to derive the existence of the corrector by following the construction of [MP07,CFP13].
Note that u i is clearly antisymmetric in the sense of Definition 7 and that u i ∈ L 2 (µ). Actually, by the Cauchy-Schwarz inequality and assumption (PM), one obtains: Consider now the orthogonal decomposition of the form , it is antisymmetric and ϕ i also (as a difference of antisymmetric functions). Hence, it follows from Corollary 11 that ϕ i is harmonic at 0.

Polynomial growth
Let us define: Proposition 12. For every β > d + 1, one has: Proof. For n fixed, let us cover [−2n, 2n] d with disjoint boxes of side log n and denote by A n the event 'each of these boxes contains at least one point of ξ'. Note that, thanks to assumption , we only need to show that n P[R n 1 An ≥ n β ] < ∞. The result then follows using the Borel-Cantelli lemma.
. Consider the simple Delaunay-path (x 0 , . . . , x m ) from x 0 = x to x m = y obtained by connecting the nuclei of successive Voronoi cells which intersect the line segment [x, y]. Observe that, on A n , any point of [−2n, 2n] d is within a distance at most √ d log n from the nucleus of its Voronoi cell. In particular, Recall that the corrector χ is shift-covariant. Hence, for i = 0, . . . , m − 1, one has: and using that χ(τ x ξ, 0) = 0: We deduce that on A n : Together with Markov inequality and Campbell formula, the inequality above leads to: Since β > d + 1 and div χ ∈ L 2 (P 0 ), this completes the proof.

Sublinearity along coordinate directions in G ∞ ( ξ)
In this section, we adapt the arguments of [CFP13, §7.2] which consist in an adaptation of the 'lattice method' developed in [BB07,BP07].
Given a unit vector e in the direction of one of the coordinate axes of R d and ξ ∈ N , let us define: Recall the definition of τ from assumption (Er) and consider the shift τ * induced on Thanks to assumption (Er) and the fact that P [0 ∈ G ∞ ] > 0, standard arguments (see e.g. [CFP13, Lemma 7.3] or [BB07, Theorem 3.2]) lead to: Lemma 13. The probability measure P [·|0 ∈ G ∞ ] is stationary and ergodic w.r.t. τ * .
Next, for ξ ∈ N * , we write w i for the point of ξ whose Voronoi cell contains the center of the box B n i ( ξ)Ke .
Lemma 14. It holds that: Proof. For ξ ∈ N * , let d( ξ) ≥ n 1 ( ξ) denote the chemical distance between 0 and n 1 ( ξ)e in the infinite cluster G ∞ . On the event {d( ξ) = j}, there exists a path z 0 = 0, z 1 , . . . , z j = n 1 ( ξ)e in G ∞ . For i = 0, . . . , j − 1, thanks to the definition of the good boxes and assumption (SD), the nuclei of the Voronoi cells intersecting the line segment [Kz i , Kz i+1 ] are within a distance at most √ ds from this line segment. By connecting the successive nuclei of the Voronoi cells intersecting the broken line [Kz 0 , Kz 1 , . . . , Kz j ], we obtain a simple path w 0 = x 0 , x 1 , . . . , x m = w 1 between w 0 and w 1 in G ∞ ( ξ) which is contained in [−K(j + 1 2 ), K(j + 1 2 )] d and has length m ≤ (j + 1)D. Thanks to the shift-covariance of χ, as in the proof of Proposition 12, one obtains: Together with the Cauchy-Schwarz inequality, this leads to:

QUENCHED INVARIANCE PRINCIPLE FOR RANDOM WALKS ON DELAUNAY TRIANGULATIONS 11
It follows from Campbell formula and formula (3) that Since points of G ∞ ( ξ) have degrees bounded by D (see Lemma 3) and #(ξ ∩ [0, 1] d ) admits a moment of order 2, one has: Thanks to [BB07,Lemma 4.4], we know that P d( ξ) = j 0 ∈ G ∞ has an exponential decay. Hence, collecting bounds, we obtain that the sum in the r.h.s. of (7) is finite.
Since χ ∈ L 2 pot (µ), it is the L 2 -limit of gradients of bounded measurable functions (g n ) n∈N defined on N 0 . Let us define χ n := ∇g n . By the same arguments as above, one obtains that: Note that, for all i, w i is a deterministic function of ξ and that w 1 ( ξ) = w 0 (τ * ξ) + n 1 ( ξ)Ke. The conclusion follows since, due to the stationarity of P[·|0 ∈ G ∞ ] with respect to τ * applied to the function Combining Lemmas 13 and 14, one obtains the sublinearity along the direction e in G ∞ ( ξ).
Proof. Thanks to the shift-covariance of the corrector, one has: Ke and recall that, by Lemma 13, P[·|0 ∈ G ∞ ] is ergodic with respect to τ * . Thanks to Birkhoff's theorem, the last The second part of the lemma then follows from equality : which is due to the shift-covariance of the corrector.

Sublinearity on average in G ∞ ( ξ)
We derive the sublinearity on average of the corrector in G ∞ ( ξ) from Lemma 15. Our approach is close in spirit to [CFP13,§7.3]. where Let us describe roughly the method which is an alternative to the one of [BB07, §5.2] and relies on multiscale arguments. The first idea is to extend the directional sublinearity result of Proposition 15 dimension by dimension. For ν ∈ {1, . . . , d}, we denote by Λ ν L the set: Assume that we have a 'good' (sublinear) control of χ(τ x 0 ξ, x − x 0 ) for x 0 whose Voronoi cell intersects B K 0 and x whose Voronoi cell intersects B K z for some z ∈ Λ ν L ∩ G ∞ . Then, using Proposition 15, one obtains a sublinear control on χ(τ x ξ, x − x) for x whose Voronoi cell intersects B K z and x whose Voronoi cell intersects B K z for any z ∈ Λ ν+1 L ∩ G ∞ which differs from z only on the (ν + 1)-th coordinate. By the shift-covariance, this gives a control on χ(τ x 0 ξ, x − x 0 ). As noticed in [BB07], we can not deduce directly Proposition 16 from this argument because G ∞ covers only a fraction of order p = P [0 ∈ G ∞ ] of the ν-dimensional section Λ ν L . The idea is then to work at a larger scale, say mK, m ≥ 1. The interest of using the mK scale is that the process of good mK-boxes stochastically dominates a percolation process with parameter as close to one as we wish for m large enough (recall assumption (SD)). We follow this strategy at the mK scale and we show in Lemmas 18 and 19 that it is possible to obtain a good control of the corrector for points in a large fraction of the mK-boxes. Finally, we go back to the K scale by finding a K-box contained in a suitable mK-box from which we can extend the control on the corrector.
In the rest of the section we add the superscripts K and mK to indicate the considered scale.
Let us denote by e 1 , . . . , e d the vectors of the standard basis of R d . In order to control the behavior of the corrector at the scale mK, for fixed C, m, ε, let us define the mesurable sets: = ∅ one has: For ν ∈ {1, . . . , d}, n ∈ N * and ξ ∈ N , let us also define: (z 1 , . . . , z ν , 0, . . . , 0). We now prove three intermediary lemmas.
Lemma 17. For each δ, ε > 0, there are C and m such that for P − a.a. ξ, there exists n 0 = n 0 ( ξ, C, m, ε, δ) < ∞ such that: Proof. Note that, thanks to the union bound, it suffices to show that for any ν ∈ {1, . . . , d}: Thanks to assumption (SD), one has for m large enough: Given δ , ε and m such that the inequality above holds, using Proposition 15 at the scale mK, we can find C large enough such that: Due to the ergodicity assumption (Er), (11) and (12), one has: This implies the result.

By ergodicity
On the other hand, denoting by π (1) the projection along the first coordinate axis, one has π (1) Γ C,m,ε n ⊂ Γ C,m,ε n and # π (1) Γ C,m,ε n ≥ #Γ C,m,ε n (2n + 1) d−1 ≥ (1 − δ)(2n + 1) by Lemma 17. It follows that Due to the choice of δ, (1 − δ)(2L + 1) + (2L + 1)p/2 > (2L + 1) = #Λ 1 L which implies that Fix j in the intersection above, thanks to the sublinearity in the direction e 1 at scale K (see Proposition 15), there exists x ∈ ξ with Vor ξ (x) ∩ B K je 1 = ∅ satisfying: Together with Lemma 19 applied with a = j m e 1 and the shift-covariance of the corrector, this allows us to conclude that for any y ∈ ξ whose Voronoi cell intersects an mK-box with index b in Γ C,m,ε n , for any x 0 ∈ ξ with Vor ξ (x 0 ) ∩ B K 0 = ∅: Thanks to the choice of ε, the last quantity is smaller than ε 0 L when L is large enough. For any x 0 ∈ ξ with Vor ξ (x 0 ) ∩ B K 0 = ∅, one has for L large enough: where we used that: and Lemma 17.

Random walks on G ∞ ( ξ)
In order to derive the strong sublinearity of the corrector in G ∞ ( ξ) from its sublinearity on average, we need to obtain heat-kernel estimates and bounds on the expected distance between positions of the walker at time t and 0 (see equations (21) and (35)). Such estimates cannot be obtained directly for the random walk on the (full) Delaunay triangulation generated by ξ in which the degree is not bounded. Nevertheless, since G ∞ ( ξ) has good regularity properties, these bounds will be established in Sections 7 and 8 for restricted random walks described below.
For ξ ∈ N , let us consider the Markov chain ( Y n ) n∈N = ( Y ξ n ) n∈N on G ∞ ( ξ) induced from the original (discrete time) random walk (X n ) n∈N on DT(ξ). In other words, ( Y n ) n∈N is the time-homogeneous Markov chain on G ∞ ( ξ) with jump probabilities given by: where T 1 := inf{j ≥ 1 : X j ∈ G ∞ ( ξ)}. Note that the holes (i.e. the (DT(ξ)-)connected components of ξ \ G ∞ ( ξ)) are a.s. finite. Hence, T 1 is a.s. finite and ( Y n ) n∈N is well defined. Moreover, for any z ∈ G ∞ ( ξ), applying the optional stopping theorem to the martingale (X n − z − χ(τ z ξ, X n − z)) n∈N starting from z, it appears that, ( Y n − z − χ(τ z ξ, Y n − z)) n∈N is a martingale. We also consider a continuous-time version of the random walk defined above ( Y t ) t≥0 := ( Y N (t) ) t≥0 where N (t) is the intensity 1 Poisson process on the half-line R + . It has infinitesimal generator: It is not difficult to see that ( Y t − z − χ(τ z ξ, Y t − z)) t≥0 is also a martingale.
We denote by P ξ x the (quenched) law of this walk starting from x. Note that this walk has speed 'at most 1'. Observe also that the measure deg DT(ξ) is reversible w.r.t. both ( Y n ) n∈N and ( Y t ) t≥0 . Actually, standard computations show that the detailed balance condition: is satisfied.
7. Heat-kernel estimates for ( Y ξ t ) t>0 The aim of this section is to prove the following heat-kernel bound.
The proof of this bound relies on isoperimetric inequalities and the technics developed in [MP05]; it is completed in Subsection 7.4. Precise definitions are given in Subsection 7.1. Isoperimetric inequalities for random walks confined in large boxes are established in Subsection 7.2. Additional technical results are isolated from the proof of Proposition 20 and given in Subsection 7.3. 7.1. Precise definitions. We will state isoperimetric inequalities for random walks confined in large boxes. We need to introduce additional notations and random walks confined in boxes of side L.
Recall the definitions of G (L) and G (L) ( ξ) from the introduction and denote by G ( where P ξ,(L) x stands for the law of X (L) n n∈N . The associated isoperimetric profile is: The advantage of Y is that this walk coincides with Y t t≥0 as long as they do not leave (the interior of) G (L) ( ξ). Nevertheless, we are not able to obtain directly a bound on the isoperimetric profile ϕ (L) . In a similar way as in [BBHK08], we compare it with the isoperimetric profile of the constant speed random walk on G (L) ( ξ), that is the walk Y (L) t t≥0 with generator: where deg L, ξ (·) denotes the degree in the restriction of DT(ξ) to G (L) ( ξ). The measure . The associated conductance of the set A ⊂ G (L) ( ξ) is given by: and the corresponding isoperimetric profile is: 7.2. Isoperimetric inequality. The goal of this section is to obtain a lower bound on the isoperimetric profile ϕ (L) ; it is stated in Corollary 24.
7.2.1. Comparison between ϕ (L) and ϕ (L) . First, note that for x ∈ G (L) ( ξ): and that for x, y ∈ G (L) ( ξ) Hence, for A ⊂ G (L) ( ξ), we have: With deg L, ξ (A) ≤ D deg L, ξ (A), this implies that: Using (27) and (29), one deduces: 7.2.2. Lower bounds for ϕ (L) . Our aim is to show the following bound on the isoperimetric Proposition 22. There exists c = c(d, K, α) > 0 such that P − a.s. for L large enough: As in [CF07], we use as much as possible an isoperimetric inequality for the percolation cluster G (L) .
Proposition 23 (see [CF07], eq. (2.5)). There exists κ > 0 such that almost surely for L large enough, for A ⊂ G (L) with 0 < #(A) ≤ 1 2 #(G (L) ): This result can be proved by adapting the arguments given in [BBHK08,Appendix] to the context of supercritical site percolation (see also the proof of [BM03, Lemma 2.6] for p p site c (Z d )). We adapt the proof of [CF07, Theorem 1.1] to the present setting. It is worth noting that the arguments of [CF07] can be used to derive isoperimetric bounds for the Delaunay triangulation confined in cubic boxes at least when the underlying point process is a PPP. This does not lead to sharp enough heat-kernel bounds for the random walk on the full Delaunay triangulation due to the unboundedness of the degree.
Proof of Proposition 22. For A ⊂ G (L) ( ξ), we define: Let us observe that, thanks to the definition of the good boxes, for any A ⊂ G (L) ( ξ): In the first inequality, we used that, for x ∈ G (L) ( ξ), Vor ξ (x) does not intersect more than 2 d good boxes since the diameter of the cell is less than K. From now on we assume that deg L, ξ (A) ≤ 1 2 deg L, ξ (G (L) ( ξ)). We are going to discuss separately the cases when #(L(A)) is large or small with respect to #(G (L) ). Roughly, if #(L(A)) is large then #(L(G (L) ( ξ) \ A)) is not too small and I (L) A is easily bounded from below by some constant. When #(L(A)) is small, a bound is obtained using the isoperimetric inequality for G (L) given in Proposition 23. The case #(L(A)) > 1 − 1 2 d+2 D 2 #(G (L) ). Using the general bound (31) and inequality deg L, ξ (A) ≤ 1 2 deg L, ξ (G (L) ( ξ)), one obtains: It follows that L(A) and L(G (L) ( ξ) \ A) have a large intersection in this case: This allows us to bound from below the numerator in I (L) A as follows. If z ∈ L(A) ∩ L(G (L) ( ξ) \ A), one can choose x ∈ A and y ∈ G (L) ( ξ) \ A whose respective Voronoi cells intersect B z . Hence, connecting the nuclei of the Voronoi cells which intersect the line segment [x, y], it is easy to find an edge between a point of A and a point of G (L) ( ξ) \ A which is included in B z = z : z −z ∞≤1 B z thanks to the definition of good boxes. Since a specific edge is associated to at most 3 d boxes by this procedure, it follows using (32) that: , we obtain that: ). Let us show that, in this case, the numerator in I Since deg L, ξ (A) ≤ D 2 #L(A) ≤ D 2 (2 d+2 D 2 − 1)# G (L) \ L(A) , we deduce that: Proposition 23 then implies that almost surely for L large: Using that #L(A) ≤ 2 d deg L, ξ (A), we obtain that: Since deg L, ξ (G (L) ( ξ)) ≤ D 2 #G (L) ≤ D 2 (2L + 1) d , the conclusion of Proposition 22 follows.

Other technical results.
7.3.1. Volume growth for deg L, ξ G (L) ( ξ) . We briefly check that there exist constants c and C such that a.s. for L large enough: The upper bound is very simple since it suffices to write: Since any box with index in G (L) contains at least a point of G (L) ( ξ) which has degree at least 1: The lower bound follows by using that a.s. for L large enough: which is a consequence of the ergodic theorem.
7.3.2. Size of the holes and connectivity of G ∞ ( ξ) in large boxes. In order to compare , we need to control the size of the holes (i.e. DT(ξ)-connected components of ξ \ G ∞ ( ξ)) and to establish connectivity properties of G ∞ ( ξ) in large boxes. More precisely, for C, γ > 1 and t > 0, let define the events: A t = A t,γ,C := any hole contained in −K t γ + 1 2 , K t γ + 1 2 d has diameter smaller than C log t , and where: We prove that: Lemma 25. Assuming that K is large enough, for each γ > 1, there exists C < ∞ such that almost surely for t large enough A t and B t are realized.
Proof. First, observe that any DT(ξ)-connected component A of ξ \ G ∞ ( ξ) is contained in the union of K-boxes with indices in some discrete hole A (i.e. a connected component of Z d \ G ∞ ). Hence, in order to show that A t holds almost surely, it suffices to verify that a.s. any discrete hole contained in [− t γ , t γ ] d ∩ Z d has diameter at most C log t, for C suitably chosen. Denote by A z the (possibly empty) hole at z ∈ Z d for an independent percolation process of parameter p. Recall that assumption (SD) ensures that the process of 'good boxes' dominates such a process with p as close to 1 as we wish whenever K is fixed large enough. Assuming that p is large enough, a standard Peierls argument shows that there exists c 6 > 0 such that: Thus, Our first claim then follows by the Borel-Cantelli lemma if C is well chosen. As above, in order to show that B t is a.s. realized for t large enough, we only need to check the corresponding claim for the percolation process, that is: almost surely for t large enough, Let us justify that, almost surely for t large enough, G ( t γ ) coincides with the largest connected component of G ∞ ∩ [− t γ , t γ ] d . Thanks to [CF07, Lemma B.1] and the Borel-Cantelli lemma, we know that there exists L 0 a.s. finite such that, for L ≥ L 0 , the maximal open cluster G (L) in [−L, L] d ∩ Z d is the only open cluster in this box with diameter larger than L/10 and crosses this box in every coordinate direction (see also [CF07,Remark 7]). In particular, G (L) has diameter 2L ≥ (L + 1)/10 and is thus included in G (L+1) . So, for L ≥ L ≥ L 0 , G (L) is contained in G (L ) . Hence, it is included in an open cluster with infinite diameter which is G ∞ .
It remains to verify that any two vertices z and z of G ∞ ∩ Q t are connected by an open path whithin [− t γ , t γ ] d . Call B t this event and write d G∞ (·, ·) for the graph distance in G ∞ . Assume that B t fails for some large t and fix z, z ∈ G ∞ ∩ Q t which are not connected in [− t γ , t γ ] d . Considering a shortest path from z to z one can find z in G ∞ ∩ Q 3t \ Q 2t such that d G∞ (z , z ) ≥ t γ − t(3C log(t)/2 + 1). Hence, for any κ, for t large enough: From [AP96], we know that P [z , z ∈ G ∞ , d G∞ (z , z ) ≥ κ z − z ] decreases exponentially with z − z which implies that P [(B t ) c ] decreases exponentially with t. One finally concludes thanks to the Borel-Cantelli lemma. 7.4. Proof of Proposition 20. For t ≥ 0, let us denote by N (t) the number of jumps of ( Y s ) s≥0 up to time t and by C t the event: ' N (t) ≤ 3t/2'. Since ( Y s ) s≥0 has speed at most 1, N (·) is dominated by a Poisson process of intensity 1 on R + and P ξ x [C c t ] ≤ c 7 exp(−c 8 t) for some c 7 , c 8 > 0. This implies that, almost surely for t large enough, C t is realized and we only need to obtain the heat-kernel bound on this event.
Recall the definitions of A t and B t from the previous subsection. On A t ∩ B t ∩ C t , starting from a point of G ∞ ( ξ) ∩ [−t, t] d , ( Y s ) s≥0 does at most 3/2t jumps of length at most C log t up to time t. In particular, it has visited only points of G ( t γ ) ( ξ) ∩ Q t and does not depend on ξ \ G ( t γ ) ( ξ) up to time t. Hence, we can find a coupling of ( Y s ) s≥0 and ( Y ( t γ ) s ) s≥0 such that these two coincide up to time t. Thus, for t large enough, we can write: It then remains to bound P To this end, we will rely on the isoperimetric inequality stated in Corollary 24 and apply the strategy developed by Morris and Peres in [MP05].
Theorem 26 (see [MP05,Theorem 13]). Let (X t ) t≥0 be an irreducible continuous-time Markov chain on a finite state space X with reversible probability measure π and isoperimetric profile ϕ.
Here the reversible probability measure is given by: Choosing ε of the form ε = c 9 t d(γ− 1 2 ) (c 9 will be chosen large enough), the conclusion of Theorem 26 reads: Using Lemma 3 and (33), one deduces that P It remains to check the validity of (34) with ε = c 9 t d(γ− 1 2 ) for t large enough when c 9 is well chosen.
Assuming that t satisfies 4t d( 1 2 −γ) ≤ c 9 /(2D), one obtains with Corollary 24 and inequality (31) that: If c 9 has been fixed large enough, the last expression is smaller than t for every t large enough.
To summarize, we have just proved that for a.a. ξ, there exist c 15 = c 15 ( ξ) and T = T ( ξ) such that for any t ≥ T , for any x ∈ G ∞ ( ξ) ∩ [−t, t] d and any y ∈ G ∞ ( ξ): This implies the required result.
8. Expected distance bound for ( Y ξ t ) t>0 It is known that bounds on the expected distance between the position of the walk at time t and its starting point can be derived from the heat-kernel estimate (21) as soon as the volume grows regularly (see for example [Bas02,Bar04,BP07]). In this section, we use this strategy to prove the following proposition.
Proposition 27. For a.a. ξ ∈ N : Proof. Proposition 20 shows that the assumption of [BP07, Proposition 6.2] is satisfied in the present setting. Hence, there exist constants c 16 and c 17 such that for a.a. ξ, for every x ∈ G ∞ ( ξ), for t large enough: where d (x, y) = d G∞( ξ) (x, y) denotes the 'natural' distance between x and y for ( Y t ) t>0 (i.e. the minimal number of jumps that the random walk needs to do in order to go from x to y).
At this point, we need to compare d with the Euclidean distance and to check that the r.h.s. of (36) is uniformly bounded. Let us denote by d = d G∞ the chemical distance in G ∞ in which we add an edge between every two points on the boundary of a (shared) discrete hole. For x ∈ G ∞ ( ξ), we choose z(x) ∈ G ∞ such that Vor ξ (x) intersects B z(x) to be the minimal one in the lexicographic order. It is not difficult to see that the definitions of G ∞ and G ∞ ( ξ) imply that there are constants c 18 and c 19 such that: By the same arguments as in the proof of [BP07, Lemma 3.1], one obtains that: for suitable constants c 20 and c 21 . Using the estimate above and the Borel-Cantelli lemma, one deduces that there exists a constant C such that, for n ≥ Then, (36), (37) and (38) imply that for every Finally, observe that once 1/s ≥ t The conclusion then follows thanks to (39). 9. Almost sure sublinearity in G ∞ ( ξ) The aim of this section is to prove the 'strong' sublinearity of the corrector in G ∞ ( ξ).
Proof of Lemma 29. We adapt the arguments given in [BP07,§5]. Let us define: Recall that these two quantities are a.s. finite thanks to Propositions 20 and 27. For large n, we choose y 0 = y 0 (n) with Vor ξ (y 0 ) ∩ B 0 = ∅ and y = y(n) ∈ [−n, n] d such that R n = χ(τ y 0 ξ, y − y 0 ) and we define the stopping time: For n is large enough, holes have sizes of logarithmic order (see Lemma 25) and thus Y t∧Sn − y ≤ 3n for all t. Due to the harmonicity of ϕ, the optional stopping theorem gives: By the shift-covariance of the corrector, one has: thus χ (τ y 0 ξ, y − y 0 ) = E ξ y χ τ y 0 ξ, Y t∧Sn − y 0 − Y t∧Sn + y . It follows that: Let us fix ε > 0 and define: Note that, by Proposition 16, #O n = o(n d ). Restricting our attention to t = t(n) ≥ 4n (whose value will be specified at the end), we will decompose the expectation above as: We first deal with the term E 1 . Since t ≥ 4n, Markov inequality shows that: Observe that Y 2t − y ≤ 3n/2, S n < t ⊂ Y 2t − Y Sn ≥ n/2, S n < t . On {S n < t}, since s := 2t − S n ∈ [t, 2t], one has: with z = Y Sn , this implies by the strong Markov property that: Recall that Y t∧Sn − z ≤ 3n for n large enough. It follows that: Thanks to the definitions of C 2 and O n and to the fact that t ≥ n, one has: But, using that deg DT(ξ) is reversible w.r.t Y s s≥0 and bounded by D on G ∞ ( ξ), we obtain by the Markov property and the Cauchy-Schwarz inequality that: Combining bounds (43)-(46), we get: The conclusion then follows by choosing t = c 28 n 2 for some c 28 = c 28 ( ξ, ε, δ) small enough and using that #O n = o(n d ).

Proof of main results
10.1. Proof of Theorem 2. We first show that Proposition 28 and the control of the diameter of the holes imply that for P [·|0 ∈ G ∞ ]-a.a. ξ: Recall that, almost surely for n large enough, holes intersecting [−n, n] d have diameters smaller than C log n (see Lemma 25). Let H ⊂ ξ be a hole intersecting [−n, n] d and denote by ∂ ext H ⊂ G ∞ ( ξ) its external boundary, that is the set of the points of ξ \ H which are neighbors of a point of H in DT(ξ). We can assume that ∂ ext H is contained in [−2n, 2n] d . Let us define S := inf{k ≥ 0 : X k ∈ H}. Thanks to the harmonicity of ϕ and the optional stopping theorem, for x ∈ H, one has: But the shift-covariance of the corrector implies that for any x 0 ∈ ξ: Together with Proposition 28, this implies that (47) holds for P [·|0 ∈ G ∞ ]-a.a. ξ. We now prove that (47) actually holds for P-a.a. ξ. Note that this is the step where we eliminate the coupling. Observe that: Hence, the event: is shift invariant w.r.t. τ = τ K,e 1 . Thanks to the ergodicity assumption (Er), it is a 0-1 event and we already know that P[A] ≥ P[0 ∈ G ∞ ] > 0. Thus, (47) holds P-a.s.. In particular, (47) holds P[·|ξ ∩ B 0 = ∅]-a.s.. The conclusion then follows by using e.g. [CFP13, Lemma 7.14] wich state that a P[·|ξ ∩ B 0 = ∅]-almost sure event is also P 0 -almost sure.
10.2. From Theorem 2 to Theorem 1. Thanks to [CFP13, Lemma B.2], Theorem 1 is a direct consequence of the following one which is a rewritting of Theorem 1 under P 0 .
Theorem 30. Under the assumptions of Theorem 1, for P 0 −a.e. ξ 0 , under P ξ 0 0 , the rescaled process (X ε t ) t≥0 = (εX ε −2 t ) t≥0 converges in law as ε tends to 0 to a Brownian motion with covariance matrix σ 2 I where σ 2 is positive and does not depend on ξ.
As mentioned in the introduction, the arguments to deduce this result from Theorem 2 are now quite standard and we only sketch the main lines of the proof. The reader is refered to [CFP13,§3.3], [BP07, p. 1340[BP07, p. -1341 or [BB07, §6.1 and §6.2] for more details.
Recall that, for P 0 -a.e. ξ 0 , ϕ(ξ 0 , ·) is harmonic. Hence, (M n ) n∈N := (ϕ(ξ 0 , X n )) n∈N is a martingale under P ξ 0 0 . By the same arguments as in [BB07,, one can show that (M n · e i ) n∈N satisfies the assumptions of the Lindeberg-Feller theorem for martingales (see [Dur96,Theorem (7.3), p. 414]). It follows with the Cramér-Wold device (see [Dur96, Theorem (9.5), p. 170]) that t → εM ε −2 t converges weakly to a Brownian motion with explicit covariance matrix proportional to the identity due to the isotropy of the point process. The diffusion coefficient does not depend on the particular realization ξ 0 of the point process and is positive. If it were zero, it would hold that x = χ(ξ 0 , x) for P 0 -a.e. ξ 0 , for all x ∈ ξ 0 , which contradicts the sublinearity of the corrector. The sublinearity of the corrector also implies that max k≤n X k − M k = max k≤n χ(ξ 0 , X k ) = o( √ n) in P ξ 0 0 -probability. The 'discrete time version' of Theorem 30 then follows. One concludes in the continuous-time case by arguing as in [CFP13,p. 666] and by showing that: where N (t) denotes the number of jumps of (X s ) s≥0 up to time t. This also proves the relation between σ 2 VSRW and σ 2 DTRW . One finally deduces Theorem 1 using [CFP13, Lemma B.2].
11. Bounds for moments of deg DT(ξ 0 ) (0) and max x∼0 x The method developed in [Zuy92,§2] can be used to derive exponential moments for deg DT(ξ 0 ) (0) and max x∼0 in DT(ξ 0 ) x when the point process has a finite range of dependence. More precisely, we show the following lemma.
Lemma 31. Assume that P 0 is isotropic and satisfies (V'), then there exists ρ 1 > 0 such that: Assume moreover that P 0 has a finite range of dependence m and satisfies (EM), then there exists ρ 2 > 0 such that: Proof. We use the method of [Zuy92,§2]. Let us recall some definitions, notations, and facts from this article. The fundamental region of a point x ∈ ξ 0 is the union of the balls centered at the vertices of its Voronoi polygon and having the nucleus x on their boundaries. Let Γ 0 be the union of 2d open balls of radii 1 centered in points ±e i . Let Φ 0 1 , . . . , Φ 0 2 d be the intersection of exactly d such balls.
Let us fix β > 1 and consider the sequence of sets Γ n , Φ n 1 , . . . , Φ n 2 d obtained by the homothetic transformation of center 0 and coefficient β n from Γ 0 , Φ 0 1 , . . . , Φ 0 2 d . The important point is that simple geometric arguments show that: Fact 32. If each d-faced lens Φ n i , i = 1, . . . , 2 d , contains a point of ξ 0 , then the fundamental region of the particle at 0 is fully included in Γ n . In particular, any neighbor of 0 in DT(ξ 0 ) is in Γ n .
where we used the isotropy of the point process in the last inequality. Now, there exists a constant c 29 > 0 such that Φ n−1 1 contains a cube C n−1 of side c 29 β n−1 . Hence, with (V') and (50), we obtain: This concludes the proof since f (ρ) goes to 0 with ρ by assumption.