Geodesics in first-passage percolation cross any pattern

In first-passage percolation, one places nonnegative i.i.d. random variables (T (e)) on the edges of Z d. A geodesic is an optimal path for the passage times T (e). Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. Under mild conditions, we show that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic.


Introduction and main result 1.Settings
Fix an integer d ≥ 2. In this article, we consider the model of first passage percolation on the hypercubic lattice Z d . We denote by 0 the origin of Z d and by E the set of edges in this lattice. The edges in E are those connecting two vertices x and y such that x− y 1 = 1. A finite path π = (x 0 , . . . , x k ) is a sequence of adjacent vertices of Z d , i.e. for all i = 0, . . . , k − 1, x i+1 − x i 1 = 1. We say that π goes from x 0 to x k . Sometimes we identify a path with the sequence of the edges that it visits, writing π = (e 1 , ..., e k ) where for i = 1, . . . , k, e i = {x i−1 , x i }. We say that k is the length of π and we denote |π| = k.
The basic random object consists of a family T = {T (e) : e ∈ E} of i.i.d. non-negative random variables defined on a probability space (Ω, F , P), where T (e) represents the passage time of the edge e. Their common distribution is denoted by F . The passage time T (π) of a path π = (e 1 , . . . , e k ) is the sum of the variables T (e i ) for i = 1, . . . , k.
For two vertices x and y, we define the geodesic time t(x, y) = inf{T (π) : π is a path from x to y}. (1.1) A self-avoiding path γ such that T (γ) = t(x, y) is called a geodesic between x and y. For the following and for the existence of geodesics, we need some assumptions on F . Let t min denote the minimum of the support of F . We recall a definition introduced in [8]. A distribution F with support in [0, ∞) is called useful if the following holds: F (t min ) < p c when t min = 0, where p c denotes the critical probability for the Bernoulli bond percolation model on Z d and − → p c the critical probability for the oriented Bernoulli bond percolation.
In the whole article, we assume that F has support in [0, ∞), is useful, and that where T 1 , . . . , T 2d are independent with distribution F . As F is useful, F (0) < p c . By Proposition 4.4 in [2], we thus know that geodesics between any points exist with probability one.

Patterns
For a set B of vertices, we denote by ∂B its boundary, this is the set of vertices which are in B and which can be linked by an edge to a vertex which is not in B. We say that an edge e = {u, v} is contained in a set of vertices if u and v are in this set.
Let L 1 , . . . , L d be non-negative integers. To avoid trivialities we assume that at least one of them is positive.
{0, . . . , L i } and two distinct vertices u Λ and v Λ on the boundary of Λ.
These points u Λ and v Λ are called endpoints. Then we fix an event A Λ , with positive probability, only depending on the passage time of the edges joining two vertices of Λ. We say that P = (Λ, u Λ , v Λ , A Λ ) is a pattern. Let x ∈ Z d . Define: • for y ∈ Z d , θ x y = y − x, • for e = {u, v} an edge connecting two vertices u and v, θ x e = {θ x u, θ x v}.
Similarly, if π = (x 0 , . . . , x k ) is a path, we define θ x π = (θ x x 0 , . . . , θ x x k ). Then θ x T denotes the environment T translated by −x, i.e. the family of random variables indexed by the edges of Z d defined for all e ∈ E by (θ x T ) (e) = T (θ −x e) .
Let π be a self-avoiding path and x ∈ Z d . We say that x satisfies the condition (π; P) if these two conditions are satisfied: Figure 1: Example of a pattern which can not be taken by a geodesic. The passage times of the edges in red are equal to 4 and those of the edges in green are equal to 1.
1. θ x π visits u Λ and v Λ , and the subpath of θ x π between u Λ and v Λ is entirely contained in Λ, Note that, if x satisfies the condition (π; P) when π is a geodesic, then (θ x π) u Λ ,v Λ is one of the optimal paths from u Λ to v Λ entirely contained in Λ in the environment θ x T . When the pattern is given, we also say "π takes the pattern in θ −x Λ" for "x satisfies the condition (π; P)". We denote: Note that the number of terms in this sum is actually bounded from above by the number of vertices in π. If N P (π) ≥ 1, we say that π takes the pattern. The aim of the article is to investigate, under reasonable conditions on P, the behavior of N P (γ) for all geodesics γ from 0 to x with x 1 large. The first step is to determine these reasonable conditions, that is why we define the notion of valid patterns.

Definition 1.2. We say that a pattern is valid if the following two conditions hold:
• A Λ has a positive probability, • the support of F is unbounded or there exist two distinct external normal unit vectors, one associated with u Λ and one associated with v Λ . , v Λ = (0, 1), and A Λ the event on which for all edges e ∈ Λ such that e is adjacent to u Λ or v Λ , T (e) = 4 and for all other edges e of Λ, T (e) = 1. This is the pattern of Figure 1 and in what follows in this remark, we use the notations of this figure. The only geodesic from u Λ to v Λ entirely contained in the pattern is (u Λ , v Λ ). However, neither (a, u Λ , v Λ ), nor (b, u Λ , v Λ ), nor (u Λ , v Λ , e), nor (u Λ , v Λ , d) is a geodesic. Hence, every geodesic taking the pattern would contain the path (c, u Λ , v Λ , f ) but this path is not a geodesic.

Main result
The main result of this paper is the following. Theorem 1.4. Let P = (Λ, u Λ , v Λ , A Λ ) be a valid pattern and assume that F is useful and satisfies (1.2). Then there exist α > 0, β 1 > 0 and β 2 > 0 such that for all x ∈ Z d , P ∃ a geodesic γ from 0 to x such that N P (γ) < α x 1 ≤ β 1 e −β2 x 1 .
Showing the existence of a constant c > 0 such that, for all large n, E[N P (π(n))] ≥ cn (1.4) where P is a properly designed pattern and where π(n) is the first geodesic from 0 to nε 1 (geodesics are ordered in an arbitrary way), has been a key intermediate result to show several properties in firstpassage percolation. The first result of this kind appears in an article by van den Berg and Kesten [8].
Let us recall their setting. Assume that F is a finite mean distribution on [0, +∞). Denote by µ(F ) the time constant associated to F , that is LetF be another finite mean distribution on [0, +∞). Assume F useful, F =F and d ≥ 2. IfF is more variable 1 than F , then µ(F ) < µ(F ). This is the main result of [8] and estimate (1.4) is the content of their Proposition 5.22. The proof relies on a modification argument. In [7], Nakajima proves a version of (1.4) to show that the number of geodesics between two vertices has an exponential growth if the distribution has an atom. The result can be deduced from Theorem 1.4 as follows. Denote by κ an atom of the distribution. Consider the pattern P = (Λ, u Λ , v Λ , A Λ ) where: • u Λ = (0, . . . , 0) and v Λ = (1, 1, 0, . . . , 0), • A Λ the event on which the passage time of every edge of Λ is equal to κ.
The key fact about this pattern is the following: each time a geodesic takes the pattern, the geodesic can chose any of the two optimal paths between the endpoints.
Then, one of the most recent result of this kind appears in an article by Krishnan, Rassoul-Agha and Seppäläinen in [5] (see Theorem 5.4 and Theorem 6.2). They use (1.4) for some specific geodesics in order to get results about the Euclidean length of geodesics and the strict concavity of the expected passage times as a function of the weight shifts.
We explain in this section the differences between our result and the ones in [5], [7] and [8] but we give more details about [5] below in Section 1.4 as we wish to strengthen some of their results to illustrate the use of Theorem 1.4. Theorem 1.4 is stronger on three aspects that are commented below: 1. It deals with general patterns while the results in [5], [7] and [8] are stated for specific patterns.
2. In the case of non-uniqueness of geodesics, it gives the result for all geodesics and not only for a specific one.
3. It provides an at least linear growth of the number of crossed patterns out of an event of exponentially small probability.
Since the proof given by van den Berg and Kesten, it has been clear that (1.4) should be true for any reasonable pattern. As explained above, (1.4) has indeed been proven for several specific patterns in [5], [7] and [8]. In some part of the proof of (1.4), one needs to design a new environment in which the geodesics have to cross the pattern. When the support of F is unbounded, the argument is relatively straightforward. However, when the support of F is bounded, this is more involved. Actually, in [5], [7] and [8], when the support of F is bounded, each of the proofs is technical and makes use of specific properties of the considered pattern. The extension to any reasonable pattern, while naturally expected, actually requires new arguments and is a significant difficulty in the proof of Theorem 1.4. Thanks to Theorem 1.4, we can for example generalize Theorem 6.2 in [5]. See Theorem 1.8 below. Let us note however that the strategy developed in the bounded case in Section 4.1 to remove the restriction in Assumption 6.1 in [5] could be used in the proof of Theorem 6.2 in [5].
In [5], [7] and [8], (1.4) is proven only for a specific geodesic. This has no consequence on the main results of [7] and [8]. However obtaining a result for all geodesics enables to strengthen one of the main results of [5] in the bounded case. See Remark 1.10. Dealing with all geodesics is obtained thanks to a new idea using concentric annuli to define and localize good boxes (see Section 2.1).
The last difference with the results of [5], [7] and [8] is that our result is stronger than a result in expectation. However, notice that the result in expectation is sufficient for the applications in [7] and [8]. We refer to Section 1.4 for comments on [5].
A result fulfilling items 2 and 3 above appears in an article by Andjel and Vares [1] for the number of edges with large time crossed by a geodesic. (1.5) Theorem 1.4 is a generalization of this theorem since, to get this result, we can take the pattern (reduced to one edge) P = ({u Λ , v Λ }, u Λ , v Λ , A Λ ) where u Λ = (0, . . . , 0), v Λ = (1, 0, . . . , 0) and A Λ is the event on which the passage time of the only edge of the pattern is greater than M . The proof of Theorem 1.4 is partly inspired by the proof of this theorem and by the proof of (1.4) in [8].
Even if it is stated for distributions with unbounded support, one can check that Theorem 2.3 in [1] holds for F with bounded support with the same proof. As we need this extension in the proof of Theorem 1.4 we state it below.

Some applications
Several of the main results recently obtained in [5] are based on modification arguments leading to results of the type (1.4). We take advantage of Theorem 1.4 to slightly improve some of these results. The purpose of this section is primarily to illustrate the use of Theorem 1.4, the details of the proofs are postponed to Section 4.

Euclidean length of geodesics
Consider the following two assumptions on the distribution F : (H1) There exist strictly positive integers k and ℓ and atoms r ′ 1 , . . . , r ′ k+2ℓ , s ′ 1 , . . . , s ′ k (not necessarily distinct) such that (1.6) (H2) There exist strictly positive integers k and ℓ and atoms r < s such that (k + 2ℓ)r = ks, or F has an atom in 0.
Theorem 1.7 (Theorem 6.2 in [5]). Assume that P(T (e) = t min ) < p c and E min [T p 1 , . . . , T p 2d ] < ∞ with p > 1. Furthermore, assume one of the following two assumptions: • the support of F is unbounded and (H1) is satisfied, • the support of F is bounded and (H2) is satisfied.
Then, there exist constants 0 < D, δ, M < ∞ such that (1.7) We use Theorem 1.4 to prove the following result. It generalizes in a way Theorem 1.7 since in the case of bounded support, we have a less restrictive assumption and since the lower bound in (1.8) is exponentially close to one in the distance instead of the uniform lower bound in (1.7). However, the assumption on the moment is less restrictive in Theorem 1.7.

Strict concavity of the expected passage times as a function of the weight shifts
For b ∈ R, define the b-shifted weights by Following the notations of [5] (see Section 2.2 in [5]), all the quantities associated with the passage times T (b) acquire the superscript. For example, t (b) (x, y) is the geodesic time between x and y defined at (1.1), where the infimum is only on self-avoiding paths. Theorem A.1 in [5] gives the existence of a constant ε 0 > 0 with which we have an extension of the Cox-Durett shape theorem for the shifted weights T (−b) for b < t min + ε 0 (note that here the weights can be negative). Note that (ii) in Theorem A.1 in [5] guarantees that Theorem 1.9. Assume F useful. Furthermore, assume that the support of F is bounded and that it contains at least two strictly positive reals. Then, there exists a finite positive constant M and a function D(b) > 0 of b > 0 such that the following bounds hold for all b ∈ (0, t min + ε 0 ) and all x 1 ≥ M :

Sketch of the proof
In what follows, we give an informal sketch of proof of Theorem 1.4. Fix a pattern P and x ∈ Z d with x large. Consider the event: M = {there exists a geodesic from 0 to x which does not take the pattern}.
The aim is to prove that M has a probability small enough in x . More precisely, we want to prove (1.10) From this result, by a standard re-normalization argument, we easily get that, out of a very low probability event, every geodesic from 0 to x takes a number of patterns linear in x (see Proposition 1.11 in Section 1.6 for a formal statement of (1.10)).
General idea. The idea is to define a suitable sequence of events M(ℓ) for 0 ≤ ℓ ≤ q such that, for some positive constant c < 1, If the above holds, we get P(M) ≤ c c x 1 + P(B), which allows us to conclude. The event M(ℓ) is approximately "there exists a geodesic from 0 to x which does not take the pattern until a distance of order ℓ", where we have to precise the sense of "distance of order ℓ". The complementary event of B is approximately "each geodesic crosses enough good boxes" and these good boxes are the boxes in which the environment and the geodesics behave in a typical way. This enables us to try to modify the environment to ensure that all geodesics from 0 to x take the pattern inside. We take a good definition for the event M(ℓ) to have M(ℓ) ⊂ M(ℓ − 1) and thus (1.11) is equivalent to the existence of a constant η > 0 (by taking η = 1 c − 1) such that To get (1.12), we would like to make a modification in an environment where M(ℓ) occurs to get a new environment in which the event M(ℓ − 1) \ M(ℓ) occurs. This requires some stability in the definition of the events M(· · · ) under the modification. This will be made clearer later.
Associated geodesics. We need the notion of associated paths. For the remaining of the sketch of the proof, "geodesic" means "geodesic from 0 to x". Let B be a set of vertices (it is intended to be the "selected box", i.e. the box where me make the modification). Two paths π 1 and π 2 from 0 to x are B-associated if there exists a, b ∈ B such that : 1. π 1 and π 2 visit successively a and b.
3. π 1 a,b and π 2 a,b are entirely contained in B. In particular the two paths coincide outside B. With this definition, we can clearly enumerate the properties we want after the modification. Imagine we have a geodesic γ "selected" in a certain way and a "good" box B (where we want to make the modification) such that γ crosses B. The aim is to modify the environment in B such that: 1. Every geodesic in the new environment takes the pattern in B.
2. Every geodesic in the new environment is B-associated with a geodesic in the original environment.
3. The geodesic γ is B-associated with at least one geodesic in the new environment.
If we identify two geodesics B-associated, the last two properties can be rephrased as follows: we have not won new geodesics, we have not lost the geodesic γ. It is one of the keys of the stability. Getting these properties in the unbounded case is elementary but it is a significant difficulty in the bounded case (see Section 3.4.1 where we give the main ideas of the modification).
Definition of the sequence M(ℓ). We would like to get a result on every geodesic (in the case where there is no uniqueness, which is a case we do not want to eliminate). A definition of the type: M(ℓ) ≈ {the first geodesic (in the lexicographical order) does not take the pattern in its first ℓ good boxes} does not provide a result for all geodesics. However, it is possible to get a result for the first geodesic in the lexicographical order with this definition.
If every box crossed were a good box, we could define a sequence of large concentric annuli centered in the origin and use a definition of the type: M(ℓ) ≈ {there exists a geodesic which does not take the pattern in the ℓ first annuli}.
Thus, we could choose one of these geodesics (let us denote it by γ) and choose a box crossed by γ in the ℓ-th annulus (let us denote it by B.) By making a modification giving the three properties stated above in the paragraph about the associated geodesics in the box B, in the new environment, the event M(ℓ−1)\M(ℓ) would occur. Indeed, M(ℓ) would not occur since every geodesic in the new environment would take the pattern in B and thus in the ℓ-th annulus. However, M(ℓ − 1) would occur: this crucially uses the fact that γ is B-associated with a geodesic in the new environment (and the fact that the environment outside B, and thus outside the ℓ-annulus, is not modified).
We could use the above definition if every geodesic crossed a good box in every annulus. Since it is not the case, we have to use a definition of the type: M(ℓ) ≈ {there exists a geodesic γ which does not take the pattern in the union of the first a ℓ (γ) annuli} where a ℓ (γ) is the index of the ℓ-th annulus in which γ crosses a good box (see Section 2.1 and Section 3.2).
Modification. Fix a positive integer ℓ. Now, the aim is to define a general plan to prove (1.12). Here we do not discuss the modification itself (see Section 3.4.1 in the bounded case). Recall that we define the sequence M(ℓ) to have We denote by T the environment (i.e. the family of passage times on the edges) and by T ′ an independent copy of T . The basic idea consists in creating a modified environment T * = ϕ(T, T ′ ) where T * (e) = T (e) for some edges (whose passage times do not change) and T * (e) = T ′ (e) for the other edges (whose passage times are re-sampled). In other words, we define a random set of edges R(T ) (the edges we want to re-sample) and we set where, for every set of edges r, ϕ r (T, T ′ )(e) is equal to T ′ (e) if e ∈ r and to T (e) else. In a utopian situation, imagine that we could define a new environment T * = ϕ(T, T ′ ) and an event A (ensuring the success of the modification) such that: • T * and T have the same distribution, • η := P(T ′ ∈ A) > 0, • and {T ∈ M(ℓ) and T ′ ∈ A} ⊂ {T * ∈ M(ℓ − 1) \ M(ℓ)}.

Then we would have
which is (1.12).
However, this situation is unrealistic since the set of re-sampled edges R(T ) depends on T , and thus the distribution of T * is different from the distribution of T . But when r is fixed, ϕ r (T, T ′ ) and T have the same distribution. It is possible to rely on this fact as soon as, observing only the modified environment, we can guess approximately in which box we performed the modification (see the use of the S 1 -variables and S 2 -variables in Lemma 2.3 and Lemma 3.3).
In the case where the passage times are bounded, we use two independent copies of T and we make a two-steps modification. The way we actually perform the modification in the bounded case is sketched in Section 3.4.1. The ideas described in this paragraph can be adapted to this two-steps modification without difficulties.
Comparison with the plan of the proof of Proposition 5.22 in [8]. Let us compare the above strategy with the plan used in [8] to prove Proposition 5.22. Fix x in Z d . In [8], van den Berg and Kesten also start by associating with some specific geodesic γ some sequence of q = C x 1 good boxes. By simple geometric arguments, they then get some family B of boxes such that E[number of boxes of B which are good and crossed by γ] ≥ c x 1 where c is a positive constant. Fix some box B ∈ B. Then they also define a new environment T * by resampling the times of the edges in B. It is then sufficient (this is the technical part of the proof in the bounded case) to prove P(every geodesic in T * crosses the pattern in B|in the environment T , γ crosses B and B is good) ≥ η for some positive constant η > 0. In particular, and contrary to what happens in our framework, it is not necessary in this setting to control what happens to geodesic(s) outside the considered box when we resample the times of the edges in the box. This is not a problem if the geodesic in the new environment completely changes.
Comparison with the plan of the proof of Theorem 2.3 in [1]. In [1], the main difference with the strategy described above is the use of penalized geodesics. Indeed, Andjel and Vares only consider geodesics which do not take edges whose passage time is greater than M and it allows them to get a result on all geodesics from 0 to x thanks to the modification argument. However, it seems difficult to use penalized geodesics with the patterns, that is why we use the strategy of concentric annuli developped in Section 2.1.

Reduction
One can check that, using a standard re-normalization argument, Theorem 1.4 is a simple consequence of the following proposition (see for example the proof of Theorem 2.3 in [1]). Proposition 1.11. Let P = (Λ, u Λ , v Λ , A Λ ) be a valid pattern and assume that F is useful and satisfies (1.2). Then there exist C > 0 and D > 0 such that for all n ≥ 0, for all x such that x 1 = n, (1.14) Thus, the aim of the paper is now to prove Proposition 1.11. Although they share some similarities, the proofs of Proposition 1.11 differ according to whether the support of F is bounded or unbounded. As the proof is easier in the unbounded case, we decide to first give the proof in the unbounded case in Section 2 and then give the proof in the bounded case in Section 3.

Some tools and notations
In this subsection, we recall some results and fix some notations. First, we denote by N the set of all non-negative integers, by N * the set N \ {0}, and by R + the set of all x ∈ R such that x ≥ 0.
For a self-avoiding 2 path π = (x 0 , ..., x k ) going from x 0 to x k , we say that x i is visited by π before For a set B of vertices, we denote by ∂B its boundary, this is the set of vertices of B which can be linked by an edge to a vertex which is not in B. Note that when we define a set of vertices of Z d , sometimes we also want to say that an edge is contained in this set. So, we say that an edge e = {u, v} is contained in a set of vertices if u and v are in this set. Since now a subset B of Z d can be seen as a set of vertices or as a set of edges, we denote by |B| v the number of vertices of B and by |B| e its number of edges.
Then, we define different balls in Z d or R d . For all c ∈ Z d and r ∈ R + , we denote and for n ∈ N * , we denote by Γ n the boundary of B 1 (0, n), i.e.
(1.15) Also for c ∈ Z d and r ∈ R + , we denote by B(c, r) the random ball Then, for x and y in R d , we define t(x, y) as t(x ′ , y ′ ) where x ′ is the unique vertex in Z d such that x ∈ x ′ + [0, 1) d (similarly for y ′ ). For c ∈ Z d and r ∈ R + , we denote byB(c, r) the random ball Furthermore, µ is a norm on R d and describes the first order of approximation ofB(0, r) when r goes to infinity. For c ∈ Z d and r ∈ R + , we denote Fix B = B µ (0, 1), then the Cox-Durett shape theorem (Theorem 2.16 in [2]) guarantees that for each ε > 0, One can easily prove that the result (1.17) is equivalent to Since µ is a norm, we can fix two constants c µ > 0 and C µ > 0 such that for all y in R d , Finally, since F is useful, by Lemma 5.5 in [8], there exist δ = δ(F ) > 0 and D 0 = D 0 (F ) fixed for the remaining of the article such that for all u, v ∈ Z d , Notice that, using the Borel-Cantelli lemma with this result, we get that for all u ∈ Z d different from 0, Even if it means replacing A Λ by A Λ ∩ {∀e ∈ Λ, T (e) ≤ M Λ }, we can assume that which is an upper bound for the travel time of an optimal path (for the passage time) going from u Λ to v Λ and entirely contained in Λ on the event A Λ . For i ∈ {1, 2, 3}, B i,s,N is the ball in Z d of radius r i N for the norm . 1 centered at the point sN where the constants r i are defined as follows. We fix r 1 = d. Denote by K the number of edges in B ∞ (0, ℓ Λ + 3). Then, fix r 2 an integer such that Let r 2,3 be a positive real such that then we fix r 3 an integer such that We use the word "box" to talk about B 3,s,N . Recall that we denote by ∂B i,s,N the boundary of B i,s,N , that is the set of points z ∈ Z d such that z − sN 1 = r i N . For u and v two vertices contained in B 3,s,N , we denote by t 3,s,N (u, v) the minimum of the times of all paths entirely contained in B 3,s,N and going from u to v.
Crossed boxes and weakly crossed boxes. We say that a path Paths associated in a box. We say that two paths γ and γ ′ from 0 to the same vertex x are associated in a box B 3,s,N if there exist two distinct vertices s 1 and s 2 such that the following conditions hold: • γ and γ ′ visit s 1 and s 2 , • γ s1,s2 and γ ′ s1,s2 are entirely contained in B 3,s,N , In particular, these two paths coincide outside B 3,s,N .
Typical boxes. We define a sequence (ν(N )) N ∈N * such that for all N ∈ N * , ν(N ) > M Λ and Note that F ((ν(N ), +∞)) > 0 for all N ∈ N * since the support of F is unbounded.
We can now define typical boxes. A box B 3,s,N is typical if it verifies the following properties: (ii) every path π entirely contained in B 3,s,N from u π to v π with u π − v π 1 ≥ (r 2 − r 1 )N has a passage time verifying: This lemma guarantees that the properties of a typical box are indeed typical ones and that they are also local ones. Its proof is in Section 2.2. Let us now introduce some further definitions.
Successful boxes. For a fixed x ∈ Z d , a box B 3,s,N is successful if every geodesic from 0 to x takes a pattern which is entirely contained in B 2,s,N , i.e. if for every geodesic γ going from 0 to x, there exists x γ ∈ Z d satisfying the condition (γ; P) and such that B ∞ (x γ , ℓ Λ ) is contained in B 2,s,N . Note that the notion of successful box depends on some fixed x ∈ Z d .
Annuli. Now, we define the annuli mentioned in Section 1.5. Fix r = 2(r 1 + r 3 + 1), (2.6) and for all integers i ≥ 1, let us define For any annulus A i,N , we call {y ∈ Z d : y 1 = (i − 1)rN } its inner sphere and {y ∈ Z d : y 1 = irN } its outer sphere. Then, we give two definitions about these annuli which are useful in the proof of Lemma 2.3.
• For i > 1, we say that a path from 0 to a vertex of Z d crosses (resp. weakly crosses) a box B 3,s,N in the annulus A i,N if the two following conditions are satisfied: it crosses (resp. weakly crosses) this box before it visits for the first time the outer sphere of -B 3,s,N is entirely contained in the annulus, i.e. every vertex of B 3,s,N is in A i,N .
• We also say that a path takes a pattern in the annulus A i,N if it takes a pattern which is entirely contained in A i,N , i.e. if every vertex of this pattern is in A i,N . Here, we do not require that the path takes a pattern before it visits the outer sphere of A i,N for the first time.
Note that the choice of r guarantees that every path passing through an annulus has to cross a box in this annulus. For all integer p ≥ 2 and all N ∈ N * , we denote by G p (N ) the event on which for all x in the outer sphere of the p-th annulus, every geodesic from 0 to x crosses a typical box in at least p 2 annuli. The following lemma, whose proof is given in Section 2.2, gives us an exponential decrease of the probability of the complement of G p (N ).

Lemma 2.2.
There exist two positive constants C 1 and D 1 , and an integer N 0 ≥ 1 such that for all p ≥ 1 and N ≥ N 0 , Setup for the proof of Proposition 1.11. For the rest of the proof, we fix C 1 , D 1 and N 0 given by Lemma 2.2. Recall that K is the number of edges in B ∞ (0, ℓ Λ + 3). Then we fix δ ′ > 0 such that Note that it is possible since we have taken r 2 large enough (see (2.3)). Then, fix (where Γ n is defined at (1.15)). Fix p = n rN and q = p 2 . Note that x belongs to the (p + 1)-th annulus.
M -sequences. Let us now define some random sets and variables which are useful for stability questions for the modification argument. Unless otherwise specified, in the remaining of this section, we write geodesic as a shorthand for geodesic from 0 to x.
First, let us associate a sequence of 0 to p − 1 typical boxes to every geodesic from 0 to x. For a geodesic γ, the deterministic construction is what follows.
Initialize the sequence as an empty sequence. For j from 1 to p − 1, do: • let a j (γ) be the index of the first annulus such that γ crosses a typical box in this annulus and such that a j (γ) > a j−1 (γ) (where a 0 (γ) = 1). If there is no such annulus, then we stop the algorithm.
• Add the first typical box crossed 4 by γ in the annulus A aj (γ),N to the sequence. Note that, since this typical box is crossed before γ leaves A aj (γ),N by the outer sphere for the first time, the j-th box of the sequence is crossed by γ after the (j − 1)-th one.
So, we get a sequence of at most p − 1 boxes crossed by the geodesic. These boxes are all in different annuli. Furthermore, every box of this sequence is crossed by γ before γ leaves the annulus containing it for the first time by the outer sphere. If the event G p (N ) occurs, we know that all these sequences have at least q elements. For j ∈ {1, . . . , p − 1}, we define a set of geodesics Γ j . A geodesic γ from 0 to x belongs to Γ j if: • the length of its sequences defined above is greater than or equal to j, • γ does not take the pattern in any annuli A k,N with k ≤ a j (γ).
We call the sequences defined above the M -sequences.
Selected geodesic and S 1 -variables. Then, for j ∈ {1, . . . , p − 1}, if Γ j is not empty, we define the selected geodesic among the geodesics of Γ j as the one which minimizes the index of the annulus containing the j-th box of its sequence. If there are several such geodesics, the selected one is the first in the lexicographical order. Then, the random variable S 1 j is equal to the vertex s such that the box B 3,s,N is the j-th box in the M -sequence of the selected geodesic. When j is fixed, we say that the box B 3,S 1 j ,N is the selected box. Finally, if Γ j is empty, set S 1 j = 0 and there is no selected geodesic.
S 2 -variables. After a modification in the box of the selected geodesic, we have to select a new box in the new environment T * defined below. Since we can not guarantee that the selected box in the initial environment is still typical in the new environment, we have to select it in a different way. That is why we introduce the S 2 -variables. Let j ∈ {1, . . . , p − 1}, we define S 2 j as follows. For every geodesic γ ∈ Γ j−1 , we define, if it is possible, S 2 j (γ) as the vertex s corresponding to the box B 3,s,N where B 3,s,N is the first successful box crossed by γ in an annulus (in the sense given with the definition of the annuli above). If it is not possible, S 2 j (γ) = 0. We denote by a ′ j (γ) the index of the annulus containing B 3,S 2 j (γ),N . Then, S 2 j is equal to the vertex S 2 j (γ) where γ satisfies the three following conditions: • for all geodesic γ ∈ Γ j−1 such that a ′ j (γ) = 1, we have a ′ j (γ) ≥ a ′ j (γ), • γ is the first geodesic in the lexicographical order among the geodesics γ such that a ′ j (γ) = a ′ j (γ).
If it is not possible, S 2 j = 0.
Modification argument. Finally, for j ∈ {1, ..., p − 1}, we define M(j) as the event on which every geodesic from 0 to x has at least j typical boxes in its M -sequence and there exists one geodesic which does not take the pattern in any annuli A k,N with k ≤ a j (γ). We also define M(0) as the event on which there exists a geodesic from 0 to x. Its probability is equal to 1 (see Section 1.1). Now, the aim is to bound from above P(M(q)) independently of x since we have: P there exists a geodesic γ from 0 to x such that N P (γ) = 0 ≤ P(T ∈ G p (N ) c ) + P(T ∈ M(q)). (2.9) In the sequel, we introduce an independent copy T ′ of the environment T , the two being defined on the same probability space. It is thus convenient to refer to the considered environment when dealing with the objects defined above. To this aim, we shall use the notation {T ∈ M(j)} to denote that the event M(j) holds with respect to the environment T . In other words, M(j) is now seen as a subset of (R + ) E , where E is the set of all the edges. Similarly, for i ∈ {1, 2} we denote by S i j (T ′ ) the random variables defined above but in the environment T ′ .
Fix ℓ ∈ {1, . . . , q}. On {T ∈ M(ℓ)}, Γ ℓ = ∅ and B 3,S 1 ℓ (T ),N is a typical box crossed by the selected geodesic. We get a new environment T * defined for all edge e by: For y and z in Z d , we denote by t * (y, z) the geodesic time between y and z in the environment T * . Note that T and T * do not have the same distribution.
3 is a consequence of Lemma 2.5 below whose proof is given in Section 2.3. Recall the definition of associated paths given page 11.
it is sufficient to prove that we have the four following points in the environment T * : 1. every geodesic from 0 to x has at least ℓ − 1 typical boxes in its M -sequence, 2. there exists a geodesic from 0 to x which does not take the pattern in the annuli until the one containing its (ℓ − 1)-th box, 3. every geodesic from 0 to x whose M -sequence contains at least ℓ elements takes the pattern in an annulus whose index is smaller than or equal to the one containing its ℓ-th box, (c) Let γ and γ * be as in item (ii) of Lemma 2.5. Then γ and γ * coincide except maybe for the part between s 1 and s 2 . To sum up: (2.10) Furthermore, by remark (b), we have that for any i = a ℓ , γ takes the pattern in the annulus A i,N in the environment T if and only if γ * takes the pattern in the annulus A i,N in the environment T * . The same property holds for the selected geodesic γ and for the associated geodesic γ * (in the environment T * ) given by item (iii) of Lemma 2.5.
(d) Let again γ and γ * be as in item (ii) of Lemma 2.5. Let us compare the M -sequence of γ (which is built in the environment T ) with the M -sequence of γ * (which is built in the environment T * ). By (a) and (2.10), we get that any box which belongs to the M -sequence of γ, with the possible exception of a box contained in A a ℓ ,N , also belongs to the M -sequence of γ * . The same property holds for the selected geodesic γ and for the associated geodesic γ * (in the environment T * ) given by item (iii) of Lemma 2.5. In particular the first ℓ − 1 elements of the M -sequence of γ (which is built in the environment T ) are the same as the first ℓ − 1 elements of the M -sequence of γ * (which is built in the environment T * ).
Let us now proceed to the proof of item 1. We assume ℓ ≥ 2 otherwise there is nothing to prove. Let γ * be a geodesic from 0 to x in the environment T * . Let γ be the associated geodesic in the environment T given by item (ii) of Lemma 2.5. Since the event {T ∈ M(ℓ)} occurs, the M -sequence (in the environment T ) of γ contains at least ℓ typical boxes. By remark (d) above, the M -sequence (in the environment T * ) of γ * contains at least ℓ − 1 typical boxes.
Let us consider item 2. We can again assume ℓ ≥ 2. Let γ * be the geodesic given by item (iii) of Lemma 2.5. Recall that γ is the selected geodesic and that γ ∈ Γ ℓ . In particular, we have the following properties: its M -sequence contains at least ℓ boxes; the ℓ-th box of its M -sequence belongs to A a ℓ ,N ; γ does not take the pattern in any annulus whose index is smaller than or equal to a ℓ . Therefore, by remark (d) above, the first ℓ − 1 boxes of the M -sequence of γ and γ * are the same. By remark (c) above, we conclude that γ * does not take the pattern (in the environment T * ) in any annulus whose index is smaller than the one containing its (ℓ − 1)-th box.
Let us prove item 3. Let γ * be such a geodesic. Assume that the ℓ-th box of the M -sequence of γ * is in an annulus whose index is strictly smaller than a ℓ . Let γ be a geodesic in the environment T given by item (ii) of Lemma 2.5. Assume, aiming at a contradiction, that γ * does not take the pattern in an annulus until the one containing its ℓ-th box. By remark (d), the ℓ first boxes of the M -sequences of γ * and γ are the same. By remarks (b) and (c), γ does not take the pattern until the annulus containing its ℓ-th box. This contradicts the definition of S 1 ℓ , so it is impossible. Thus the ℓ-th box of the M -sequence of γ * is in an annulus whose index is greater than or equal to a ℓ . By item (i) of Lemma 2.5, γ * takes the pattern in the annulus whose index is a ℓ . Therefore it takes the pattern in an annulus whose index is smaller than or equal to the one containing its ℓ-th box and the third point is satisfied.
Finally, let us prove item 4. Note that since B 3,s,N is a successful box, S 2 ℓ (T * ) = 0. There are two steps. First, we prove that N S 2 ℓ (T * ) is the center of a box contained in the annulus A a ℓ ,N and then we prove that B 3,s,N ∩ B 3,S 2 (T * ),N = ∅, which gives the result. Assume that N S 2 ℓ (T * ) is not the center of a box contained in A a ℓ ,N . Let again γ * be the geodesic given by item (iii) of Lemma 2.5 and recall that γ is the selected geodesic in the environment T . Since B 3,s,N is successful, we have a ′ ℓ (γ) ≤ a ℓ and thus the definition of S 2 ℓ (T * ) implies that N S 2 ℓ (T * ) is the center of a box contained in an annulus A k,N such that k < a ℓ . So γ * takes the pattern in this annulus in the environment T * . By remark (c), γ takes the pattern in this annulus in the environment T * and then in the environment T by remark (b). This is impossible since γ does not take the pattern in any annuli A k,N with k ≤ a ℓ .
To conclude, assume, aiming at a contradiction, that B 3,s,N ∩ B 3,S 2 (T * ),N = ∅. It implies that γ * takes a pattern outside B 3,s,N in the environment T * and the portion of γ * taking this pattern is a portion of γ * 0,s ′ 1 or γ * s ′ 2 ,x . By remark (b), the portion of γ * taking this pattern also takes this pattern in the environment T . By remark (c), it implies that γ takes the pattern in the environment T in the annulus A a ℓ ,N , which is a contradiction. Now, thanks to Lemma 2.3, we can adapt Lemma 3.8 from [1]. Lemma 2.6. There exists λ ∈ (0, 1), which does not depend on n and on x ∈ Γ n , such that Proof. Let ℓ be in {1, ..., q}. For every s ∈ Z d , let us consider the environment T * s defined for all edge e by: Thus T * s and T have the same distribution and on the event So, using this environment and writing with indicator functions the result of Lemma 2.3, we get: We compute the expectation on both sides. The right side yields For the left side, we have Since on the event {T ∈ M(ℓ)} ∩ {S 1 ℓ (T ) = s}, we have P (T ′ ∈ B * (T )|T ) ≥ η, the left side is bounded from below by ηP(T ∈ M(ℓ), S 1 ℓ (T ) = s). Then, by summing on all s ∈ Z d and writing K a constant which bounds from above for all s ′ ∈ Z d the number of vertices s ∈ Z d such that s ′ ≈ s, we get Thus, Hence, using P(T ∈ M(0)) = 1, we get by induction Proof of Proposition 1.11. Recall that N , x (and then n and p) are fixed at (2.8) but that C 1 , D 1 and λ does not depend on x, n and p. Then, by Lemma 2.2 and Lemma 2.6, using the inequality (2.9), As C 1 > 0 and λ ∈ (0, 1), and as this inequality holds for any n ≥ 2rN and any x ∈ Γ n , we get the existence of two constants C > 0 and D > 0 such that for all n, for all x ∈ Γ n , P there exists a geodesic γ from 0 to x such that N P (γ) = 0 ≤ D exp(−Cn 1 d ).

Typical boxes crossed by geodesics
Let us first begin with the proof of the lemma stated in the paragraph of typical boxes in Section 2.1.
Proof of Lemma 2.1.
1. Let B 3,s,N be a typical box. Then the event T (s, N ) occurs. Let u 0 and v 0 be two vertices in Let π 0 be a path from u 0 to v 0 which is not entirely contained in B 3,s,N . Let z 0 denote the first vertex on the boundary of B 3,s,N visited by π 0 . Then Indeed, by (1.17), Note that, for the first inequality, we use the fact that for all z ∈ B 2,0,N , t 3,0,N (0, z) = t(0, z) thanks to the first point of Lemma 2.1 proved above.
The probability that (iii) is satisfied by B 3,0,N goes to 1 by (2.4). Then, let us prove that the probability that (ii) is satisfied by B 3,0,N goes to 1. Let |B 3,0,N | denote the number of vertices in B 3,0,N and Π 0 denote the set of self-avoiding paths entirely contained in B 3,0,N . Then, using (2.3), we have that r 2 > r 1 , and by (1.19), is not satisfied by a path of Π 0 whose endpoints are u π and v π ) is not satisfied by a path whose endpoints are u π and v π ) Proof of Lemma 2.2. To begin this proof, one need an upper bound on the Euclidean length of geodesics. Using Theorem 4.6 in [2], we have two positive constants K 1 and C 2 such that for all y ∈ Z d , where m(y) = max {|σ| e : σ is a geodesic from 0 to y} and where for a path σ, |σ| e means the number of different edges taken by σ. For all p ∈ N * , we define the event N p (N ) on which every geodesic from 0 to the outer sphere of the p-th annulus takes less than K 1 prN distinct edges. Note that r = 2(r 1 + r 3 + 1) is fixed at (2.6) and rN corresponds to the widths of the annuli. Then, Hence, we obtain two positive constants C 3 and D 3 only depending on r, d and F such that for all p ∈ N * , for all N ∈ N * , Now, we assume that the event N p (N ) ∩ G p (N ) c occurs. So, every geodesic from 0 to the outer sphere of the p-th annulus takes a number of distinct edges which is between prN and K 1 prN . Let us consider a re-normalized model. We introduce the meta-cubes These meta-cubes form a partition of Z d . Furthermore, the meta-cubes and the boxes defined above have the same centers (which are the vertices N s for s ∈ Z d ), and for all s For a geodesic γ from 0 to the outer sphere of the p-th annulus, we associate the set of meta-cubes visited by γ, that is This set can be identify with the subset of the re-normalized graph N Z d : ) forms a lattice animal, denoted by A R (γ). Recall that a lattice animal A in N Z d is a finite connected sub-graph of N Z d that contains 0. We denote by A R the set of lattice animals in N Z d associated with a geodesic going from 0 to the outer sphere of the p-th annulus.
Let us bound the size of these lattice animals. By the size of a lattice animal A R , denoted by |A R | v , we mean its number of vertices in the re-normalized model. Recall that, since the event N p (N ) occurs, every geodesic from 0 to the outer sphere of the p-th annulus takes a number of distinct edges which is between prN and K 1 prN . Then, in the meta-cube set A(γ) associated to such a geodesic γ, since r 1 = d and thanks to the choice of r, there are p − 1 meta-cubes associated to boxes crossed by γ in distinct annuli. In particular (considering also the meta-cube centered at the origin), the size of every lattice animal A R ∈ A R is bounded from below by p. For an upper bound, we use the inequality for all geodesic γ from 0 to the outer sphere of the p-th annulus, where |γ| e still denotes the number of edges taken by γ and where |A R (γ)| v is the number of vertices of A R (γ). Let us prove this inequality. Let γ be a geodesic from 0 to the outer sphere of the p-th annulus and denote by γ = (v 0 , . . . , v m ) the sequence of vertices visited by γ. For all v ∈ Z d , denote by s(v) the unique s ∈ Z d such that v belongs to B ∞ s,N . We define by induction a strictly increasing sequence i 0 , . . . , i κ by setting κ = 0 and i 0 = 0 and then applying the following algorithm: , we denote by i κ+1 the smallest of these i, then we increment κ and go back to (a).
(b) Otherwise we stop the algorithm.
Then, we necessarily have and (2.11) is proved. Now, using (2.11), writing K 2 = ⌈3 d (K 1 r + 1)⌉ (which does not depend on p and N ), for every lattice animal A R ∈ A R , |A R | v is bounded from above by K 2 p. Furthermore, for j ∈ {p, . . . , K 2 p}, using (4.24) in [4], we have that (2.12) Let us consider the random variables (X N ℓ ) ℓ∈Z d such that X N ℓ = 1 if the meta-cube B ∞ ℓ,N is typical and X N ℓ = 0 otherwise. By Lemma 2.1, there exists a positive constant K 3 such that X N ℓ is independent from the sigma-algebra generated by Thus, by Corollary 1.4 in [6], there exists η 1 = η 1 (N ) > 0 such that Indeed, on G p (N ) c , there exists a geodesic γ from 0 to the outer sphere of the p-th annulus which crosses a typical box in strictly less than p 2 annuli, and thus there are strictly more than p 2 − 1 annuli A i,N with i > 1 such that γ does not cross a typical box in them. Furthermore, there are p − 1 meta-cubes in A(γ) such that each of them is associated to a box crossed by γ in one of the p − 1 distinct annuli A i,N with 1 < i ≤ p. Thus, there are strictly more than p 2 − 1 of these specified meta-cubes which are not typical. Hence the number of typical meta-cubes in A(γ) is strictly smaller than |A R | v − p 2 + 1. Then, using the random variables (Y N ℓ ) ℓ∈Z d , Then, for p ≥ 4 and N large enough to have η 1 (N ) < 1 4K 2 , using a Chernov bound for the binomial distribution (see Section 2.2 in [3]), we get where for x ∈ (η 1 , 1), Thus, since we can take η 1 as small as we want by taking N large enough, Finally, we have a constant N 0 such that for all p ≥ 4, for all N ≥ N 0 , So, there exist two positive constants C 1 and D 1 such that for all p ≥ 1, for all N ≥ N 0 ,
So assume that this event occurs. On the event {T ∈ M(ℓ)}, Γ ℓ is not empty and thus there is a selected geodesic. We denote this selected geodesic by γ. We define the entry point (resp. the exit point) of a self-avoiding path in a set of vertices as the first (resp. the last) vertex of this path belonging to this set. Let u denote the entry point of γ in B 2,s,N and v the exit point.
We call entry point and exit point of the pattern (centered at 0) the endpoints denoted by u Λ and v Λ in the introduction. Note that, if a self-avoiding path takes the pattern, its entry and exit points in the set B ∞ (0, ℓ Λ ) are not necessarily the entry and exit points of the pattern (as it can visit the set before and after taking the pattern).
Here, we want to put the pattern centered at sN . The vertex s being fixed, we keep the notation u Λ and v Λ to designate the entry and the exit points of the pattern centered at sN .

Construction of π.
We have the following inclusions: For the modification, we need a path π, constructed in a deterministic way and satisfying several properties, whose existence is guaranteed by the following lemma.
Lemma 2.7. We can construct a path π in a deterministic way such that : way for the norm . 1 (and thus being contained in B ∞ (sN, ℓ Λ )) and then goes (ii) π is entirely contained in B 2,s,N and does not have vertices on the boundary of B 2,s,N except u and v, The proof of this lemma is given in Appendix A.2 but the idea is to construct two paths, one from u to sN and the other from sN to v which minimize the distance for the norm . 1 and such that the only vertex belonging to both paths is sN . Then, we denote by u 0 the first vertex of B ∞ (sN, ℓ Λ + 3) visited by the path from u to sN and v 0 the last vertex of B ∞ (sN, ℓ Λ + 3) visited by the path from sN to v. We construct two paths entirely contained in B ∞ (sN, ℓ Λ + 3) from u 0 to u Λ and from v Λ to v 0 which do not take vertices of B ∞ (sN, ℓ Λ ) except u Λ and v Λ and which have no vertices in common and we consider the concatenation of the path from u to u 0 , the one from u 0 to u Λ , a path from u Λ to v Λ in a shortest way, the path from v Λ to v 0 and the one from v 0 to v (see Figure 2).
Let π be the path given by Lemma 2.7.
Definition of E * + , E * − and B * . Define E * − (T ) as the set of edges e such that e ∈ π \ B ∞ (sN, ℓ Λ ) and E * + (T ) as the set of edges which are in B 2,s,N but which are not in ). Thus, η only depends on F , the pattern and N and we have Figure 2: Example of the construction of π in dimension 2.
Consequences of the modification. We denote by γ * the path γ 0,u ∪ π ∪ γ v,x . Note that γ * is a self-avoiding path.
Proof. We have that γ u,v visits at least one vertex in B 1,s,N . Denote by w the first of these vertices. Then, γ u,w and γ w,v are two geodesics, both between two vertices in B 2,s,N . Using item 1 in Lemma 2.1, γ u,w and γ w,v are entirely contained in B 3,s,N . Thus, since B 3,s,N is a typical box, using (2.5) and the fact that Then, by the construction of π and of B * (T ), where τ Λ is fixed at (2.2). Thus, Lemma 2.9. Let γ * be a geodesic from 0 to x in the environment T * . Then γ * weakly crosses the box B 3,s,N and the first vertex of B 2,s,N visited by γ * is u and the last is v. Furthermore, γ * takes the pattern Proof. Let γ * be a geodesic from 0 to x in the environment T * . By Lemma 2.8, T * (γ * ) < T (γ * ). Thus γ * takes an edge of B 2,s,N and by item (iii) of the definition of a typical box and since there is no edge whose time has been modified outside B 2,s,N , γ * can not take any edge of time greater than ν(N ) in B 2,s,N . Indeed, assume that γ * takes an edge e such that T * (e) ≥ ν(N ). Then, denoting by E(γ * ) the edges of γ * and using (2.4), which is impossible. Hence, γ * has to take edges of π or of the pattern and can not take other edges of B 2,s,N .
Since π does not visit any vertex on the boundary of B 2,s,N except u and v, γ * has to visit u and v and to follow π between u and u Λ and between v Λ and v. If γ * u Λ ,v Λ leaves the pattern, it takes an edge whose time is greater than ν(N ), which is impossible. So, γ * u Λ ,v Λ is a path entirely contained in B ∞ (sN, ℓ Λ ) and is optimal for the passage time since γ * is a geodesic.
To conclude, let us show that u is visited by γ * before v. Assume that it is not the case. Then, there exists γ * 1 a geodesic from 0 to v and γ * 2 a geodesic from u to x in the environment T * which does not take any edge in B 2,s,N . Thus, there are also geodesics in the environment T . Then, By concatenating γ 0,u and γ * 2 , we obtain a path from 0 to x. Thus, Using Lemma 2.9, we can prove Lemma 2.5. Indeed, by this previous lemma, every geodesic from 0 to x takes the pattern inside B 2,s,N and the first item holds. For the third item, one can check that the concatenation of γ 0,u , π u,u Λ , one of the optimal paths for the passage time between u Λ and v Λ entirely contained in B ∞ (sN, ℓ Λ ), π v Λ ,v and γ v,x gives a geodesic γ * which is associated to γ in B 3,s,N (with s 1 = u and s 2 = v). Finally, let us prove the second item. If γ * is a geodesic from 0 to x in the environment T * , then γ * u,v is contained in B 2,s,N . Furthermore, so T (γ * 0,u ) = T (γ 0,u ) and γ * 0,u is a geodesic in the environment T . Similarly, γ * v,x is a geodesic in the environment T . Hence, we get a not necessarily self-avoiding optimal path for the passage time by We get a geodesic γ ′ which satisfies the properties of this item by cutting the loops of π ′ using a standard process 5 .

Bounded case
In this section, we assume that the support of F is bounded. In this case, Theorem 1.4 also follows from Proposition 1.11. Our proof of Proposition 1.11 still follows the strategy given in the preceding section, but the modification argument is more involved. Let P = (Λ, u Λ , v Λ , A Λ ) be a valid pattern. We set t max = sup(support(F )). Remark that, because of (1.19), we have t min + δ < t max . 5 Note that γ * 0,u , γu,v and γ * v,x are three self-avoiding paths and that γ * 0,u and γ * v,x do not have vertices in common. So we can consider s 1 the last vertex belonging to both path γ * 0,u and γu,v in the order in which they are visited by γu,v and s 2 the first vertex belonging to both paths γs 1 ,v and γ * v,x in the order in which they are visited by γu,v. Thus, since γu,v is entirely contained in B 3,s,N , s 1 and s 2 are two vertices contained in B 3,s,N and we can take γ ′ = γ * 0,s 1 ∪ γs 1 ,s 2 ∪ γ * s 2 ,x .

Oriented pattern
The proof in the bounded case uses a modification argument in which we have to connect the pattern to a straight path in a given direction. It is convenient to show the feasibility of this construction before starting the modification. The following lemma, whose proof is in Appendix B, shows that it is indeed feasible by proving that a pattern can be associated to d patterns (with a larger size), each having endpoints aligned in a distinct direction, and each having the original pattern as a sub-pattern. By direction, we mean one of the d directions of the canonical basis which is denoted by • on A Λ0 j , any path from −ℓ 0 ε j to ℓ 0 ε j optimal for the passage time among the paths entirely inside Λ 0 contains a subpath from u Λ to v Λ entirely inside Λ, We fix ℓ 0 , Λ 0 and the patterns P j , for j = 1, . . . , d, for the remaining of the proof. By definition, for all j ∈ {1, . . . , d}, N P (π) ≥ N P j (π), and actually . . , d}, x satisfies the condition (π;P j )} .
From now on, we forget the original pattern and only consider the oriented patterns P j for all directions j ∈ {1, . . . , d}. We talk about oriented pattern when its orientation is specified and we simply say "pattern" when talking about one of the oriented patterns P j . Consistently with these conventions and to lighten notations, we write ℓ Λ instead of ℓ Λ 0 , Λ instead of Λ 0 and A Λ j instead of A Λ0 j . Now, several parameters related to the distribution F have to be introduced. First, we fix a positive real ν such that : • F ([ν, +∞)) > 0, • the event A Λ ∩ {∀e ∈ Λ, T (e) ≤ ν} has a positive probability.
Notice that, if F has an atom, one could have ν = t max . Even if it means replacing A Λ j by A Λ j ∩ {∀e ∈ Λ, T (e) ≤ ν}, we can assume that A Λ j ⊂ {∀e ∈ Λ, T (e) ≤ ν}. Further, we set τ Λ = 2ℓ Λ ν, which can be interpreted as an upper bound for the passage time of an optimal path from ℓ Λ ε j to −ℓ Λ ε j on the event A Λ j . Finally, we denote by T Λ the constant K Λ (t max − t min ) where K Λ is the number of edges in an oriented pattern. We will use it as an upper bound for the time that a path can save using edges of a pattern after a modification.

Proof of Proposition 1.11 in the bounded case
We keep the overall plan of the unbounded case. Unlike in the unbounded case, we cannot use edges of prohibitive time and thus the modification argument is more elaborate here. This section follows the structure of Section 2.1 but the one step modification is replaced by a two-steps modification. To this aim, we slightly change the structure of our boxes and our definition of typical boxes. Let us begin by fixing some constants.

Constants.
Note that we keep the notations introduced in Section 1.7 and that τ Λ and T Λ are fixed in Section 3.1. In the next two paragraphs, we introduce the constants used in the proof. Before looking at their definitions below, one can keep in mind that we fix ε ≪ 1 and r 1 ≪ 1 ε ≪ ∇ ≪ r 2 ≪ r 3 ≪ r 4 , where "≪" means that the ratio is large enough and only depends on the dimension d and on the distribution F .
• Using Theorem 1.6 with M = t min + δ, we get two constants α > 0 and C, fixed for the rest of the proof, such that for all n ∈ N * and u, v ∈ Z d such that u − v 1 = n, We give here other lower bounds for ∇ that we need for the sequel and which are consequences of (3.2).
-Using the fact that Boxes. With theses constants, we can now define boxes. For i ∈ {1, 2, 3, 4}, as in the unbounded case, B i,s,N is the ball of radius r i for the norm . 1 centered at the point sN with: • r 1 = d, • r 2 an integer such that • r 3 an integer such that Note that r 3 ≥ r 2 + 1.
• r 4 an integer such that Note that r 4 ≥ r 3 + 1.
We use the word "box" to talk about B 4,s,N . Recall that we denote by ∂B i,s,N the boundary of B i,s,N , that is the set of points z ∈ Z d such that z − sN 1 = r i N . Paths associated in a box. We say that two paths γ and γ ′ from 0 to the same vertex x are associated in a box B 4,s,N if there exist two distinct vertices s 1 and s 2 such that the following conditions hold: • γ and γ ′ visit s 1 and s 2 , • γ s1,s2 and γ ′ s1,s2 are entirely contained in B 4,s,N , In particular, these two paths coincide outside B 4,s,N .
Typical boxes. B 4,s,N is called a typical box if it verifies the following properties: (i) every geodesic γ u,v from u to v entirely contained in B 3,s,N with u−v 1 ≥ N has at least α u−v 1 edges whose time is greater than or equal to t min + δ, As in the unbounded case, we need properties which are guaranteed with the definition of typical boxes. We state them in the following lemma whose proof is given in Section 3.3.
Annuli. Following the proof in the unbounded case, we define the annuli A i,N with r = 2(r 1 + r 4 + 1) and G p (N ) as in Section 2.1 but with the definitions of crossed and typical boxes defined here in Section 3.2. The bound on P(G p (N ) c ) of Lemma 2.2 also holds here. The proof is exactly the same in this case thanks to Lemma 3.2. For the rest of the proof, we fix C 1 , D 1 and N 0 given by Lemma 2.2.
Modification argument. Fix K ′ = T Λ + 2(C µ L 1 + t max (ℓ Λ + 1)). Then, fix and M(j) as in Section 2.1 but with the notions of typical and successful boxes defined here in Section 3.2. As in the unbounded case, the aim is to bound from above P(T ∈ M(q)) independently of x. For the sequel, we use a two-steps modification. So we introduce two independent copies T ′ and T ′′ of the environment T , the three being defined on the same probability space. Fix ℓ ∈ {1, ..., q}. On {T ∈ M(ℓ)}, B 4,S 1 ℓ (T ),N is a typical box crossed by the selected geodesic. From this configuration, as a first step, we shall associate a set of edges E * + (T ) which is contained in It corresponds to the edges for which we want to reduce the time. Then, we get a new environment T * defined for all edge e by: (3.7) From this environment, as a second step, we get three new subsets E * * + (T, T ′ ), E * * − (T, T ′ ) and E * * P (T, T ′ ) of edges of B 2,S 1 ℓ (T ),N which are respectively the edges for which we want to reduce the time, to increase the time and the edges of the location where we want to put the pattern. We get a third environment T * * defined for all edge e by: (3.8) Note that T * * and T do not have the same distribution. For y and z in Z d , we denote by t * (y, z) (resp. t * * (y, z)) the geodesic time between y and z in the environment T * (resp. T * * ). Similarly, we define for c ∈ Z d and t ∈ R + : We formalize this modification in the next lemma and we will describe precisely the construction of E * + , E * * + , E * * − and E * * P in the next subsection.   (i) in the environment T * * , every geodesic from 0 to x takes the pattern inside B 2,S 1 ℓ (T ),N , (ii) for all geodesic γ * * from 0 to x in the environment T * * , there exists a geodesic γ from 0 to x in the environment T such that γ and γ * * are associated in B 4,S 1 ℓ (T ),N , (iii) there exists a geodesic γ * * in the environment T * * from 0 to x such that γ * * and the selected geodesic γ in the environment T are associated in B 4,S 1 ℓ (T ),N . The proof of Lemma 3.4 is the aim of Section 3.4. We now conclude the proof of Proposition 1.11 in the bounded case. The following lemma is the counterpart of Lemma 2.6 in this case. Lemma 3.5. There exists λ ∈ (0, 1) which does not depend on n and on x ∈ Γ n such that Proof. Let ℓ be in {1, ..., q}. For every s ∈ Z d and E * * subset of edges of B 3,s,N , let us consider the environment T * * s,E * * defined for all edge e by: We define E * * (T, We take the expectation on both sides. The right side yields P T * * s,E * * ∈ M(ℓ − 1) \ M(ℓ), For the left side, we have Since on the event {T ∈ M(ℓ)}∩ S 1 ℓ (T ) = s ∩{T ′ ∈ B * (T )}∩{E * * (T, T ′ ) = E * * }, P ( T ′′ ∈ B * * (T, T ′ )| T, T ′ ) is bounded from below by η, the left side is bounded from below by Then, by summing on all E * * subsets of edges of B 3,s,N and writing K a constant which does not depend on s and which bounds from above the number of different subsets of edges of B 3,s,N , we get Then, by summing on all s ∈ Z d and writing K ′ a constant which bounds from above for all s ∈ Z d the number of vertices s ′ ∈ Z d such that s ′ ≈ s, we get η 2 KK ′ P(T ∈ M(ℓ)) ≤ P(T ∈ M(ℓ − 1) \ M(ℓ)).

Properties of a typical box
In this section, we state and prove the following lemma, which gives us properties of typical boxes useful for the modification argument, and the proof of Lemma 3.2.
Lemma 3.6. If B 4,s,N is a typical box, we have the following properties.
(ii) For all z ∈ B 3,s,N , for all r > 0, for all N ∈ N * , (iii) For all z ∈ B 3,s,N , for all r > 0, for all N ∈ N * , The requirement on u and v implies (ii) and (iii) In both cases, it is easy to check that the first inclusion follows from property (iii) of a typical box and an easy computation using (i) shows the second inclusion.

Proof of Lemma 3.2.
1. Let B 4,s,N be a typical box and u 0 and v 0 two points of B 3,s,N . Then, taking paths minimizing the distance for the norm . 1 between u 0 and sN and between v 0 and sN , we have t(u 0 , v 0 ) ≤ 2r 3 N t max . Then, if a geodesic γ u0,v0 takes an edge which is not in B 4,s,N , since r 4 ≥ r 3 + 1, using the item (ii) of the definition of a typical box leads to which is impossible since r 4 > r 3 (t min + δ + t max ) t min + δ . Note that we only use item (ii) of the definition of a typical box to prove this property. To prove that recall that ε is fixed in Section 3.2 and fix ρ = 1 2d((1 + ε)C µ + t max ) . Let us consider the following property for N large enough to have ⌊ρN ⌋ = 0: Finally, the proof is completed by showing that (3.10) implies that B 4,0,N satisfies (iii). Assume (3.10) and let u and v be two vertices in B 3,0,N . The aim is to show We get (3.11) thanks to the choice of ρ.

Modification argument
In this section, we prove Lemma 3.4.

Framework.
Before proving in detail Lemma 3.4, we give the main ideas of the proof of the modification argument. We consider a geodesic γ crossing a typical box. Recall that this box is composed of four concentric balls of different size, denoted 6 by B 1 , B 2 , B 3 and B 4 , and that γ visits at least one vertex in B 1 . Recall also that the radii of the balls satisfy r 1 ≪ r 2 ≪ r 3 ≪ r 4 . The aim is to modify the environment to get the following properties: 1. Every geodesic in the new environment takes the pattern in B 2 .
2. Every geodesic in the new environment is associated in B 4 with a geodesic in the first environment.
3. The geodesic γ is associated in B 4 with at least one geodesic in the new environment.

First modification.
Let us begin with some notations.
• We denote by D before the set of edges e verifying the following three conditions: γ takes e between its first entry in B 3 and its first entry in B 2 .
e belongs to B 3 but e does not belong to B 2 .
We denote by s 1 the first vertex in D before visited by γ.
• D after and s 2 are symmetrically defined. In particular, s 2 is the last vertex of D after visited by γ.
• We set D = D before ∪ D after .
The first modification simply consists in reducing the passage times of edges in D below t min + δ ′ (with 0 < δ ′ < δ properly chosen). This provides a localization of the geodesics in the new environment T * : they all take all edges of D (see Lemma 3.7 where D is called E * + (T )). Note also that γ remains a geodesic in the environment T * . One could easily deduce that any geodesic in this new environment visits s 1 and s 2 and then that properties 2 and 3 about associated geodesics hold. The point is that these properties will be preserved by the second modification whose influence on time is negligible with respect to the first modification.

Second modification.
Framework and overall plan. The selected geodesic γ visits at least one vertex of B 1 , let us denote by c 0 the first of them. We denote by u 1 the first vertex of γ such that T (γ u1,c0 ) ≤ ∇N and v 1 the last vertex of γ such that T (γ c0,v1 ) ≤ ∇N (see (3.18)). Since the box is typical and since r 1 and ∇ are small enough compared with r 2 , all that we consider (here and in what follows) takes place in B 2 where the environments T and T * coincide (see Lemma 3.8).
The rough plan is to modify the environment between u 1 and v 1 : 1. We consider an oriented path π from u 1 to v 1 and we make the passage times on its edges very small: this is our highway.
2. We put the pattern somewhere on π.
3. We make the passage times on the other edges in a certain neighborhood (to be defined) very large: these are our walls.
Requirements for π. It is quite simple and it is stated in Lemma 3.11. We want it to be close to the segment [u 1 , v 1 ] of R d : this will allow us to have good estimates on the relevant times. We also require that the path contains towards its middle a sequence of steps in the same direction: this will allow us to place the pattern there. We call this sequence of steps the central segment of π. There is, however, a small difficulty: we can not choose the orientation of the central segment. That is why we need to be able to place the original pattern in an overlapping pattern of arbitrary orientation (this is the purpose of Lemma 3.1).
Requirements for the neighborhood of the walls. This point is less simple. We do not want to put walls on edges of γ \ γ u1,v1 but we also do not want that these edges affect our estimates. Ideally, we would like the only relevant edges, apart those from π and the pattern-location, to be walls. This can be done in a non technical way as follows. Set B * (0) = B * (0, T * (0, u 1 )) and B * (x) = B * (x, T * (x, v 1 )).
There are three types of edges: • The "before" edges: the ones belonging to B * (0).
• The "after" edges: the ones belonging to B * (x).

• The "intermediate" edges: all other edges.
Since γ is a geodesic in the environment T * , γ takes "before" edges, then "intermediate" edges and then "after" edges. Note that the edges of D before are "before" edges and that the edges of D after are "after" edges.
We can then define the neighborhood on which we put walls: this is the set of "intermediate" edges of B 2 which do not belong to π and to the pattern-location. We call these edges the "wall" edges. The idea is that, as explained at (3.12), if π * * is a part of a geodesic from 0 to x in the environment T * * linking two vertices outside B * (0) and B * (x), then π * * only takes "intermediate" edges. Hence, if in addition π * * does not take edges of π and is entirely contained in B 2 , then π * * only takes "wall" edges or edges of the pattern-location: this will ensure in the end that every geodesic takes the pattern.

It makes sense.
Recall that γ is still a geodesic in the environment T * and that it visits u 1 and then c 0 . Using the definition of B * (0) and the triangle inequality, we get In particular, the distance for the norm · 1 between B * (0) and c 0 is at least of order ∇N . The same applies to the distance for the norm · 1 between B * (x) and c 0 . Thus there is enough room for the central segment and the pattern since their size is of order 1 (see Lemma 3.13).

The second modification.
• We put the pattern somewhere on the "central segment".
• With the exception of the edges of π connected to B * (0) or B * (x), we put the time of every "intermediate" edge of π which is not in the pattern below t min + δ ′ .
• We put the time of every other "intermediate" edge of B 2 which does not belong to π and whose time is lower than ν greater than or equal to ν (recall that in particular we take ν such that, in the pattern, the passage time of every edge is lower than or equal to ν).

Consequences.
In what follows, we denote by γ * * a geodesic in the environment T * * .
1. Since the passage times of the "before" and "after" edges have not been modified and since the passage times of the "intermediate" edges touching B * (0) or B * (x) have not been reduced, B * (0) remains a ball centered in 0 for the passage times of the environment T * * and similarly for B * (x) (see again Lemma 3.13). Hence, γ * * takes "before" edges, then "intermediate" edges and then "after" edges. (3.12) 2. The creation of the highway on π makes the passage time from 0 to x lower (as stated in Lemma 3.14, it allows to save a time of order ∇N ; it can be seen by taking a geodesic from 0 to the last vertex of π belonging to B * (0), then following π, then taking a geodesic from the first vertex of π belonging to B * (x) to x). Hence γ * * takes an edge whose time has been reduced during the second modification and thus γ * * takes an edge of π whose time has been reduced or an edge of the pattern (and thus an "intermediate" edge of B 2 ). (3.13) 3. The time saved during the first modification (of order r 3 ) is such that every geodesic in the environment T * * from 0 to a vertex w in B 2 has to take an edge of D (i.e. an edge whose time has been reduced during the first modification). If it was not the case, a path following γ until B 2 and then taking edges of B 2 to go to w would have a passage time smaller (we use here that r 3 ≫ r 2 ). This is similar for a geodesic between a vertex of B 2 and x. Hence, as stated in Lemma 3.17, γ * * takes an edge of D before and after visiting B 2 . (3.14) 4. Recall that D = D before ∪ D after , that the edges of D before are "before" edges and that those of D after are "after" edges. Combining (3.12), (3.13) and (3.14) we get that γ * * takes an edge of D before then an edge of π whose time has been reduced or an edge of the pattern and then an edge of D after .

(3.15)
We can easily deduce from this that γ * * visits s 1 just before taking the first edge whose time has been modified and visits s 2 just after taking the last edge whose time has been modified (see Lemma 3.18). Then we can deduce the desired properties about associated geodesics. It remains to prove that γ * * takes the pattern in B 2 .

Let us now prove that
γ * * takes an edge of π whose time has been reduced before (with respect to π) the pattern and after (with respect to π) the pattern. (3.16) This is the purpose of Lemma 3.20. By symmetry it is sufficient to prove the first part of this property. To this aim, it is sufficient to consider a vertex w in the pattern or on π between the pattern and B * (x) and to prove Thus (since the three vertices are roughly aligned in the correct order): and the result holds.
6. By (3.16) and the remark made after the definition of walls, we can easily deduce that γ * * takes the pattern we have put on π (see Lemma 3.21).

First modification
Let us go back to the proof. For the rest of Section 3.4, we fix ℓ ∈ {1, . . . , q} and s ∈ Z d , and we assume that the event {T ∈ M(ℓ)} ∩ S 1 ℓ (T ) = s occurs. In particular, B 4,s,N is a typical box in the environment T .
As explained above, the first modification is used to get that any geodesic from 0 to x in the new environment goes from 0 to a specific vertex in B 4,s,N , then follows γ to another specific vertex in B 4,s,N and then goes to x. It is useful to get the properties about associated geodesics after the second modification. We denote by γ the selected geodesic in the environment T . Recall the definition of entry and exit points given at the beginning of Section 2.  (i) There are at least α(r 3 − r 2 )N edges of γ u,u0 and α(r 3 − r 2 )N edges of γ v0,v that belong to E * + (T ). Thus, (ii) In the environment T * , every geodesic from 0 to x visits every edge of E * + (T ). (iii) In the environment T * , γ is a geodesic from 0 to x.

Proof. (i) This item follows from the first property of a typical box applied to the portion of γ u,u0
entirely contained in B 3,s,N going from ∂B 3,s,N to u 0 and to the portion of γ v0,v entirely contained in B 3,s,N going from v 0 to ∂B 3,s,N (note that these two portions does not have edges in common).
(ii) To prove the second point, let γ * be a geodesic from 0 to x in the environment T * . Assume that there exists an edge of E * + (T ) which is not an edge of γ * . Then Since γ is a geodesic from 0 to x in the environment T , we have T (γ) ≤ T (γ * ), which implies Thus T * (γ) < T * (γ * ), which contradicts the fact that γ * is a geodesic from 0 to x in the environment T * .
(iii) Let us now assume that γ is not a geodesic in the environment T * . Hence, if we denote by γ * a geodesic from 0 to x, we have T * (γ * ) < T * (γ). By item (ii), Thus, T (γ * ) < T (γ), which contradicts the fact that γ is a geodesic in the environment T .

Construction of π
Here, we shall identify an oriented path π between two vertices of γ in B 2,s,N . This oriented path is later used to place the pattern. Let c 0 denote the entry point of γ in B 1,s,N and recall the definition of ∇ in Remark 3.9. The idea of the proof is that µ(u 1 − v 1 ) is roughly equal to t(u 1 , v 1 ). Furthermore, since γ is a geodesic visiting u 1 , c 0 and v 1 in this order, we have t( We have a sufficient control over the approximations to guarantee a lower bound by N ∇. Proof. Using item (ii) of Lemma 3.6 with z = c 0 and r = ∇(1 − ε) − 1 leads to Since u 1 is the entry point of a path in B µ (c 0 , N ∇), there exists a vertex u 1 ∈ B 3,s,N such that u 1 − u 1 1 = 1 and The same argument holds for t(c 0 , v 1 ). Hence Thus, since ∇ > 3 + 2t max 1 − 3ε , by the third item of the definition of a typical box, For the second part of the proof, using the third item of Lemma 3.6 with z = c 0 and r = ∇ leads to Then, by (3.3) we have r 2 > r 1 + 2(∇ + 2) c µ . So, for all z ∈ B µ (c 0 , 2N (∇ + 2)), we have 2N (∇ + 2)).
Thus, by the second item of Lemma 3.6 with z = c 0 and r = ∇, we get To sum up, Now, assume that γ u1,v1 visits a vertex which is not in B 2,s,N . Let denote by z such a vertex and assume for example that z is visited by γ u1,c0 . Then, we have, thanks to these inclusions, t(c 0 , z) > t(c 0 , u 1 ), which is impossible since γ u1,c0 is a geodesic.
(i) We have the following inclusions: x)).
Now, we can make the construction of π, which is the path on which we would like to put the pattern that the geodesics have to take. We begin by two definitions: • We say that a path between two vertices y 1 and y 2 is oriented if its number of edges is equal to y 1 − y 2 1 .
• A step of length ℓ is a path of ℓ consecutive edges in the same direction.
We state the following lemma whose proof is left to the reader and where [u 1 , v 1 ] is the segment in R d and . 1 is the norm on R d .
Lemma 3.12. The path π is contained in B 2,s,N .
Proof. Let z be a vertex of π. Using Lemma 3.11, there exists y ∈ [u 1 , v 1 ] such that z − y 1 ≤ L 1 . The vertices u 1 and v 1 belong to B µ (c 0 , N ∇) which is convex, thus y ∈ B µ (c 0 , N ∇). So, we have which proves that z ∈ B 2,s,N .

Second modification
Now, let us define E * * P , E * * + , E * * − and O. Let c 1 ∈ R d denote the midpoint of [u 1 , v 1 ] and let us consider the set of all vertices of π u2,v2 at distance at most L 1 for the norm . 1 from c 1 . This set is not empty, hence we can choose one such vertex with a deterministic rule. Recall that L 2 = L 1 + (10 + d)ℓ Λ . Since ∇ ≥ 6dL 2 C µ (see (3.2)), we have that the distance for the norm . ∞ between the chosen vertex and v 1 is greater than 10ℓ Λ . So there exist one or two steps of the path π of length 10ℓ Λ that contain the chosen vertex. We chose one step (between these one or two steps) with a deterministic rule to put the pattern. We denote the midpoint of this chosen step by c P . Then, O(T, T ′ ) is equal to the direction of this step, (3.23) and E * * P (T, T ′ ) = edges connecting vertices belonging to B ∞ (c P , ℓ Λ ) . (3.24) Note that by the direction of a step, we mean the integer j such that for every distinct vertices z 1 and z 2 of this step, z 1 − z 2 = ± z 1 − z 2 1 ε j . Note also that for all vertex z in B ∞ (c P , ℓ Λ ), We define u 3 and v 3 as the endpoints of the oriented pattern such that π visits u 2 , u 3 , v 3 and v 2 in this order. Recall that for a pattern, the entry and exit points correspond to the endpoints given by the definition of this pattern. Note that, thanks to the construction, u 3 and v 3 are two vertices of the chosen step of length 10ℓ Λ . We denote by γ π the path composed by the first geodesic in the lexicographic order from 0 to u 2 in the environment T * , then π u2,v2 and then the first geodesic in the lexicographic order from v 2 to x in the environment T * . Then, we have to define E * * + (T, T ′ ) and E * * − (T, T ′ ): For all the sequel, we assume that the event Recall the definition of the environment T * * given at ( In what follows, when we talk about edges whose time has been reduced, it means the edges e such that T * * (e) < T (e). Before proving item (ii) of Lemma 3.3, we state the following lemma which completes the vision of the sets E * * + (T, T ′ ), E * * − (T, T ′ ) and E * * P (T, T ′ ). Recall the definition of B * * given at (3.9).

Proof of the second and third items of Lemma 3.4
Recall that we assume that The aim of this section is to prove the following properties (which are the second and third items of Lemma 3.4): (ii) for all geodesic γ * * from 0 to x in the environment T * * , there exists a geodesic γ from 0 to x in the environment T such that γ and γ * * are associated in B 4,S 1 ℓ (T ),N , (iii) there exists a geodesic γ * * in the environment T * * from 0 to x such that γ * * and the selected geodesic γ in the environment T are associated in B 4,S 1 ℓ (T ),N . To prove this, we use the following sequence of lemmas. Proof. Let γ * * be a geodesic from 0 to x in the environment T * * . Since γ is a geodesic in the environment T * , we have the following inequalities So, using Lemma 3.14, It means that γ * * has to take at least one edge whose time has been reduced during the second modification which is not in E * * P (T, T ′ ). Hence, since the only edges which are not in E * * P (T, T ′ ) whose time has been reduced are edges of π u2,u3 ∪ π v3,v2 , the result follows. Lemma 3.17. Every geodesic from 0 to x in the environment T * * takes at least one edge of γ whose time has been reduced before taking its first edge of B 2,s,N , and takes at least one edge of γ whose time has been reduced after taking its last edge of B 2,s,N .
Proof. The idea is to use that the time saved by the geodesics from 0 to x after the first modification is much greater than the geodesic time between any two vertices of B 2,s,N in any environment. Let γ * * be a geodesic from 0 to x in the environment T * * . Let u * * be the first vertex in B 2,s,N that γ * * visits. Its existence is guaranteed by Lemma 3.16. The aim of the proof is to show T * * (γ * * 0,u * * ) < T (γ * * 0,u * * ).
Indeed, the definition of u * * and the fact that the only edges whose time has been reduced which are in B 3,s,N but not in B 2,s,N are edges of γ gives us the result. Recall that u 0 is the entry point of γ in B 2,s,N . First, since γ * * is a geodesic in the environment T * * , Then, using the first item of Lemma 3.7, we obtain Finally, using the fact that γ is a geodesic in the environment T leads to To conclude, it is sufficient to observe that the condition r 3 > 7r 2 (4t max + αδ) αδ implies that α(r 3 − r 2 )N (δ − δ ′ ) − 4r 2 N t max > 0, so we have the desired strict inequality. The same proof gives us the second part of the lemma. Lemma 3.18. Let γ * * be a geodesic from 0 to x in the environment T * * . Then the first edge of γ * * T (γ * * 0,z * * ) = T * * (γ * * 0,z * * ).
Since γ * * is a geodesic, T * * (γ * * 0,z * * ) ≤ T * * (γ 0,z * * ). Now, let us assume z * * is not s 1 . Then, by the definition of s 1 , z * * is visited by γ u,u1 after s 1 and thus, Combining these three inequalities yields which is impossible because γ 0,z * * is a geodesic in the environment T . So, the result is proved and the same proof leads to the second part of this lemma.
We can now prove the two properties stated at the beginning of this subsection. For item (ii), let γ * * be a geodesic from 0 to x in the environment T * * . By Lemma 3.17 and Lemma 3.18, like γ, γ * * visits s 1 and before that γ * * does not visit any edge whose time has been changed when replacing T by T * * . Hence, T (γ 0,s1 ) = T * * (γ 0,s1 ) ≥ T * * (γ * * 0,s1 ) = T (γ * * 0,s1 ) ≥ T (γ 0,s1 ). (3.27) This proves that T (γ 0,s1 ) = T (γ * * 0,s1 ) and γ * * 0,s1 is a geodesic in the environment T . Similarly, γ * * s2,x is a geodesic in the environment T and the path γ = γ * * 0,s1 ∪ γ s1,s2 ∪ γ * * s2,x is a geodesic in the environment T that satisfies (ii) in Lemma 3.4 if we prove that it is a self-avoiding path and it is contained in B 4,s,N . Assume, aiming at a contradiction that γ * * 0,s1 visits a vertex of γ s1,s2 which is not s 1 and denote it by s 3 . Then γ * * 0,s1 ∪ γ s1,x is an optimal path for the passage time in the environment T since γ * * 0,s1 is a geodesic in this environment. It implies that γ * * s3,s1 ∪ γ s1,s3 has at least one edge which is e 1 and that T (γ * * s3,s1 ∪ γ s1,s3 ) = 0, which is impossible since T (e 1 ) > 0. The same proof gives that γ * * s2,x does not visit a vertex of γ s1,s2 and thus γ is self-avoiding. To get item (ii), it remains to prove that γ * * s1,s2 is contained in B 4,s,N . This comes from the fact that, in any environment, the geodesic time between two vertices of B 3,s,N is bounded by 2r 3 t max , and, since the edges in B 4,s,N \ B 3,s,N have the same times in the environments T or T * * , by property (i) of a typical box, the geodesic time to reach a vertex outside B 4,s,N and to come back in B 3,s,N is bounded from below by 2(r 4 − r 3 )(t min + δ). The condition on r 4 insures that (r 4 − r 3 )(t min + δ) > r 3 t max .

Every geodesic takes the pattern.
Recall that we assume that The aim of this last subsection is to show that every geodesic in the environment T * * takes the pattern in E * * P (T, T ′ ). The proof is decomposed in two steps. The first step is to show that every geodesic takes an edge of π u2,u3 and an edge of π v3,v2 . The second step is to show that every geodesic verifying this property takes the pattern in E * * P (T, T ′ ). We begin with a technical lemma. Lemma 3.19. For all w vertex of π u2,u3 ∪ π v3,v2 , Proof. Let w be a vertex of π u2,u3 ∪π v3,v2 . Then, by the construction of π, there exists a Lemma 3.20. Let γ * * be a geodesic from 0 to x in the environment T * * . Then γ * * visits a vertex of π u2,u3 and one of π v3,v2 . More precisely, the first vertex of γ * * that belongs to π u2,u3 ∪ π v3,v2 belongs to π u2,u3 and the last to belongs to π v3,v2 .
Proof. Let γ * * be a geodesic from 0 to x in the environment T * * . By Lemma 3.16, there exists at least one vertex of π u2,u3 ∪ π v3,v2 visited by γ * * . Let w be the first vertex of γ * * that belongs to π u2,u3 ∪ π v3,v2 and assume that w belongs to π v3,v2 . Note that γ * * 0,w does not visit any other vertex of π u2,u3 ∪ π v3,v2 . The aim of the proof is to show that which is impossible since γ * * is a geodesic in the environment T * * . We start with since, by Lemma 3.7, γ is a geodesic in the environment T * . By construction, there is no edge of γ u1,v1 whose time has been changed at the first modification, thus t * (u 1 , v 1 ) = t(u 1 , v 1 ). Furthermore, since there is no edge whose time has been increased at the first modification, t * (v 1 , w) ≤ t(v 1 , w), so We now want to bound from below T * * (γ * * 0,w ). The only edges whose time has been reduced at the second modification are among those of π u2,u3 ∪ π v3,v2 and of E * * P (T, T ′ ). So, since γ * * 0,w does not take any edge of π u2,u3 ∪ π v3,v2 , it can only save time taking edges of E * * P (T, T ′ ). So, Then, using the definition of a typical box, Lemma 3.19 and the inequality t * (0, u 1 ) ≥ t * (0, u 2 ) (which comes from the definition of u 2 ) leads to On the other hand, note that, π being an oriented path, u 1 − w 1 ≥ u 2 − w 1 , so, using the knowledge of T * * on edges of π, To conclude, let us show that we have the inequality (3.28) First, combining Lemma 3.8 and Lemma 3.15 leads to Then, by (1.20), we have Then, by Lemma 3.15, By (3.29) and since 1 < δ∇ 24C µ (by (3.2)) and N > 12C µ K ′ δ∇ (by (3.6)), the condition δ ′ ≤ δ 4 gives us (3.28). Hence T * * (γ π 0,w ) < T * * (γ * * 0,w ). Finally, let us prove the following lemma which completes the proof of Lemma 3.4. Lemma 3.21. Any geodesic from 0 to x takes the pattern at the pattern-location E * * P (T, T ′ ). Proof. Let γ * * be a geodesic from 0 to x in the environment T * * . By Lemma 3.20, γ * * visits a vertex of π u2,u3 and one of π v3,v2 . As a consequence, there exist a vertex u 4 of π u2,u3 and a vertex v 4 of π v3,v2 such that γ * * goes from u 4 to v 4 without taking edges of π u2,u3 ∪ π v3,v2 . Let us remember that for all edge e of E * * P (T, T ′ ), we have T * * (e) ≤ ν. We prove successive properties. • The edges of γ * * u4,v4 which are not in E * * P (T, T ′ ) have a passage time greater than or equal to ν.
Assume that it is not the case. Let us show that T * * (γ * * u4,v4 ) − T * * (π u4,v4 ) > 0, which is a contradiction since γ * * u4,v4 is a geodesic in the environment T * * . To this aim, let us compare the time that can save each path compared with the other in any direction. Recall the notation introduced in the proof of Lemma 4.1: for i ∈ {1, . . . , d} and a pathπ, T * * i (π) denotes the sum of the passage times of the edges ofπ which are in the direction ε i . In the direction O(T, T ′ ), since u 4 − u 3 1 > 4ℓ Λ or v 4 − v 3 1 > 4ℓ Λ , and by the construction, we have that where the term 2ℓ Λ t min − 2ℓ Λ ν comes from the time potentially saved by γ * * u4,v4 by taking edges of E * * P (T, T ′ ). Then, in any other direction, γ * * u4,v4 can save a time lower than or equal to 2ℓ Λ δ ′ compared with π u4,v4 thanks to the edges in E * * P (T, T ′ ). Hence, • We have that u 4 = u 3 and v 4 = v 3 .
Thus γ * * u4,v4 is a path entirely contained in E * * P (T, T ′ ), going from u 3 to v 3 and with an optimal time. So, we have the result.

Modification proof for the Euclidean length of geodesics
To prove Theorem 1.8, we begin by defining the valid pattern in three different cases. Recall that k and ℓ are given by the assumptions of this theorem and that (ε i ) i∈{1,...,d} are the vectors of the canonical basis.
Case where zero is an atom. In addition to the assumptions of Theorem 1.8, we assume that zero is an atom for F . We set L 1 = k + 2, L 2 = ℓ + 2 and if d ≥ 3, for all i ∈ {3, . . . , d}, L i = 2. We define a pattern in Λ = d i=1 {0, . . . , L i }. We take the endpoints u = d i=2 ε i and v = u + (k + 2)ε 1 . We denote by π + the path going from u to v by k + 2 steps in the direction ε 1 and by π ++ the path going from u to u + ε 1 by one step in the direction ε 1 , then to u + ε 1 + ℓε 2 by ℓ steps in the direction ε 2 , then to u + (k + 1)ε 1 + ℓε 2 by k steps in the direction ε 1 , then to u + (k + 1)ε 1 by ℓ steps in the direction −ε 2 and then to v by one step in the direction ε 1 . We define A Λ as follows: • for all e ∈ π + ∪ π ++ , T (e) = 0, • for all e ∈ Λ which is not in π + ∪ π ++ , T (e) > 0.
Note that A Λ has a positive probability. Then, π + and π ++ are the only two optimal self-avoiding paths from u to v entirely contained in the pattern. Furthermore, for every vertex z ∈ π + ∪ π ++ different from u and v, there exists no path from ∂Λ \ {u, v} to z whose passage time is equal to 0.

Unbounded case.
Here, in addition to the assumptions of Theorem 1.8, we assume that zero is not an atom and that the support of F is unbounded. We set L 1 = k, L 2 = ℓ and if d ≥ 3, for all i ∈ {3, . . . , d}, L i = 0. We define a pattern in Λ = d i=1 {0, . . . , L i }. We take the endpoints u = 0 and v = kε 1 . We denote by π + the path going from 0 to kε 1 by k steps in the direction ε 1 and by π ++ the path going from 0 to ℓε 2 by ℓ steps in the direction ε 2 , then to kε 1 + ℓε 2 by k steps in the direction ε 1 and then to v by ℓ steps in the direction −ε 2 . Then, we index the edges of π + and the ones of π ++ in the order in which they are taken by theses paths. We respectively denote them by (e 1 i ) i∈{1,...,k} and (e 2 i ) i∈{1,...,k+2ℓ} . We fix M > k j=1 s ′ j and we define A Λ as follows: • for all i ∈ {1, . . . , k}, T (e 1 i ) = s ′ i and for all i ∈ {1, . . . , k + 2ℓ}, T (e 2 i ) = r ′ i , • for all e ∈ Λ which are not in π + or π ++ , T (e) > M .
Since M > T (π + ), the optimal paths from u to v entirely contained in the pattern can not take other edges than those in π + ∪ π ++ . Hence π + and π ++ are the only two optimal paths from u to v entirely contained in the pattern.
Bounded case. In addition to the assumptions of Theorem 1.8, we assume that zero is not an atom and that the support of F is bounded. We set t max = sup(support(F )). We denote a max = max(r ′ 1 , . . . , r ′ k+2ℓ , s ′ 1 , . . . , s ′ k ). Then, there are at least 2ℓ integers j ∈ {1, . . . , k +2ℓ} such that r ′ j < a max . Indeed, assume that this is not the case. Then, we have and this contradicts (1.6). Thus, even if it means changing the indexes, we can assume that for all i ∈ {1, . . . , 2ℓ}, r ′ i < a max and we denote t w = a max − max(r ′ 1 , . . . , r ′ 2ℓ ) > 0. We fix α > 0 an integer such that: {0, . . . , L i }. We take the endpoints u = 0 and v = k ′ ε 1 . Let π + be the path going from u to v by k ′ steps in the direction ε 1 and π ++ be the path going from u to u ′ = ℓ ′ ε 2 by ℓ ′ steps in the direction ε 2 , then to v ′ = k ′ ε 1 + ℓ ′ ε 2 by k ′ steps in the direction ε 1 and then to v by ℓ ′ steps in the direction −ε 2 . Then we index the edges of π + , π ++ u,u ′ , π ++ u ′ ,v ′ and π ++ v ′ ,v in the order in which they are taken by these paths. We respectively denote them by e 1 i i∈{1,...,k ′ } , e 2 i i∈{1,...,ℓ ′ } , e 3 i i∈{1,...,k ′ } and e 4 i i∈{1,...,ℓ ′ } . The idea for the event A Λ is just to alternate the atoms on every boundary of the rectangle whose vertices are u, u ′ , v ′ and v. It allows us a better control of the time of a path taking both vertices of π + and vertices of π ++ u ′ ,v ′ (see Figure 5). So, we define A Λ as follows : • for all other edges e ∈ Λ, T (e) = a max .
Note that A Λ has a positive probability and that on this event, T (π + ) = T (π ++ ).
Lemma 4.1. The paths π + and π ++ belong to the family of optimal paths from u to v which are entirely contained in the pattern.
Proof of Lemma 4.1. Let us begin by introducing some notations. For i ∈ {1, . . . , d} and a path π, T i (π) denotes the sum of the passage times of the edges of π which are in the direction ε i . So we have T (π) = T 1 (π) + · · · + T d (π). Furthermore, for j ∈ {0, . . . , ℓ ′ − 1}, we denote by S j 2 the set of edges of Λ which can be written {x, x + ε 2 } where the second coordinate of x is equal to j. Now, to prove the lemma, assume for a contradiction that there exists an optimal self-avoiding path π entirely contained in Λ such that T (π) < T (π + ) = T (π ++ ).
(4.2) Λ Conclusion in the three cases.
Proof of Theorem 1. 8. For x ∈ Z d , we denote by π(x) the first geodesic from 0 to x in the lexicographical order 7 among those who have the minimal number of edges. We say that two patterns are disjoint if they have no vertex in common. We denote by N P (π(x)) the maximum number of disjoint patterns defined above in the three cases visited by π(x). Simple geometric considerations provide a constant c > 0 such that for all path π, N P (π) ≥ cN P (π). Further, note that, in each pattern visited by π(x), π(x) takes the π + segment since the π + segment belongs to the set of optimal paths entirely contained in the pattern and is the only optimal path for the norm . 1 . We can define a self-avoiding path π(x) from 0 to x by replacing each π + segment of π(x) with the π ++ segment in each disjoint pattern visited by π(x). This path π(x) has the same passage time and hence both π(x) and π(x) are geodesics. Note that in the case where zero is not an atom, π is obviously self-avoiding (since every geodesic is self avoiding), and in the case where zero is an atom, π is self-avoiding since in the patterns, the passage times of all edges which are not in the π + or π ++ segments are strictly positive, and thus π(x) can not visit vertices in the π ++ segments except those which also belong to the corresponding π + segment. In the three cases, we have: Finally, we get: The hypothesis of Theorem 1.4 are satisfied, thus there exist α > 0, β 1 > 0 and β 2 > 0 such that for all x ∈ Z d , P ∃ a geodesic γ from 0 to x such that N P (γ) < α x 1 ≤ β 1 e −β2 x 1 .

Modification proof for the strict concavity of the expected passage times as a function of the weight shifts
Proof of Theorem 1.9. Recall that we assume that the support of F is bounded and that there are two different positive points in this support. We set t max = sup(support(F )). Let 0 < r < s be two points in the support of F . Fix b ∈ (0, r). Applying Lemma 5.5 in [5], we get positive integers k, ℓ fixed for the rest of the proof such that holds for sufficiently small δ > 0. Fix L = k + ℓ + 1, With this definition, every path from the boundary of Λ to the boundary of Λ 0 has to take at least L edges of Λ \ Λ 0 . We take π + the path going from u Λ to v Λ by k + 2L steps in the direction ε 1 , and π ++ the one going from u Λ to u 1 = u Λ + Lε 1 ∈ Λ 0 by L steps in the direction ε 1 , then going to u 2 = u 1 + ℓε 2 ∈ Λ 0 by ℓ steps in the direction ε 2 , then to u 3 = u 2 + kε 1 ∈ Λ 0 by k steps in the direction ε 1 , then to u 4 = u 3 − ℓε 2 ∈ Λ 0 by ℓ steps in the direction −ε 2 , and then to v Λ by L steps in the direction ε 1 . For a deterministic family (t e ) e∈E Λ of passage times on the edges of Λ and for a path π, we use the abuse of writing T (π) to denote e∈π t e . For all δ ≥ 0, we consider the set G(δ) of families (t e ) e∈E Λ of passage times on the edges of Λ which satisfy the following two conditions: 7 The lexicographical order is based on the directions of the consecutive edges of the geodesics.
Since the only edges in the direction ε 1 whose passage time is smaller than s are in π ++ u2,u3 , π takes an edge of π ++ u2,u3 and thus T 2 (π) ≥ 2ℓr. Furthermore Hence we get where the strict inequality comes from (4.5). Thus, it contradicts the fact that π is an optimal path and (P1) holds. Now, to prove that (P2) holds, let π 1 be a path from a vertex w 1 of the boundary of Λ 0 to a vertex of the boundary of Λ and let w 2 a vertex of Λ 0 . Let π 2 be an optimal path for the norm . 1 going from w 1 to w 2 . Then, by the definition of Λ 0 (see (4.6)), π 1 has to take at least L edges whose passage time is equal to s although π 2 takes at most k + ℓ edges whose passage time is smaller than or equal to s. Thus where the strict inequality comes from the fact that L = k + ℓ + 1. Hence (P2) holds and G(0) ⊂ H. Furthermore, H is an open set since for a family (t e ) e∈E Λ to belong to H, it is required that the time of a finite family of paths is strictly smaller than the time of every path of another finite family of paths. Hence, for δ > 0 small enough, we have G(δ) ⊂ H. (4.7) Fix δ > 0 such that (4.5) and (4.7) hold. Consider the pattern P = (Λ, u Λ , v Λ , A Λ ) with A Λ = {(T (e)) e∈E Λ ∈ G(δ)}. Now, we denote by π(x) the first geodesic from 0 to x in the lexicographical order among those of maximal Euclidean length and we denote by N P (π(x)) the maximum number of disjoint patterns visited by π(x). Recall the existence of a constant c > 0 small enough such that for all path π, N P (π) ≥ cN P (π). Since P(A Λ ) > 0, we can apply Theorem 1.4. Let α, β 1 , β 2 > 0 be the constants given by Theorem 1.4. Then, we have Now, let us follow the end of Stage 3 of the proof of Theorem 5.4 in [5]. By (P1), π(x) takes the π + segment in each pattern that it takes. Furthermore, by (P2), π(x) does not take any edge in the π ++ segment which is not in the π + segment. So, we can define a self-avoiding path π(x) from 0 to x by replacing each π + segment with the π ++ segment in each pattern visited by π(x). Reduce the weights on each edge e from T (e) to T (−b) (e) = T (e) − b. By the definition of the pattern, the T (−b) -passage times of the segments π + and π ++ obey the inequality: Then, following the proof of Theorem 5.4 in [5], we get • for every self-avoiding path π, N P (π) ≥ N P0 (π).
• u Λ 0 (resp. v Λ 0 ) one vertex of ∂Λ 0 such that there exists a path π u (resp. π v ) fixed for the remaining of the proof linking u Λ 0 and u Λ (resp. v Λ 0 and v Λ ) which satisfies the following two conditions: π u (resp. π v ) does not visit any vertex of Λ except u Λ (resp. v Λ ), π u (resp. π v ) only takes edges in the direction of one external normal unit vector associated with u Λ (resp. v Λ ) chosen arbitrarily if there are several such vectors, • A Λ 0 the event such that: for all e belonging to π u ∪ π v , we have T (e) ≤ M Λ , -for all e which does not belong to Λ ∪ π u ∪ π v , where |Λ 0 | e is the number of edges in Λ 0 .
Note that, since u Λ and v Λ are distinct, π u and π v are disjoint. We get: • P(A Λ 0 ) is positive since P is valid and since the support of F is unbounded, and then P 0 is a valid pattern.
• On A Λ 0 , any path from u Λ 0 to v Λ 0 optimal for the passage time among the paths entirely inside Λ 0 contains a subpath from u Λ to v Λ entirely inside Λ. Indeed, let π 0 be a path from u Λ 0 to v Λ 0 which does not contain a subpath from u Λ to v Λ entirely inside Λ. It implies that π 0 takes an edge whose time is greater than |Λ 0 | e M Λ . Let π ′ 0 be a path following π u , then going from u Λ to v Λ inside Λ, and then following π v . We have T (π ′ 0 ) ≤ |Λ 0 | e M Λ < T (π 0 ) and thus π 0 is not an optimal path. Hence, for every path π, if a vertex x ∈ Z d satisfies the condition (π; P 0 ), x satisfies the condition (π; P).

A.2 Construction of the path π for the modification in the unbounded case
Recall that, here, u Λ and v Λ are the vertices defined at the beginning of Section 2.3. The aim is to prove that we can construct a path π in a deterministic way such that: (i) π goes from u to u Λ without visiting a vertex of B ∞ (sN, ℓ Λ ), then going from u Λ to v Λ in a shortest way for the norm . 1 (and thus being contained in B ∞ (sN, ℓ Λ )) and then goes from v Λ to v without visiting a vertex of B ∞ (sN, ℓ Λ ), (ii) π is entirely contained in B 2,s,N and does not have vertices on the boundary of B 2,s,N except u and v, (iii) π is self-avoiding, (iv) the length of π u,u Λ ∪ π v Λ ,v is bounded from above by 2r 2 N + K, where K is the number of edges in B ∞ (0, ℓ Λ + 3).
As it is said in Section 2.3, we want to construct a path from u to sN and a path from sN to v which have no vertex in common except sN and such that their lengths are bounded from above by r 2 N . To get them, we use the following lemma whose proof is left to the reader.
Lemme A.2. Let m ∈ N * and x, y two vertices of Z d such that x 1 = y 1 = m and x = y. Then we can build in a deterministic way two paths π x and π y linking respectively x and y to 0 and such that their length is equal to m, they have only 0 as a common vertex and they have respectively only x and y as vertices whose norm . 1 is greater than or equal to m.
Using this lemma and replacing 0 by sN using a translation, we get two paths linking u to sN and v to sN with the stated properties. Recall that B ∞ (sN, ℓ Λ + 3) ⊂ B 2,s,N and let u 0 (resp. v 0 ) denote the first vertex in B ∞ (sN, ℓ Λ + 3) visited by the path going from u to sN (resp. the one going from v to sN ). Then we get two paths π u,u0 and π v0,v respectively from u to u 0 and from v 0 to v both constructed in a deterministic way such that π u,u0 and π v0,v do not have any vertex in common, are entirely contained in B 2,s,N , have only u or v as points on the boundary of B 2,s,N , and their lengths are bounded from above by r 2 N − (ℓ Λ + 3).
Then we build two paths π u0,u Λ and π v Λ ,v0 respectively from u 0 to u Λ and from v Λ to v 0 contained in B ∞ (sN, ℓ Λ +3)\B ∞ (sN, ℓ Λ ) except for u Λ and v Λ , such that they do not have any vertex in common. By the definition of K, this implies that the sum of their lengths is bounded from above by K (recall that K is the number of edges in B ∞ (0, ℓ Λ +3)). To get these paths, assume first that u 0 −v Λ 1 > 3 or v 0 −u Λ 1 > 3. Assume that we have v 0 − u Λ 1 > 3, the other case being the same. We begin by considering a pathπ going in a shortest way from v 0 to a vertex of the boundary of B ∞ (sN, ℓ Λ + 2), denoted by v ′ 0 . This path has at most d edges. Then, let u Λ,0 be a vertex on the boundary of B ∞ (sN, ℓ Λ + 3) such that u Λ − u Λ,0 1 = 3. We get π u0,u Λ by going from u 0 to u Λ,0 in a shortest way on the boundary of B ∞ (sN, ℓ Λ + 3) avoiding all vertices ofπ, and then by going from u Λ,0 to u Λ by three steps. This is possible since v 0 − u Λ 1 > 3. To get π v0,v Λ , we begin by followingπ. Let v Λ,0 be a vertex on the boundary of B ∞ (sN, ℓ Λ + 2) such that v Λ − v Λ,0 1 = 2. Then, π v0,v Λ goes from v ′ 0 to v Λ,0 in a shortest way on the boundary of B ∞ (sN, ℓ Λ +2) avoiding the unique vertex of π u0,v Λ belonging to B ∞ (sN, ℓ Λ +2). Finally, π v0,v Λ goes from v Λ,0 to v Λ by two steps (see the left side of Figure 6 for an example of this construction in dimension 2). Now, if u 0 − v Λ 1 = 3 and v 0 − u Λ 1 = 3, the construction has to be slightly different. From u 0 , π u0,u Λ goes to a vertex u ′′ 0 belonging to the boundary of B ∞ (sN, ℓ Λ + 3) and such that u ′′ 0 − v 0 1 = 1 in a shortest way on the boundary of B ∞ (sN, ℓ Λ + 3). After, it makes one step to go to the boundary of B ∞ (sN, ℓ Λ + 2) and goes to u Λ in a shortest way by taking only one vertex in B ∞ (sN, ℓ Λ + 1). Then, π v0,v Λ goes on the boundary of B ∞ (sN, ℓ Λ +3) to a vertex v ′′ 0 belonging to the boundary of B ∞ (sN, ℓ Λ +3) and such that v ′′ 0 − u 0 1 = 1 by avoiding every vertex which belongs to π u0,u Λ , and then makes two steps to go to the boundary of B ∞ (sN, ℓ Λ + 1) and goes to v Λ in a shortest way (see the right side of Figure 6 for an example of this construction in dimension 2). In this case, we also have that the sum of their lengths is bounded from above by K.
Finally, π is the path obtained by concatenating π u,u0 , π u0,u Λ , a path going from u Λ to v Λ in a shortest way for the norm . 1 , π v Λ ,v0 and π v0,v in this order. We have that π is a self-avoiding path contained in B 2,s,N and has only u and v on the boundary of B 2,s,N .

B Overlapping pattern in the bounded case
Proof of Lemma 3.1. Recall that ν is fixed at the beginning of Section 3 and that δ ′ = min δ 8 , δ 1 + d .
Notice that, if F has an atom, one could have ν 0 = ν, or even ν 0 = ν = t max . Then, fix ℓ Λ = max(L 1 , . . . , L d ), (B.1) and ℓ 0 > ℓ Λ ((2d + 1)ν 0 + (2d − 1)t min + 2dδ ′ ) + ℓ 1 (ν 0 + 3t min + 4δ ′ ) Let j ∈ {1, ..., d}. Let us construct the overlapping pattern in Λ 0 = {−ℓ 0 , ..., ℓ 0 } d with endpoints ℓ 0 ε j and −ℓ 0 ε j . We denote by u 3 and v 3 the endpoints of the original pattern. The first step is the construction of an oriented path going from the face of Λ 0 containing ℓ 0 ε j to the face containing −ℓ 0 ε j and visiting u 3 and v 3 . We denote by Λ 1 the set d i=1 {−ℓ 1 , . . . , L i + ℓ 1 }. Then we define u 2 (resp. v 2 ) the vertex of ∂Λ 1 which can be linked to u 3 (resp. v 3 ) by a path using exactly ℓ 1 edges in only one direction (the direction of the external normal vector associated to u 3 (resp. v 3 )). If u 3 or v 3 are associated to several external normal unit vectors, then the choice of the direction is not unique. We choose one external normal unit vector in an arbitrary way but we ensure that the two normal unit vectors chosen to build u 2 and v 2 are different. Note that this is possible since the pattern is valid (recall Definition 1.2). For a vertex z ∈ Z d , denote by z(j) its j-th coordinate. Even if it means exchanging the roles of u 3 and v 3 , we can assume that u 2 (j) ≥ v 2 (j). Then, we define u 1 = u 2 + (ℓ 0 − u 2 (j))ε j and v 1 = v 2 − (ℓ 0 + v 2 (j))ε j . Note that u 1 (resp. v 1 ) belongs to the face of Λ 0 containing ℓ 0 ε j (resp. −ℓ 0 ε j ). We define a pathπ going from ℓ 0 ε j to u 1 in a shortest way, then to u 2 in the shortest way, then to u 3 in the shortest way, then to v 3 in a shortest way, then to v 2 in the shortest way, then to v 1 in the shortest way and to v 0 in a shortest way. Note thatπ u1,v1 is an oriented path. Indeed,π u3,v3 is an oriented path. Extending it outside Λ following the direction of an external normal unit vector preserves the fact that it is oriented. Thusπ u2,v2 is an oriented path. Then, we assume that u 2 (j) ≥ v 2 (j) on purpose to guarantee that, with the definitions of u 1 and v 1 , the pathπ u1,v1 remains an oriented path. The event (whose probability is positive) of the overlapping pattern, denoted by A Λ0 j is the following: • for all e ∈π ℓ0εj ,u3 and e ∈π v3,−ℓ0εj , T (e) < t min + δ ′′ , • the event A Λ ∩ {∀e ∈ Λ, T (e) ≤ ν 0 } is realized, • for all e in Λ 0 \ (π ∪ Λ), ν 0 ≤ T (e) ≤ ν.
On this event, let γ be one of the fastest path from ℓ 0 ε j to −ℓ 0 ε j among the path entirely contained in Λ 0 and let us show that γ visits u 3 and v 3 and that γ u3,v3 is entirely contained in Λ. We proceed by proving successive properties.
Indeed, let a 1 be such a vertex and let a 0 be the vertex of the preceding property. We only have to prove that γ a0,a1 =π a0,a1 and the proof is the same as the one for γ ℓ0εj ,a0 =π ℓ0εj ,a0 .
Among the vertices visited by γ, we denote by a (resp. b) the last vertex ofπ ℓ0εj ,u3 (resp. the first vertex ofπ v3,−ℓ0εj ). Then, γ a,b does not visit any vertex ofπ ℓ0εj ,u3 ∪π v3,−ℓ0εj (except a and b). Indeed, by the two properties above, we have that γ ℓ0εj ,a =π ℓ0εj ,a and γ b,−ℓ0εj =π b,−ℓ0εj and γ a,b can not visit a vertex ofπ a,u3 ∪π v3,b thanks to the definition of a and b.
Then, since the edges whose time is smaller than ν 0 taken by γ a,b are those in Λ, Using (B.2), we get T (γ a,b ) > T (π a,b ), which is impossible.
• We have that a = u 3 and b = v 3 .
Assume that a = u 3 , the other case being the same. We have T (π a,b ) ≤ ( a − b 1 − u 3 − v 3 1 )(t min + δ ′ ) + u 3 − v 3 1 ν 0 , and thus γ a,b takes at least one edge of Λ otherwise T (γ a,b ) ≥ a − b 1 ν 0 > T (π a,b ) since a − b 1 − u 3 − v 3 1 > 0. Let u 0 be the first entry point of γ a,b in Λ and let us consider the path π following π a,u3 , then going from u 3 to u 0 in a shortest way and then following γ u0,b . Then, the number of edges of π a,u0 is lower than or equal to the number of edges of γ a,u0 , for all e ∈ π a,u0 , T (e) ≤ ν 0 , there exists e ′ ∈ π a,u0 such that T (e ′ ) ≤ t min + δ ′ and for all e ∈ γ a,u0 , T (e) ≥ ν 0 . So, we have T (π) < T (γ a,b ) which is impossible since γ is an optimal path among paths entirely contained in Λ 0 .
• γ a,b takes the original pattern.
Assume that γ a,b is not entirely contained in Λ. Let v 0 be the first exit point from Λ of γ a,b and u 0 the first entry point after v 0 . Let us consider the shortcut π going from v 0 to u 0 in a shortest way. Then, using the same argument as in the proof of Lemma 3.21, we have that γ v0,u0 has strictly more edges than π. Furthermore, all edges of γ v0,u0 have a time greater than or equal to ν 0 although all edges of π have a time lower than or equal to ν 0 . So, T (π) < T (γ v0,u0 ), which is impossible since γ is an optimal path among paths entirely contained in Λ 0 . Since γ a,b is a path entirely contained in Λ, going from u 3 to v 3 and with an optimal time, so we have the result.