Existence and continuity of the flow constant in first passage percolation

We consider the model of i.i.d. first passage percolation on Z^d, where we associate with the edges of the graph a family of i.i.d. random variables with common distribution G on [0, +$\infty$] (including +$\infty$). Whereas the time constant is associated to the study of 1-dimensional paths with minimal weight, namely geodesics, the flow constant is associated to the study of (d--1)-dimensional surfaces with minimal weight. In this article, we investigate the existence of the flow constant under the only hypothesis that G({+$\infty$})<p c (d) (in particular without any moment assumption), the convergence of some natural maximal flows towards this constant, and the continuity of this constant with regard to the distribution G.

from one side to the other. This question was adressed notably in [23] and [27] where one can find laws of large numbers and large deviation principles when the dimensions of the box grow to infinity. We refer to section 1 for a more precise picture of the background, but let us stress for the moment that in those works, moment assumptions were made on the capacities. It is however interesting for modelling purposes to remove this assumption, allowing even infinite capacities which would represent microscopic defects where capacities are of a different order of size than elsewhere. The first achievement of the present work, Theorem 1.4, is to prove a law of large numbers for maximal flows without any moment assumption, allowing infinite capacities under the assumtion that the probability that an edge has infinite capacity is less than the critical parameter of percolation in Z d .
Once such a result is obtained, one may wonder in which way the limit obtained in this law of large numbers, the so-called flow constant, depends on the capacity distribution put on the edges. The second achievement of this article, Theorem 1.6, is to show the continuity of the flow constant. One application of this continuity result could be the study of maximal flows in an inhomogeneous environment when capacities are not identically distributed but their distribution depends smoothly (at the macroscopic scale) on the location of the edges.
The rest of the paper is organized as follows. In section 1, we give the necessary definitions and background, state our main results and explain in detail the strategy of the proof. The law of large numbers is proved in section 2 and the continuity result is shown in section 4. Between those two sections lies in section 3 a technical intermezzo devised to express the flow constant as the limit of a subbadditive object. The reason why we need it will be decribed at length in section 1.4.

Definition of the maximal flows
We use many notations introduced in [23] and [27]. Given a probability measure G on [0, +∞], we equip the graph (Z d , E d ) with an i.i.d. family (t G (e), e ∈ E d ) of random variables of common distribution G. Here E d is the set of all the edges between nearest neighbors in Z d for the Euclidean distance. The variable t G (e) is interpreted as the maximal amount of water that can cross the edge e per second. Consider a finite subgraph Ω = (V Ω , E Ω ) of (Z d , E d ) (or a bounded subset of R d that we intersect with (Z d , E d ) to obtain a finite graph), which represents the piece of rock through which the water flows, and let G 1 and G 2 be two disjoint subsets of vertices in Ω: G 1 (resp. G 2 ) represents the sources (resp. the sinks) through which the water can enter in (resp. escapes from) Ω. A possible stream inside Ω between G 1 and G 2 is a function f : E d → R d such that for all e ∈ E d , • f (e) 2 is the amount of water that flows through e per second, • f (e)/ f (e) 2 is the direction in which the water flows through e.
For instance, if the endpoints of e are the vertices x and y, which are at Euclidean distance 1, then f (e)/ f (e) 2 can be either the unit vector xy or the unit vector yx. A stream f inside Ω between G 1 and G 2 is G-admissible if and only if it satisfies the following constraints: "surface" of plaquettes is thus very similar to a path in the dual graph of Z 2 in dimension 2. The study of maximal flows in first passage percolation is equivalent, through the max-flow min-cut theorem, to the study of the minimal capacities of cutsets. When we compare this to the classical interpretation of first passage percolation, the study of geodesics (which are paths) is replaced by the study of minimal cutsets (which are rather hypersurfaces). In this sense, the study of maximal flow is a higher dimensional version of classical first passage percolation.
We now define two specific maximal flows through cylinders that are of particular interest. Let A be a non-degenerate hyperrectangle, i.e., a rectangle of dimension d − 1 in R d . Let v be one of the two unit vectors normal to A.  We denote by H d−1 the Hausdorff measure in dimension d − 1: We can expect that φ G (A, h) grows asymptotically linearly in H d−1 (A) when the dimensions of the cylinder go to infinity, since H d−1 (A) is the surface of the area through which the water can enter in the cylinder or escape from it. However, φ G (A, h) is not easy to deal with. Indeed, by the max-flow min-cut theorem, φ G (A, h) is equal to the minimal capacity of a set of edges that cuts B 1 (A, h) from B 2 (A, h) in the cylinder. The dual of this set of edges is a surface of plaquettes whose boundary on the sides of cyl(A, h) is completely free. This implies that the union of cutsets between the top and the bottom of two adjacent cylinders is not a cutset itself between the top and the bottom of the union of the two cylinders.
Thus the maximal flow φ G (A, h) does not have a property of subadditivity, which is the key tool in the study of classical first passage percolation. This is the reason why we define another maximal flow through cyl(A, h), for which subadditivity is recovered. The set cyl(A, h) A has two connected components, denoted by C 1 (A, h) and C 2 (A, h). For i = 1, 2, we denote by C i (A, h) the discrete boundary of C i (A, h) defined by (1.2) We denote by τ G (A, h) the maximal flow from the upper half part of the boundary of the cylinder to its lower half part, i.e., By the max-flow min-cut theorem, τ G (A, h) is equal to the minimal capacity of a set of edges that cuts C 1 (A, h) from C 2 (A, h) inside the cylinder. To such a cutset E corresponds a dual set of plaquettes E * whose boundary has to be very close to ∂A, the boundary of the hyperrectangle A. We say that a cylinder is straight if v = v 0 := (0, 0, . . . , 1) and if there exists k i , l i , c ∈ Z such that k i < l i for all i and A = A( k, l) = In this case, for c = 0 and k i ≤ 0 < l i , the family of variables (τ G (A( k, l), h)) k, l is subadditive, since the minimal cutsets in adjacent cylinders can be glued together along the common side of these cylinders.

Background on maximal flows
A straightforward application of ergodic subadditive theorems in the multiparameter case (see Krengel and Pyke [25] and Smythe [28]) leads to the following result. Proposition 1.1. Let G be a probability measure on [0, +∞[ such that R + x dG(x) < ∞.
Let h : N → R + such that lim p→∞ h(p) = +∞. Then there exists a constant ν G ( v 0 ), that does not depend on A and h, such that lim p→∞ τ G (pA, h(p)) H d−1 (pA) = ν G ( v 0 ) a.s. and in L 1 .
This result has been stated in a slightly different way by Kesten in [23]. He considered there the more general case of flows through cylinders whose dimensions goes to infinity at different speeds in each direction, but in dimension d = 3. The constant ν G ( v 0 ) obtained here is the equivalent of the time constant µ G (e 1 ) defined in the context of random distances (see Section 1.4), and by analogy we call it the flow constant.
As suggested by classical first passage percolation, a constant ν G ( v) can be defined in . This is not that trivial, since a lack of subadditivity appears when we look at tilted cylinders, due to the discretization of the boundary of the cylinders. Moreover, classical ergodic subadditive theorems cannot be used if the direction v is not rational, i.e., if there does not exist an integer M such that M v has integer coordinates. However, these obstacles can be overcome and the two authors proved in [27] the following law of large numbers.
If moreover the origin of the graph belongs to A, or If the cylinder is flat, i.e., if lim p→∞ h(p)/p = 0, then the same convergences hold also for φ G (pA, h(p)).
When the origin of the graph belongs to A, and for an increasing function h for instance, the cylinder cyl(pA, h(p)) is completely included in the cylinder cyl((p+1)A, h(p+ 1)). The mean of the capacities of the edges inside cyl(pA, h(p)) converges a.s. when p goes to infinity as soon as R + x dG(x) < ∞ by a simple application of the law of large numbers, and Theorem 1.2 states that τ G (pA, h(p))/H d−1 (pA) converges a.s. under the same hypothesis. On the other hand, when the origin of the graph does not belong to A, the cylinders cyl(pA, h(p)) and cyl((p + 1)A, h(p + 1)) may be completely disjoint. The a.s. convergence of the mean of the capacities of the edges included in cyl(pA, h(p)) when p goes to infinity is thus stated by some result about complete convergence of arrays of random variables, see for instance [18,19]. This kind of results requires a stronger moment condition on the law of the random variables we consider, namely we need that Existence and continuity of the flow constant The asymptotic behavior of the maximal flows φ G (pA, h(p)) in non-flat cylinders (i.e., when h(p) is not negligible in comparaison with p) is more difficult to study since these flows are not subadditive. In the case of straight cylinders (and even in a non isotropic case, i.e., when the dimensions of the cylinders go to infinity at different speed in every directions), Kesten [23] and Zhang [30] proved that φ G (pA, h(p))/H d−1 (pA) converges a.s. towards ν G ( v 0 ) also, under some moment condition on G. The behavior of φ G (pA, h(p)) is different in tilted and non-flat cylinders, we do not go into details and refer to [26] (for d = 2) and to [5,7,6,8] in a more general setting.
We stress the fact that for all the results mentioned above, a moment assumption is required on the probability measure G on [0, +∞[: G must at least have a finite mean.

Main results
Our first goal is to extend the previous results to probability measures G on [0, +∞[ that are not integrable, and even to probability measures G on [0, +∞] under the hypothesis that G({+∞}) < p c (d).
For any probability measure G on [0, +∞], for all K > 0, we Throughout the paper, we shall say that a function h : We prove the following law of large numbers for cylinders with mild height functions. Theorem 1.4. For any probability measure G on [0, +∞] such that G({+∞}) < p c (d), for any v ∈ S d−1 , for any non-degenerate hyperrectangle A normal to v, for any mild function h, we have We also want to establish the continuity of the function G → ν G ( v) when we equip the set of probability measures on [0, +∞] with the topology of weak convergence -in fact these two questions are linked, as we will see in Section 1.4. More precisely, let (G n ) n∈N and G be probability measures on [0, +∞]. We say that G n converges weakly towards G when n goes to infinity, and we write G

About the existence and the continuity of the time constant
First passage percolation was introduced by Hammersley and Welsh [20] in 1965 with a different interpretation of the variables associated with the edges. We consider the graph (Z d , E d ) and we associate with the edges of the graph a family of i.i.d. random variables (t G (e), e ∈ E d ) with common distribution G as previously, but we interpret now the variable t G (e) as the time needed to cross the edge e (we call it the passage time of e). If γ is a path, we define the passage time of γ as T G (γ) = e∈γ t G (e). Then the passage time between two points x and y in Z d , i.e., the minimum time needed to go from x to y for the passage times (t G (e), e ∈ E d ), is given by This defines a random pseudo-distance on Z d (the only property that can be missing is the separation property). This random distance has been and is still intensively studied. A reference work is Kesten's lecture notes [22]. Auffinger, Damron and Hanson wrote very recently the survey [2] that provides an overview on results obtained in the 80's and 90's, describes the recent advances and gives a collection of old and new open questions. Fix e 1 = (1, 0, . . . , 0). Thanks to a subadditive argument, Hammersley and Welsh [20] and Kingman [24] proved that if d = 2 and F has finite mean, then lim n→∞ T F (0, ne 1 )/n exists a.s. and in L 1 , the limit is a constant denoted by µ F (e 1 ) and called the time constant. The moment condition was improved some years later by several people independently, and the study was extended to any dimension d ≥ 2 (see for instance Kesten's Saint-Flour lecture notes [22]). The convergence to the time constant can be stated as follows.
a.s. and in L 1 .
This convergence can be generalized by the same arguments, and under the same hypothesis, to rational directions: there exists a homogeneous function µ F : Q d → R + such that for all x ∈ Z d , we have lim n→∞ T F (0, nx)/n = µ F (x) a.s. and in L 1 . The function µ F can be extended to R d by continuity (see [22]). These results can be extended by considering a law F on [0, +∞[ which does not satisfy any moment condition, at the price of obtaining weaker convergence. This work was performed successfully by Cox and Durrett [11] in dimension d = 2 and then by Kesten [22] in any dimension d ≥ 2. More precisely, they proved that there always exists a functionμ F : The functionμ F is built as the a.s. limit of a more regular sequence of timesT F (0, nx)/n that we now describe roughly. They consider an M ∈ R + large enough so that F ([0, M ]) is very close to 1. Thus the percolation (1 {t F (e)≤M } , e ∈ E d ) is highly supercritical, so if we denote by C F,M its infinite cluster, each point x ∈ Z d is a.s. surrounded by a small contour S(x) ⊂ C F,M . They defineT F (x, y) = T F (S(x), S(y)) for x, y ∈ Z d . The timesT F (0, x) have good moment properties, thusμ F (x) can be defined as the a.s. and L 1 limit of T F (0, nx)/n for all x ∈ Z d by a classical subadditive argument; thenμ F can be extended to Q d by homogeneity, and finally to R d by continuity. The convergence of T F (0, nx)/n towardsμ F (x) in probability is a consequence of the fact that T F andT F are close enough.
It is even possible to consider a probability measure F on [0, +∞] under the hypothesis that F ([0, +∞[) > p c (d). This was done first by Garet and Marchand in [14] and then by Cerf and the second author in [9]. We concentrate on [9], where the setting is closer to the one we consider here. To prove the existence of a time constant for a probability measure F on [0, +∞] such that F ([0, +∞[) > p c (d), Cerf and the second author exhibit a quite intuitive object that is still subadditive. For x ∈ Z d ,μ F (x) is defined by a subadditive argument as the limit of T F (f M (0), f M (nx))/n a.s. and in L 1 , where M is a real number large enough such that F ([0, M ]) > p c (d), and for z ∈ Z d , f M (z) is the points of C F,M which is the closest to z. The convergence of T F (0, nx)/n towardsμ F (x) still holds, but in a very weak sense: T F (0, nx)/n converges in fact in distribution towards where θ F is the probability that the connected component of 0 in the percolation (1 t F (e)<∞ , e ∈ E d ) is infinite. For short, all these constants (μ F ,μ F and µ F ) being equal when they are defined, we denote all of them by µ F .
Once the time constant is defined, a natural question is to wonder if it varies continuously with the distribution of the passage times of the edges. This question has been answered positively by Cox and Kesten [10,12,22] for probability measures on [0, +∞[.
Cox [10] proved first this result in dimension d = 2 with an additional hypothesis of uniform integrability: he supposed that all the probability measures F n were stochastically dominated by a probability measure H with finite mean. To remove this hypothesis of uniform integrability in dimension d = 2, Cox and Kesten [12] used the regularized passage times and the technology of the contours introduced by Cox and Durrett [11].
Kesten [22] extended these results to any dimension d ≥ 2. The key step of their proofs is the following lemma. Lemma 1.9. Let F be a probability measure on R + , and let F K = 1 [0,K) F +F ([K, +∞))δ K be the distribution of the passage times t F (e) truncated at K. Then for every x ∈ R d , To prove this lemma, they consider a geodesic γ from 0 to a fixed vertex x for the truncated passage times inf(t F (e), K). When looking at the original passage times t F (e), some edges along γ may have an arbitrarily large passage time: to recover a path γ from 0 to x such that T F (γ ) is not too large in comparison with T F K (γ), they need to bypass these bad edges. They construct the bypass of a bad edge e inside the contour S(e) ⊂ C F,M of the edge e, thus they bound the passage time of this bypass by M card e (S(e)) where card e (S(e)) denotes the number of edges in S(e). More recently, Garet, Marchand, Procaccia and the second author extended in [15] these results to the case where the probability measures considered are defined on [0, +∞] as soon as the percolation of edges with finite passage times are supercritical. To this end, they needed to perform a rescaling argument, since for M large enough the percolation of edges with passage times smaller than M can be choosen supercritical but not highly supercritical as required to use the technology of the contours.
The study of the existence of the time constant without any moment condition and the study of the continuity of the time constant with regard to the distribution of the passage times of the edges are closely related. Indeed, in the given proofs of the continuity of the time constant, the following results are used: • the time constant µ F is the a.s. limit of a subadditive process, • this subadditive process is integrable (for any distribution F of the passage times, even with infinite mean), • this subadditive process is monotonic with regard to the distribution of the passage times.
Moreover, the technology used to prove the key Lemma 1.9 (using the contours) is directly inspired by the study of the existence of the time constant without any moment condition.
The proof of the continuity of the flow constant, Theorem 1.6, we propose in this paper is heavily influenced by the proofs of the continuity of the time constant given in [12,22,15]. The real difficulty of our work is to extend the definition of the flow constant to probability measure with infinite mean -once this is done, it is harmless to admit probability measures F on [0, +∞] such that F ({+∞}) < p c (d), we do not even have to use a renormalization argument. We choose to define the flow constant ν F via (1.4) so that the result equivalent to Lemma 1.9 in our setting is given by the precise definition of ν F . However, two major issues remain: (i) prove that ν F is indeed the limit of some quite natural sequence of maximal flows, (ii) prove that ν F can be recovered as the limit of a nice subadditive process.
The first point, (i), is precisely the object of Theorem 1.4, that we prove in Section 2.
With no surprise, the difficulties we do not meet to prove the result equivalent to Lemma 1.9 for the flow constant are found in the proof of this convergence, see Proposition 2.5. The maximal flows that converge towards ν G are maybe the most natural ones, i.e., maximal flows from the top to the bottom of flat cylinders, and the convergence holds a.s., i.e., in a strong sense, which is quite satisfying. It is worth noticing that in fact, to prove the a.s. convergence in tilted cylinders when ν F = 0 (see Proposition 2.11), we use the continuity of the flow constant -without this property, we obtain only a convergence in probability. However, to obtain a convergence (at least in probability) of these maximal flows towards ν F , we do not have to exhibit a subadditive process converging towards ν F . The existence of such a nice subadditive process, i.e., the point (ii) above, is nevertheless needed to prove the continuity of the flow constant. In Section 3, we define such a process and prove its convergence towards ν F (see Theorem 3.1).
Finally in Section 4 we prove the continuity of the flow constant, Theorem 1.6.
Before starting these proofs, we give in the next section some additional notations.

More notations
We need to introduce a few more notations that will be useful.
Given a unit vector v ∈ S d−1 and a non-degenerate hyperrectangle A normal to v, hyp(A) denotes the hyperplane spanned by A defined by where · denotes the usual scalar product on R d . For a positive real h, we already defined cyl(A, h) as the cylinder of height 2h with base A − h v and top A + h v, see Equation (1.1). It will sometimes be useful to consider the cylinder cyl v (A, h) with height h, base A and top A + h v, i.e., Some sets can be seen as sets of edges or vertices, thus when looking at their cardinality it is convenient to specify whether we count the number of edges or the number of vertices in the set. The notation card e (·) denotes the number of edges in a set whereas card v (·) denotes the number of vertices.
Given a probability measure G on [0, +∞], a constant K ∈]0, +∞[ and a vertex x ∈ Z d (respectively an edge f ∈ E d ), we denote by C G,K (x) (resp. C G,K (f )) the connected component of x (resp. the union of the connected components of the two endpoints of f ) in the percolation (1 t G (e)>K , e ∈ E d ), which can be seen as an edge set and as a vertex set. For any vertex set C ⊂ Z d , we denote by diam(C) the diameter of C, x ∈ C , y / ∈ C and there exists a path from y to infinity in Z d C , and by ∂ v C its exterior vertex boundary defined by Given a set E of edges, we can define also its diameter diam(E) as the diameter of the vertex set made of the endpoints of the edges of E. We also define its exterior ext(E) by Notice that by definition, C ⊂ int(∂ e C) and if C is bounded and x ∈ int(∂ e C), then ∂ e C separates x from infinity. For any vertices x and y, for any probability measure G on [0, +∞] and any K ∈]0, +∞], one of the three following situation occurs: Case (i) corresponds to the case where x and y are connected in the percolation (1 t G (e)>K , e ∈ E d ), whereas cases (ii) and (iii) correspond to the case where x an y are not connected, thus their connected components for this percolation are disjoint.
)) or conversely, whereas case (ii) corresponds to the case where x ∈ ext(∂ e C G,K (y)) and y ∈ ext(∂ e C G,K (x)).
For any subset C of R d and any h ∈ R + , we denote by E G,K (C, h) the following event and by E G,K (C, h) the corresponding event involving edges instead of vertices In what follows c d denotes a constant that depends only on the dimension d and may change from one line to another. Notice that for any finite and connected set C of vertices, card e (∂ e C) ≤ c d card v (C).
For two probability measures H and G on [0, +∞], we define the following stochastic domination relation: (1.9)

Convergence of the maximal flows
This section is devoted to the proof of Theorem 1.4.

Properties of ν G
First we investigate the positivity ν G as defined by (1.4).
Proof. By the above coupling, see Equation (1.9), for any such probability G, for any 0 < K 1 ≤ K 2 , for any v ∈ S d−1 , for any hyperrectangle A and any h ∈ R + , we have We now state a stochastic domination result, in the spirit of Fontes and Newman [13], which will be useful to prove that ν G is finite, and will be used again in section 2.2.
Let also (X i , i ∈ {1, . . . , n}) be a family of i.i.d. random variables distributed as Z 1 = card v (C(x 1 )). Then, for all a, a 1 , . . . , a n in R, Proof. For any i, let F i be the sigma-field generated by the successive exploration of We now state that the constant ν G is finite.
Proof. Let G be a probability measure on [0, +∞] such that G({+∞}) < p c (d). Let v ∈ S d−1 be a unit vector, let A be a non-degenerate hyperrectangle normal to v containing the origin 0 of the graph, and let h : N → R + be mild.
To every x ∈ B 2 (pA, h(p)), the bottom of the cylinder cyl(pA, h(p)), we associate S(x) = ∂ e C G,K0 (x). Some of the sets S(x) may be equal, thus we denote by (S i ) i=1,...,r the collection of disjoint edge sets we obtain (notice that by construction for every i = j, S i ∩ S j = ∅). For every i ∈ {1, . . . , r}, let z i ∈ B 2 (pA, h(p)) be such that S i = S(z i ). We consider the set of edges On the event E G,K0 (cyl(pA, h(p)), h(p)), the set E(p) is a cutset that separates the top B 1 (pA, h(p)) from the bottom B 2 (pA, h(p)) of cyl(pA, h(p)). Indeed, let γ = (x 0 , e 1 , x 1 , . . . , e n , x n ) be a path from the bottom to the top of cyl(pA, h(p)). There exists i ∈ {1, . . . , r} such that For every β > 0, we obtain that We now want to use the stochastic comparison given by Lemma 2.2. Consider the set . We put an order on W and build the variables where κ i are constants depending only on d and G(]K 0 , +∞]), see for instance Theorems (6.1) and (6.75) in [17]. Thus there exists λ(G, d) > 0 such that E[exp(λX 1 )] < ∞, and we get Since lim p→∞ h(p)/ log p = +∞, the first term of the right hand side of (2.2) vanishes when p goes to infinity. We can choose β(G, d) large enough such that the second term of the right hand side of (2.2) vanishes too when p goes to infinity, and we get Since for every K ∈ R + , φ G K (pA, h(p)) ≤ φ G (pA, h(p)) by coupling (see Equation (1.9)), we get for the same β that By Theorem 1.2, we know that for every K ∈ R + , This ends the proof of Proposition 2.3.
Finally we state that ν G satisfies some weak triangular inequality.
As a consequence, the homogeneous extension of ν G to R d , defined by This proposition is a direct consequence of the corresponding property already known for G K for all K, see Proposition 4.5 in [27] (see also Proposition 11.6 and Corollary 11.7 in [4]).

Truncating capacities
We first need a new definition. Given a probability measure G on [0, +∞], a unit vector v ∈ S d−1 , a non-degenerate hyperrectangle A normal to v and a height function h : N → R + , we denote by E G (pA, h(p)) the (random) cutset that separates the top from the bottom of the cylinder cyl(pA, h(p)) with minimal capacity, i.e., φ G (pA, h(p)) = T G (E G (pA, h(p))), with minimal cardinality among them, with a deterministic rule to break ties.
Furthermore, in this section, if E ⊂ E d is a set of edges and C = cyl(A, h) a cylinder, we shall say that E cuts C efficiently if it cuts the top of C from its bottom and no subset of E does. Notice that E G (pA, h(p)) cuts cyl(pA, h(p)) efficiently.
Then, for any ε > 0 and α > 0, there exist constants K 1 and C < 1 such that for every K ≥ K 1 , every unit vector v ∈ S d−1 , every non-degenerate hyperrectangle A normal to v, every mild height function h : N → R + , and for every p ∈ N + large enough, we have Let us say a few words about the proof before starting it. Proposition 2.5 is the equivalent of Lemma 1.9 in the study of the time constant. The proof of Proposition 2.5 is thus inspired by the proof of Lemma 1.9. The spirit of the proof is the following: we consider a cutset E which is minimal for the truncated G K -capacities. Our goal is to construct a new cutset E whose G-capacity is not much larger than the G K -capacity of E. To obtain this cutset E , we remove from E the edges with huge G-capacities, and replace them by some local cutsets whose G-capacity is well behaved. In fact, the construction of these local modifications of E is in a sense more natural when dealing with cutsets rather than geodesics.
Before embarking to the proof of Proposition 2.5, let us state a lemma related to renormalization of cuts. For a fixed L ∈ N * , we define Λ L = [−L/2, L/2] d , and we define the family of L-boxes by setting, for i ∈ Z d , be the set of all L-boxes that E intersects.
Proof. Let us prove that Γ(E) is a lattice animal. Since E cuts cyl(A, h) efficiently, we know that E is somehow connected. More precisely, let us associate with any edge e ∈ E d a small plaquette that is a hypersquare of size length 1, that is normal to e, that cuts e in its middle and whose sides are parallel to the coordinates hyperplanes. We associate with E the set E * of all the plaquettes associated with the edges of E, and we can see E * as a subset of R d . Then E * is connected in R d (see [23] Lemma 3.17 in dimension 3, but the proof can be adapted in any dimension). Thus Γ(E) is Z d -connected. Now, let us prove (2.6). We shall denote by Λ L the enlarged box First of all we prove that for there exists a path γ = (x 0 , e 1 , x 1 , . . . , e n , x n ) in cyl(A, h) from the top to the bottom of cyl(A, h) such that γ does not intersect E {e}. Since E is a cutset, this implies that e ∈ γ. We will prove that locally, inside Λ L (i) Λ L (i), the set E must contain at least L/2 edges. To do so, we shall remove e from γ and construct of order L possible bypaths of e for γ inside Λ L (i) Λ L (i), i.e., L/2 disjoint paths γ such that γ ⊂ Λ L (i) Λ L (i) and the concatenation of the two parts of γ {e} and of γ creates a path in cyl(A, h) from the top to the bottom of cyl(A, h), see Figure 1.
For all k ∈ [L/2, 3L/2] ∩ N, let V k be the set of vertices that lies on the faces of Λ 2k (i), i.e., and let E k be the set of edges between vertices in V k , When looking at Figure 1, one sees that the graph (V k , E k ) forms a kind of shell that surrounds the box Λ L (i). Then any two points x, y ∈ V k are connected by a path in (V k , E k ), and if x, y also belong both to cyl(A, h) and h is at least c d L for some constant c d depending only on the dimension, x and y are also connected by a path in We claim that the set (γ {e}) ∪ (E k ∩ cyl(A, h)) contains a path from the top to the bottom of cyl(A, h). Let us assume this for the moment, and finish the proof of the lemma. Since the set (γ {e}) ∪ (E k ∩ cyl(A, h)) contains a path from the top to the bottom of cyl(A, h), we know that E k must intersect the cutset E. Since the sets E k are disjoint, we conclude that card e (E ∩ Λ L (i)) ≥ card([L/2, 3L/2] ∩ N) ≥ L/2 .  This implies that It remains to prove the claim we have left aside, i.e., that the set (γ {e})∪(E k ∩cyl(A, h)) contains a path from the top to the bottom of cyl(A, h). Suppose first that x 0 and x n do not belong to Λ 2k (i). Then let l 1 = min{l : x l ∈ Λ 2k (i)} and l 2 = max{l : x l ∈ Λ 2k (i)} , see Figure 1. There exists a path γ from x l1 to x l2 in (V k ∩ cyl(A, h), E k ∩ cyl(A, h)).
We can now concatenate the paths (x 0 , e 1 , . . . , x l1 ), γ and (x l2 , . . . , x n ) to obtain a path from the top to the bottom of cyl(A, h). Suppose now that x 0 ∈ Λ 2k (i). Thus, if l(A, h) is at least c d L for some constant c d depending only on the dimension, x n / ∈ Λ 2k (i) and there exists a vertex y ∈ V k ∩ B 1 (A, h) (B 1 (A, h) is the top of the cylinder). We define as previously l 2 = max{l : x l ∈ Λ 2k (i)} .
There exists a path γ from y to x l2 in (V k ∩ cyl(A, h), E k ∩ cyl(A, h)), and we can concatenate γ with (x l2 , . . . , x n ) to obtain a path from the top to the bottom of cyl(A, h). We can perform the symmetric construction if x n ∈ Λ 2k (i). Thus for every k ∈ [L/2, 3L/2] ∩ N the set (γ {e}) ∪ (E k ∩ cyl(A, h)) contains a path from the top to the bottom of cyl(A, h).
Proof of Proposition 2.5. Let G be a probability measure on [0, +∞] such that G({+∞}) < p c (d). We use the natural coupling t G K (e) = min(t G (e), K) for all e ∈ E d . Let K 0 be such that G(]K 0 , +∞]) < p c (d). We shall modify E G K (pA, h(p)) around the edges having too large G-capacities in order to obtain a cut whose capacity is close enough to φ G K (pA, h(p)) (for K large enough). We recall that C G,K0 (f ) is the connected component of the edge f in the percolation (1 t G (e)>K0 , e ∈ E d ). For short, we write S(e) = ∂ e C G,K0 (e), the edge-boundary of C G,K0 (e) separating e from infinity, see Figure 2.
Define also We collect all the sets (S(e), e ∈ F (p)). As in the proof of Proposition 2.3, from this collection we keep only one copy of each distinct edge set. We obtain a collection (S i ) i=1,...,r of disjoint edge sets. For every i ∈ {1, . . . , r}, let f i ∈ F (p) such that S i = S(f i ).

Let us define
We consider the event First, we claim that on the event E G,K0 (cyl(pA, h(p)), h(p)), the set E (p) cuts the top from the bottom of cyl(pA, h(p)). Indeed, suppose that γ is a path in cyl(pA, h(p)) joining its bottom to its top. Since E G K (pA, h(p)) is a cutset, there is an edge e in E G K (pA, h(p)) ∩ γ. If e does not belong to F (p), then e belongs to E (p) and thus γ intersects E (p). If e belongs to F (p), denote by x (resp. y) a vertex belonging to γ and the top of cyl(pA, h(p)) (resp. to γ and the bottom of cyl(pA, h(p))), and let i ∈ {1, . . . , r} such that e ∈ int(S i ) = int(S(f i )). On the event E G,K0 (cyl(pA, h(p)), h(p)), x and y cannot belong both to int S(f i ), otherwise diam C G,K0 (f i ) would be at least 2h(p) − 2 ≥ h(p) (at least for p large enough). Thus, γ contains at least one vertex in ext S(f i ) and one vertex (any endpoint of e) in int(S(f i )). Thus, at least one edge e of γ must be in S(f i ), and since γ is included in cyl(pA, h(p)), e must be in cyl(nA, h(p)) ∩ S(f i ). Thus e ∈ E (p) and this proves that E (p) cuts the top from the bottom of cyl(pA, h(p)). Now, on the event E G,K0 (cyl(pA, h(p)), h(p)) we get Moreover, still on the event E G,K0 (cyl(pA, h(p)), h(p)), we notice that if we replace a single edge e of F (p) by (S(e) ∩ cyl(pA, h(p))) in E G K (pA, h(p)) we obtain a new set of edges that is still a cutset between the top and the bottom of cyl(pA, h(p)) (this could be proved by a similar but simpler argument than the one presented to prove that E (p) is a cutset). By minimality of the capacity of E G K (pA, h(p)) among such cutsets, we deduce that ∀e ∈ F (p) , K 0 card e (S(e)) ≥ K . (2.9) We recall that card e (S(e)) ≤ c Let us denote by B the event whose probability we want to bound from above: and for positive β, γ and for x 1 , . . . , x k in Z d , let If E ⊂ E d and x ∈ Z d , we say that x ∈ E if and only if x is the endpoint of an edge e that belongs to E. We obtain this way As in the proof of the continuity of the time constant given by Cox and Kesten in [12], we need a renormalization argument to localize these vertices x 1 , . . . , x k in a region of the space whose size can be controlled. For a given i ∈ Z d and k ∈ N * , we denote by A(i, k) the set of all lattice animals of size k ∈ N * containing i. If k ∈ R + , then we write A(i, k) instead of A(i, k ) for short, where k ∈ N and satisfies k ≤ k < k + 1. Let E ⊂ E d such that E cuts cyl(pA, h(p)) efficiently. Let us denote by u ∈ R d one of the corners of A. We can find a path from the top to the bottom of cyl(pA, h(p)) that is located near any of the vertical sides of cyl(pA, h(p)), more precisely there exists a constant c d depending only on d such that the top and the bottom of cyl(pA, h(p)) are connected in V (u, h(p)) := {u + l v + w : l ∈ [−h(p), h(p)] , w 2 ≤ c d } ∩ cyl(pA, h(p)). Thus any custset E must contain at least one edge in V (u, h(p)). We denote by I(pA, h(p)) the set of L-boxes that intersect V (u, h(p)): Then Γ(E) must intersect I(pA, h(p)), and card v (I(pA, h(p))) ≤ c d h(p)/L. Furthermore, Lemma 2.6 ensures that for p large enough, From all these remarks, we conclude that if E cuts cyl(pA, h(p)) efficiently and if card e (E) ≤ α p d−1 , then Thus if card e (E) ≤ α p d−1 , we obtain that there exists i ∈ I(pA, h(p)) and Γ ∈ A(i, c d αp d−1 /L) such that Γ(E) ⊂ Γ. For any lattice animal Γ, we denote by Γ L the uion of the boxes associated to this vertex set, i.e., We now use the stochastic comparison given by Lemma 2.2. We consider the set of vertices W = {x 1 , . . . , x k }, the percolation (1 t G (e)>K0 , e ∈ E d ) and associate to each vertex x i the variable Z i = Z(x i ) = card v (C G,K0 (x i )). We build the variables (Y i , 1 ≤ i ≤ k) as in Lemma 2.2 and let (X i , 1 ≤ i ≤ k) be i.i.d. random variables with distribution card v (C G,K0 (e)). Then by Lemma 2.2 we obtain for λ > 0, where we used the Cauchy-Schwartz inequality. Thus, we get And finally we choose K 1 = K 1 (G, d, α, ε) such that: Combining (2.10) and (2.11) we get for some C(G, d, α, ε) < 1, for every K ≥ K 1 (G, d, α, ε) and for every p large enough, since lim p→∞ h(p)/p = 0. For every edge e = x, y , since for some positive constants κ i (see (2.1)) and for p large enough since lim p→∞ h(p)/ log p = +∞. Since lim p→∞ h(p)/p = 0, this ends the proof of Proposition 2.5.
Remark 2.7. The result of Proposition 2.5 could apply, with the same constants depending on G, to any probability measure H on [0, +∞] such that H G (we recall that the stochastic comparison H G is defined in (1.8)).

Proof of the convergence I: case G({0}) < 1 − p c (d)
To prove Theorem 1.4, we shall consider the two situations G({0}) < 1 − p c (d) and G({0}) ≥ 1 − p c (d). The purpose of this section is to prove the following proposition, that corresponds to the statement of Theorem 1.4 in the case where G({0}) < 1 − p c (d).
Proof. Let v ∈ S d−1 , let A be a non-degenerate hyperrectangle normal to v, let h : N * → R + be mild. Let G be a probability measure on [0, +∞] such that G({+∞}) < p c (d).
Since d, G, v, A, h are fixed, we will omit in the notations a lot of dependences in these parameters.
In this section, we suppose that G({0}) < 1 − p c (d). For any fixed K ∈ R + , we know by Theorem 1.2 that a.s.
It remains to prove that a.s., We claim that it is sufficient to prove that and Proposition 2.8 is proved. Now for every ε > 0, for every α > 0 and every β > 0, we ) by coupling (see Equation (1.9)) and lim p→∞ h(p)/ log p = +∞, we know that we can choose β = β(G, d) such that for any K ∈ R + , the last term of the right hand side of (2.13) is summable in p.
Given this β(G, d), by Zhang's Theorem 2 in [30], as adapted in Proposition 4.2 in [27], we know that since all the probability measures G K coincide on a neighborhood of 0, we can choose a constant α(G, d) such that for any K ∈ R + the second term of the right hand side of (2.13) is summable in p.

Proof of the convergence II: case
It remains to prove that the convergence in Theorem 1.4 holds when G({0}) ≥ 1−p c (d), i.e., when ν G = 0. We first deal with straight cylinders. . . . , 0, 1). We recall the definition of the event E G,K (C, h), for any subset C of R d and any h ∈ R + , that was given in (1.6): This result is in fact a generalization of Zhang's Theorem 1.3, and the strategy of the proof is indeed largely inspired by Zhang's proof. However, we need to work a little bit harder, because we do not have good integrability assumptions. We thus re-use here some ideas that appeared in the proof of Proposition 2.3. Notice that φ G (pA, h(p)) itself may not be integrable in general (it can even be infinite with positive probability).
Proof. We shall construct a particular cutset with an idea quite similar to the one we used in the proof of Proposition 2.3. Let K 0 be large enough to have G(]K 0 , +∞]) < p c (d).
For any x ∈ Z d−1 × {0}, we define the event x, occurs, we associate with x the set ∂ e C G,0 (x), that is by definition made of edges with null capacity. If F x, occurs, we associate with x the set ∂ e C G,K0 (x), see We consider the good event We claim that on E G,K0 (cyl v0 (A, ), ), the set E (A, ) cuts the top B v0 1 (A, ) from the bottom B v0 2 (A, ) in the cylinder cyl v0 (A, ). Let γ = (x 0 , e 1 , x 1 , . . . , e n , x n ) be a path from the bottom to the top of the cylinder cyl v0 (A, ). If F c x0, occurs, since x n ∈ Z d−1 × { }, then x n ∈ ext(∂ e C G,0 (x 0 )), thus γ has to use an edge in ∂ e C G,0 (x) ∩ cyl v0 (A, ). If F x0, occurs, on E G,K0 (cyl v0 (A, ), ) we know that x n ∈ ext(∂ e C G,K0 (x 0 )), thus γ must contain an edge in ∂ e C G,K0 (x 0 ) ∩ cyl v0 (A, ). We conclude that on E G,K0 (cyl v0 (A, ), ), E (A, ) is indeed a cutset from the top to the bottom of cyl v0 (A, ). Thus on E G,K0 (cyl v0 (A, ), ), For every , the process (X D , D ∈ J ) is a discrete additive process. By classical multiparameter ergodic Theorems (see for instance Theorem 2.4 in [1] and Theorem 1.1 in [28]), if E[R 0 ] < ∞, then there exists an integrable random variable X such that for Using the fact that k → P 0 It is known that at criticality, there is no infinite cluster in the percolation in half space, see [17], Theorem (7.35).
Thus for all η > 0 we can choose η large enough so that for every ≥ η , E[R x ] < η. For every height function h : N → R + such that lim n→∞ h(n) = +∞, let p 0 be large enough such that for all p ≥ p 0 , h(p) ≥ η . The function → R x is non-increasing, thus for every We turn back to the study of φ v0 G (pA, h(p)). We recall that we supposed lim p→∞ h(p)/ log p = +∞. As in the proof of Proposition 2.3 (see (2.1)), since G(]K 0 , +∞]) < p c (d), we thus by Borel-Cantelli we know that a.s., for all p large enough, E G,K0 (cyl v0 (pA, h(p)), h(p)) occurs. (2.19) Proposition 2.9 is proved by combining (2.15), (2.18) and (2.19).
We now extend Proposition 2.9 to the study of any tilted cylinder. We will bound the maximal flow through a tilted cylinder by maximal flows through straight boxes at an intermediate level. Unfortunately, at this stage, we could not prove that the convergence holds almost surely. However, we prove that the convergence holds in a weaker sense, namely in probability. We will upgrade this convergence in Proposition 2.11.
. By the max-flow min-cut Theorem, i.e., roughly speaking, φ G (L, i) is the minimal capacity of a cutset in the annulus Λ L (i) Λ L (i). For every i ∈ Z d , let E G (L, i) be a minimal cutset for φ G (L, i). We choose L = L(p) such that h(p) is large in comparison with L, in the sense that no 3L-box can intersect both the top and the bottom of cyl(pA, h(p)). Thus we can choose L(p) = 2 h(p)/c d for some constant c d depending only on the dimension. Let J(pA, L) be the indices of all the L-boxes that are intersected by the hyperrectangle pA (see Figure 4): Let us prove that inequality (2.20) holds: by proving that ∪ i∈J E G (L, i) is a cutset for φ G (pA, h(p)). Let γ = (x 0 , e 1 , x 1 , . . . , e n , x n ) be a path from the top to the bottom of φ G (pA, h(p)). Since pA ⊂ ∪ i∈J Λ L (i) and γ (seen as a continuous curve) must intersect pA, then there exists i ∈ J such that γ ∩ Λ L (i) = ∅. Since h(p) is large in comparison with L, γ cannot be included in Λ L (i), thus γ contains a path from ∂Λ L (i) ∩ Z d to ∂Λ L (i) ∩ Z d in Λ L (i) Λ L (i), thus by definition it must intersect E G (L, i) (see Figure 4). This proves that ∪ i∈J E G (L, i) is a cutset for φ G (pA, h(p)), thus  : The dual sets of the cutsets:  We claim that for every L ∈ N * , for We now prove this claim. Let γ = (x 0 , e 1 , x 1 , . . . , e n , x n ) be a path from ∂( Then there exists k ∈ {1, . . . , d}, l ∈ {+, −} such that Then by continuity of γ we know that x j ∈ Λ L (i, k, l) but x j has a neighbor outside Λ L (i, k, l). By definition of j, since j < j, x j can be only on one side of the boundary of Thus the subset of γ between x j and x j is a path from the bottom to the top of Λ L (i, k, l), thus it must intersect E G (L, i, k, l). This proves that ∪ l=+, Combining (2.20) and (2.21), we obtain that for every L, for every p large enough, φ G (L, i, k, l) .
On the other hand, let A 1 by a hyperrectangle a bit larger than A, namely We recall that the event E G,K0 (cyl(pA 1 , h(p)), L) is defined by Notice that for all i ∈ J, we have Λ L (i) ⊂ cyl(pA 1 , h(p)) at least for p large enough since we choose L = L(p) = 2 h(p)/c d for some constant c d depending only on the dimension.
and it remains to notice that L = L(p) goes to infinity and card(J)L(p) d−1 /p d−1 remains bounded when p goes to infinity to conclude by (2.23) that For every η > 0, we obtain as in (2.1) that that goes to zero when p goes to infinity since h(p) ≤ p, L(p) = 2 h(p)/c d and lim p→∞ h(p)/ log(p) = +∞.

Proof of the convergence III: end of the proof of Theorem 1.4
At this stage, what remains to prove to finish the proof of Theorem 1.4 is to strengthen the mode of convergence in Proposition 2.10. This can be done easily using the continuity of G → ν G , i.e., Theorem 1.6. Proposition 2.11. We suppose that G → ν G ( v) is continuous, i.e., if G and (G n ) n∈N are probability measures on [0, +∞] such that G({+∞}) < p c (d) and for all n ∈ N, For any ε > 0, we denote by ε G the distribution of the variables t G (e) + ε. Obviously ε G({0}) = 0 thus Proposition 2.8 states that Moreover, by coupling (see Equation (1.9)), φ G (pA, h(p)) ≤ φε G (pA, h(p)), thus To conclude the proof, we will use the continuity of G → ν G : since ε G d −→ G when ε goes to 0 we obtain that lim Thenh is mild, thus we just proved that Moreover, sinceh(p) ≤ h(p) for all p ∈ N * , any cutset from the top to the bottom of cyl(pA,h(p)) is also a cutset from the top to the bottom of cyl(pA, h(p)), thus by the max-flow min-cut Theorem we obtain that φ G (pA,h(p)) ≥ φ G (pA, h(p)) for all p ∈ N * .
This allows us to conclude that lim sup This ends the proof of Proposition 2.11.

Remark 2.12.
It is worth noticing that this proof does not use Propositions 2.9 or Proposition 2.10 directly. However, we need these intermediate results to prove the continuity of G → ν G that we use here.

Subadditivity
As mentioned in section 1.4, expressing the flow constant as the limit of a subadditive and integrable object is crucial to prove its continuity. This is the purpose of the present section. The first idea is to take the capacity of a cut which in a sense separates a hyperrectangle A from infinity in a half-space. This will ensure subadditivity. However, in order to have a chance to compare it to the flows that we used so far, one needs the cut to stay at a small enough distance from A so that it will be flat in the limit. In addition, to ensure good integrability properties, one needs this distance to be large enough so that one may find enough edges with bounded capacity to form a cutset. These constraints lead to searching for a cutset in a slab of random height, which height is defined in (3.1).
Let v ∈ S d−1 , and let A be any non-degenerate hyperrectangle normal to v. For any h, we denote by slab(A, h, v) the cylinder whose base is the hyperplane spanned by A and of height h (possibly infinite), i.e., the subset of R d defined by Let V (A) be the following set of vertices in Z d , which is a discretized version of A: ∃y ∈ Z d ∩ slab(A, ∞, v) , x, y ∈ E d and x, y intersects A .
Let W (A, h, v) be the following set of vertices in Z d , which is a discretized version of hyp(A + h v): We say that a path γ = (x 0 , e 1 , x 1 , . . . , e n , x n ) goes from A to hyp( and for all i ∈ {1, . . . , n}, x i ∈ slab(A, h, v) (see Figure 6). We  Finally, we say that a direction v ∈ S d−1 is rational if there exists M ∈ R + such that M v has rational coordinates. Now, we will prove that these flows φ G,F,K0 (A) properly rescaled converge for large hyperplanes towards ν G ( v) (as defined in (1.4)), and thus obtain an alternative definition of ν G ( v). This will be done in two steps. First we prove the convergence of φ G,F,K0 (pA)/H d−1 (pA) towards some limit ν G,F,K0 ( v) by some subadditive argument in Theorem 3.1. Then, we compare φ G,F,K0 (pA) with φ G (pA, h(p)) to prove that ν G,F,K0 ( v) = ν G ( v), and this is done in Proposition 3.3. For any probability measure F on [0, +∞] such that F ({+∞}) < p c (d) and G F , for any K 0 ∈ R such that F (]K 0 , +∞]) < p c (d), for any rational v ∈ S d−1 , there exists a non-degenerate hyperrectangle A (depending on v but neither on G, F nor on K 0 ) which is normal to v and contains the origin of the graph Z d such that We consider a fixed rational v ∈ S d−1 and H, the hyperplane normal to v containing 0. Since v is rational, there exists a orthogonal basis of H of vectors with integer coordinates, let us call it ( f 1 , . . . , f d−1 ). Then, we take A to be the hyperrectangle built on the origin and this basis: also an hyperrectangle, we claim that Indeed, first notice that if B 1 , B 2 are hyperrectangles normal to v such that B 1 ⊂ B 2 , then by definition any set of edges E that cuts B 2 from hyp( , we know that m := inf{p ∈ {1, . . . , n} : , v) for some j ∈ {1, . . . , k}, and we conclude also that γ contains an edge of E j . Inequality (3.3) follows by optimizing on T G (E i ) for all i.
We now prove that φ G,F,K0 (A) has good integrability properties. For any x ∈ V (A) we consider the connected component of x in slab(A, ∞, v) for the percolation (1 t F (e)>K0 ), i.e., y is connected to x by a path γ = (x 0 , e 0 , . . . , x n ) s.t. ∀i ∈ {1, . . . , n}, By definition of H F,K0 (A), we know that any path γ in slab(A, H F,K0 (A), v) from A to hyp(A+H F,K0 (A) v) must contain at least one edge e such that t F (e) ≤ K 0 . Thus γ cannot be included in ∪ x∈V (A) C v F,K0 (x), and this implies that γ must contain at least one edge e that belongs to ∂ e (∪ x∈V (A) C v F,K0 (x)). This edge e satisfies (by the coupling relation (1.9)) t G (e) ≤ t F (e) ≤ K 0 . Thus comparing clusters in the slab with clusters in the full space, we obtain We can thus apply a multi-parameter ergodic theorem (see for instance Theorem 2.4 in [1] and Theorem 1.1 in [28]) to deduce that there exists a constant ν G,F,K0 ( v) (that depends on v but not on A itself) such that a.s. and in L 1 .
We now state that the limit ν G,F,K0 ( v) appearing in Theorem 3.1 is in fact equal to ν G ( v). We want to clarify the fact that in the proof of Proposition 3.3 below, we will use the convergence in probability of rescaled flows in tilted cylinders towards the flow-constant, stated above in Propositions 2.8 and 2.10.
Proof. We first prove that ν G ( v) ≤ ν G,F,K0 ( v). We associate with a fixed rational v ∈ S d−1 the same hyperrectangle A as in the proof of Theorem 3.1. We consider the function h(p) = H d−1 (pA) Moreover, let γ = (x 0 , e 0 , . . . , e n , x n ) be a path from the bottom to the top of cyl(pA, h(p)) inside cyl(pA, h(p)). Let k = max{j ≥ 0 : x j / ∈ slab(pA, ∞, v)}. Then x j ∈ V (pA) and the truncated path γ = (x k , e k+1 , . . . , x n ) is a path from pA to hyp(pA + h(p) v) in slab(pA, h(p), v). On the event {H F,K0 (pA) = h(p)}, we conclude that any set of edges E that cuts pA from hyp(pA + H F,K0 (pA) v) in slab(pA, H F,K0 (pA), v) also cuts any path from the bottom to the top of cyl(pA, h(p)), thus on the event {H F,K0 (pA) = h(p)} we have φ G (pA, h(p)) ≤ φ G,F,K0 (pA) .
We now prove that ν G ( v) ≥ ν G,F,K0 ( v) for a fixed rational v ∈ S d−1 . We associate with a fixed rational v ∈ S d−1 the same hyperrectangle A as in the proof of Theorem 3.1, it is an orthonormal basis of the orthogonal complement of v made of rational vectors. We want to construct a set of edges that cuts pA from hyp(pA + H F,K0 (pA) v) in slab(pA, H F,K0 (pA), v) by gluing together cutsets from the top to the bottom of different cylinders. For any fixed η > 0, we slightly enlarge the hyperrectangle A by considering Let h(p) = H d−1 (pA) 1 2(d−1) as previously. We consider the cylinder cyl v (pA η , h(p)), and a minimal cutset E 0 (p, A, η) between the top B v 1 (pA η , h(p)) and the bottom B v 2 (pA η , h(p)) of this cylinder. To obtain a set of edges that cuts pA from hyp(pA + H F,K0 (pA) v) in slab(pA, H F,K0 (pA), v), we need to add to E 0 (p, A, η) some edges that prevent some flow to escape from cyl v (pA η , h(p)) by its vertical sides, see Figure 7. For i ∈ {1, . . . , d}, let D + i (p, A, η) and D − i (p, A, η) be the two d − 1 dimensional sides of ∂(cyl(pA η , h(p))) that are normal to w i , and such that D + i (p, A, η) is the translated of D − i (p, A, η) by a translation of vector f (i, p, A, η) w i for some f (i, p, A, η) > 0. We consider the cylinder cyl − wi (D + i (p, A, η), p 1/4 ) (resp. cyl + wi (D − i (p, A, η), p 1/4 )) and a minimal cutset E + i (p, A, η) (resp. E − i (p, A, η)) from the top to the bottom of cyl − wi (D + i (p, A, η), p 1/4 ) (resp. from the top to the bottom of cyl + wi (D − i (p, A, η), p 1/4 )) in the direction w i . We emphasize the fact that the lengths of the sides of cyl − wi (D + i (p, A, η), p 1/4 ) and cyl + wi (D − i (p, A, η), p 1/4 ) do no grow to infinity at the same rate in p. We shall prove the three following properties: (i) For every η > 0, at least for p large enough, the set of edges F (p, A, η) defined by Figure 7: The construction of a cutset that separates pA from hyp(pA + H F,K0 (pA) v) in slab(pA, H F,K0 (pA), v) (here d = 2 and the cutsets E 0 (p, A, η), E + 1 (p, A, η) and E + 1 (p, A, η) are represented via their dual as surfaces).
speed h(p) (of order p 1 2(d−1) ) in one direction. We did not take into account this kind of anisotropic growth in our study (contrary to Kesten in [22] and Zhang in [29,30]). We can conjecture that φ −l wi (D l i (p, A, η), p 1/4 ) grows linearly with p d−2 h(p), with a multiplicative constant given precisely by ν G ( w i ), but this cannot be deduced easily from what has already been proved. However we do not need such a precise result. We recall that the definition of the event E G,K0 (·, ·) was given in (1.6). Mimicking the proof of inequality (2.2) in the proof of Proposition 2.3, we obtain that for a constant K(A, d, η), for variables (X i ) that are i.i.d. with the same distribution as card e (C G,K0 (0)), we have ≤ P E G,K0 cyl −l wi (D l i (p, A, η), p 1/4 ), , the first term of the right hand side of (3.8) vanishes when p goes to infinity. We can choose β(G, d, A) large enough such that the second term of the right hand side of (3.8) vanishes too when p goes to infinity. This is enough to conclude that property (iii) holds.

Continuity of G → ν G
This section is devoted to the proof of Theorem 1.6. To prove this theorem we mimick the proof of the corresponding property for the time constant, see [10], [12], [22] and [15]. We stress the fact that the proof relies heavily on these facts: (i) ν G ( v) can be seen without any moment condition as the limit of a subadditive process that has good properties of monotonicity, We stated Theorem 3.1 to get (i). Property (ii) is a direct consequence of the definition (1.4) of ν G itself, but we had consequently to work to prove that the constant ν G defined this way is indeed the limit of some rescaled flows. As a consequence, the following proof of Theorem 1.6 is quite classical and easy, since we already have in hand all the appropriate results to perform it efficiently.

Preliminary lemmas
Let G (resp. G n , n ∈ N) be a probability measure on [0, +∞] such that G({+∞}) < p c (d) (resp. G n ({+∞}) < p c (d)). We define the function G : t ∈ [0, +∞[ → G([t, +∞]) (respectively G n (t) = G n ([t, +∞])) that characterizes G (resp. G n ). Notice that the stochastic domination between probability measures can be easily characterized with these functions: We recall that we always build the capacities of the edges for different laws by coupling, using a family of i.i.d. random variables with uniform law on ]0, 1[ and the pseudo-inverse of the distribution function of these laws. Thanks to this coupling, we get this classical result of convergence (see for instance Lemma 2.10 in [15]).
In what follows, it will be useful to be able to exhibit a probability measure that dominates stochastically (or is stochastically dominated by) any probability G n of a convergent sequence of probability measures, thus we recall this known result (see for instance Lemma 5.3 in [15]).

Upper bound
This is the easy part of the proof. It relies on the expression of ν G ( v) as the infimum of a sequence of expectations. Proof. Let v ∈ S d−1 be a rational vector, and let A be the non-degenerate hyperrectangle normal to v given by Theorem 3.1. By Lemma 4.3 we know that there exists a probability measure F + on [0, +∞] such that F + ({+∞}) < p c (d), G n F + for all n ∈ N and G F + . Let K 0 be large enough such that F + (]K 0 , +∞]) < p c (d). Let k ∈ N * . We recall that the definition of H F + ,K0 (kA) is given in (3.1). Let E k be a set of edges that cuts kA from hyp(kA + H F + ,K0 (kA) v) in slab(kA, H F + ,K0 (kA), v). By coupling (see Equation (1.9)) we know that φ G,F + ,K0 (kA) ≤ φ Gn,F + ,K0 (kA), and by Lemma 4.1 we have a.s. Moreover, for all k ∈ N * , we have also by coupling (see Equation (1.9)) that φ Gn,F + ,K0 (kA) ≤ φ F + ,F + ,K0 (kA) which is integrable. The dominated convergence theorem implies that ∀k ∈ N * , lim n→∞ E φ Gn,F + ,K0 (kA) = E φ G,F + ,K0 (kA) .   For any k, let us denote by E n (k) a minimal cutset for τ G n (kA, k) -if there is more than one such cutset we choose (with a deterministic rule) one of those cutsets with minimal cardinality. Then, there exists positive constants C, D 1 , D 2 and an integer n 0 such that ∀n ≥ n 0 , ∀β > 0, ∀k ∈ N, P card e (E n (k)) ≥ βk d−1 and τ Gn (kA, k) ≤ βCk d−1 ≤ D 1 e −D2k d−1 .
Proof of Proposition 4.5. Let v ∈ S d−1 be a rational direction. Let G, (G n ) n∈N be probability measures on [0, R] for some R ∈ [0, +∞[. We define G n = min(G, G n ) (resp. G n = max(G, G n )), and we denote by G n (resp. G n ) the corresponding probability measure on [0, R]. Then G n G G n and G n G n G n for all n ∈ N, G n d → G and G n d → G. To conclude that lim n→∞ ν Gn ( v) = ν G ( v), it is thus sufficient to prove that Inequality (i) is a straightforward consequence of Proposition 4.4. If ν G ( v) = 0, then inequality (ii) is trivial and we can conclude the proof. From now on we suppose that ν G ( v) > 0. By [29] (see Theorem 1.3 above), we know that ν G ( v) > 0 ⇐⇒ G({0}) < 1 − p c (d). Thanks to the coupling (see equation (1.9)), we know that for every edge e ∈ E d , we have t G n (e) ≤ t G (e).
Let A be a non-degenerate hyperrectangle normal to v that contains the origin of the graph and such that H d−1 (A) = 1. We recall that τ G (kA, k) is defined in Equation (1.3). It denotes the maximal flow for the capacities (t G (e)) from the upper half part C 1 (kA, k) to the lower half part C 2 (kA, k) of the boundary of cyl(kA, k) as defined in Equation (1.2), and it is equal to the minimal G-capacity of a set of edges that cuts the upper half part from the lower half part of the boundary of cyl(kA, k) in this cylinder. Since we work with integrable probability measures G and G n , we know by Theorem 1.2 that a.s.  Now, let us denote by E n (k) a minimal cutset for τ G n (kA, k) as in Lemma 4.6. According to Kesten's Lemma 3.17 in [22], any such minimal cutset with minimal cardinality is associated with a set of plaquettes which is a connected subset of R d -we will say that E n (k) is •-connected. Let x ∈ ∂A. There exists a constantĉ d , depending only on the dimension, such that for every k ∈ N * there exists a path from the upper half part to the lower half part of the boundary of cyl(kA, k) that lies in the Euclidean ball of center kx and radiusĉ d . We denote by F (k) the set of the edges of E d whose both endpoints belong to this ball. Then E n (k) must contain at least one edge of F (k), and card e (F (k)) ≤ĉ d for some constantĉ d . Moreover, given a fixed edge e 0 , the number of •-connected sets of m edges containing e 0 is bounded byc m d for some finite constantc d (see the proof of Lemma 2.1 in [22], that uses (5.22) in [21]). Thus for every k ∈ N * , for every β, C, ε > 0  We can easily bound τ G (kA, k) by e∈E k t G (e) ≤ R card e (E k ), where E k is a deterministic cutset of cardinality smaller than c d k d−1 -choose for instance E k as the set of all edges in cyl(kA, k) that are at Euclidean distance smaller than 2 from kA. For any fixed C > 0, since for every edge e we have t G (e) ≤ R there exists a constant β such that ∀k ∈ N * , P[τ G (kA, k) > βCk d−1 ] = 0 . By Markov's inequality, for any α > 0 we have For any fixed ε > 0 and β < ∞, we can choose α = α(ε) large enough so that c d exp(−αε/β) ≤ 1/4. Then, by Lemma 4.1 we know that lim n→∞ t G n (e) = t G (e) a.s., and since t G (e) ≤ R we can use the dominated convergence theorem to state that for n large enough, E exp 2α(t G (e) − t G n (e)) ≤ 2 .