Olivier Garet

Régine Marchand
email: regine.marchand@univ-lorraine.fr

Eviatar B Procaccia
email: procaccia@math.tamu.edu

Marie Théret
email: marie.theret@univ-paris-diderot.fr

CONTINUITY OF THE TIME AND ISOPERIMETRIC CONSTANTS IN SUPERCRITICAL PERCOLATION

Keywords: Subject Classification. 60K35, 82B43 -passage percolation, time constant, isoperimetric constant

We consider two different objects on supercritical Bernoulli percolation on the edges of Z d : the time constant for i.i.d. first-passage percolation (for d ≥ 2) and the isoperimetric constant (for d = 2). We prove that both objects are continuous with respect to the law of the environment. More precisely we prove that the isoperimetric constant of supercritical percolation in Z 2 is continuous in the percolation parameter. As a corollary we obtain that normalized sets achieving the isoperimetric constant are continuous with respect to the Hausdorff metric. Concerning first-passage percolation, equivalently we consider the model of i.i.d. first-passage percolation on Z d with possibly infinite passage times: we associate with each edge e of the graph a passage time t(e) taking values in [0, +∞], such that P[t(e) < +∞] > pc(d).

 for first-passage percolation with finite passage times.

2010 Mathematics

Introduction

We consider supercritical bond percolation on Z d , with parameter p > p c (d), the critical parameter for this percolation. Almost surely, there exists a unique infinite cluster C ∞ -see for instance Grimmett's book [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]. We study the continuity properties of two distinct objects defined on this infinite cluster: the isoperimetric (or Cheeger) constant, and the asymptotic shape (or time constant) for an independent first-passage percolation. In this section, we introduce briefly the studied objects and state the corresponding results: more precise definitions will be given in the next section.

1.1. Isoperimetric constant of the infinite cluster in dimension 2. For a finite graph ג = (V ,)ג( E(,))ג the isoperimetric constant is defined as

ϕ ג = min |∂ ג A| |A| : A ⊂ V ,)ג( 0 < |A| ≤ |V |)ג( 2 ,
where ∂ ג A is the edge boundary of A in ,ג ∂ ג A = {e = (x, y) ∈ E()ג : x ∈ A, y / ∈ A}, and |B| denotes the cardinal of the finite set B.

We consider the isoperimetric constant ϕ n (p) of C ∞ ∩ [-n, n] d , the intersection of the infinite component of supercritical percolation of parameter p with the box [-n, n] d :

ϕ n (p) = min |∂ C∞∩[-n,n] d A| |A| : A ⊂ C ∞ ∩ [-n, n] d , 0 < |A| ≤ |C ∞ ∩ [-n, n] d | 2 ,
In several papers (e.g. [START_REF] Benjamini | On the mixing time of a simple random walk on the super critical percolation cluster[END_REF], [START_REF] Mathieu | Isoperimetry and heat kernel decay on percolation clusters[END_REF], [START_REF] Pete | A note on percolation on Z d : isoperimetric profile via exponential cluster repulsion[END_REF], [START_REF] Berger | Anomalous heat-kernel decay for random walk among bounded random conductances[END_REF]), it was shown that there exist constants c, C > 0 such that c < nϕ n (p) < C, with probability tending rapidly to 1. This led Benjamini to conjecture the existence of lim n→+∞ nϕ n (p). In [START_REF] Procaccia | Concentration estimates for the isoperimetric constant of the supercritical percolation cluster[END_REF], Rosenthal and Procaccia proved that the variance of nϕ n (p) is smaller than Cn 2-d , which implies nϕ n (p) is concentrated around its mean for d ≥ 3. In [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF], Biskup, Louidor, Procaccia and Rosenthal proved the existence of lim n→+∞ nϕ n (p) for d = 2. This constant is called the Cheeger constant. In addition, a shape theorem was obtained: any set yielding the isoperimetric constant converges in the Hausdorff metric to the normalized Wulff shape W p , with respect to a specific norm given in an implicit form, see Proposition 2.4 below. For additional background and a wider introduction on Wulff construction in this context, the reader is referred to [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF]. Our first result is the continuity of the Cheeger constant and of the Wulff shape in dimension d = 2: 1.2. First-passage percolation on the infinite cluster in dimension d ≥ 2. Consider a fixed dimension d ≥ 2. First-passage percolation on Z d was introduced by Hammersley and Welsh [START_REF] Hammersley | First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory[END_REF] as a model for the spread of a fluid in a porous medium. To each edge of the Z d lattice is attached a nonnegative random variable t(e) which corresponds to the travel time needed by the fluid to cross the edge. When the passage times are independent identically distributed variables with common distribution G, with suitable moment conditions, the time needed to travel from 0 to nx behaves like nµ G (x) for large n, where µ G is a semi-norm associated to G called the time constant; Cox and Durrett [START_REF] Cox | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF] proved this result under necessary and sufficient integrability conditions on the distribution G of the passage times. Kesten in [START_REF] Kesten | Aspects of first passage percolation[END_REF] proved that the semi-norm µ G is a norm if and only if G({0}) < p c (d). In casual terms, the asymptotic shape theorem (in its geometric form) says that in this case, the random ball of radius n, i.e. the set of points that can be reached within time n from the origin, asymptotically looks like nB µG , where B µG is the unit ball for the norm µ G . The ball B µG is thus called the asymptotic shape associated to G.

A natural extension is to replace the Z d lattice by a random environment given by the infinite cluster C ∞ of a supercritical Bernoulli percolation model. This is equivalent to allow t(e) to be equal to +∞. The existence of a time constant in firstpassage percolation in this setting was first proved by Garet and Marchand in [START_REF] Garet | Asymptotic shape for the chemical distance and firstpassage percolation on the infinite Bernoulli cluster[END_REF], in the case where (t(e)1 1 t(e)<+∞ ) is a stationary integrable ergodic field. Recently, Cerf and Théret [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF] focused of the case where (t(e)1 1 t(e)<+∞ ) is an independent field, and managed to prove the existence of an appropriate time constant without any integrability assumption. In the following, we adopt the settings of Cerf and Théret: the passage times are independent random variables with common distribution G i.e., G K is the law of the truncated passage time t K G (e) = min(t G (e), K). We have the following control on the effect of these truncations on the time constants: Theorem 1.6. Let G be a probability measure on [0, +∞] such that G([0, +∞)) > p c (d).

Then ∀x ∈ Z d lim K→∞ µ G K (x) = µ G (x) .
Remark 1.7. In fact the convergence in Theorem 1.6 (and in its Corollary 1.8) is uniform with respect to direction. This can be obtained as a consequence of Theorem 1.2 applied to G n = G (n) (or by adapting the proof of the uniformity of the convergence in Theorem 1.2). However, Theorem 1.6, as it is written, is a key step in the proof of Theorem 1.2.

As a consequence of these results, we can approximate the time constants for the chemical distance in supercritical percolation on Z d by the time constants for some finite passage times: Corollary 1.8. Let p > p c (d), and consider

G = pδ 1 + (1 -p)δ +∞ . Then G K = pδ 1 + (1 -p)δ K for all K ≥ 1 and ∀x ∈ Z d lim K→∞ µ G K (x) = µ G (x) .
1.3. Idea of the proofs. Obviously, the two main theorems of the paper, Theorems 1.1 and 1.2, state results of the same nature. Beyond this similarity, their proofs share a common structure and a common renormalisation step. The idea of the delicate part of both proofs is inspired by Cox and Kesten's method in [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF].

Consider that some edges of Z d are "good" (i.e. open, or of passage time smaller than some constant), and the others are "bad", for a given law of the environment (a parameter p for the percolation, or a given law G of passage times), and look at a path of good edges in this setting. Then change a little bit your environment : decrease p to pε, or increase the passage times of the edges. Some edges of the chosen path become bad. To recover a path of good edges, you have to bypass these edges. The most intuitive idea is to consider the cluster of bad edges around each one of them, and to bypass the edge by a short path along the boundary of this cluster. This idea works successfully in Cox and Kesten's paper. Unfortunately in our setting the control we have on these boundaries, or on the number of new bad edges we create, is not good enough. This is the reason why we cannot perform our construction of a modified good path at the scale of the edges. Thus we need to use a coarse graining argument to construct the bypasses at the scale of good blocks.

In section 2, we give more precise definitions of the studied objects and state some preliminary results. In Section 3, we present the renormalization process and the construction of modified paths that will be useful to study both the time constant and the isoperimetric constant. Sections 4 and 5 are devoted to the study of first-passage percolation. In Section 4, we use the renormalization argument to study the effect of truncating the passage times on the time constant. We then use it in Section 5 to prove the continuity of the time constant. Finally Section 6 is devoted to the proof of the continuity of the isoperimetric constant, using again the renormalization argument.

Definitions and preliminary results

In this section we give a formal definition of the objects we briefly presented in the introduction. We also present the coupling that will be useful in the rest of the paper, and prove the monotonicity of the time constant.

2.1. Lattice and passage times. Let d ≥ 2. We consider the graph whose vertices are the points of Z d , and we put an edge between two vertices x and y if and only if the Euclidean distance between x and y is equal to 1. We denote this set of edges by E d . We denote by 0 the origin of the graph. For

x = (x 1 , . . . , x d ) ∈ R d , we define x 1 = d i=1 |x i |, x 2 = d i=1 x 2 i and x ∞ = max{|x i | : i ∈ {1, . . . , d}}.
Let (t(e), e ∈ E d ) be a family of i.i.d. random variables taking values in [0, +∞] with common distribution G. We emphasize that +∞ is a possible value for the passage times, on the contrary to what is assumed in classical first-passage percolation. The random variable t(e) is called the passage time of e, i.e., it is the time needed to cross the edge e. If x, y are vertices in Z d , a path from x to y is a sequence r = (v 0 , e 1 , . . . , e n , v n ) of vertices (v i ) i=0,...,n and edges (e i ) i=1,...,n for some n ∈ N such that v 0 = x, v n = y and for all i ∈ {1, . . . , n}, e i is the edge of endpoints v i-1 and v i . We define the length |r| of a path r as its number of edges and we define the passage time of r by T (r) = e∈r t(e). We obtain a random pseudo-metric T on Z d in the following way (the only possibly missing property is the separation of distinct points):

∀x, y ∈ Z d , T (x, y) = inf{T (r) : r is a path from x to y} ∈ [0, +∞] .
Since different laws appear in this article, we put a subscript G on our notations to emphasize the dependence with respect to the probability measure G : t G (e), T G (r) and T G (x, y).

As we are interested in the asymptotic behavior of the pseudo-metric T G , we will only consider laws G on [0, +∞] such that G([0, +∞)) > p c (d). Here and in the following, p c (d) denotes the critical parameter for bond Bernoulli percolation on (Z d , E d ). Thus there a.s. exists a unique infinite cluster C G,∞ in the super-critical percolation (½ {tG(e)<∞} , e ∈ E d ) that only keeps edges with finite passage times. Our generalized first-passage percolation model with time distribution G is then equivalent to standard i.i.d. first-passage percolation (where the passage time of an edge e is the law of t G (e) conditioned to be finite) on a super-critical Bernoulli percolation performed independently (where the parameter for an edge to be closed is G({+∞})). For instance, if we take G p = pδ 1 + (1p)δ +∞ with p > p c (d), the pseudo-distance T Gp is the chemical distance in supercritical bond percolation with parameter p.

To get round the fact that the times T G can take infinite values, we introduce some regularized times T C G , for well chosen sets C. These regularized passage times have better integrability properties. Let C be a subgraph of (Z d , E d ). Typically, C will be the infinite cluster of an embedded supercritical Bernoulli bond percolation. For every x ∈ Z d , we define the random vertex x C as the vertex of C which minimizes xx C 1 , with a deterministic rule to break ties. We then define the regularized passages times We collect in this subsection the definitions and properties of the Cheeger constant obtained in [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF]. The Cheeger constant can be represented as the solution of a continuous isoperimetric problem with respect to some norm. To define this norm, we first require some definitions. We fix p > p c (2), we denote by C p the P p -a.s. unique infinite cluster C Gp,∞ and we set θ p = P p (0 ∈ C p ). For a path r = (v 0 , e 1 , . . . , e n , v n ), and i ∈ {2, . . . , n -1}, an edge e = (x i , z) is said to be a right-boundary edge if z is a neighbor of x i between x i+1 and x i-1 in the clockwise direction. The right boundary ∂ + r of r is the set of right-boundary edges. A path is called right-most if it uses every edge at most once in every orientation (thus an edge can be used twice, but in different orientations) and it doesn't contain right-boundary edges. See Figure 1; the solid lines represent the path, dashed lines represent the right-boundary edges, and the curly line is a path in the medial graph which shows the orientation (see [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF] for a thorough discussion). For x, y ∈ Z 2 , let R(x, y) be the set of right-most paths from x to y. For a path r ∈ R(x, y), define b(r) = |{e ∈ ∂ + r : e is open}|. For x, y ∈ C p we define the right boundary distance, b(x, y) = inf{b(r) : r ∈ R(x, y), r is open}. The next result yields uniform convergence of the right boundary distance to a norm on R 2 . Proposition 2.1 (Definition of the norm, Theorem 2.1 in [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF] ). For any p > p c [START_REF] Benjamini | On the mixing time of a simple random walk on the super critical percolation cluster[END_REF], there exists a norm β p on R 2 such that for any x ∈ R 2 ,

T C G by ∀x, y ∈ Z d , T C G (x, y) = T G ( x C , y C ) .
β p (x) := lim n→∞ b( 0 Cp , nx Cp ) n P p -a.s. and in L 1 (P p ).
Moreover, the convergence is uniform on

S 1 = {x ∈ R 2 : x 2 = 1}.
We will require the following control on the length of right-most paths.

Lemma 2.2 (Proposition 2.9 in [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF]). There exist C, C ′ , α > 0 (depending on p) such that for all n,

P ∃γ ∈ x∈Z 2 R(0, x) : |γ| > n , b(γ) ≤ αn ≤ Ce -C ′ n .
The connection between the Cheeger constant and the norm β p goes through a continuous isoperimetric problem. For a continuous curve λ : [0, 1] → R 2 , and a norm ρ, let the ρ-length of λ be

len ρ (λ) = sup N ≥1 sup 0≤t0<...<tN ≤1 N i=1 ρ(λ(t i ) -λ(t i-1 )).
A curve λ is said to be rectifiable if len ρ (λ) < ∞ for any norm ρ. A curve λ is called a Jordan curve if λ is rectifiable, λ(0) = λ(1) and λ is injective on [0, 1). For any Jordan curve λ, we can define its interior int(λ) as the unique finite component of R 2 \ λ([0, 1]). Denote by Leb the Lebesgue measure on R 2 . The Cheeger constant can be represented as the solution of the following continuous isoperimetric problem:

Proposition 2.3 (Theorem 1.6 in [4]). For every p > p c (2), lim n→+∞ nϕ n (p) = ( √ 2 θ p ) -1 inf{len βp (λ) : λ is a Jordan curve, Leb(int(λ)) = 1}.
Moreover one obtains a limiting shape for the sets that achieve the minimum in the definition of ϕ n (p). This limiting shape is given by the Wulff construction [START_REF] Wulff | Xxv. zur frage der geschwindigkeit des wachsthums und der auflösung der krystallflächen[END_REF]. Denote by ( 1)

W p = n: n 2=1 {x ∈ R 2 : n • x ≤ β p (n)} and W p = W p Leb(W p )
, where At this stage, a few words should be said about the Cheeger constant in higher dimensions. Very recently, Gold extended in [START_REF] Gold | Isoperimetry in supercritical bond percolation in dimensions three and higher[END_REF] the results of Biskup, Louidor, Procaccia and Rosenthal [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF] to dimensions 3 and higher. To this purpose, he generalizes the definition of the norm β p to higher dimensions, see Proposition 3.2 in [START_REF] Gold | Isoperimetry in supercritical bond percolation in dimensions three and higher[END_REF]. However, in a general dimension d ≥ 2, this norm β p is defined as the limit of the infimum of the weight (properly rescaled) of a large hypersurface belonging to a certain class of hypersurfaces. Here by hypersurface we mean an object of dimension d -1. For d = 2, we recover an object of dimension 1, i.e., a path, and β p is defined as the infimum of the rescaled weight of a large path belonging to the class of the right-most paths. The study of the norm β p in dimension 2 is thus closely related to the study of geodesics in first-passage percolation. However, in higher dimension, the study of the norm β p is no longer related to the study of geodesics in first-passage percolation, but it is more naturally related to the study of minimal cutsets in first-passage percolation as defined in [START_REF] Kesten | Surfaces with minimal random weights and maximal flows: a higher dimensional version of first-passage percolation[END_REF]. This connects the question of the continuity of the Cheeger constant in dimensions 3 or higher to the question of the continuity of the flow constant (rather than the time constant) in first-passage percolation. For this reason, the technics developed in this article, when we deal with modifications of geodesics, cannot be used straightforwardly to prove the continuity of the Cheeger constant in dimensions 3 and higher.

2.3. Definition and properties of the time constant. As announced in the introduction, we follow the approach by Cerf and Théret in [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF], which requires no integrability condition on the restriction of G to [0, +∞). We collect in this subsection the definition and properties of the time constants obtained in their paper.

Let G be a probability measure on [0, +∞] such that G([0, +∞)) > p c (d), and let M > 0 be such that G([0, M ]) > p c (d). We denote by C G,M the a.s. unique infinite cluster of the percolation (½ {tG(e)≤M} , e ∈ E d ), i.e. the percolation obtained by keeping only edges with passage times less than or equal to M . For any x, y ∈ Z d , the (level M ) regularized passage time

T CG,M G (x, y) is then T CG,M G (x, y) = T G ( x CG,M , y CG,M ) .
The parameter M only plays a role in the choice of x CG,M and y CG,M . Once these points are chosen, the optimization in T CG,M G (x, y) is on all paths between x CG,M and y CG,M , paths using edges with passage time larger than M included. But as x CG,M ∈ C G,M and y CG,M ∈ C G,M , we know that exists a path using only edges with passage time less than or equal to M between these two points. To be more precise, we denote by D C (x, y) the chemical distance (or graph distance) between two vertices x and y on C: The regularized passage time T CG,M G enjoys then the same good integrability properties as the chemical distance on a supercritical percolation cluster (see [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF]): Proposition 2.5 (Moments of T , [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF]). Let G be a probability measure on [0, +∞]

such that G([0, +∞)) > p c (d). For every M ∈ [0, +∞) such that G([0, M ]) > p c (d), there exist positive constants C 1 , C 2 and C 3 such that ∀x ∈ Z d , ∀l ≥ C 3 x 1 , P T CG,M G (0, x) > l ≤ C 1 e -C2l .
We denote by C G,∞ the a.s. unique infinite cluster of the percolation obtained by keeping only edges with finite passage time, i.e. the percolation (½ {tG(e)<∞} , e ∈ E d ). Property 2.5 implies in particular that the times T CG,M G (0, x) are integrable. A classical application of a subadditive ergodic theorem gives the existence of a time constant: Proposition 2.6 (Convergence to the time constant, [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF]). Let G be a probability measure on [0, +∞] such that G([0, +∞)) > p c (d). There exists a deterministic function µ G : Z d → [0, +∞) such that for every M ∈ [0, +∞) satisfying G([0, M ]) > p c (d), we have the following properties:

∀x ∈ Z d µ G (x) = inf n∈N * E T CG,M G (0, nx) n = lim n→+∞ E T CG,M G (0, nx) n , (2) ∀x ∈ Z d lim n→∞ T CG,M G (0, nx) n = µ G (x) a.s. and in L 1 , (3) ∀x ∈ Z d lim n→∞ T CG,∞ G (0, nx) n = µ G (x) in probability , (4) ∀x ∈ Z d lim n→∞ T G (0, nx) n = θ 2 G δ µG(x) + (1 -θ 2 G )δ +∞ in distribution, ( 5
)
where θ G = P[0 ∈ C G,∞ ].
Note that even if the definition (2) of the time constants µ G (x) requires to introduce a parameter M in the definition of the regularized passage times T CG,M G (0, nx), these time constants µ G (x) do not depend on M . Note also that if instead of taking the x CG,M in the infinite cluster C G,M of edges with passage time less than M , we take the x CG,∞ in the infinite cluster C G,∞ of edges with finite passage time, the almost sure convergence is weakened into the convergence in probability (4). Without any regularization, the convergence in ( 5) is only in law.

As in the classical first-passage percolation model, the function µ G can be extended, by homogeneity, into a pseudo-norm on R d (the only possibly missing property of µ G is the strict positivity): Proposition 2.7 (Positivity of µ G , [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF]). Let G be a probability measure on [0, +∞] such that G([0, +∞)) > p c (d). Then either µ G is identically equal to 0 or µ G (x) > 0 for all x = 0, and we know that

µ G = 0 ⇐⇒ G({0}) ≥ p c (d) .
Proposition 2.5 gives strong enough integrability properties of T CG,M G (0, x) to ensure that the convergence to the time constants is uniform in the direction: Proposition 2.8 (Uniform convergence, [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF]). Let G be a probability measure on

[0, +∞] such that G([0, +∞)) > p c (d). Then for every M ∈ [0, +∞) such that G([0, M ]) > p c (d), we have lim n→∞ sup x∈Z d , x 1≥n T CG,M G (0, x) -µ G (x)
x 1 = 0 a.s.

When µ G > 0, this uniform convergence is equivalent to the so called shape theorem, that we briefly present now. We define B G,t (resp.

B CG,M G,t , B CG,∞ G,t
) as the set of all points reached from the origin within a time t, enlarged by adding a small unit cube around each such point:

B G,t = {z + u : z ∈ Z d , T G (0, z) ≤ t , u ∈ [-1/2, 1/2] d } , (resp. T CG,M G , T CG,∞ G
), and when µ G is a norm we denote by B µG its closed unit ball. Roughly speaking, the shape theorem states that the rescaled set B G,t /t (respectively

B CG,M G,t /t, B CG,∞ G,t /t) converges towards B µG .
The convergence holds in a sense that depends on the regularity of times considered (see [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF] for more precise results).

Coupling.

To understand how µ G depends on G, it is useful to consider passage times (t G (e)) with common distribution G, that also have good coupling properties. For any probability measure G on [0, +∞], we denote by G the function

G : [0, +∞) → [0, 1] t → G([t, +∞]),
which characterizes G. For two probability measures G 1 , G 2 on [0, +∞], we define the following stochastic domination relation:

G 1 G 2 ⇔ ∀t ∈ [0, +∞) G 1 (t) ≥ G 2 (t).
This is to have this equivalence that we choose to characterize a probability measure G by G instead of the more standard distribution function t → G([0, t]).

Given a probability measure G on [0, +∞], we define the two following pseudoinverse functions for G:

∀t ∈ [0, 1] , Ĝ(t) = sup{s ∈ [0, +∞) : G(s) ≥ 1 -t} and G(t) = sup{s ∈ [0, +∞) : G(s) > 1 -t} .
These pseudo-inverse functions can be used to simulate random variable with distribution G:

Lemma 2.9. Let U be a random variable with uniform law on (0, 1). If G is a probability measure on [0, +∞], then Ĝ(U ) and G(U ) are random variables taking values in [0, +∞] with distribution G, and G(U ) = Ĝ(U ) a.s.

Proof. The function Ĝ has the following property

(6) ∀t ∈ [0, 1] , ∀s ∈ [0, +∞) , Ĝ(t) ≥ s ⇐⇒ G(s) ≥ 1 -t .
Then for all s ∈ [0, +∞), we have

P[ Ĝ(U ) ≥ s] = P[U ≥ 1 -G(s)] = G(s), thus Ĝ(U ) has distribution G.
The function G does not satisfy the property (6). However, Ĝ(t) = G(t) only for t ∈ [0, 1] such that G -1 ({1 -t}) contains an open interval, thus the set {t ∈ [0, 1] : Ĝ(t) = G(t)} is at most countable. This implies that Ĝ(U ) = G(U ) a.s., thus G(U ) has the same law as Ĝ(U ).

In the following, we will always build the passage times of the edges with this lemma. Let then (u(e), e ∈ E d ) be a family of i.i.d. random variables with uniform law on (0, 1). For any given probability measure G on [0, +∞], the family of i.i.d passage times with distribution G will always be [START_REF] Couronné | Surface order large deviations for 2D FKpercolation and Potts models[END_REF] ∀e ∈ E d , t G (e) = Ĝ(u(e)) .

Of course the main interest of this construction is to obtain couplings between laws:

if G 1 and G 2 are probability measures on [0, +∞],

G 1 G 2 ⇒ t G1 (e) ≤ t G2 (e) for all edges e.
In particular in the case of Bernoulli percolation, if

p ≤ q, G q = qδ 1 + (1 -q)δ ∞ G p = pδ 1 + (1 -p)δ ∞ thus C p ⊂ C q .
Moreover, we have the following pleasant property:

Lemma 2.10. Let G, (G n ) n∈N be probability measures on [0, +∞]. We define the passage times t G (e) and t Gn (e) as in equation [START_REF] Couronné | Surface order large deviations for 2D FKpercolation and Potts models[END_REF].

If G n d → G, then a.s. , ∀e ∈ E d , lim n→∞ t Gn (e) = t G (e) . Proof. (i) Let us prove that if G n G for all n, then ∀t ∈ [0, 1] lim n→∞ Ĝn (t) = Ĝ(t) . (8) Consider t ∈ [0, 1], let x = Ĝ(t) and x n = Ĝn (t). Since G n G, we have G n ≥ G thus x n ≥ x. Suppose that lim n→+∞ x n := x > x.
Up to extraction, we suppose

that lim n→+∞ x n = x. Choose β ∈ (x, x) such that G is continuous at β, thus lim n→∞ G n (β) = G(β).
On one hand, by the definition of Ĝ and the monotonicity of G, we have G(β) < 1t. On the other hand, β < x n for all n large enough, thus G n (β) ≥ 1-t for all n large enough, and we conclude that As G n G for all n and t G (e) = G(u(e)) almost surely, (ii) implies that a.s.

G(β) = lim n→∞ G n (β) ≥ 1 -t, which is a contradiction, and (8) is proved. (ii) Similarly, if G n G for all n, then ∀t ∈ [0, 1] lim n→∞ Ĝn (t) = G(t) . (iii) We define G n = min{G, G n } (resp. G n = max{G, G n }),
lim n→∞ t G n (e) = t G (e) .
Finally, as G n G n G n for all n, we know by coupling that t G n (e) ≤ t Gn (e) ≤ t Gn (e), which gives the desired convergence.

Stabilization of the point x and monotonicity of the time constant.

We need to extend the monotonicity of the time constant to first-passage percolation on the infinite cluster of supercritical percolation. Since we work with different probability measures, the fact that, in the regularization process, the point x CG,M depends on G may be disturbing. We get round this problem by considering an alternative probability measure H:

Lemma 2.11. Let G and H be probability measures on [0, +∞] such that G H. For all M ∈ [0, +∞) satisfying H([0, M ]) > p c (d), we have µ G (x) = inf n∈N * E T CH,M G (0, nx) n = lim n→∞ T CH,M G (0, nx) n a.s. and in L 1 . Proof. Since G H we get by coupling that t G (e) ≤ t H (e) for all e ∈ E d . Let M ∈ [0, +∞) satisfying H([0, M ]) > p c (d), then G([0, M ]) > p c (d) and C H,M ⊂ C G,M . The proof of the convergence of T CH,M G (0, nx)/n is a straightforward adaptation of the proof of the convergence of T CG,M G (0, nx)/n: by the subadditive ergodic theorem, there exists a function µ ′ G,H : R d → [0, +∞) such that for all x ∈ Z d we have µ ′ G,H (x) = inf n∈N * E T CH,M G (0, nx) n = lim n→∞ T CH,M G (0, nx) n a.s. and in L 1 .
It remains to prove that µ ′ G,H = µ G . For any x ∈ Z d , for any ε > 0, we have

P T CH,M G (0, nx) -T CG,M G (0, nx) > nε ≤ P T G ( 0 CG,M , 0 CH,M ) + T G ( nx CG,M , nx CH,M ) > nε ≤ 2 P T G ( 0 CG,M , 0 CH,M ) > nε/2 . (9) Since 0 CG,M ∈ C G,M ⊂ C G,∞ and 0 CH,M ∈ C H,M ⊂ C G,M ⊂ C G,∞ , the time T G ( 0 CG,M , 0 CH,M
) is finite a.s. thus the right hand side of inequality ( 9) goes to 0 as n goes to infinity. This concludes the proof of Lemma 2.11.

As a simple consequence of the coupling built in section 2.4 and the stabilization Lemma 2.11, we obtain the monotonicity of the function G → µ G : Lemma 2.12. Let G, H be probability measures on [0, +∞]. we have Lemma 2.13. For any p, p 0 such that p c (2) < p 0 ≤ p ≤ 1, for any x ∈ R 2 , we have

G H =⇒ µ G ≤ µ H . Proof. By construction of µ G and µ H , it suffices to prove that µ G (x) ≤ µ H (x) for all x ∈ Z d .
µ G (x) = lim n→∞ T CH,M G (0, nx) n ≤ lim n→∞ T CH,M H (0, nx) n = µ H (x) .
β p (x) = lim n→∞ b p ( 0 Cp 0 , nx Cp 0 ) n P p -a.s. and in L 1 (P p ).
Proof. Exactly as in the proof of Lemma 2.11, since the convergence of b p ( 0 Cp , nx Cp )/n follows from a subadditive argument, the proof can be adapted straightforwardly to prove the existence of

β p,p0 (x) := lim n→∞ b p ( 0 Cp 0 , nx Cp 0 ) n P p -a.s. and in L 1 (P p ).
The only thing we have to prove is the equality β p,p0 (x) = β p (x). By the almostsubbadditivity of b p (see equation (2.27) in [START_REF] Biskup | Isoperimetry in two-dimensional percolation[END_REF]), we have

b p ( 0 Cp , nx Cp ) ≤ b p ( 0 Cp 0 , nx Cp 0 ) + b p ( 0 Cp 0 , 0 Cp ) + b p ( x Cp 0 , nx Cp ) + 4 .
Since C p0 ⊂ C p , there exists a finite simple (thus also right-most) path γ which is p-open between 0 Cp 0 and 0 Cp , and by [4, Lemma 2.5] we know that |∂ + γ| < 3|γ|, thus b p ( 0 Cp 0 , 0 Cp ) ≤ 3|γ| < +∞. The same holds for b p ( nx Cp 0 , nx Cp ). As in the proof of Lemma 2.11, this is enough to conclude that β p,p0 (x) = β p (x).

Notice that Lemma 2.13 does not imply the monotonicity of the Cheeger constant. Indeed, consider p c (2) < p ≤ q, then • a q-open path γ may not be p-open, • a p-open path γ is q-open by coupling, but b q (γ) may be strictly bigger than b p (γ), thus no trivial comparison between b p ( 0 Cp , nx Cp ) and b q ( 0 Cp , nx Cp ) holds.

Renormalization

In this section we present the renormalization process and the construction of modified paths that will be useful to study both the time constant and the isoperimetric constant. We consider coupled bound i.i.d. Bernoulli percolations of different parameters. As we will see in Section 4, the construction of modified paths in the model of first passage percolation reduces to this simplest case.

3.1. Definition of the renormalization process. Consider parameters p 0 and q such that p c (d) < p 0 ≤ q ≤ 1. Consider i.i.d. Bernoulii percolation on the edges of Z d for these two parameters that are coupled, i.e. any p 0 -open edge is q-open. Denote as before by C p0 the a.s. unique infinite cluster of the supercritical Bernoulli field of parameter p 0 . We call this field the p 0 -percolation and its clusters the p 0 -clusters.

We use a renormalization process in the spirit of the work of Antal and Pisztora [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF]. For a large integer N , that will be apropriately chosen later, we chop Z d into disjoint N -boxes as follows: we set B N to be the box [-N, N ] d ∩ Z d and define the family of N -boxes by setting, for i ∈ Z d ,

B N (i) = τ i(2N +1) (B N ),
where τ b stands for the shift in Z d with vector b ∈ Z d . We will also refer to the box B N (i) as the N -box with coordinate i. The coordinates of N -boxes will be denoted in bold and considered as macroscopic sites, to distinghish them from the microscopic sites in the initial graph Z d . We also introduce larger boxes: for i ∈ Z d ,

B ′ N (i) = τ i(2N +1) (B 3N
). As in [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF], we say that a connected cluster C is a crossing cluster for a box B, if for all d directions there is an open path contained in C ∩ B joining the the two opposite faces of the box B.

Let C ′ p0 = (Z d , {e ∈ E d : e is p 0 -open}) be the graph whose edges are opened for the Bernoulli percolation of parameter p 0 . We recall that C p0 is the infinite cluster of C ′ p0 , and we have D Cp 0 (x, y) = D C ′ p 0 (x, y) for every vertices x and y in C p0 , and D Cp 0 (x, y) = +∞ if x or y are not in C p0 . Let us recall the following result, obtained by Antal and Pisztora [1, Theorem 1.1], that says that the chemical distance D C ′ p 0 can't be too large when compared to • 1 or • ∞ (or any other equivalent norm): there exist positive constants Â, B, β such that [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF] ∀x ∈

Z d P(β x 1 ≤ D C ′ p 0 (0, x) < +∞) ≤ Â exp(-B x 1 ) , ( 11
) ∀x ∈ Z d P(β x ∞ ≤ D C ′ p 0 (0, x) < +∞) ≤ Â exp(-B x ∞ ) , and (12) ∀x ∈ Z d P(β x 2 ≤ D C ′ p 0 (0, x) < +∞) ≤ Â exp(-B x 2 ) .
In fact Antal and Pisztora proved [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF], but different norms being equivalent in R d , we can obtain [START_REF] Fontes | First passage percolation for random colorings of Z d[END_REF] and ( 12) by changing the constants.

Definition 3.1. We say that the macroscopic site i is (p 0 , q)-good (or that the box B N (i) is (p 0 , q)-good) if the following events occur: (i) There exists a unique p 0 -cluster C in B ′ N (i) which has more than N vertices; (ii) This p 0 -cluster C is crossing for each of the

3 d N -boxes included in B ′ N (i); (iii) For all x, y ∈ B ′ N (i), if x and y belong to this p 0 -cluster C, then D C ′ p 0 (x, y) ≤ 6βN ; (iv) If π is a q-open path in B ′ N (i) such that |π| ≥ N , then π intersects this p 0 -cluster C in B ′ N (i), i.e.
, they share a common vertex. We call this cluster C the crossing p 0 -cluster of the (p 0 , q)-good box B N (i).

Otherwise, B N (i) is said to be (p 0 , q)-bad. For short, we say that B is good or bad if there is no doubt about the choice of (p 0 , q).

On the macroscopic grid Z d , we consider the same standard nearest neighbour graph structure as on the microscopic initial grid Z d . Moreover we say that two macroscopic sites i and j are * -neighbors if and only if ij ∞ = 1. If C is a connected set of macroscopic sites, we define its exterior vertex boundary

∂ v C = i ∈ Z d \ C : i has a neighbour in C and i is connected to infinity by a Z d -path in Z d \ C .
For a bad macroscopic site i, denote by C(i) the connected cluster of bad macroscopic sites containing i. [START_REF] Timár | Boundary-connectivity via graph theory[END_REF]. For a good macroscopic site i, we define ∂ v C(i) to be {i}.

If C(i) is finite, the set ∂ v C(i) is then a * -connected set of good macroscopic sites, see Lemma 2 in

Modification of a path.

Let p c (d) < p 0 ≤ p ≤ q and N be fixed. Let now γ be a path in Z d . What we want to do is to remove from γ the edges that are p-closed, and to look for bypasses for these edges using only edges that are p 0 -open.

To γ, we associate the connected set Γ ⊂ Z d of N -boxes it visits: this is a lattice animal, i.e. a connected finite set of Z d , containing the box that contains the starting point of γ. We decompose γ into two parts, namely γ a = {e ∈ γ : e is p-open} and γ b = {e ∈ γ : e is p-closed}. We denote by Bad the (random) set of bad connected components of the macroscopic percolation given by the states of the N -boxes. Lemma 3.2. Assume that y ∈ C p0 , that z ∈ C p0 , that the N -boxes containing y and z are (p 0 , q)-good and belong to an infinite cluster of (p 0 , q)-good boxes. Let γ be a path between y and z. Then there exists a p-open path γ ′ between y and z that has the following properties :

(1) γ ′ \ γ is a collection of disjoint self avoiding p 0 -open paths that intersect γ ′ ∩ γ only at their endpoints;

(2)

|γ ′ \ γ| ≤ ρ d   N C∈Bad: C∩Γ =∅ |C| + N d |γ b |  
, where ρ d is a constant depending only on the dimension d.

The proof of Lemma 3.2 is in the spirit of Antal and Pisztora's proof in [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF]. However, our construction is a bit more involved, since what we do can be seen as a refinement of Antal and Pisztora's arguments, in order to obtain more precise estimates. In fact, we will use Antal and Pisztora's result itself to prove that the property (iii) of good blocks is typical (see Lemma 3.5).

Before proving Lemma 3.2, we need a simpler estimate on the cardinality of a path inside a set of good blocks.

Lemma 3.3.

There exists a constant ρd , depending only on d, such that for every fixed N , for every n ∈ N * , if (B N (i)) i∈I is a * -connected set of n (p 0 , q)-good N -blocks, if x ∈ B N (j) for j ∈ I and x is in the crossing p 0 -cluster of B N (j), if y ∈ B N (k) for k ∈ I and y is in the crossing p 0 -cluster of B N (k), then there exists a p 0 -open path from x to y of length at most equal to ρd (N n + N d ).

Proof of Lemma 3.3. Since I is a * -connected set of macroscopic sites, there exists a self-avoiding * -connected path (ϕ i ) 1≤i≤r from j and k in I. We notice that r ≤ 3 d |I| ≤ 3 d n. Since B N (i) is a good block for all i ∈ I, the definition of good boxes ensures that there exists a p 0 -cluster

C in C ′ p0 ∩ ∪ i∈I {e ∈ B ′ N (i)} which is crossing for every N -box included in ∪ i∈I B ′ N (i) (see Proposition 2.
1 in Antal and Pisztora [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF]), and by hypotheses x and y are in C. We now consider a sequence of points (z i ) 1≤i≤r such that for each i ∈ {1, . . . , r}, z i ∈ B N (ϕ i ) and z i belongs to the p 0 -crossing cluster of B N (ϕ i ). For every i ∈ {2, . . . , r}, we have

z i ∈ B N (ϕ i ), z i-1 ∈ B N (ϕ i-1 ) ⊂ B ′ N (ϕ i )
, and these two points belong to the crossing p 0 -cluster of B ′ N (ϕ i ). The fact that B N (ϕ i ) is good ensures that there exists a p 0 -open path from z i-1 to z i of length at most equal to 6βN (see property (iii) of good boxes). By concatenating these paths, we obtain a p 0 -open path from z 1 to z r of length at most equal to 6βN r ≤ 6βN 3 d n. Proof of Lemma 3.2. To the path γ, we associate the sequence ϕ 0 = (ϕ 0 (j)) 1≤j≤r0 of N -boxes it visits. Note that ϕ is not necessarily injective, and that the previously defined lattice animal Γ is equal to ϕ 0 ({1, . . . , r 0 }).

From the sequence ϕ 0 , we extract the subsequence (ϕ 1 (j)) 1≤j≤r1 , with r 1 ≤ r 0 , of N -boxes B such that γ ∩ B contains at least one edge that is p-closed (more precisely, we keep the indices of the boxes B that contain the smallest extremity, for the lexicographic order, of an edge of γ that is p-closed). Notice that r 1 ≤ |γ b |. The idea is the following:

(1) If ϕ 1 (j) is good, we add to γ all the p 0 -open edges in B ′ : there will be enough such edges in the good N -box to find a by-pass for the edge of γ that is p-closed. (2) If ϕ 1 (j) is bad, we will look for such a by-pass in the exterior vertex boundary ∂ v C(ϕ 1 (j))) of the connected component of bad boxes of ϕ 1 (j). In the second case, we use Lemma 3.3 to control the length of the by-pass we create. We recall that if i is good, then ∂ v C(i) = {i}. Note that some ∂ v C(ϕ 1 (j))) may coincide or be nested one in another or overlap or be * -connected. In order to define properly the modification of our path, we need thus to extract a subsequence once again, see Figure 2. We first consider the * -connected components (S ϕ2(j) ) 1≤j≤r2 , with r 2 ≤ r 1 , of the union of the (∂ v C(ϕ 1 (j))) 1≤j≤r1 , by keeping only the smallest index for each connected component. Thus, for all i = j, i, j ∈ {1, . . . , r 2 }, S ϕ2(i) and S ϕ2(j) cannot be * -connected. Next, in case of nesting, we only keep the largest connected component. We denote by (S ϕ3(j) ) 1≤j≤r3 , with r 3 ≤ r 2 , the remaining hypersurfaces of good N -boxes. Finally it may happen that γ visits several times : the boxes (ϕ 1 (j)) 1≤j≤r 1 : the sets of good boxes in (S ϕ 2 (j) ) 1≤j≤r 2 that do not belong to (S ϕ 4 (j) ) 1≤j≤r 4 y z γ : the sets of good boxes (S ϕ 4 (j) ) 1≤j≤r 4 the same S ϕ3(j) for some j: in this situation we can remove the loops that γ makes between its different visits in S ϕ3(j) . Thus by a last extraction we obtain (S ϕ4(j) ) 1≤j≤r4 , where S ϕ4(1) = S ϕ3 [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF] and for all k ≥ 1, ϕ 4 (k + 1) is the infimum of the indices (ϕ 3 (j)) 1≤j≤r3 such that γ visits S ϕ3(j) after it exits S ϕ4(k) for the last time (if such a j exists).

Note that the path γ must visit each (S ϕ4(j) ) 1≤j≤r4 . We now cut γ in several pieces. Let Ψ in (1) = min{k ≥ 1 : γ k ∈ ∪ i∈S ϕ 4 (1) B N (i)} and i in (1) be the macroscopic site such that γ Ψin(1) ∈ B N (i in (1)), let Ψ out (1) = max{k ≥ Ψ in [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF] : γ k ∈ ∪ i∈S ϕ 4 (1) B N (i)} and i out (1) be the macroscopic site such that γ Ψout(1) ∈ B N (i out (1)) (see Figure 3). By recurrence, for all 1 ≤ j ≤ r 4 , we define

γ ′ j,link γ ′ j,in γ ′ j,out γ j γ j-1 γ y j γ ψ in (j) y ′ j x ′ j γ ψout(j) x j S ϕ 4 (j) Figure 3. Construction of the path γ ′ -step 2. Ψ in (j) = min{k ≥ Ψ out (j -1) : γ k ∈ ∪ i∈S ϕ 4 (j) B N (i)}, γ Ψin(j) ∈ B N (i in (j)), and Ψ out (j) = max{k ≥ Ψ in (j) : γ k ∈ ∪ i∈S ϕ 4 (j) B N (i)}, γ Ψout(j) ∈ B N (i out (j))
. For all 1 ≤ j ≤ r 4 -1, let γ j be the part of γ from γ Ψout(j) to γ Ψin(j+1) . By construction γ j contains no p-closed edge, and has at least N vertices in B ′ N (i out (j)) (resp. in B ′ N (i in (j + 1))) since i out (j) and i in (j + 1) cannot be * -connected, thus γ j intersects the crossing p 0 -cluster of B ′ N (i out (j)) (resp. B ′ N (i in (j + 1))) by property (iv) of the good boxes; let us denote by x j (resp. y j+1 ) the last (resp. first) intersection of γ j with the crossing p 0 -cluster of B ′ N (i out (j)) (resp. B ′ N (i in (j + 1))). The vertex x j (resp. y j+1 ) is not inside B N (i out (j)) (resp. B N (i in (j + 1))), but it is connected inside the p 0 -cluster of

B ′ N (i out (j)) (resp. B ′ N (i in (j + 1))) to a vertex x ′ j (resp. y ′ j+1 ) of B N (i out (j)) (resp. B N (i in (j + 1))) by a path γ ′ j,out (resp. γ ′ j+1,in ) of length at most equal to 2d3 d N d ≤ ρd N d .
Let us study more carefully the beginning of the path γ. Since the N -box containing y belongs to an infinite cluster of good boxes, it cannot be in the interior of S ϕ4 [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF] . Let i 0 be the macroscopic site such that y ∈ B N (i 0 ). We have to consider three cases. [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF] , then Ψ in (1) = 1 and γ Ψin(1) = y. As y ∈ C p0 and B N (i 0 ) is good, y is in the crossing p 0 -cluster of B N (i 0 ), thus we can define

• If i 0 is in S ϕ4
y 1 = y ′ 1 = y, γ 0 = ∅ and γ ′ 1,in = ∅. • If i 0 is not in S ϕ4(1) (thus it is outside S ϕ4(1) ), then Ψ in (1) > 1 and γ Ψin(1)
does not belong to B N (i 0 ). Let i 1 = i in (1) be the macroscopic site such that γ Ψin(1) ∈ B N (i 1 ).

-If i 0 and i 1 are * -connected, then we can choose y 1 = y, y ′ 1 ∈ B N (i 1 ) such that y ′ 1 belongs to the crossing p 0 -cluster of the good box B N (i 1 ), and we can connect y = y 1 to y ′ 1 inside B ′ N (i 1 ) by a path γ ′ 1,in of length at most equal to 2d3 d N d ≤ ρd N d . We define γ 0 = ∅. -If i 0 and i 1 are not * -connected, let γ 0 be the part of γ between y and γ Ψin(1) , then by construction γ 0 does not contain any p-closed edge and has at least N vertices in B ′ N (i 1 ), thus by property (iv) γ 0 intersects the crossing p 0 -cluster of the good box B ′ N (i 1 ). We denote by y 1 the first intersection of γ 0 with the crossing p 0 -cluster of B ′ N (i 1 ). As previously, we know that y 1 is connected inside the crossing p 0cluster of

B ′ N (i 1 ) to a vertex y ′ 1 of B N (i 1 ) by a path γ ′ 1,in of length at most equal to 2d3 d N d ≤ ρd N d .
Similarly, we define x r4 , x ′ r4 , γ r4 and γ ′ r4,out depending on the fact that the box containing z belongs to S ϕ4(r4) or not, and is * -connected to the boxe containing γ Ψout(r4) or not.

For all 1 ≤ j ≤ r 4 , we can apply Lemma 3.3 to state that there exists a p 0 -open path γ ′ j,link from y ′ j to x ′ j of length at most ρd (

N d + N |S ϕ4(j) |). For all 1 ≤ j ≤ r 4 , define γ ′ j = γ ′ j,in ∪ γ ′ j,link ∪ γ ′ j,out . By construction each γ ′ j is p 0 -open.
We can glue together the paths γ 0 , γ ′ 1 , γ 1 , γ ′ 2 , . . . , γ ′ r4 , γ r4 in this order to obtain a p-open path γ ′ from y to z. Up to cutting parts of these paths, we can suppose that each γ ′ i is a self-avoiding path, that the γ ′ i are disjoint and that each γ ′ i intersects only γ i-1 and γ i , and only with its endpoints.

Finally we need an estimate on |γ

′ γ|. Obviously γ ′ γ ⊂ ∪ r4 i=1 γ ′ i , thus |γ ′ γ| ≤ 2r 4 ρd N d + r4 j=1 ρd (N d + N |S ϕ4(j) |) ≤ 3r 4 ρd N d + ρd N r4 j=1 |S ϕ4(j) | ≤ 4 ρd N d |γ b | + ρd N C∈Bad: C∩Γ =∅ |∂ v C| ,
where we have used the fact that

r4 j=1 |S ϕ4(j) | ≤ |γ b | + C∈Bad: C∩Γ =∅ |∂ v C| .
To conclude, we just have to remark that |∂ v C| ≤ 2d|C|.

Probabilistic estimates.

We want to bound the probability that |γ ′ γ| is big for qp 0 small enough. Lemma 3.2 makes appear the connected set Γ ⊂ Z d of N -boxes visited by the path γ. To control |γ ′ γ|, we need to have a deterministic control on |Γ|. This is the purpose of the following Lemma.

Lemma 3.4.

There exists a constant Cd , depending only on d, such that for every path γ of Z d , for every N ∈ N * , if Γ is the animal of N-blocks that γ visits, then

|Γ| ≤ Cd 1 + |γ| + 1 N .
Proof. Let γ = (γ i ) i=1,...,n be a path of Z d for a n ∈ N * (γ i is the i-th vertex of γ, n = |γ| + 1), and fix N ∈ N * . Let Γ be the animal of N-blocks that γ visits. We will include Γ in a bigger set of blocks whose size can be controlled. Let p(1) = 1 and i 1 be the macroscopic site such that γ 1 ∈ B N (i 1 ). If p(1), . . . , p(k) and i 1 , . . . , i k are constructed, define p(k + 1) = inf{j ∈ {p(k), . . . , n} : γ j / ∈ B ′ N (i k )} if this set is not empty and let i k+1 be the macroscopic site such that γ p(k+1) ∈ B N (i k+1 ), and stop the process if for every j ∈ {p(k), . . . , n} , γ j ∈ B ′ N (i k ). We obtain two finite sequences (p(1), . . . , p(r)) and (i 1 , . . . , i r ). First notice that

Γ ⊂ r k=1 B ′ N (i k )
by construction, thus |Γ| ≤ 3 d r. Moreover for every k ∈ {1, . . . , r -1}, γ p(k+1)γ p(k) 1 ≥ N , thus p(k+1)-p(k) ≥ N . This implies that N (r-1) ≤ p(r)-p(1) ≤ n, and we conclude that

|Γ| ≤ 3 d 1 + n N .
Then we need a control on the probability that a block is good.

Lemma 3.5.

(i) For every q > p c (d), there exists δ 0 (q) > 0 such that if p 0 ∈ (p c (d), q] satisfy qp 0 ≤ δ 0 , then for every p < 1, there exists an integer N (p 0 , q, p) such that the field (1 1 {BN (i) is (p0, q)-good} ) i∈Z d stochastically dominates a family of independent Bernoulli random variables with parameter p.

(ii) For every p 0 > p c (d), there exists δ 1 (p 0 ) > 0 such that if q 1 ∈ [p 0 , 1] satisfy q 1p 0 ≤ δ 1 , then for every p < 1 there exists an integer N ′ (p 0 , q 1 , p) such that for any q ∈ [p 0 , q 1 ] the field (1 1 {BN (i) is (p0, q)-good} ) i∈Z d stochastically dominates a family of independent Bernoulli random variables with parameter p.

Proof. Obviously, the states of (B N (i)) i∈Z d have a finite range of dependance and are identically distributed. Then, by the Liggett-Schonmann-Stacey Theorem [START_REF] Liggett | Domination by product measures[END_REF], it is sufficient to check that lim N →+∞ P(B N good) = 1, for B N = B N (0). Consider first the properties (i) and (ii) of the Definition 3.1, that depend only on p 0 . For any p 0 > p c (d), when d ≥ 3, the fact that lim N →+∞ P(B N satisfies (i) and (ii)) = 1 follows from the Pisztora coarse graining argument (see Pisztora [START_REF] Pisztora | Surface order large deviations for Ising, Potts and percolation models[END_REF] or the coarse graining section in Cerf [START_REF] Cerf | The Wulff crystal in Ising and percolation models[END_REF]), see also for instance Grimmett [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] Lemma (7.104). When d = 2, see Couronné and Messikh [START_REF] Couronné | Surface order large deviations for 2D FKpercolation and Potts models[END_REF]. We now study the property (iii) in the Definition 3.1, that also depends only on p 0 . Let us define the property (iii') by (iii') For all x, y ∈ B ′ N , if xy ∞ ≥ N and x and y belong to the crossing p 0 -cluster of B N , then D C ′ p 0 (x, y) ≤ 3βN . Notice that if x and y belong to B ′ N , there exists z ∈ B ′ N such that xz ∞ ≥ N and yz ∞ ≥ N , thus if B N satisfies property (iii') it also satisfies property (iii). Using Antal and Pisztora's estimate [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF], for any fixed p 0 > p c (d), we have for all N P[B N does not satisfy (iii)]

≤ P[B N does not satisfy (iii')] ≤ x∈B ′ N y∈B ′ N ½ x-y ∞≥N P x C ′ p 0 ←→ y , D C ′ p 0 (x, y) ≥ 3βN ≤ x∈B ′ N y∈B ′ N ½ x-y ∞≥N P x C ′ p 0 ←→ y , D C ′ p 0 (x, y) ≥ β x -y ∞ ≤ x∈B ′ N y∈B ′ N ½ x-y ∞≥N Âe -B x-y ∞ ≤ (3N ) d .(3N ) d Âe -BN
that goes to 0 when N goes to infinity. The delicate part of the proof is the study of the property (iv) in the Definition 3.1. For q = p 0 , we are done since property (iv) is implied by the uniqueness of the p 0 -crossing cluster in B ′ N . We want to deduce from this that property (iv) is asymptotically typical. We follow the proof of Russo's formula, see for instance Theorem 2.25 in [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]. For given parameters p c (d) < p 0 < p ≤ 1, we denote by P p0,p the probability of the corresponding coupled Bernoulli percolation, and we declare that

• an edge e is in state 0 if e is p-closed,

• an edge e is in state 1 if e is p 0 -closed and p-open,

• an edge e is in state 2 if e is p 0 -open. We define A N as the event that there exists a crossing cluster C of edges of state 2 in B ′ N , and a path γ ⊂ B ′ N of edges of state 1 or 2 such that |γ| = N and γ does not intersect C. Let us fix p 0 . When p vary, the edges of state 2 remain unchanged, we only change the state of edges from 0 to 1 and conversely. For a given p 0 , the event A N is increasing in p. We denote by N (A N ) the random number of edges that are 0 -1-pivotal for A N , i.e., the number of edges e such that if e is in state 1 then A N occurs, and if e is in state 0 then A N does not occur. Following the proof of Russo's formula, we obtain that

∂ ∂p P p0,p (A N ) = 1 p E p0,p [N (A N ) | A N ] P p0,p (A N ) .
We remark that when A N occurs, N (A N ) ≤ N , the length of the desired path, thus

E p0,p [N (A N ) | A N ] = E p0,p [½ AN N (A N ) | A N ] ≤ N .
We obtain that

P p0,q (A N ) = P p0,p0 (A N ) exp q p0 1 p E p0,p [N (A N ) | A N ] dp ≤ P p0,p0 (A N ) exp N q p0 1 p dp ≤ P p0,p0 (A N ) exp N log q p 0 ≤ P p0,p0 (A N ) exp N log 1 + q -p 0 p 0 (13)
It comes from the coarse graining arguments previously cited to study property (i) that P p0,p0 (A N ) decays exponentially fast with N : there exists κ 1 (p 0 ), κ 2 (p 0 ) such that [START_REF] Garet | Moderate deviations for the chemical distance in Bernoulli percolation[END_REF] P p0,p0 (A N ) ≤ κ 1 (p 0 )e -κ2(p0)N .

Part(ii) of Lemma 3.5: When p 0 is fixed, combining ( 13) and ( 14) is enough to conclude that there exists δ 1 (p 0 ) > 0 such that if q 1 < p 0 + δ 1 , then [START_REF] Gold | Isoperimetry in supercritical bond percolation in dimensions three and higher[END_REF] lim

N →∞ P p0,q1 (A N ) = 0 .
We conclude that for every p 0 > p c (d), there exists δ 1 (p 0 ) > 0 such that if q 1 ∈ [p 0 , 1] satisfy q 1p 0 ≤ δ 1 , then for every p < 1 there exists an integer N ′ (p 0 , q 1 , p) such that for q = q 1 the field (1 1 {BN (i) is (p0, q1)-good} ) i∈Z d stochastically dominates a family of independent Bernoulli random variables with parameter p. The only property of a good block that depends on q is property (iv), and if p 0 ≤ q ≤ q 1 then any q-open path is also a q 1 -open path, thus if a block is (p 0 , q 1 )-good then it is (p 0 , q)good for any parameter q ∈ [p 0 , q 1 ] . We conclude that for N ′ = N ′ (p 0 , q 1 , p), for any q ∈ [p 0 , q 1 ], the field (1 1 {B ′ N (i) is (p0, q)-good} ) i∈Z d stochastically dominates a family of independent Bernoulli random variables with parameter p.

Part(i) of Lemma 3.5:

If q is fixed, we need to replace ( 14) by a control on P p0,p0 (A N ) which is uniform for p 0 in a left neighborhood of q. Let us have a closer look at the proof of [START_REF] Garet | Moderate deviations for the chemical distance in Bernoulli percolation[END_REF]. In dimension d ≥ 3, we refer to the proof of Lemma 7.104 in Grimmett [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] : the constants κ 1 (p 0 ), κ 2 (p 0 ) of ( 14) appearing in Grimmett's book are explicit functions of the parameters δ(p 0 ) and L(p 0 ) chosen in Lemma 7.78 in [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF]. The probability controlled in Lemma 7.78 in [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF] is clearly non decreasing in the parameter p of the percolation, thus the choice of δ(p) and L(p) made for a given p > p c (d) can be kept unchanged for any p ′ ≥ p. Fixing p ′ 0 = (qp c (d))/2, we obtain that for any p 0 ∈ [p ′ 0 , q], (16)

P p0,p0 (A N ) ≤ κ 1 (p ′ 0 )e -κ2(p ′ 0 )N .
Combining ( 13) and ( 16) we can conclude that in dimension d ≥ 3, when q is fixed, there exists δ 0 (q) such that if p 0 ∈ [p ′ 0 , q] satisfies qp 0 ≤ δ 0 , then (15) still holds. In dimension 2, ( 14) is obtained by Couronné and Messikh [START_REF] Couronné | Surface order large deviations for 2D FKpercolation and Potts models[END_REF], Theorem 9, in a more general setting. The constants appearing in this theorem are explicit functions of the constants appearing in Proposition 6 in [START_REF] Couronné | Surface order large deviations for 2D FKpercolation and Potts models[END_REF], and the same remark as in dimension d ≥ 3 leads to the uniform control [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF], and the proof is complete.

We can now use Lemma 3.5 to bound the probability that C∈Bad: C∩Γ =∅ |C| is big. Denote by Animals the set of lattice animals containing 0, and Animals n the subset of those having size n. Lemma 3.6. Let ε > 0. Let p site c (d) be the critical parameter for independent Bernoulli site percolation on Z d . Choose α = α(ε) > 0 and then p = p(ε) ∈ (p site c (d), 1), such that

7 d exp(-αε) ≤ 1 3 , ( 17
) p + e α 7 d (1 -p) 1 -e α 7 d (1 -p) ≤ 3 2 . ( 18
)
For a given q > p c (d) (resp. p 0 > p c (d)), for a fixed p 0 ∈ (p c (d), q] such that qp 0 ≤ δ 0 (q) (resp. q 1 ≥ p 0 such that q 1p 0 ≤ δ 1 (p 0 ) and any q ∈ [p 0 , q 1 ]), let finally N = N (p 0 , q, p(ε)) (resp. N = N ′ (p 0 , q 1 , p(ε))) be large enough to have the stochastic comparison of Lemma 3.5 with this parameter p(ε). Then for all m ∈ N, we have

P   ∃Γ ∈ Animals, |Γ| ≥ m N , C∈Bad: C∩Γ =∅ |C| ≥ ε|Γ|   ≤ e -m N +1 .
Proof. We have

P(m) def = P   ∃Γ ∈ Animals, |Γ| ≥ m N , C∈Bad: C∩Γ =∅ |C| ≥ ε|Γ|   ≤ n≥ m N Γ∈Animalsn P   C∈Bad: C∩Γ =∅ |C| ≥ ε|Γ|   ≤ n≥ m N Γ∈Animalsn P p   C∈Bad: C∩Γ =∅ |C| ≥ ε|Γ|   .
For the last inequality, we use the coupling Lemma 3.5 to replace the locally dependent states of our N -boxes by an independent Bernoulli site percolation with parameter p chosen in [START_REF] Kesten | Aspects of first passage percolation[END_REF]; by analogy, we still denote by Bad the random set of closed connected components for this independent Bernoulli site percolation. From now on, we work with this Bernoulli site percolation with parameter p. Denote by C(0) the connected component of closed sites containing 0 (with the convention that if 0 is open, then C(0) = ∅). Let ( C(i)) i∈Z d be independent and identically distributed random sets of Z d with the same law as C(0). Fix a set Γ = (Γ(i)) 1≤i≤n of sites; we first prove that, for the independent Bernoulli site percolation, the following stochastic comparison holds:

(19)

C∈Bad: C∩Γ =∅ |C| n i=1 | C(i)|.
The idea is to build algorithmically the real clusters from the sequence of preclusters ( C(i)) i∈Z d , as in the work of Fontes and Newman [START_REF] Fontes | First passage percolation for random colorings of Z d[END_REF], proof of Theorem 4. Note however that in our sum [START_REF] Kesten | Percolation theory for mathematicians, volume 2 of Progress in Probability and Statistics[END_REF], each visited cluster is only counted once, while they count each cluster the number of times it is visited, which explains the difference between our stochatic domination and their one. We proceed by induction on j ∈ {1, . . . , n} to build a new family (C(i)) 1≤i≤n such that

A j def = C∈Bad: C∩{Γ(i): 1≤i≤j} =∅ C law ⊂ j i=1 C(i) ⊂ j i=1 (Γ(i) + C(i)) ,
where for two random sets A and B, A [START_REF] Antal | On the chemical distance for supercritical Bernoulli percolation[END_REF]. Assume now that (C(i)) 1≤i≤j are built for some j < n:

law ⊂ B means that the field (½ A (x), x ∈ Z d ) is stochastically dominated by the field (½ B (x), x ∈ Z d ). Set C(1) = Γ(1) + C
• if Γ(j + 1) ∈ A j , then A j+1 = A j , so we set C(j + 1) = ∅; • if Γ(j + 1) ∈ ∂ v A j (
the exterior vertex boundary of A j ), then it is a good site, so we set C(j + 1) = ∅; • otherwise, the conditional distribution of the bad cluster C containing the site Γ(j + 1), given A j , is that of the percolation cluster of Γ(j + 1) in a site percolation model where Z d is replaced by Z d \(A j ∪ ∂ v A j ); thus, it has the same law as the connected component of Γ(j + 1) in

C(j + 1)= Γ(j + 1) + C(j + 1) \ (A j ∪ ∂ v A j ) ,
which ends the construction and proves [START_REF] Kesten | Percolation theory for mathematicians, volume 2 of Progress in Probability and Statistics[END_REF]. As the number of lattice animals containing 0 with size n is bounded from above by (7 d ) n (see Kesten [START_REF] Kesten | Percolation theory for mathematicians, volume 2 of Progress in Probability and Statistics[END_REF], p 82. or Grimmett [START_REF] Grimmett | Percolation, volume 321 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences[END_REF], p.85), we have, by the Markov inequality,

P(m) ≤ n≥ m N (7 d ) n exp(-αεn) (E p (exp(α|C(0)|))) n . But E p (exp(α|C(0)|)) = p + k≥1 exp(αk)P p (|C(0)| = k) ≤ p + k≥1 exp(αk)P p (|C(0)| ≥ k) ≤ p + k≥1 exp(αk)(7 d ) k (1 -p) k = p + e α 7 d (1 -p) 1 -e α 7 d (1 -p) .
With the choices ( 17) and ( 18) we made for α and p, this ensures that

P(m) ≤ n≥ m N 2 -n ≤ 2 -m N +1 .

Truncated passage times, proof of Theorem 1.6

Let G be a probability measure on [0, +∞] such that q := G([0, +∞)) > p c (d). Let δ 0 (q) be given by Lemma 3.5. Fix M 0 large enough so that p 0 := G([0, M 0 ]) > p c (d) and qp 0 ≤ δ 0 . For a K ∈ [M 0 , +∞), define p = p(K) = G([0, K]). We define the following i.i.d. Bernoulli bond percolations :

• an edge e is declared p 0 -open if and only if t G (e) ≤ M 0 ,

• an edge e is declared p-open if and only if t G (e) ≤ K,

• an edge e is declared q-open if and only if t G (e) < ∞.

These percolations are naturally coupled, thus we can use the modification of paths presented in the previous section. Denote as before by C G,M0 the a.s. unique infinite cluster of the supercritical Bernoulli field {½ tG(e)≤M0 : e ∈ E d }. We call this field the M 0 -percolation and its clusters the M 0 -clusters. They correspond exactly to the p 0 -percolation and the p 0 -clusters.

4.1. Estimation for the passage time of the modified path.

Lemma 4.1.

There exists a positive constant ρ ′ d (depending only on d and M 0 ) such that the following holds: Assume that y ∈ C G,M0 , that z ∈ C G,M0 , that the N -boxes containing y and z are good and belong to an infinite cluster of good boxes. Then for every

K ≥ M 0 , T G (y, z) ≤ T G K (y, z) 1 + ρ ′ d N d K + ρ ′ d N C∈Bad: C∩Γ =∅ |C|,
where Γ is the lattice animal of N -boxes visited by an optimal path between y and z for the passage times with distribution G K . Remark 4.2. In this Lemma, Γ can be the lattice animal of N -boxes visited by any arbitrary geodesic between y and z for the passage times with distribution G K if there is more than one.

Proof. As y ∈ C G,M0 and z ∈ C G,M0 , the quantities T G (y, z) and T G K (y, z) are bounded by M 0 times the chemical distance in C G,M0 between y and z, and are thus finite. Let γ be an optimal path between y and z for T G K (y, z). We can consider the modification γ ′ given by Lemma 3.2. Since γ ′ is a path between y and z, and γ ′ γ is p 0 -open, we have

T G (y, z) ≤ e∈γ ′ t G (e) = e∈γ∩γ ′ t G (e) + e∈γ ′ \γ t G (e) ≤ e∈γa t G (e) + M 0 |γ ′ \ γ| .
On one hand, since γ is an optimal path between y and z for T G K (y, z), we have

e∈γa t G (e) = e∈γa t G K (e) ≤ e∈γ t G K (e) = T G K (y, z) .
On the other hand, using the estimate on the cardinality of γ ′ \γ given in Lemma 3.2, and noticing that the number of edges in γ b is less than T G K (γ)/K, we obtain

|γ ′ \ γ| ≤ ρ d   N d T G K (γ) K + N C∈Bad: C∩Γ =∅ |C|   . Lemma 4.3. Suppose that G({0}) < p c (d).
For every ε > 0 there exists p 1 (ε) > 0 and A(ε) > 0 such that for every K ≥ M 0 , for all x large enough,

P T CG,M 0 G (0, x) ≤ T CG,M 0 G K (0, x) 1 + A(ε) K + ε x 1 ≥ p 1 (ε).
Proof. Let ε > 0 be fixed. Let p site c (d) be the critical parameter for independent Bernoulli site percolation on Z d . Choose α = α(ε) > 0 and then p = p(ε) ∈ (p site c (d), 1), such that

7 d exp(-αε) ≤ 1 3 , ( 20
) p + e α 7 d (1 -p) 1 -e α 7 d (1 -p) ≤ 3 2 (21)
as in Lemma 3.6. Let N = N (p 0 , q, p(ε)) be large enough to have the stochastic comparison of Lemma 3.5 with this parameter p(ε). Let K ≥ M 0 . Fix a large x, at least large enough so that x 1 ≥ 12dN .

Let F x be the following good event: the N -boxes containing 0 and x and all the adjacent boxes belong to an infinite cluster of good boxes (in particular they are good). For any y in the same (3N )-box as 0, for any z in the same (3N )-box as x, let E y,z be the event that y ∈ C G,M0 , z ∈ C G,M0 and the N -boxes containing y and z and belong to an infinite cluster of good boxes (in particular they are good). For any such (y, z), we have yz 1 ≤ x 1 + 6dN ≤ 2 x 1 . For any given β ′′ , we have

P T CG,M 0 G (0, x) ≥ T CG,M 0 G K (0, x) 1 + ρ ′ d N (ε) d K + 4 ε Cd ρ ′ d β ′′ x 1 ≤ P[F c x ] + P F x ∩ T CG,M 0 G (0, x) ≥ T CG,M 0 G K (0, x) 1 + ρ ′ d N (ε) d K + 4 ε Cd ρ ′ d β ′′ x 1 = P[F c x ] + y,z P F x ∩ { 0 CG,M 0 = y , x CG,M 0 = z} ∩ T G (y, z) ≥ T G K (y, z) 1 + ρ ′ d N (ε) d K + 4 ε Cd ρ ′ d β ′′ x 1 ≤ P[F c x ] + y,z P E y,z ∩ T G (y, z) ≥ T G K (y, z) 1 + ρ ′ d N (ε) d K + 2 ε Cd ρ ′ d β ′′ y -z 1 , (22)
where the sum is over every y in the same (3N )-box as 0, and every z in the same (3N )-box as x -indeed, on the event F x , we know that C G,M0 intersects the box of 0 (resp. x) thus 0 CG,M 0 (resp. x CG,M 0 ) belongs to the same (3N )-box as 0 (resp.

x). Note that the stochastic comparison and the FKG inequality ensure that ( 23)

P(F x ) ≥ θ 2•3 d site,p (
ε) > 0 , where θ site,p(ε) denotes the density of the infinite cluster in a supercritical vertex i.i.d. Bernoulli percolation of parameter p(ε).

Consider a couple (y, z) as in [START_REF] Liggett | Domination by product measures[END_REF]. On the event E y,z , we have

T G K (y, z) ≤ M 0 D CG,M 0 (y, z) < ∞ .
Let γ y,z be a geodesic for T G K (y, z), and let Γ y,z be the lattice animal of the Nboxes visited by this geodesic. By Lemma 4.1, we have

P E y,z ∩ T G (y, z) ≥ T G K (y, z) 1 + ρ ′ d N (ε) d K + 2 ε Cd ρ ′ d β ′′ y -z 1 ≤ P   E y,z ∩    C∈Bad: C∩Γy,z =∅ |C| ≥ 2 ε Cd β ′′ y -z 1 N (ε)      .
Note that by construction, on the event E y,z , we have

|Γ y,z | ≥ y -z 1 /N . On the other hand Lemma 3.4 implies that |Γ y,z | ≤ Cd (1+(|γ y,z |+1)/N ) ≤ 2 Cd |γ y,z |/N at least for x large enough (remember that |γ y,z | ≥ y -z 1 ≥ x 1 -6dN ≥ x 1 /2).
Thus we obtain

P E y,z ∩ T G (y, z) ≥ T G K (y, z) 1 + ρ ′ d N (ε) d K + 2 ε Cd ρ ′ d β ′′ y -z 1 ≤ P (E y,z ∩ {|γ y,z | > β ′′ y -z 1 }) + P   E y,z ∩    C∈Bad: C∩Γy,z =∅ |C| ≥ ε|Γ y,z |      . ( 24
)
Since G K ({0}) = G({0}) < p c (d), there exist positive constants A ′ , B ′ , β ′ such that for all k ∈ N * (see Proposition 5.8 in Kesten [START_REF] Kesten | Aspects of first passage percolation[END_REF]):

G K (r) ≤ β ′ k] ≤ A ′ exp(-B ′ k).
Let β be given by Antal and Pisztora's estimate [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF]. By [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF] we have

P(E y,z ∩ {D CG,M 0 (y, z) ≥ β y -z 1 }) ≤ P(β y -z 1 ≤ D C ′ G,M 0 (y, z) < +∞) ≤ Â exp(-B y -z 1 ) . ( 26
)
Fix β ′′ = βM0 β ′ > 0. Combining ( 25) and ( 26) we obtain the existence of positive constants A ′′ , B ′′ such that

P (E y,z ∩ {|γ y,z | > β ′′ y -z 1 }) ≤ P(E y,z ∩ {D CG,M 0 (y, z) ≥ β y -z 1 }) + P (E y,z ∩ {T G K (y, z) ≤ M 0 β y -z 1 } ∩ {|γ y,z | > β ′′ y -z 1 }) ≤ Âe -B y-z 1 + A ′ e -B ′ β ′′ y-z 1 ≤ A ′′ e -B ′′ y-z 1 . ( 27
)
By Lemma 3.6, with the choices ( 20) and ( 21) we made for α and p, we know that

P   E y,z ∩    C∈Bad: C∩Γy,z =∅ |C| ≥ ε|Γ y,z |      ≤ P   ∃Γ ∈ Animals, |Γ| ≥ y -z 1 N (ε) , C∈Bad: C∩Γ =∅ |C| ≥ ε|Γ|   = P( y -z 1 ) ≤ 2 -y-z 1 N (ε) +1 . ( 28
)
Combining ( 22), ( 23), ( 24), ( 27) and ( 28), we obtain that

P T CG,M 0 G (0, x) ≥ T CG,M 0 G K (0, x) 1 + ρ ′ d N (ε) d K + 4 ε Cd ρ ′ d β ′′ x 1 ≤ 1 -θ 2 3d site,p(ε) + y,z A ′′ e -B ′′ y-z 1 + 2 -y-z 1 N (ε) +1 ≤ 1 -θ 2 3d site,p(ε) + 2(3N (ε)) d A ′′ e -B ′′ x 1/2 + 2 -x 1 2N (ε) +1 ≤ 1 -p 1 (ε),
for a well-chosen p 1 (ε) > 0 and every x large enough.

4.2. Proof of Theorem 1.6. If G({0}) ≥ p c (d), then µ G K (x) = µ G (x) = 0
, so there is nothing to prove. Suppose from now on that G({0}) < p c (d).

For any ε > 0, consider p 1 (ε) and A(ε) as given by Lemma 4.3, and define, for

K ≥ M 0 , Ψ(K) = inf ε>0 A(ε) K + ε. It is easy to see that lim K→+∞ Ψ(K) = 0. Fix ε > 0, δ > 0, K ≥ M 0 and x ∈ Z d .
With the convergence (2) in Proposition 2.6 and Lemma 4.3, we can choose n large enough such that

P µ G (x) -δ ≤ T CG,M 0 G (0, nx) n ≥ 1 - p 1 (ε) 3 , P T CG,M 0 G K (0, nx) n ≤ µ G K (x) + δ ≥ 1 - p 1 (ε) 3 , P T CG,M 0 G (0, nx) ≤ T CG,M 0 G K (0, nx) 1 + A(ε) K + εn x 1 ≥ p 1 (ε).
For every ε > 0, for every δ > 0, on the intersection of these 3 events, that has positive probability, we obtain

∀K ≥ M 0 , x ∈ Z d µ G (x) -δ ≤ (µ G K (x) + δ) 1 + A(ε) K + ε x 1 ,
and by letting δ going to 0 we get

∀ε > 0, K ≥ M 0 , x ∈ Z d µ G (x) ≤ µ G K (x) 1 + A(ε) K + ε x 1 .
It follows that for every ε > 0,

0 ≤ µ G K (x) -µ G (x) ≤ µ G K A(ε) K + ε x 1 ≤ (µ G (x) + x 1 ) A(ε) K + ε , thus, by optimizing ε, 0 ≤ µ G K (x) -µ G (x) ≤ (µ G (x) + x 1 )Ψ(K) .
Theorem 1.6 is proved by using the fact that lim To prove Theorem 1.2, we follow the general structure of Cox and Kesten's proof of the continuity of the time constant in first-passage percolation with finite passage times in [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF]. We first deduce Theorem 1.2 from Theorem 1.6 and Lemmas 5.1 and 5.2. Lemmas 5.1 and 5.2 will be respectively proved in subsections 5.3 and 5.4. We define

G n = min{G, G n } (resp. G n = max{G, G n }),
G K = ½ [0,K) G + G([K, +∞])δ K (resp. G K n = ½ [0,K) G n + G n ([K, +∞])δ K ), the distribution of t K G (e) = min(t G (e), K) (resp. t K G n
(e) = min(t G n (e), K)). Using Lemmas 2.12 and 5.2, since

G K n d → G K , we obtain for all K lim n→∞ µ G n (x) ≥ lim n→∞ µ G K n (x) = µ G K (x) ,
and by Theorem 1.6 we have lim K→∞ µ G K (x) = µ G (x). This concludes the proof of (i), and of (29). By homogeneity, (29) also holds for all x ∈ Q d . We know that |µ Gn (x)µ Gn (y)| ≤ µ Gn (e 1 ) xy 1 , where e 1 = (1, 0, . . . , 0). Moreover lim n→∞ µ Gn (e 1 ) = µ G (e 1 ), thus for all n ≥ n 0 large enough we have |µ Gn (x)µ Gn (y)| ≤ 2µ G (e 1 ) xy 1 for all x, y ∈ R d . This implies that for any fixed ε > 0, there exists η > 0 such that for all x, y ∈ R d such that xy 1 ≤ η, we have

sup{|µ G (x) -µ G (y)|, |µ Gn (x) -µ Gn (y)|, n ≥ n 0 } ≤ ε .
There exists a finite set (y 1 , . . . , y m ) of rational points of R d such that

S d-1 ⊂ m i=1 {x ∈ R d : y i -x 1 ≤ η} .
A i < +∞ such that G i (A i ) < 1 -p c (d). Fix A ′ = max(A, A 0 , . . . , A n0-1 ) < +∞. We conclude that Ĥ+ (A ′ ) = max G 0 (A ′ ), . . . , G n0-1 (A ′ ), sup n≥n0 G n (A ′ ) ≤ max G 0 (A 0 ), . . . , G n0-1 (A n0-1 ), sup n≥n0 G n (A) < 1 -p c (d) , thus H + ([0, +∞)) = 1 -lim +∞ H + > p c (d).
(ii) We define H -= inf n∈N G n . Then H -is non-increasing, defined on [0, +∞) and it takes values in [0, 1]. Fix t 0 ∈ [0, +∞). Let us prove that H -is left continuous at t 0 . By definition of H -, for any ε > 0, there exists n 0 such that H -(t 0 ) ≥ G n0 (t 0 )-ε. Since G n0 is left continuous, there exists η > 0 such that for all t ∈ (t 0η, t 0 ] we have G n0 (t) ≤ G n0 (t 0 ) + ε. Thus for all t ∈ (t 0η, t 0 ], we obtain

H -(t) ≤ G n0 (t) ≤ G n0 (t 0 ) + ε ≤ H -(t 0 ) + 2ε , thus H -is right continuous. By construction H -≤ G n ,
for all n ∈ N. Moreover H -(t) = 1 for all t ≤ 0. Thus there exists a probability measure H -on [0, +∞] such that H -(t) = H -([t, +∞]) for all t ∈ [0, +∞). It remains to prove that H -({0}) < p c (d). Since G({0}) < p c (d), there exists η > 0 such that G([0, η)) < p c (d), i.e., G(η) > 1p c (d). Let ε > 0 such that G(η) ≥ 1p c (d) + 2ε. There exists δ ∈ [0, η) such that G is continuous at δ. Then lim n→∞ G n (δ) = G(δ), thus there exists n 0 such that for all n ≥ n 0 , G n (δ) ≥ G(δ)ε ≥ 1p c (d) + ε. For any i ∈ {1, . . . , n 0 -1}, there exists δ i > 0 such that G i (δ i ) > 1p c (d). Fix δ ′ = min(δ, δ 0 , . . . , δ n0-1 ) > 0. We conclude that

H -(δ ′ ) = min G 0 (δ ′ ), . . . , G n0-1 (δ ′ ), inf n≥n0 G n (δ ′ ) ≥ min G 0 (δ 0 ), . . . , G n0-1 (δ n0-1 ), inf n≥n0 G n (δ) > 1 -p c (d) ,
and

H -({0}) = 1 -lim t→0,t>0 H -(t) ≤ 1 -H(δ ′ ) < p c (d) .
5.3. Proof of Lemma 5.1. We follow the structure of Cox and Kesten's proof of Lemma 1 in [START_REF] Cox | On the continuity of the time constant of first-passage percolation[END_REF].

We take H + as given in Lemma 5.3 (i), and we fix M ∈ [0, +∞) such that H + ([0, M ]) > p c (d). We work with the stabilized points x C H + ,M . We consider a point x ∈ Z d , and k ∈ N * . For any path r from 0 C H + ,M to kx C H + ,M , using Lemma 2.10 we have a.s.

T G (r) = e∈r t G (e) = lim n→+∞ e∈r t Gn (e) ≥ lim n→+∞ T C H + ,M Gn (0, kx) .
Taking the infimum over any such path r, we obtain

T C H + ,M G (0, kx) ≥ lim n→+∞ T C H + ,M Gn (0, kx) .
Conversely, since G G n , thanks to the coupling of the laws we get T

C H + ,M G (0, kx) ≤ T C H + ,M Gn
(0, kx) for all n, thus ∀k ∈ N * , a.s., lim

n→∞ T C H + ,M Gn (0, kx) = T C H + ,M G (0, kx) .
Since for all n we have T

C H + ,M Gn (0, kx) ≤ T C H + ,M H +
(0, kx) that is integrable by Proposition 2.5, the dominated convergence theorem implies that, for all k ∈ N * , (30) lim

n→∞ E T C H + ,M Gn (0, kx) = E T C H + ,M G (0, kx) . By Lemma 2.11, we know that µ G (x) = inf k∈N * E T C H + ,M G (0, kx) /k. For any ε > 0, there exists K(G, ε) such that (31) µ G (x) ≥ E T C H + ,M G (0, Kx) K -ε ,
and using (30) we know that there exists n 0 (ε, K) such that for all n ≥ n 0 we have (32)

E T C H + ,M G (0, Kx) K ≥ E T C H + ,M Gn (0, Kx) K -ε . Since µ Gn (x) = inf k∈N * E T C H + ,M Gn
(0, kx) /k, combining equations (31) and (32), we obtain that for any ε > 0, for all n large enough,

µ G (x) ≥ µ Gn (x) -2ε .
This concludes the proof of Lemma 5.1.

A ∈ N * , for all C ∈ [0, +∞), P[T G n (0, kx) ≤ T G (0,
P[T G (0, kx) > ACk] = 0 .
If we prove that there exists n 0 (G, (G n ), ε) such that for all n ≥ n 0 ,

(2d) Ak P Ak i=1 t G (e i ) -t G n (e i ) ≥ εk < +∞ , (33) k>0
then for all n ≥ n 0 we have k P[T G n (0, kx) ≤ T G (0, kx)εk] < +∞. By Borel-Cantelli's lemma we obtain that for all n ≥ n 0 , a.s., for all k ≥ k 0 (n) large enough, We choose α(ε) large enough so that 2d ≤ exp αε 4A , and then n 0 (G, (G n ), ε) large enough so that for all n ≥ n 0 , we have

T G n (0, kx) > T G (0, kx) -εk , thus for all n ≥ n 0 we get µ G n (x) ≥ µ G (x) -ε . We conclude that lim n→∞ µ G n (x) ≥ µ G (x
E exp α(t G (e) -t G n (e)) ≤ exp αε 4A .
Thus for all n ≥ n 0 , we have The definition of the objects used in this section are given in section 2.2. The main step in the proof of Theorem 1.1 is the following lemma: Lemma 6.1. For every p > p c (2), lim

(2d) Ak P Ak i=1 t G (e i ) -t G n (e i ) ≥ εk ≤ exp - αε 2A , so (
p ′ →p sup x∈S 1 |β p ′ (x) -β p (x)| = 0 . Proof. Let x ∈ S 1 . Let p c (2) < p 0 ≤ p ≤ q
, and define δ = qp. We couple the percolations with different parameters in the usual way using uniform variables. We extend the definition of ỹC to any y ∈ R d . For a path r ∈ R(x, y), let us define b p (r) = |{e ∈ ∂ + r : e is p -open}|. For x, y ∈ C p , we define b p (x, y) = inf{b p (r) : r ∈ R(x, y), r is p -open}.

Step (i). By Lemma 2.2 there exist C, C ′ , α > 0 (depending on p 0 ) such that ∀p ≥ p 0 , ∀n, (34)

P ∃γ ∈ x∈Z 2 R(0, x) : |γ| > n , b p (γ) ≤ αn ≤ Ce -C ′ n .
Let F p0 be the event {0 ∈ C p0 } ∩ {nx ∈ C p0 }. On the event F p0 , by [4, Lemma 2.5], we have b p (0, nx) ≤ 3D Cp (0, nx) ≤ 3D Cp 0 (0, nx), thus using [START_REF] Garet | Asymptotic shape for the chemical distance and firstpassage percolation on the infinite Bernoulli cluster[END_REF] we know that there exist positive constants Â, B, β (depending only on p 0 ) such that for all p > p 0 , for all x ∈ S 1 , (35)

P [F p0 ∩ {b p (0, nx) ≥ 3βn}] ≤ Â exp(-Bn) .
For any p-open path γ, γ is also q-open. However some additional right-boundary edges may be open. To bound the difference between b q (γ) and b p (γ), note that if |γ| < α ′ n by [4, Lemma 2.5] |∂ + γ| < 3α ′ n. We can bound b q (γ)b p (γ) by Cramér's theorem. For every fixed path γ such that |γ| < α ′ n, for every ε > 0 and δ < ε,

(36) P [b q (γ) -b p (γ) > 3εα ′ n] ≤ e -3α ′ n(ε log ε δ +(1-ε) log 1-ε 1-δ ) .
Fix α ′ = 3β/α. Since there are at most 4 α ′ n paths of length smaller than α ′ n containing 0, we obtain that for all p 0 ≤ p < q, for all x ∈ S 1 ,

P b q ( 0Cq , nx Cq ) > b p ( 0Cp , nx Cp ) + 3εα ′ n ≤ P[F c p0 ] + Q p [F p0 ∩ {b p (0, nx) > 3βn}] + P [F p0 ∩ {∃γ ∈ R(0, nx) : |γ| > α ′ n , b p (γ) ≤ 3βn}] + P [F p0 ∩ {∃γ ∈ R(0, nx) : |γ| ≤ α ′ n , b q (γ) > b p (γ) + 3εα ′ n}] ≤ (1 -θ 2 p0 ) + Âe -Bn + Ce -C ′ α ′ n + 4 α ′ n e -3α ′ n(ε log ε δ +(1-ε) log 1-ε 1-δ ) .
For every p 0 > p c (d), for every ε > 0, there exists δ(ε) > 0 and p 2 (ε) > 0 such that for every x ∈ S 1 , for every p 0 ≤ p < q satisfying qp < δ, we have

P b q ( 0Cq , nx Cq ) > b p ( 0Cp , nx Cp ) + 3εα ′ n ≤ 1 -p 2 (ε) ,
thus for every p 0 > p c (d), for every ε > 0, there exists δ(ε) > 0 such that for every x ∈ S 1 , for every p 0 ≤ p < q satisfying qp < δ, we have

β q (x) < β p (x) + 3α ′ ε.
Step (ii). Given a q-open path γ, γ may not be p-open. Thus we use the results of Section 3 to modify the path to a p-open path which does not gain too many extra right-boundary edges. We mimic the proof of Lemma 4.3. Fix ε > 0. Choose α(ε) and p(ε) as in ( 17) and [START_REF] Kesten | Aspects of first passage percolation[END_REF].

• If p is fixed and we let q goes to p, we choose p 0 = p, q 1 ∈ (p 0 , p 0 + δ 1 (p)) as defined in Lemma 3.5, and we consider only values of q such that q ∈ [p 0 , q 1 ]. Then we choose N = N ′ (p 0 , q 1 , p(ε)) as given in Lemma 3.5.

• If q is fixed and we let p goes to q, we choose p 0 ∈ (p c (d), q] such that qp 0 ≤ δ 0 (q) as defined in Lemma 3.5, and we consider only values of p in the interval [p 0 , q]. Then we choose N = N (p 0 , q, p(ε)) as given in Lemma 3.5. Let x ∈ S 1 , we denote by ⌊nx⌋ the point y of Z d which minimizes nxy 1 (with a deterministic rule to break ties). Let F ′ be the following good event: the N -boxes containing 0 and ⌊nx⌋ and all the adjacent boxes are good and belong to an infinite cluster of good boxes. By the FKG inequality and the stochastic comparison, we have (37) P(F ′ ) ≥ θ 18 site,p(ε) . Fix α ′′ = 6β/α = 2α ′ as defined in step (i). We have

P b p ( 0Cp 0 , ⌊nx⌋ Cp 0 ) > b q ( 0Cp 0 , ⌊nx⌋ Cp 0 ) + 12α ′′ Cd ρ d εn ≤ P[F c ] + y,z P E y,z ∩ {b p (y, z) > b q (y, z) + 12α ′′ Cd ρ d εn} , ( 38
)
where the sum is over every y in the same (3N )-box as 0 and z in the same (3N )box as ⌊nx⌋, and E y,z is the event that y ∈ C p0 , z ∈ C p0 and the N -boxes containing y and z are good and belong to an infinite cluster of good boxes. For any such (y, z), on E y,z , let γ y,z ∈ R(y, z) be a q-open right-most path from y to z such that b q (y, z) = b q (γ y,z ), and let Γ y,z be the lattice animal of N -boxes it visits. For short, we write γ for γ y,z . As previously we define Note that b p (γ ′ ) ≤ b q (γ ′ ) ≤ b q (γ) + 3|γ ′ \ γ|.

Moreover, since a simple path is also a right-most path, we have for all y, z ∈ C p0 , (39) b q (y, z) ≤ 3D Cp 0 (y, z) .

Using Equation (39), Proposition 2.3, Antal and Pizstora's estimate (10), Cramér's theorem again and Lemma 3.6, for all x ∈ S 1 and for all n large enough (in particular such that yz 1 ≤ ⌊nx⌋ + 12N ≤ 2n and yz 1 ≥ ⌊nx⌋ -12N ≥ n/2), we have (40)

P
Combining Equations (37), (38) and (40), we conclude that for every fixed ε > 0 and every fixed p > p c (d) (thus p 0 , q 1 and N are fixed), there exists δ(ε, p) ∈ (0, q 1p] and p 3 (ε, p) > 0 such that for every q > p satisfying qp < δ, for every x ∈ S 1 , for every n large enough, we have P b p ( 0Cp , nx Cp ) > b q ( 0Cq , nx Cq ) + 12α ′′ Cd ρ d εn ≤ 1p 3 , thus for every ε > 0 and for every p > p c (d), there exists δ(ε, p) > 0 such that for every q > p satisfying qp < δ, for every x ∈ S 1 , β p (x) < β q (x) + 12α ′′ Cd ρ d ε.

Similarly, for every fixed ε > 0 and every fixed q > p c (d) (thus p 0 and N are fixed), there exists δ ′ (ε, q) ∈ (0, qp 0 ] and p 4 (ε, q) > 0 such that for every p < q satisfying qp < δ ′ , for every x ∈ S 1 , for every n large enough, we have

P b p ( 0Cp , nx Cp ) > b q ( 0Cq , nx Cq ) + 12α ′′ Cd ρ d εn ≤ 1 -p 4 ,
thus for every ε > 0 and for every q > p c (d), there exists δ ′ (ε, q) > 0 such that for every p < q satisfying qp < δ, for every x ∈ S 1 ,

β p (x) < β q (x) + 12α ′′ Cd ρ d ε.
This ends the proof of Lemma 6.1. By Lemma 6.1 for every ε > 0 there exists a δ > 0 such that for every q > p c (2) satisfying |p -q| < δ we have sup The infimum in Theorem 2.3 is achieved (by compactess of the set of Lipschitz curves), so let us denote by λ p (resp. λ q ) a Jordan curve such that Leb(int(λ p )) = 1 and len βp (λ p ) = √ 2θ p lim n→∞ nϕ n (p) (resp. Leb(int(λ q )) = 1 and len βq (λ q ) = √ 2θ q lim n→∞ nϕ n (q)). All norms in R 2 are equivalent thus we know that len • 2 (λ p ) < ∞ and len • 2 (λ q ) < ∞. From (41) we deduce that for every ε > 0 there exists δ > 0 such that if |p -q| < δ then √ 2θ p lim n→∞ nϕ n (p) = len βp (λ p ) ≥ len βq (λ p )εlen • 2 (λ p ) ≥ √ 2θ q lim n→∞ nϕ n (q)εlen • 2 (λ p ) (42) and √ 2θ p lim n→∞ nϕ n (p) ≤ len βp (λ q ) ≤ len βq (λ q ) + εlen • 2 (λ q ) ≤ √ 2θ q lim n→∞ nϕ n (q) + εlen • 2 (λ q ) . (43) Let β min q = inf x∈S 1 β q (x), for all q. By Lemma 6.1 again we know that for every q satisfying |p -q| < δ we have β min q ≥ β min p ε, which is positive for ε small enough (β min p is not zero since β p is a norm), thus len • 2 (λ q ) ≤ len βq (λ q ) β min q ≤ len βq (λ q ) β min As previously let δ > 0 satisfy sup x∈S 1 |β q (x)β p (x)| < ε for all q > p c (2) such that |p -q| < δ. For every x ∈ W q we have by definition of W q that for every n ∈ S 1 , n • x ≤ β q (n). Thus for all q > p c (2) such that |p -q| < δ,

Proof of Theorem

n • x ≤ β q (n) ≤ β p (n) + ε ≤ (1 + η)β p (n),
where the last inequality comes from (45), thus x ∈ (1 + η)W p . We obtain that for all p > p c [START_REF] Benjamini | On the mixing time of a simple random walk on the super critical percolation cluster[END_REF], for all η > 0, there exists δ > 0 such that for every q > p c (2) satisfying |p -q| < δ, (46) W q ⊂ (1 + η)W p .

For every q > p c (2) satisfying |p -q| < δ, we also have β min q ≥ β min p ε ≥ β min p /2 ≥ ε/η by (45), thus by the same method we obtain that for every x ∈ W p , for every n ∈ S 1 , n • x ≤ β p (n) ≤ β q (n) + ε ≤ (1 + η)β q (n), thus (47) W p ⊂ (1 + η)W q .

For every x ∈ W p , x 2 = x•x/ x 2 ≤ β p (x) ≤ β max p , where β max p = sup x∈S 1 β p (x) < ∞, thus (1 + η)xx 2 ≤ ηβ max p . Similarly, for all q > p c (2) satisfying |p -q| < δ, x 2 ≤ β max q ≤ 2β max p and (1 + η)xx 2 ≤ 2ηβ max p . With (46) and (47), we conclude that for every p > p c (2), for every η > 0, there exists δ > 0 such that for every q > p c (2) satisfying |p -q| < δ, d H (W p , W q ) ≤ 2ηβ max p , thus lim q→p d H (W p , W q ) = 0. This implies that lim q→p Leb(W q ) = Leb(W p ), and since

W p = Wp √ Leb(Wp)
we deduce from ( 46) and (47) by a similar argument that lim q→p d H ( W p , W q ) = 0. This concludes the proof of Theorem 1.1.

Remark 6.2. To deduce the continuity of the

Theorem 1 . 1 .

 11 For d = 2, the applications p ∈ (p c (2), 1] → lim n→+∞ nϕ n (p) and p ∈ (p c (2), 1] → W p are continuous, the last one for the Hausdorff distance between non-empty compact sets of R 2 .

Figure 1 . 2 .

 12 Figure 1. A right most path

  ∀x, y ∈ Z d , D C (x, y) = inf{|r| : r is a path from x to y , r ⊂ C} , where inf ∅ = +∞. The event that the vertices x and y are connected in C is denoted by {x C ←→ y}. Then, for any x, y ∈ Z d , T CG,M G (x, y) ≤ M D CG,M ( x CG,M , y CG,M ) .

  and we denote by G n (resp. G n ) the corresponding probability measure on [0, +∞]. Notice that G n d → G and G n d → G. Fix an edge e. Then G n G for all n, and (i) implies that a.s. lim n→∞ t Gn (e) = t G (e) .

2. 6 .

 6 Stabilization of the point x for the Cheeger constant. Concerning the Cheeger constant, we need a stabilization result similar to Lemma 2.11. For a path r ∈ R(x, y), let us define b p (r) = |{e ∈ ∂ + r : e is p -open}|. For x, y ∈ C p , we define b p (x, y) = inf{b p (r) : r ∈ R(x, y), r is p -open}.

  Finally, since x and z 1 belong to the crossing p 0 -cluster of B N (j) = B N (ϕ 1 ), there exists a p 0 -open path from x to z 1 of length at most |{e ∈ B ′ N (j)}| ≤ 2d 3 d N d . The same holds for z r and y in B N (k) = B N (ϕ r ). By glueing these paths, we obtain a p 0 -open path from x to y of length at most equal to 3βN 3 d n + 4d 3 d N d .

Figure 2 .

 2 Figure 2. Construction of the path γ ′ -step 1.

( 25 )

 25 P [∃r s.a. path starting at y s.t. |r| ≥ k and T

5 . 2 We first state two properties: Lemma 5 . 1 .Lemma 5 . 2 .

 525152 Continuity of the time constant, proof of Theorem 1.Suppose that G, (G n ) n∈N are probability measures on[0, +∞] such that G([0, +∞)) > p c (d) and G n ([0, +∞)) > p c (d) for all n ∈ N. If G n d → G and G n G for all n, then ∀x ∈ Z d , lim n→+∞ µ Gn (x) ≤ µ G (x) . Suppose that G, (G n ) n∈N are probability measures on [0, R] for some common and finite R ∈ [0, +∞). If G n d → G, then ∀x ∈ Z d , lim n→∞ µ Gn (x) = µ G (x) .

5. 1 .

 1 Proof of Theorem 1.2. Let G, (G n ) n∈N be probability measures on [0, +∞]. We first prove that for all fixed x ∈ Z d , we have (29) lim n→∞ µ Gn (x) = µ G (x) .

1 |µ 5 . 2 .Lemma 5 . 3 .

 15253 Gn (x)-µ G (x)| ≤ 2ε+ lim n→+∞ max i=1,...,m |µ Gn (y i )-µ G (y i )| = 2ε. Since ε was arbitrary, Theorem 1.2 is proved. Bound on sequences of probability measures. Suppose that G and (G n ) n∈N are probability measures on [0, +∞]such that G n d → G. (i) If G([0, +∞)) > p c (d) and G n ([0, +∞)) > p c (d) for all n ∈ N, then there exists a probability measure H + on [0, +∞] such that G n H + for all n and H + ([0, +∞)) > p c (d). (ii) If G({0}) < p c (d) and G n ({0}) < p c (d) for all n ∈ N, then there exists a probability measure H -on [0, +∞] such that G n H -for all n and H -({0}) < p c (d).Proof. (i) We define Ĥ+ = sup n∈N G n , and H + (x) = inf{ Ĥ+ (y) : y < x} for all x ∈ [0, +∞). Then Ĥ+ and H + are non-increasing functions defined on [0, +∞) and they take values in [0, 1]. By construction H + is left continuous and H + ≥ G n , for all n ∈ N. Moreover we have Ĥ+ (x) = H + (x) = 1 for all x ≤ 0. Thus there exists a probability measureH + on [0, +∞] such that H + (t) = H + ([t, +∞]) for all t ∈ [0, +∞). It remains to prove that H + ([0, +∞)) > p c (d). Since G([0, +∞)) > p c (d), i.e. lim +∞ G < 1p c (d), there exist A ∈ [0, +∞) and ε > 0 such that G is continuous at A and G(A) ≤ 1p c (d) -2ε. Moreover G n d → G and G is continuous at A, thus there exists n 0 such that for all n ≥ n 0 we have G n (A) ≤ G(A) + ε ≤ 1p c (d)ε. For any i ∈ {1, . . . , n 0 -1}, G i ([0, +∞)) > p c (d) thus there exists

33) is proved. 6 .

 6 Continuity of the Cheeger constant, proof of Theorem 1.1

  γ a = {e ∈ γ : e is p-open} γ b = {e ∈ γ : e is p-closed} .By Lemma 3.2, on the event E y,z , there exists a path γ ′ with the following properties:(1) γ ′ is a path from y to z which is p-open;(2) γ ′ \ γ is a collection of simple paths (and also right-most) that intersect γ ′ ∩ γ only at their endpoints thus γ ′ is a right-most path (see[4, Lemma 2.6]); (3) |γ ′ \ γ| ≤ ρ d N d |γ b | + N C∈Bad: C∩Γ =∅ |C| .

≤≤

  E y,z ∩ {b p (y, z) > b q (y, z) + 12α ′′ Cd ρ d εn} ≤ P[E y,z ∩ {b q (y, z) > 6βn}] + P [E y,z ∩ {∃γ ∈ R(y, z) : |γ| > α ′′ n , b p0 (γ) ≤ b q (γ) ≤ 6βn}] + P E y,z ∩ {∃γ ∈ R(y, z) : γ is q-open, |γ| ≤ α ′′ n , |γ ′ \ γ| > 4α ′′ Cd ρ d εn} ≤ P β yz 1 ≤ 2βn ≤ D Cp 0 (y, z) < ∞ + Ce -C ′ α ′′ n + P ∃γ : γ starts at y, γ is q-open, |γ| ≤ α ′′ n , |γ b | > 2α ′′ Âe -B y-z 1 + Ce -C ′ α ′′ n Âe -Bn/2 + Ce -C ′ α ′′ n

  x∈S 1 |β q (x)β p (x)| < ε, thus (41)|len βp (λ)len βq (λ)| ≤ εlen • 2 (λ).

  n→∞ nϕ n (p) . Since p → θ p is continuous on (p c (2), 1], this conludes the first part of the proof.(ii) Proof of the continuity of the Wulff shape. Next we prove that p → W p is continuous for the Hausdorff distance. Fix η > 0 and p > p c[START_REF] Benjamini | On the mixing time of a simple random walk on the super critical percolation cluster[END_REF] and let ε = ε(η, p) > 0 be small enough such that

  • denotes the Euclidean inner product. The set W p is a minimizer for the isoperimetric problem associated with the norm β p , and it gives the asymptotic shape of the minimizer sets in the definition of ϕ n (p). Denote by U n (p) be the set of minimizers of ϕ n (p); then

	Proposition 2.4 (Shape theorem for the minimizers, [4] Theorem 1.8). For every
	p > p c (2), P p almost surely,
	max U∈Un(p)	inf ξ∈R 2 d H	U n	, ξ +	√ 2 W p	-→ n→+∞	0.
	By Proposition 2.3 and Definition (1), Theorem 1.1 will follow from the conti-
	nuity of p → β p .

  Using Lemma 2.11 the conclusion is immediate, since we have a.s.

By coupling, since G H, we have t G (e) ≤ t H (e) for every edge e.

  and we denote byG n (resp. G n ) the corresponding probability measure on [0, +∞]. Then G n ≤ G ≤ G n (resp. G n ≤ G n ≤ G n ), thus by Lemma 2.12 we have µ G n (x) ≤ µ G (x) ≤ µ G n (x).

	To conclude that (29) holds, it is sufficient to prove that
	(i)	lim n→∞	µ G n (x) ≥ µ G (x) and (ii)	lim
	Notice that G n application of Lemma 5.1. For any K ∈ [0, +∞), we define d → G and G n d → G. Inequality (ii) is obtained by a straightforward

n→+∞ µ Gn (x) ≤ µ G (x) .

  kx)εk] ≤ P ∃r s.a. path starting at 0 s.t. |r| ≥ Ak and TG n (r) ≤ ACk (e i )t G n (e i ) ≥ εk ,where (e i , i = 1, . . . , Ak) is a collection of distinct edges. Since H -({0}) < p c (d), we know that we can choose C ∈ (0, +∞) (depending on d and H) such that there exist finite and positive constants D, E (depending also on d and H) satisfying, for all k ∈ N

	+ P[T G n (0, kx) > ACk] +	r s.a. path from 0	P	e∈r	t G (e) -t G n (e) ≥ εk
		s.t. |r| ≤ Ak
	≤ P [∃r s.a. path starting at 0 s.t. |r| ≥ Ak and T H -(r) ≤ ACk] Ak
	+ P[T G (0, kx) > ACk] + (2d) Ak P	t G
		i=1

* , P [∃r s.a. path starting at 0 s.t. |r| ≥ k and T H -(r) ≤ Ck] ≤ De -Ek

(see Proposition 5.8 in

[START_REF] Kesten | Aspects of first passage percolation[END_REF]

). Since the support of G is included in [0, R] for some finite R, we know that T G (0, kx) ≤ Rk x 1 , thus we choose A large enough (depending on F , d and C) so that

  Wulff crystal from Lemma 6.1, we can also consider a more general setting. Consider β * p the dual norm of β p , defined by ∀x ∈ R d , β * p (x) = sup{x • y : β p (y) ≤ 1} . Then β * p is a norm, and what we did is equivalent to deduce from Lemma 6.1 the same result concerning β * Notice that W p , the Wulff crystal associated to β b , is in fact the unit ball associated to β * b , then (48) implies the continuity of p → W p according to the Hausdorff distance.

		p :
	(48)	lim q→p	sup x∈S 1	|β * q (x) -β * p (x)| = 0 .

Research supported by the ANR project PPPP (ANR-16-CE40-0016).

Research supported by NSF grant 1407558

Remark 5.4. The domination we use to prove (30) is free, since whatever the probability measure H + on [0, +∞] we consider, the regularized times T C H + ,M H + (0, x) are always integrable. In [START_REF] Cox | The time constant of first-passage percolation on the square lattice[END_REF], Cox considered the (non regularized) times T Gn (0, x) for probability measures G n on [0, +∞). By Lemma 5.3 it is easy to obtain T Gn (0, x) ≤ T H (0, x) for some probability measure H on [0, +∞). However, without further assumption, T H (0, x) may not be integrable. This is the reason why Cox supposed that the family (G n , n ∈ N) was uniformly integrable. In [START_REF] Cox | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF], Cox and Kesten circumvent this problem by considering some regularized passage times that are always integrable. There is no straigthtforward generalization of their regularized passage times to the case of possibly infinite passage times, but the T introduced in [START_REF] Cerf | Weak shape theorem in first passage percolation with infinite passage times[END_REF] plays the same role.

5.4. Proof of Lemma 5.2. Of course, Lemma 5.2 can be seen as a particular case of the continuity result by Cox and Kesten. But, as noted by Kesten in his Saint-Flour course [START_REF] Kesten | Aspects of first passage percolation[END_REF], the Cox-Kesten way makes use of former results by Cox in [START_REF] Cox | Some limit theorems for percolation processes with necessary and sufficient conditions[END_REF] and is not the shortest path to a proof in the compact case. In [START_REF] Kesten | Aspects of first passage percolation[END_REF] Kesten also gave a sketch of a shorter proof in the compact case. We thought the reader would be pleased to have a self-contained proof, so we present a short but full proof of Lemma 5.2, quite inspired by Kesten [START_REF] Kesten | Aspects of first passage percolation[END_REF].

Let G, (G n ) n∈N be probability measures on [0, R], and consider x ∈ Z d . As in the proof of Theorem 1.2, we have

0 and the proof is complete. We suppose from now on that µ G (x) > 0, thus x = 0. Since the passage times t G (e) are finite, it is well known that µ G (x) > 0 for x = 0 if and only if G({0}) < p c (d) (see Theorem 6.1 in [START_REF] Kesten | Aspects of first passage percolation[END_REF], or Proposition 2.7 in a more general setting). We want to prove that lim n→∞ µ G n (x) ≥ µ G (x), where G n = min(G, G n ). Notice that x CG n ,M = x CG,M = x for any M ≥ R, thus we do not need to introduce regularized times T . In what follows we note s.a. for self avoiding. Since G n d → G, we have lim n→∞ G n ({0}) ≤ G({0}) < p c (d), thus we consider only n large enough so that G n ({0}) < p c (d). Applying Lemma 5.3 (ii) to the sequence of functions G n , we obtain the existence of a probability measure H -on [0, +∞] (in fact on [0, R]) such that H - G n for all n and H -({0}) < p c (d). Thanks to the coupling, we know that t H -(e) ≤ t G n (e) ≤ t G (e) for every edge e, thus we obtain that for all