Greedy Polyominoes and first-passage times on random Voronoi tilings

Let N be distributed as a Poisson random set on R^d with intensity comparable to the Lebesgue measure. Consider the Voronoi tiling of R^d, (C_v)_{v\in N}, where C_v is composed by points x in R^d that are closer to v than to any other v' in N. A polyomino P of size n is a connected union (in the usual R^d topological sense) of n tiles, and we denote by Pi_n the collection of all polyominos P of size n containing the origin. Assume that the weight of a Voronoi tile C_v is given by F(C_v), where F is a nonnegative functional on Voronoi tiles. In this paper we investigate the tail behavior of the maximal weight among polyominoes in Pi_n for some functionals F, mainly when F(C_v) is the number of faces of C_v. Next we apply our results to study self-avoiding paths, first-passage percolation models and the stabbing number on the dual graph, named the Delaunay triangulation. As the main application we show that first passage percolation has at most linear variance.


Introduction
Greedy animals on Z d have been studied notably in Cox et al. (1993). Imagine that positive weights, or awards, are placed on all vertices of Z d . A greedy animal of size n is a connected subset of n vertices, containing the origin, and which catches the maximum amount of awards. When these awards are random, i.i.d, it is shown in Cox et al. (1993) that the total award collected by a greedy animal grows at most linearly in n if the tail of the award is no too thick. This is shown in a rather strong sense, giving exponential deviations inequalities. This result has already been proved useful for the study of percolation and (dependent) First Passage Percolation, see Fontes and Newman (1993). In section 2, we extend the results of Cox et al. (1993) to some greedy polyominoes on the Poisson-Voronoi tiling, i.e in this paper, the Voronoi tiling based on a Poisson random set on R d with intensity comparable to the Lebesgue measure. Our greedy polyominoes use special dependent random awards, in the sense that the award at each Voronoi tile is a function of the number of faces of the Voronoi tile. We believe that our results may be useful to control the geometry of the Poisson-Voronoi tiling. To illustrate this, we shall give in section 3 three applications of our results on greedy polyominoes to some geometric problems on the Poisson-Voronoi tiling. The Poisson-Delaunay graph is the dual graph of Leandro P. R. Pimentel was partially supported by grants from the Swiss National Science Foundation, Fundação de Amparo a Pesquisa do Estado de São Paulo and The Netherlands Organisation for Scientific Research.
Raphaël Rossignol was supported by the Swiss National Science Foundation grants 200021-1036251/1 and 200020-112316/1. the Poisson-Voronoi tiling. The first application is the control, on the Poisson-Delaunay graph, of the number of self-avoiding paths of length n starting from the origin (section 3.1). Section 3.2 is devoted to a technical result about Bernoulli First Passage Percolation on the Poisson-Delaunay graph, which is used in our main application. Next, and this is our main application, we prove in section 3.3 that the first passage time in First Passage Percolation has (at most) linear variance on the Poisson-Delaunay graph. Finally, we show in section 3.4 that the stabbing number of the Poisson-Delaunay graph has an exponential tail. The stabbing number is related to the study of simple random walk on this graph.
In the end of this introduction, we give all the needed formal definitions to state precisely our main results on greedy polyominoes.
Voronoi tilings and polyominoes. To any locally finite subset N of R d one can associate a partition of the plane as follows. To each point v ∈ N corresponds a polygonal region C v , the Voronoi tile (or cell) at v, consisting of the set of points of R d which are closer to v than to any other v ′ ∈ N . Closer is understood here in the large sense, and the partition is not a real one, but the set of points which belong to more than one Voronoi tile has Lebesgue measure 0. The collection {C v } v∈N is called the Voronoi tiling (or tessellation) of the plane based on N . From now on, N is understood to be distributed like a Poisson random set on R d with intensity measure µ. We shall always wuppose that µ is comparable to Lebesgue's measure on R d , λ d , in the sense that there exists a positive constant c µ such that for every Lebesgue-measurable subset A of R d : 1 For each positive integer number n ≥ 1, a Voronoi polyomino P of size n is a connected union of n Voronoi tiles ( Figure 1). Notice that with probability one, when two Voronoi tiles are connected, they share a (d − 1)-dimensional face. We denote by Π n the set of all polyominoes P of size n and such that the origin 0 belongs to P. Assume that the "weight" of a Voronoi tile C v is given by F (C v ) = f (d N (v)), where d N (v) is the number of (d − 1)-dimensional faces of C v and f is a positive nondecreasing function on the integers. In this way we define a random weight functional on polyominoes by The maximal weight among polyominoes in Π n is: A greedy Voronoi polyomino P n is a Voronoi polyomino that attains the maximum in the definition of F n : F (P n ) = F n . To state our next theorem, we require the following notation: Definition 1.1. Given a nondecreasing function f from R + to [1, ∞), define: let g −1 denote the pseudo-inverse of any strictly increasing function g: and let: Theorem 1.2. There exists a constant c 1 ∈ (0, ∞) such that, if f is c 1 -nice then there exist constants z 1 , c 2 , c 3 ∈ (0, ∞) such that for all z ≥ z 1 and for all n ≥ 1: In particular, there also exists a constant c 4 ∈ (0, ∞) such that The Delaunay graph. An important graph for the study of a Voronoi tiling is its facial dual, the Delaunay graph based on N . This graph, denoted by D(N ) is an unoriented graph embedded in R d which has vertex set N and edges {u, v} every time C u and C v share a (d − 1)-dimensional face ( Figure 2 We shall prove an easier variant of Theorem 1.2, which will be useful in the applications. Define Ω to be the set of at most countable subsets of R d . Then, for any ω ∈ Ω and any subgraph φ of D(ω), we define Γ(ω, φ) to be the set of points in R d whose addition to ω "perturbs" φ: We get the following result for the maximal size of Γ(ω, φ) when φ belongs to SA n , the set of self-avoiding path on D(N ) starting from v(0) and of size n, where v(0) is the a.s. unique v ∈ N s.t. 0 ∈ C v : Proposition 1.3. There are constants z 1 and C 1 such that for every z ≥ z 1 , and for every n ≥ 0, P max φ∈SAn µ(Γ(N , φ)) > nz ≤ e −C 1 nz .

The renormalization trick
The main idea to obtain results for Voronoi Polyominoes, i.e our Theorem 1.2, comparable to those of Cox et al. (1993) is to combine a renormalization trick with an adaptation of the chaining techniques from Cox et al. (1993) to our setting. The renormalization trick is to consider a box in R d large enough so that it contains with a "large enough" probability some configuration of points which prevents a Delaunay cell to cross it completely. This will allow us to control the degrees of vertices along a Polyomino. The "large enough"probability alluded to is some percolation probability: we need that the "bad boxes" (those who can be crossed) do not percolate. In section 2.1, we explain this renormalization by showing how to cover animals with boxes and clusters of boxes. We also state the adaptation of Cox et al. (1993) to our setting. The proof of Theorem 1.2 is carried out in section 2.2, and the proof of Proposition 1.3 in section 2.3.
2.1. Comparison with site percolation and lattice animals. We denote by G d the d-dimensional lattice with vertex set Z d and edge set composed by pairs (z, z ′ ) such that For z in G d and r > 0, we define (This is what we call a box in R d .) Now, we define the notions of nice and good boxes. These are notions depending on a set N of points in R d , which basically says that there are enough points of N inside the box so that a Voronoi tile of a point of N outside the box cannot "cross" the box. This will be given a precise sense thanks to Lemma 2.2.
Definition 2.1. Define: Let N be a set of points in R d . We say that a box B ⊆ R d is N -nice if, cutting it regularly into α d d sub-boxes:, each one of these boxes contains at least one point of the set N . We say that a box B is N -good if, cutting it regularly into (3α d ) d sub-boxes, each one of these boxes contains at least one point of the set N . When N is understood, we shall simply say that B is nice (resp. good) if it is N -nice (resp. N -good). We say a box is N -ugly (resp. N -bad) if it is not N -nice (resp. not N -good).
To count the number of points of N which are in a subset A of R d , we define: An animal A in G d is a finite and connected subset of G d . To each real number r, and each bounded and connected subset A of R d , we associate 3 d different animals in G d : . . , f 3 d are the neighbors of 0 in G d . If V is a set of vertices of G d , define the border ∂V of V as the set of vertices which are not in V but have a neighbor in V. We also define the following subsets of R d : To each fixed positive real number r, we associate 3 d site percolation schemes on G d defined by: X r,i = (X r,i z ) z∈G d is a collection of independent Bernoulli random variables. Of course, the comparison of µ to Lebesgue's measure (1) implies that µ(B r,i z ) goes to infinity when r goes to infinity. Thus, lim When r is fixed, a vertex z is said to be (r, i)-bad (or simply bad when r and i are implicit) when X r,i z = 1. An (r, i)-bad cluster is then a maximal connected subset C of bad vertices of G d . Similarly, we define ugly vertices and ugly clusters. For any set of vertices V, we define by Cl r,i (V) the collection all (r, i)-bad clusters intersecting V. We shall make a slight abuse of notation by writing Cl r,i (z) to be the bad cluster containing z, if there is any (otherwise, we let Cl r,i (z) = ∅).
The following lemma says essentially that the Delaunay cells cannot cross the boundaries of an ugly (or bad) cluster.
Lemma 2.2. Fix r > 0 and i ∈ {1, . . . , 3 d }, assume that C is a (r, i)-ugly cluster in G d . Let ∆ be any Delaunay cell of D(N ). Then,

The same holds for bad clusters.
Proof : The idea of the proof is essentially the same as in Lemma 2.1 from Pimentel (2005). Suppose that ∆ ∩ Bl r,i (C) = ∅ but ∆ ⊂ Ad r,i (C). Let us define: Notice that the boundary of an ugly cluster is composed of nice vertices, and therefore ∂ ∞ C is composed of boxes B such that cutting B regularly into α d 2 d sub-boxes, each sub-box contains at least one point of N . This implies that one may find x 1 , x 2 and y in ∆, and a box B of side length r 2 such that: y ∈ [x 1 , x 2 ], y ∈ central(B), x 1 ∈ B, x 2 ∈ B , and cutting B regularly into α d 2 d sub-boxes, each sub-box contains at least one point of N , where central(B) is the only sub-box of B containing the center of B when one cuts B regularly into α d d sub-boxes. Since ∆ is a Delaunay cell, there is a ball B ′ containing ∆ and whose interior does not contain any point of N . This ball necessarily has diameter greater than x 1 − x 2 2 , which is larger than r/2. Thus, there is a ball B ′ of diameter larger than r/2, which contains y and whose interior does not contain any point of N . But one may see that any ball of diameter strictly larger than 2r √ d/α d necessarily contains at least one of the sub-boxes of side length r/α d . Notice that any ball of diameter strictly smaller than and since B is a nice box, we deduce that B ′ contains a sub-box of B which contains a point of N , whence a contradiction.
The following lemma transfers the problem of counting the degrees in our functional to counting the points in some boxes or clusters of bad boxes which cover the animals.
Lemma 2.3. Let G be a finite collection of bounded and connected subsets of R d . Then, for any positive real number r, t > 0.
Proof : Let γ be any member of G. Recall we say a box is ugly if it is not nice, and say it is bad when it is not good. We write z ∼ z ′ if z and z ′ are two points adjacent on G d , and Now, remark that if there exists z ′ ≃ z ∈ A r 1 such that B r z ′ is ugly, then rz + B 3r 0 is bad, and so there exists i ∈ {1, . . . , 3 d } such that Bl 3r,i (z) = 3rz + B 3r 0 + 3r f i /3 is bad. Therefore, When v belongs to some bad box Bl 3r,i (z), Lemma 2.2 shows that every Delaunay cell to which v belongs is included in Ad 3r,i (Cl 3r,i (z)). Therefore, in this case, d N (v) is bounded Lemma 2.4. Let g be a nondecreasing function from R + to [1, ∞), and define: There exists r 0 > 0 such that for all r > r 0 there exist positive and finite constants c 5 , c 6 and c 7 such that, if: then for every n ≥ m, and We shall also require the following result from Pimentel (2005): Lemma 2.5. There exist two finite and positive constants z 2 , and c 13 such that for every r ≥ 1, and every i = 1, . . . , 3 d , Let K be a positive real number, and r ≥ 1, Thus, thanks to Lemma 2.5. Now, let t be any positive number and m a positive integer. Using Lemma 2.3, then inequality (11) and Lemma 2.4 imply: Therefore, fix K = 1 c 6 and c 1 = sup{ 2 c 7 , c 5 }, and suppose that: This implies that e −c 7 l(q(x)) is integrable on R + . Thus, and then, Plugging these bounds into (9), (10) leads to the desired result.
2.3. Sketch of the proof of Proposition 1.3. The proof of Proposition 1.3 is almost entirely similar to that of Theorem 1.2, so we shall only stress the differences. First, we need to see that the notion of a nice box is still a good one to control Γ(N , φ). To do that, we need to express Γ(N , φ) differently. Let E(φ) be the set of edges of φ. Then, To any Delaunay cell ∆ of D(N ) corresponds a unique ball B(∆) in R d containing ∆ and whose interior does not contain any point of N . It may be seen that: where the intersection is taken over all Delaunay cells ∆ which contain e. Notice that the ball B(∆) is exactly the ball B appearing in the proof of Lemma 2.2. Thus, the same proof shows the following lemma.
Lemma 2.6. Fix r > 0 and i ∈ {0, . . . , 3 d }, assume that C is an (r, i)-ugly cluster in G d . Let φ be a self-avoiding path in D(N ). Then, for any vertex v in φ, and any edge e ∈ E(φ)

Applications
3.1. The connectivity constant on the Delaunay triangulation. Problems related to self-avoiding paths are connected with various branches of applied mathematics such as long chain polymers, percolation and ferromagnetism. One fundamental problem is the asymptotic behavior of the connective function κ r , defined by the logarithm of the number N r (v) of self-avoiding paths (on a fixed graph G) starting at vertex v and with r steps. For planar and periodic graphs subadditivity arguments yields that r −1 κ r (v) converges, when r → ∞, to some value κ ∈ (0, ∞) (the connectivity constant) independent of the initial vertex v. In disordered planar graphs subadditivity is lost but, if the underline graph possess some statistical symmetries (ergodicity), we may believe that the rescaled connective function still converges to some constant. When x is in R d , let us denote by v(x) the (a.s unique) point of N such that x ∈ C v . Let N r denote the number of self-avoiding paths of length r starting at v(0) on the Delaunay triangulation of a Poisson random set N . We recall that the intensity of the Poisson random set is bounded from above and below by a constant times the Lebesgue measure on R d . From Theorem 1.2 we obtain a linear upper bound for the connective function of the Delaunay triangulation: Proposition 3.1. Let κ(r) = log N r . There exist positive constants z 3 and c 2 such that, for every u ≥ rz 3 , P(κ(r) ≥ u) ≤ e −c 2 u 1/3 . In particular, E(κ(r)) = O(r) .
Proof : We shall use the following intuitive inequality which is a consequence of Lemma 3.2 below.
We now deduce the following: Now, notice that f (x) = x is c−nice for every c > 0. Thus, according to Theorem 1.2, there are positive constants z 3 and c 2 such that, for all t ≥ e rz 3 , Or, equivalently, for every u ≥ rz 3 , This implies notably that: Lemma 3.2. Let G = (V, E) be a graph with set of vertices V and set of edges E. Define, for any n ∈ N, any x ∈ V and I ⊂ V : Define: Then, for any n ≥ 1, where d(v) stands for the degree of the vertex v.
Proof : We shall prove the result by induction. When n = 1 it is obviously true. Indeed, if x ∈ I, then N 1 (x, I) = 0 and if x ∈ I, N 1 (x, I) ≤ d(x). Suppose now that the result is true until n ≥ 1. Let x ∈ V and I ⊂ V . If x ∈ I, then N n+1 (x, I) = 0. Suppose thus that x ∈ I. Then, denoting u ∼ x the fact that u and x are neighbours, and using the induction hypothesis, and the induction is proved.
3.2. Linear passage weights of self-avoiding paths in percolation. When D is a fixed Delaunay triangulation on the plane, the bond percolation model X = {X e : e ∈ D} on the Delaunay triangulation, with parameter p and probability law P p (.|D), is constructed by choosing each edge e ∈ D to be open, or equivalently X e = 1, independently with probability p and closed, equivalently X e = 0, otherwise. An open path is a path composed by open edges. We denote by P p the probability measure obtained when D is the Delaunay triangulation based on N , which is distributed like a Poisson random set on R d with intensity measure µ. We denote by C 0 the maximal connected sugraph of D composed by edges e that belong to some open path starting from v 0 , the point of N whose Voronoi tile contains the origin, and we call it the open cluster. Define the critical probability: It is extremely plausible that this critical probability coincides with the classical one, i.e sup{p s.t. P p (|C 0 | = ∞) = 0}, but we are not going to prove this here. By using Proposition 3.1 to count the number of self-avoiding paths of size n, and then evaluating the probability that it is an open path, we have that there exists a positive and finite constant B = B(d) such that if p < 1/B then the probability of the event {|C 0 | = n} decays like e −αn 1/(3d) for some constant α. Consequently: In the subsequent section we will require the following lemma on the density of open edges: Lemma 3.4. If (1 −p) <p c (d) then there exists constants (only depending on p) a 1 , a 2 > 0 such that Remark 1. We could prove Lemma 3.4 directly under (1 − p) < 1/B (again using Proposition 3.1) but we want to push forward optimality as much as we can.
To prove Lemma 3.4 we shall first obtain a control on the number of boxes intersected by a self-avoiding path in D. Recall that, in section 2.1, for fixed L > 0 and for each self-avoiding path γ in D we have defined an animal A(γ) := A L 1 (γ) on G d by taking the vertices z such that γ intersects B L/2 z . Let g L r (N ) := min{|A(γ)| : s.a. γ, v 0 ∈ γ and |γ| ≥ r} and G L r (N ) := max{|A(γ)| : s.a. γ, v 0 ∈ γ and |γ| ≤ r} . By Proposition 2.1 in Pimentel (2005) we have (this proposition is only stated for d = 2 but the method to prove it holds for all d ≥ 2): Lemma 3.5. For each L ′ ≥ 1 there exist finite and positive constants b 3 , b 4 , b 5 , b 6 such that and Proof of Lemma 3.4: Let L > 0, z ∈ G d . In this section, we call B L/2 z a good box if: Since the intensity of the underlying Poisson random set is comparable with the Lebesgue measure (1), condition (i) has probability going to one as L goes to infinity. Now, denote by E L the event that (ii) is false and (i)  z . Also, there is a point v ′ such that either v 1 or v 2 lies at distance at least L from v ′ and γ connects v 1 and v 2 to v ′ with closed edges. Divide B 5L/2 z regularly into subcubes of side length 1. This partition P L has cardinality of order L d . For each box B, let A(B) be the event that there exist a vertex v ∈ N ∩ B, v ′ ∈ N such that |v − v ′ | ≥ L, and a path γ in D from v to v ′ such that e∈γ X e = 0. Denote by A L the event A(B 1/2 0 ). The remarks we just made imply that: Now, let us bound P p A L . If p = 1, P p A L = 0, so we suppose that p < 1. Let B d denote the ball of center 0 and radius 3 √ d, and let E d be the set of edges of D lying completely inside B d . A simple geometric lemma (see Lemma 4.5 in section 4.2) shows that if v ∈ N ∩ B 1/2 0 , there is a path from v 0 to v on D whose edges are in E d . Thus, if A L holds and if all the edges in E d are closed, then there is a closed path from v 0 to some vertex v ′ such that |v ′ | ≥ L − √ 2. Let L be larger than 2 √ 2. Using (14) (with L ′ = 1), there exist constants a, b > such that, with probability greater than (1 − e −aL ), a path γ in the Poisson-Delaunay triangulation, that connects v 0 to a point v ′ such that |v ′ | ≥ L − √ 2, has at least bL edges. Now, using the FKG inequality, we have: where C cl 0 is the closed cluster containing 0. Thus: Since B d is a convex set and D is a triangulation, a simple application of Euler's formula gives |E d | ≤ 3(|N ∩ B d | − 1). Now, notice that |N ∩ B d | has Poisson distribution with parameter µ(B d ). Thus, one can find some positive constants C 1 , C 2 and C 3 , depending only on µ and d, such that, for every L > 0: Now, using Cauchy-Schwartz inequality, and inequality (16), which goes to zero as L goes to infinity. This concludes the proof of (15). Our second step is: the collection {Y z (L) : z ∈ G d } is a 5-dependent Bernoulli field. To see this, first notice that Y z (L) = Y z (L, N , X). Thus, the claim will follow if we prove that With (15) and 5-dependence in hands, we can chose L 0 ≥ 1 large enough such that for c 1 , c 2 only depending on L 0 . The proof of (17) follows the same line as the proof of Lemma 4.4 in Pimentel (2005) (by first proving (17) for i.i.d. sums, and then using Liggett-Schonmann-Stacey theorem Liggett et al. (1997) to extend it to the 5-dependent case).
Let γ be a path in D with v 0 ∈ γ and |γ| ≥ m. If B L/2 z 1 and B L/2 z 2 are good boxes in A(γ), and |z 1 − z 2 | ≥ 5, then for each j = 1, 2 there is a piece of γ, say γ j , connecting ∂B L/2 z j to ∂B 3L/2 z j and with e∈γ j X e ≥ 1. Since these good boxes are 5-distant, by Lemma 2.2, γ 1 and γ 2 can be assumed disjoint. This yields that e∈γ X e ≥ 2. By repeating this argument inductively, one gets that (We count good points in G d that are at distance 5 from each other and so we have to divide by 11 d .) On the other hand, A(γ) has at least k ≥ g m (N ) boxes, and hence j ≥ m k (Y (L 0 )) good boxes. Combining this together with (13) and (17), we finish the proof of Lemma 3.4.

3.3.
Variance bound for FPP on the Delaunay triangulation. On any graph G, one can define a First Passage Percolation model (FPP in the sequel) as follows. Assign to each edge e of G a random non-negative time t(e) necessary for a particle to cross edge e. The First Passage time from a vertex u to a vertex v on G is the minimal time needed for a particle to go from u to v following a path on G. This is of course a random time, and the understanding of the typical fluctuations of this times when u and v are far apart is of fundamental importance, notably because of its physical interpretation as a growth model.  (1990), where the graph G is the Delaunay graph of a Lebesgue-homogeneous Poisson random set in R d . Often, this Delaunay triangulation is heuristically considered to behave like a perturbation of the triangular lattice. One important question is whether the fact that G itself is random affects the fluctuations of the first passage times. Since it is not already known exactly what is the right order of these fluctuations on the triangular lattice, one does not have an exact benchmark, but the best bound known up today is that the variance of the passage time between the origin and a vertex at distance n is of order O(n/ log n). We are not able to show a similar bound for FPP on the Delaunay triangulation, but we can show the analogue of Kesten's bound of Kesten (1993): the above variance is at most of order O(n). This improves upon the results in Pimentel (2005) and answers positively to open problem 12, p. 169 in Howard (2004). We now precise our notations. Let ν be a probability measure on R + . Recall that Ω is the set of at most countable subsets of R d . Let π denote the Poisson measure on R d with intensity µ, where µ is comparable to the Lebesgue measure on R d , in the sense of (1). Let E d denote the set of pairs {x, y} of points of R d . We endow the space Ω × R E d + with the product measure π ⊗ ν ⊗E d and each element of Ω × R E d + is denoted (N , τ ). This means that N is a Poisson random set with intensity µ, and to each edge e ∈ D(N ) is independently assigned a non-negative random variable τ (e) from the common probability measure ν. The passage time T (γ) of a path γ in the Delaunay triangulation is the sum of the passage times of the edges in γ: The first-passage time between two vertices v and v ′ is defined by is the borelian tribe on R + and F is the smallest algebra on Ω which lets the coordinate applications N → 1I x∈N measurable. To see this, fix x, y ∈ R d and for each r > |x, y| define T r (x, y) to be the first-passage time restricted to the Delaunay graph D(N ∩D r (x, y)), where D r (x, y) is the ball centred at x and of radius r. Then clearly T r is measurable, since it is the infimum over a countable collection of measurable functions. On the other hand, T r (x, y) is non-increasing with r, and so T = lim r→∞ T r (x, y) is also measurable.
We may now state the announced upper-bound on the typical fluctuations of the first passage time. We denote by M k (ν) the k − th moment of ν: Theorem 3.6. For any natural number n ≥ 1, let n denote the point (n, 0, . . . , 0) ∈ R d . Assume that ν({0}) <p c and that Then, as n tends to infinity, Var(T (0, n, τ, N )) = O(n) .
Before we prove Theorem 3.6, we need to state a Poincaré inequality for Poisson random sets (for a proof, see for instance Houdré and Privault (2003), inequality (2.12) and Lemma 2.3).
Proposition 3.7. Let F : Ω → R be a square integrable random variable. Then, where: A geodesic between x and y, in the FPP model, is a path ρ(x, y) connecting v x to v y and such that T (x, y) = T ρ(x, y) = e∈ρ(x,y) τ (e) .
When a geodesic exists, we shall denote by ρ n (τ, N ) a geodesic between 0 and n with minimal number of vertices. If there are more than one such geodesics, we select one according to some deterministic rule. We shall write T n (τ, N ) for T (0, n, τ, N ).
We shall need to control the length of ρ n (τ, N ). The following result generalizes to every dimension the control obtained in Proposition 2.2 Pimentel (2005), but here the condition on ν({0}) is not optimal.
Lemma 3.8. There exist constants a 0 , C 1 , C 2 and C 3 depending only on ν({0}) and d, and a random variable Z n such that if ν({0}) <p c (d), then for every n and every m, Proof : For any a > 0, and m ∈ N, Fix ǫ > 0 such that P(τ e > ǫ) <p c and let X e = 1I{τ e > ǫ}. Then ǫX e ≤ τ e , and consequently Finally, in Pimentel (2005) and for d = 2, it is shown that there is a particular path γ n from 0 to n, that is independent of τ and whose expected number of edges has is of order n. This path is constructed by walking through neighbour cells that intersect line segment [0, n] (connecting 0 to n). Hence, the size of γ n is at most the number of cells intersecting [0, n], which turns to be of order n. This construction is easily generalized to R d and thus, denoting Z n = e∈γ(0, n) τ (e) the passage time along this path, we have: Using only the fact that the edge-times possess a finite moment of order 1, there is a constant C 3 such that: E(Z n ) = E(τ e )E(|γ n |) ≤ C 3 n .

Proof of Theorem 3.6 :
For any function f from R E d + × Ω to R, we write: and for any function g in L 2 (Ω, π), We shall use the following decomposition of the variance: To show that E π (Var ν (T n )) = O(n) is now standard. Indeed, π-almost surely (see (2.17) and (2.24) in Kesten (1993)): Thus, according to Lemma 3.8. The harder part to bound is Var π (E ν (T n )). Let us denote by F n the random variable E ν (T n ). We want to apply Proposition 3.7 to F n . First note that F n belongs to L 2 (Ω, π): this follows from (18). We claim that: and Indeed, suppose first that v belongs to N . If v ∈ ρ n (τ, N ), then ρ n (τ, N ) is still included in D(N \ {v}). This implies that T n (τ, N \ {v}) ≤ T n (τ, N ). Suppose that on the contrary, v ∈ ρ n (τ, N ). Let S 1 (N , v) be the set of edges in D(N ) which belong to a Delaunay cell containing v, but which do not contain v. Define a set of edges S 2 (N , v) containing all the edges of ρ n (τ, N ) that are still in D (N \ {v}), and all the edges of S 1 (N , v). Then, S 2 (N , v) is a set of edges in D(N \ {v}) which contains a path from 0 to n. From these considerations, we deduce that for v in N : where we used Cauchy-Schwarz inequality. This gives claim (19). To see that claim (20) is true, suppose that x does not belong to N . If x is not in Γ(N , ρ n (τ, N )), obviously N ). On the contrary, if x is in Γ(N , ρ n (τ, N )), there are two special neighbors, v in (x) and v out (x) such that ρ n (τ, N ) still connects v in (x) to 0 and v out (x) to n in D(N ∪ {x}). Let S 3 (N , x) be the set of edges containing all the edges of ρ n (τ, N ) that are still in D(N ∪ {x}), plus the two edges (v in (x), x) and (x, v out (x)). S 3 (N , x) contains a path in D(N ∪ {x}) from 0 to n. Notice that v in (x) and v out (x) depend on N and (τ (e)) e∈D(N ) , and that conditionnally on N , τ (v in (x), x) and τ (x, v out (x)) are independent from (τ (e)) e∈D(N ) . From this, we deduce that for x not in N , τ (e) 1I x∈Γ(N ,ρn(τ,N )) , which proves claim (20) via Jensen's inequality. Now, these two claims together with Proposition 3.7 applied to F n give: We shall conclude using Lemma 3.8, Proposition 1.3 and Theorem 1.2. Let z 1 be as in Theorem 1.2, and a as in Lemma 3.8. Notice that f : x → x 2 is c-nice for any c > 0.
From Lemma 3.8 and Theorem 1.2, we have: ≤ e −C 1 (nz/z 1 ) + P(Z n > anz/z 1 ) + nz z 1 e −c 2 (c 3 nz) 1/5 , and thus, using that E(Z n ) = O(n), we get: The proof that E(λ(Γ(N , ρ n (τ, N ))) = O(n) is completely similar, using Proposition 1.3 and Lemma 3.8. Theorem 3.6 follows now from (21).  and, for any α > 0, The importance of Lemma 3.9 is due to the fact that it is the essential tool to prove that simple random walk on D(N ) is recurrent in R 2 and transient in R d for d ≥ 3. Here, we show that our method allows to improve Lemma 3.9 as follows.
Proof : Since the proof is very close to the proof of Proposition 1.3, we only sketch the main differences. We divide the boundary of [0, n] d into 2 d 2 d−1 n d−1 (d − 1)-dimensional cubes of (d − 1)-volume 1, and call this collection S n . For each pair of cubes (s 1 , s 2 ) in S n , we define st(s 1 , s 2 ) as the maximum number of Delaunay cells that intersect a single line-segment going from a point in s 1 to a point in s 2 . Let V (s 1 , s 2 ) be the union of those line-segments. Notice that st(s 1 , s 2 ) is bounded from above by the number of points which belong to a Delaunay cell intersecting V (s 1 , s 2 ). Lemma 2.2 allows us to control those points. We obtain the following analogue of Lemma 2.3.
Note that the cardinals of A 3r i (V (s 1 , s 2 )), i = 1, . . . , 3 d are of order O(n), and the possible choices for the pair (s 1 , s 2 ) is of order O(n 2(d−1) ). Now, when r is chosen large enough (see section 4.1), Proposition 1.3 follows from Lemmas 2.5, 4.2 and 4.1.

Appendix
4.1. Proof of Lemma 2.4. It suffices to prove Lemma 2.4 for i = 1, so we shall omit i as a subscript or superscript. Remark first that, as already noted at the beginning of section 2.1, p r := sup i sup z P(X r,i z = 1) tends to 0 as r tends to infinity. Let us choose r 0 in such a way that p r < p c (G d ) , where p c (G d ) is the critical probability for site percolation on G d . When r ≤ r 0 , we are in the so-called "subcritical" phase for percolation of clusters of (N , 6)-bad boxes. From now on, we fix r to satisfy r > r 0 and we shall omit r as a subscript or superscript, to shorten the notations. It is well known that in this "subcritical" phase, the size of the (bad)-cluster containing a given vertex decays exponentially (see Grimmett (1989), Theorem (6.75)). For instance, there is a positive constant c, which depends only on r, such that, Remark that there is probably an exponential number of animals of size less than m, and therefore, if C∈Cl(A) g(|Ad(C)|) had an exponential moment, where g(x) = xf (x), it would be easy to bound the first summand in the right-hand term of the inequality above. When f (x) is larger than x, C∈Cl(A) g(|Ad(C)|) surely does not have exponential moments. Therefore, we have to refine the standard argument. This refinement is a chaining technique essentially due to Cox et al. (1993) and we rely heavily on that paper.
We shall prove (7), the proof of (8) being similar and easier. We begin by stating and proving two lemmas on site percolation.
Lemma 4.1. For all r > r 0 there is a constant c 9 (depending only on r) such that: where by P pr , we mean that every site is open with the same probability Proof : First, we shall condition on the value of X = (X z ). Let u be a positive real number. (0))| N > s|X) ≤ e −us E(e u|Ad(Cl(0))| N |X)) .
Remark that, X being fixed, |Ad(Cl(0))| N = |Cl(0)| N + |∂ ∞ Cl(0)| N and the two summands are independent. Through the comparison to Lebesgue's measure 1, the term |Cl(0)| N is stochastically dominated by a sum of |Cl(0)| independent random variables, each of which is a sum of independent random variables with a Poisson distribution of parameter β = β(r), conditioned on the fact that one of them at least must be zero. The term |∂ ∞ Cl(0)| N is stochastically dominated by a sum of at most c d |Cl(0)| random variables (c d only depending on d), each of which is a sum of l independent random variables with Poisson distribution, conditioned on the fact that all of them are greater than one. Obviously, the sum of l independent random variables with a Poisson distribution, conditioned on the fact that one of them at least must be zero is stochastically smaller than the sum of the same number of independent random variables with Poisson distribution, conditioned on the fact that all of them are greater than one. If Z is a Poisson random variable with parameter β, one has: Therefore, and K ′ = φ −1 (2 log K). Remark that φ is a bijection from [1, +∞) onto [0, +∞). Minimizing in u gives: This leads to: P(|Ad(Cl (0) Therefore, denoting c 9 = min{K, c 7βK ′ }, we get the desired result (notice that β and K ′ depend on r).
Proof : Let us recall Reimer's inequality (cf. Grimmett Grimmett (1989) p.39). Let n be a positive integer, let B(n) = Z 2 ∩ [−n, n] d and define Γ = {0, 1} B(n) . For ω ∈ Γ and K ⊂ B(n), define the cylinder event C(ω, K) generated by ω on K by: If A and B are two subsets of Γ, define their disjoint intersection A B as follows: Reimer's inequality states that: Remark that is a commutative and associative operation, and that, for any l subsets A 1 , . . . , A l of Γ, Now take n large enough so that Λ ⊂ B(n). Let l = |Λ|, and order the elements of Λ = {x 1 , x 2 , . . . , x l }. For any x in B(n), and any ω ∈ Γ, let C n (x, ω) be the closed cluster (for the configuration ω) in B(n) containing x, which is empty if ω(x) = 1. Define: Let k 1 , . . . , k l be nonnegative integers and: Then, we claim that: This shows that ω ∈ A 1 . . . A l , and proves the claim above. Therefore, using Reimer's inequality (which is equivalent to BK inequality in this context), Now, let f be an increasing function from R + to [1, ∞). Define f 1 as follows: f 1 (0) = 1 and ∀k ≥ 1, f 1 (k) = f (k) .
First, Lemma 1 in Cox et al. (1993) (Lemma 4.3 below) remains true in our setting, without any modification (just remark that the condition l ≤ n in their lemma is in fact not needed).
Then, we define the following box in G d , centered at x ∈ G d : Lemma 4.3. Let A be a lattice animal of G d containing 0, of size |A| = m and let 1 ≤ l.
In view of the last inequality and (26), we would be happy if we had t(n, k) ≥cl(q(n)) and c 10 c ′ t(n, k) ≥ (2 log 9). 2m l k .
Of course, we still have to check conditions (27) and (28), and we can choose c ′ as large as we need. A natural way to proceed is first to choose l k in such a way that the right-hand side in (28) is proportional to m l k , and then to take large enough. Recall that n ≥ m. Thus, define: l k = e c 9 2 l(2 k ) .
Lemma 4.5. Let N be a locally finite set of points in R d and D the Delaunay triangulation based on N . Let u and v be two distinct points in N . Then, there is a path on D going from v to u and totally included in the (closed) ball of center u and radius |u − v|.
Proof : First we show that there is a neighbour w of v in D inside the (closed) ball of diameter [u, v]. To see this, let us define by (B α ) α∈[0,1] the collection of euclidean balls such that B α has diameter [v, x α ], where x α = v + α(u − v). Notice that B α ⊂ B α ′ as soon as α ≤ α ′ . Define: This is indeed a minimum because N is locally finite. Notice also that the set is non-empty since it contains α = 1. Thus the interior of B α 0 does not intersect N , but ∂B α 0 contains (at least) two points of N (including v). This implies that there is a point on the sphere ∂B α 0 which is a neighbour of v. So we have proved that there is a neighbour w 1 of v inside the ball of diameter [u, v]. Then, as long as the neighbour obtained is different from u, we may iterate this construction to get a sequence of neighbours w 0 = v, w 1 , w 2 , . . . such that w i+1 belongs to the ball of diameter [u, w i ]. All these balls are included in the ball of center u and radius |u − v|. Since N is locally finite, this construction has to stop at some k, when the condition that w k is distinct from u is no longer satisfied. Then the desired path is constructed.