Inhomogeneous first-passage percolation

We study first-passage percolation where edges in the left and right half-planes are assigned values according to different distributions. We show that the asymptotic growth of the resulting inhomogeneous first-passage process obeys a shape theorem, and we express the limiting shape in terms of the limiting shapes for the homogeneous processes for the two weight distributions. We further show that there exist pairs of distributions for which the rate of growth in the vertical direction is strictly larger than the rate of growth of the homogeneous process with either of the two distributions, and that this corresponds to the creation of a defect along the vertical axis in the form of a `pyramid'.


Introduction
First-passage percolation is a stochastic model for spatial growth that has been widely studied by mathematicians and physicists (see e.g. [KS91,Kes03,How04]). Since the pioneering work of Eden [Ede58], both communities have benefited from intense activity connected to first-passage percolation, resulting in a rigorous theory for subadditive ergodic processes [Kin73], and far reaching predictions of KPZ-theory [KPZ86]. Typically one assigns nonnegative i.i.d. weights to the edges of the usual integer lattice in two or more dimensions, and studies the pseudo-metric T induced by the resulting weighted graph. In this paper we introduce an inhomogeneous version in which edges in the left and right half-planes of the Z 2 lattice are assigned weights according to distributions F − and F + . The first fundamental question is then whether the Shape Theorem still holds; that is, does the rescaled ball B t = {x : T (0, x) ≤ t} obey a law of large numbers as in the usual model, and if so, what does the limiting shape for B t /t look like?
Unlike in the homogeneous case, when F − = F + , we cannot rely on the usual ergodic theory for subadditive processes due to the lack of horizontal translation invariance. To establish existence of radial limits, a precursor to the Shape Theorem, we instead complement the classical approach using large deviation estimates for half-plane passage times introduced in [Ahl13a]. Our method also shows how the asymptotic shape of B t /t can be described in terms of the shapes for homogeneous first-passage percolation with F − or F + . If either of F − and F + dominates the other (in a concave stochastic ordering), then the asymptotic shape is the convex hull of the restriction to respective half-planes of the asymptotic shapes for F − and F + . When no such relation is present, the asymptotic shape equals the convex hull of the two half-shapes and a potentially wider additional line segment along the vertical axis (see Figure 1, page 5).
The behavior of first-passage percolation along the boundary of two regions with different passage times has attracted much attention due to its physical relevance and mathematical challenge. In [dBK93] it was shown that if passage times with distribution F are replaced throughout the graph by others, distributed according to F ′ , which is strictly smaller than F in a stochastic sense, then the time constant changes strictly. But how would the passage time change, for example, in the first coordinate direction e 1 if passage times are modified only along edges lying on the e 1 -axis? Will an arbitrarily small modification be detectable on a macroscopic scale? This is the well-known 'columnar defect' problem, which has been studied in many forms both numerically and rigorously; see for instance [WT90, KD92, MMM + 03]. There is no satisfactory theory which would explain why some models are 'sensitive' to any arbitrarily small perturbation and others are not; this is determined by competition between localized reinforcement, induced by an impurity, and bulk fluctuations, which in many cases are difficult to analyze. Perhaps the most prominent example is the one-dimensional totally asymmetric simple exclusion process with a 'slow bond' at the origin and particle density ρ = 1/2 [JL92,JL94], which for some initial conditions can be represented either in terms of last-passage percolation with a columnar defect or the so-called Poly-Nuclear Growth model (PNG), for which it is believed that any perturbation will be reflected at the macroscopic level in the change of the current. In line with the terminology of this article, this would mean the creation of a defect in form of a 'pyramid'. A theory of propagation of influence of impurities was proposed in [BSSV06] and later used to show that for a randomized PNG model with a columnar defect, only modifications above a certain threshold result in a 'spike' on the macroscopic profile [BSV10].
The inhomogeneous model introduced in this paper is rich enough to display defects at a macroscopic scale: We show in Theorem 1.6 that there are pairs of distributions (F − , F + ) for which the speed of growth along the vertical axis is faster than it would be for homogeneous first-passage percolation with either of F − and F + . Due to subadditivity in the model, the enhanced speed in the vertical direction does not create a 'spike' on the limiting shape (as in randomized polynuclear growth), but instead a 'pyramid' (see Theorem 1.4). As in pinning phenomena for polymers, the formation of a pyramid indicates that minimizing paths in the vertical direction are 'attracted' to the vertical axis, benefiting from low-weight edges in both half-planes. Moreover, our approach extends to the more general situation where edges in the left and right half-planes are assigned weights according to F − and F + respectively, but edges on the vertical axis are assigned weights according to a third distribution F 0 (see Remark 3.1). This covers, in particular, the above mentioned case of a columnar defect.
We conclude by mentioning two open problems. First, for which pairs of distributions (F − , F + ) is a pyramid formed on the limiting shape in the vertical direction? Our theorems give necessary conditions but not sufficient ones. Second, in the case of a columnar defect, meaning F + = F − , which distributions F 0 result in a pyramid on the limiting shape? Again, we know only of necessary conditions. In Section 7 we introduce a related construction, with defects appearing in each column independently at random. For this model we show (see Theorem 7.1) that at all intensities, the contribution of the defects to the time constant prevails, resulting in a strict change.

Convergence towards an asymptotic shape
Let F − and F + denote distribution functions of two probability measures supported on [0, ∞). For each edge e in the set of nearest-neighbour edges E of Z 2 , assign a random variable τ e , according to F − if at least one endpoint of e lies within the left half-plane (contained in {(x 1 , x 2 ) ∈ R 2 : x 1 < 0}), and according to F + otherwise. For each nearestneighbour path Γ in Z 2 we let T (Γ) := e∈Γ τ e . Distances in the induced random psuedometric we refer to as passage times, defined for u, v ∈ Z 2 as T (u, v) := inf{T (Γ) : Γ is a path from u to v} .
To describe our results on radial convergence and the shape theorem we introduce two variables: Extend the function T (x, y) to the full space R 2 by identifying T (x, y) = T (x ′ , y ′ ) when x, y ∈ R 2 and x ′ , y ′ ∈ Z 2 satisfy x ∈ x ′ + [0, 1) 2 and y ∈ y ′ + [0, 1) 2 .
Theorem 1.1. For any F − and F + with E Y − < ∞ and E Y + < ∞, there exists a function µ : R 2 → [0, ∞) such that for each x ∈ R 2 , T (0, nx) n →μ(x) almost surely and in L 1 . (1.1) The shape theorem that we will prove describes how the set W t ⊂ R 2 , given by compares asymptotically to the set We write | · | for Euclidean distance on R 2 .
If, in addition, max{F − (0), F + (0)} < p c , the critical probability for bond percolation on Z 2 , then the set W is convex and compact with non-empty interior, and for every ε > 0, almost surely, for large enough t . Remark 1.3. As in the homogeneous case, if either Y − or Y + has infinite mean, then the above almost sure and L 1 -convergence in (1.1) fails for all x in the interior of the respective half-plane. However, for arbitrary F + and F − , the convergence in (1.1) holds in probability. A similar weakening holds also for (1.2): For arbitrary F − and F + lim sup We omit the argument, but mention that a proof would follow along the lines of Cox and Durrett [CD81, Theorem 1]. One defines approximate passage timesT (x, y) between circuits of low-weight edges encircling each of x and y, and shows that {T (x, y) −T (x, y) : x, y ∈ Z 2 } is tight.

Properties of the asymptotic shape
Below, we characterize the 'time constant'μ in terms of time constants in homogeneous environments, and this gives a representation for the asymptotic shape. Generally, the shape W is the closed convex hull of the two homogeneous shapes (restricted to their respective half-planes) and a symmetric interval on the e 2 -axis of width 2μ(e 2 ) −1 , where e i denotes the ith coordinate vector. Define Further,μ is sub-additive and positive homogeneous, and W equals the closed convex hull (1.4) Figure 1: Schematic exhibiting the structure of the asymptotic shape W. The left picture is known to be the case when F + is more variable than F − .
From (1.4) we also see that if F + (0) ≥ p c but F − (0) < p c , then limit shape is a half-plane. The interval on the e 2 -axis described in the last theorem gives the possibility of an additional 'pyramid' in the coordinate direction for W. In this case, optimal paths in the e 2 -direction are able to benefit from low-weight edges in both half-planes better than if they were to remain in one of them, and will thus feel an 'attraction' towards the e 2 -axis. However, we will see below that if one of F − and F + dominates the other (in a certain concave ordering), thenμ(e 2 ) equals either µ − (e 2 ) or µ + (e 2 ), and the statement in (1.4) is reduced to with no additional pyramid. We do not completely understand the mechanism that determines whether strict inequality holds. For the statement of the next theorem, we say that for every concave non-decreasing function φ : R → R for which the two integrals above converge absolutely. In this case we write F 1 ≺ F 2 . This terminology was introduced to first-passage percolation by van den Berg and Kesten [dBK93, Definition 2.1], who used it to prove inequalities between time constants for different distributions. Note, in particular, that if where equality holds if either of F − and F + is more variable than the other.
If strict inequality holds above, then W has a pyramid in the coordinate direction e 2 . The final theorem we state here shows that there are examples that display this behavior.
In Section 7 we will prove a related result for the homogeneous model with edgeweights given by F and columnar defects given by F 0 introduced at random locations: For a large class of distributions F it suffices that F 0 ≺ F and F 0 = F for the time constant in the vertical direction to be strictly smaller than µ F , regardless of the density at which defects are introduced.

Preliminaries
To derive properties of the inhomogeneous model, we will when possible rely on known facts about the homogeneous case. Some of these facts will be recalled in this section.
Let F be a distribution function for a probability measure supported on [0, ∞) and let Y F denote the minimum of 4 independent random variables distributed as F . Let T F denote travel times on Z 2 in an homogeneous i.i.d. environment generated by F . A small extension of the arguments in [CD81, Theorem 4] (also see Sections 2.2 and A of [Ahl13b]) show that for every F there is a function µ F : If E Y F < ∞, then the convergence also holds almost surely and in L 1 , as a consequence of the Subadditive Ergodic Theorem of [Kin68]. Kesten [Kes86, Theorem 6.1] identified the condition for µ F to be non-degenerate: where p c = 1/2 denotes the critical threshold for bond percolation on the lattice Z 2 . µ F is a semi-norm on R 2 , so for x = (x 1 , x 2 ) ∈ R 2 , µ F (x) ≤ |x 1 |µ F (e 1 ) + |x 2 |µ F (e 2 ), and thus where we use · 1 to denote ℓ 1 -distance. At times we will couple the homogeneous and inhomogeneous models. Given the distribution function F , define the right-continuous inverse If ξ is uniformly distributed on the interval [0, 1], then F −1 (ξ) is distributed according to F . The inhomogeneous collection {τ e } e∈E of edge-weights of the lattice can thus be obtained by setting τ e = F −1 − (ξ e ) for edges with an endpoint in the interior of H − , and F −1 + (ξ e ) for all other edges, where {ξ e } e∈E is a collection of independent random variables uniform on [0, 1]. This implies where F sub := max{F − , F + } and F dom := min{F − , F + }. The inequalities for the pointwise coupling in (2.4) carry over to passage times between sites: In addition, (2.4) gives rise to a comparison of means.
In particular, for every The proof of the last statement consists of constructing 4 edge-disjoint paths from 0 to x and arguing as in [CD81, Lemma 3.1].
The last result we mention here will be used to bound tail probabilities while deriving both radial convergence and the shape theorem (Theorems 1.1 and 1.2). Here and below we let T + denote the passage time over paths restricted to H + and not including edges along the e 2 -axis. (2.5) can be chosen so that (2.6) The former statement in Proposition 2.2 was obtained in [GK84, (1.4)] for x = ne 1 , and extended to general directions in [Ahl13b, Theorem 3]. The latter statement is proven in [Ahl13a, Theorem 9] (see [Ahl13b,Theorem 4] for the corresponding statement for the whole lattice). We remark that the moment condition in the latter part can be replaced by the condition that E Y α + < ∞ for some α > 0, but we will not need to use that. We further remark that (2.5) and (2.6) imply that n≥1 P |T F (0, nx)−µ F (x)| > εn < ∞ for every ε > 0, under the condition E Y F < ∞. This fact has certain relevance for the results of this paper (in particular Theorem 1.1) to be obtained under minimal assumptions.
3 Radial convergence -Proof of Theorem 1.1 In this section we prove Theorem 1.1, and assume throughout that E Y − and E Y + are finite. By Lemma 2.1, E T (me 2 , ne 2 ) < ∞ for all m, n ∈ Z. This proves the claim for λ = ±1, for which the limits coincide due to symmetry. More generally, for λ > 0 and n ∈ N, The second term converges almost surely and in L 1 to λν. The other term converges to zero almost surely via Borel-Cantelli: for each ε > 0, which is finite. This proves almost sure convergence for λ > 0. L 1 convergence follows since E T (0, nλe 2 ) − T (0, ⌊nλ⌋e 2 ) is bounded. The case of λ < 0 is similar.
A key observation used in the rest of the proof of Theorem 1.1 is that (Recall that T + is the passage time among paths using edges with at least one endpoint in the interior of H + .) To prove this, let ε > 0 and choose a path Γ from 0 to z such that T (Γ) (the sum of edge-weights for edges in Γ) is no bigger than T (0, z) + ε. Γ has a terminal segment Γ t contained in the open right half-plane (except its initial and possibly its final vertex) from some ke 2 to z. Write Γ i for the initial segment of Γ up to ke 2 . Then Taking infimum over k and sending ε → 0 gives one inequality of (3.2). For the other, let k ∈ Z and choose paths Γ 1 from 0 to ke 2 and Γ 2 , contained in the interior of H + except its initial and possibly its final vertex, from ke 2 to z such that T (Γ 1 ) ≤ T (0, ke 2 ) + ε/2 and T + (Γ 2 ) ≤ T + (ke 2 , z) + ε/2. The concatenation of Γ 1 and Γ 2 , written Γ, is a path from 0 to z, so This is true for all k and ε > 0, so it proves the other inequality.
Before proving a lower bound matching Claim 2, we separate a consequence of Proposition 2.2 that we will use.
Claim 3. For every ε > 0 and x = (x 1 , x 2 ) ∈ R 2 with x 1 > 0, Proof. Note that T + (ke 2 , nx) equals T + (0, nx − ke 2 ) in distribution. Pick ε > 0 and let M = M(ε) and γ = γ(ε) be given as in the first part of Proposition 2.2. Then, for all n ∈ N and k ∈ Z. Since x 1 > 0, as n and k ranges over N and Z, respectively, nx − ke 2 will be in each unit square z + [0, 1) 2 at most a finite number (say K) of times, for each z ∈ Z 2 . Replacing nx − ke 2 by a corner of the unit square in which it is in, we find that which is finite. The claim thus follows by Borel-Cantelli.
To complete the proof of Claim 4, it remains to verify the case µ + ≡ 0 andμ(e 2 ) > 0. We will use (3.2) and we first bound its right side for large k. So, fix b > 0 such that b ·μ(e 2 ) > ν(x) .
We have now shown almost sure convergence of 1 n T (0, nx) when x 1 ≥ 0. The case x 1 < 0 is similar, and the only modifications necessary are to replace T + with T − , the passage time for paths using only edges with at least one endpoint in the interior of H − , as well as equation (3.2) with its obvious analogue and µ + with µ − .
To complete the proof of Theorem 1.1, we must prove L 1 -convergence. (This was already remarked when x 1 = 0 in Claim 1.) We use dominated convergence, bounding T (0, nx) above by T dom (0, nx), where T dom is the passage time in the homogeneous environment with edge-weights distributed as F dom . By Lemma 2.1, the assumption max{E where Y dom is as in Lemma 2.1. Therefore L 1 -convergence in homogeneous environments (mentioned below (2.1)) completes the proof of Theorem 1.1.
Remark 3.1. The reader may verify that the above proof goes through, essentially word for word, if edges in the left and right half-planes are assigned weights according to F − and F + , respectively, but edges on the vertical axis are assigned weights according to a distribution F 0 . In this case, one may require, for example, E Y − , E Y + and E Y 0 to be finite.

The time constant and the asymptotic shape
We aim in this section to prove Theorem 1.4. Assume then that E Y − and E Y + are finite, so that the limit in Theorem 1.1 exists. We begin with a simple observation.
We now begin the proof of Theorem 1.4, starting with formula (1.3).
The case x ∈ H − is analogous, so this proves part a).
Based on the characterization ofμ and its subsequent properties, we next prove some properties of the asymptotic shape W = {x ∈ R 2 :μ(x) ≤ 1}, and end the proof of Theorem 1.4 by verifying the formula (1.4). Proof. We first show that W is a closed convex set with non-empty interior. These are all easy consequences of the derived properties ofμ. By part c) of Proposition 4.2,μ is a continuous function on R 2 , and so W = {x ∈ R 2 :μ(x) ≤ 1} is closed. By parts a) and b) of Proposition 4.2, Thus if x, y ∈ W, then also tx + (1 − t)y ∈ W, proving convexity. Finally, by Lemma 4.1 and (2. showing that W contains {x ∈ R 2 : x 1 ≤ r} for small values of r > 0. This means that W has nonempty interior. We continue with part b), which we will derive from part d) of Proposition 4.2. If either F − (0) ≥ p c or F + (0) ≥ p c , thenμ(e 2 ) = 0 and therefore λe 2 ∈ W for every λ ∈ R, so W is unbounded. Suppose conversely that max{F − (0), F + (0)} < p c . Sincē µ is continuous, it attains its infimum on the unit circle {x ∈ R 2 : |x| = 1}, which has to be positive; otherwise would contradict part d) of Proposition 4.2. Consequently, for |x| > sup |y|=1 [μ(y)] −1 , we haveμ(x) = |x|μ(x/|x|) > 1 and W is bounded, hence compact.
The proof of Theorem 1.4 is now complete by virtue of (1.3) and Proposition 4.3.

Simultaneous convergence -Proof of Theorem 1.2
We proceed at this point with a proof of Theorem 1.2 -the shape theorem for inhomogeneous environments. It will follow from now standard arguments. We assume throughout that E Y 2 − and E Y 2 + are finite, and first prove (1.2). If either E Y 2 − or E Y 2 + is infinite, then (1.2) cannot hold since T (0, z) is bounded below by the minimum of the four weights of edges adjacent to z, each of which is distributed as F + when z ∈ Z + × Z. Then we apply Borel-Cantelli with Proof of (1.2). We first need to show that T (0, z) may not be 'too large' for more than a finite number of points in Z 2 . To quantify 'large', we will fix a constant M already at this stage: Recall that T dom denotes passage times in a homogeneous environment distributed as F dom . By the latter part of Proposition 2.2 (for ε = 1 and q = 3) there is a finite constant M, only depending on F dom , such that for every x ∈ R 2 and t ≥ |x| (5.1) (We have here used that passage times in a half-plane dominate those in the whole plane.) To argue for (1.2), let ε > 0 and fix δ such that 0 < δ < ε 6 min µ − (e 1 ) −1 , µ + (e 1 ) −1 , 2, (2M) −1 .
To verify (5.3), recall that T (x, y) ≤ T dom (x, y) for every x, y ∈ R 2 by (2.4). Since the choice of δ ensures that ε|z|/3 > Mδ|z| and δ|z| ≥ |z − |z|u i |, using (5.1) we arrive at where we also use that z 1 ≤ 2|z|. Since the number of points in Z 2 with ℓ 1 -norm n is 4n, the above sum is finite when E Y 2 dom is. By Lemma 2.1, the latter coincides with finiteness of E Y 2 − and E Y 2 + . As this was assumed, (5.3) follows and therewith (1.2). We proceed with the second statement of Theorem 1.2, and assume for that, in addition, max{F − (0), F + (0)} < p c . That W in this case is convex, compact and has non-empty interior was seen in Proposition 4.3. It remains to prove the concluding inclusion formula of Theorem 1.2, which will follow from (1.2) via a straightforward inversion argument. In a first step, we prove a discretized inclusion formula.
Under the assumption max{F − (0), F + (0)} < p c , the functionμ is bounded away from both zero and infinity on the unit circle {x ∈ R 2 : |x| = 1}. If follows thatμ is equivalent to the Euclidean metric on R 2 . Using (1.2) we find an almost surely finite constant K such that which is equivalent to inclusion in Z ε .

Comparisons between time constants in the vertical direction
We first aim to prove Theorem 1.5, and so we recall some notation. As in Section 2, let {ξ e } e∈E be a collection of random variables independent and uniform on [0, 1], and let F −1 (x) = min{y ∈ R : F (y) ≥ x}. Set τ ± e = F −1 ± (ξ e ) for each e ∈ E, and let τ e equal τ − e or τ + e depending on whether e has at least one endpoint in the interior of H − or not. This construction produces a coupling between three environments, two homogeneous in which each edge-weight is distributed as F ± , and one inhomogeneous.
Given an integer N ≥ 1, let E N denote the set of edges with both endpoints in [−N, N] 2 . For a set of edge weights σ = {σ e } e∈E N and path Γ ⊂ E N , let T σ (Γ) = e∈Γ σ e and define T N σ (x, y) as the minimum of T σ (Γ) over all such Γ connecting x and y. Fix an enumeration e 1 , e 2 , . . . , e |E N | of E N which goes through the edges of the right half-plane first. For each j = 0, . . . , |E N |, let σ j = {σ j e } e∈E N be the family given by σ j e i = τ + e i for i ≤ j and τ − e i otherwise. Last, let σ j,t = {σ j,t e } e∈E N equal {σ j e } except σ j,t e j = t, and let g j (t) = T N σ j,t (x, y). As a function of t, g j is the minimum of increasing linear functions, so it is non-decreasing and concave.
For N ≥ x 1 + y 1 + 1, E T N σ j (x, y) is finite from Lemma 2.1. Therefore, by Fubini's theorem, for almost every realization of {(τ + e , τ − e )} e∈E N , the integrals g j (t) dF + (t) and g j (t) dF − (t) are finite. So using the definition of more variable, In particular, , where T N and T N F are the restrictions of T and T F to [−N, N] 2 as above. By monotone convergence, E T (x, y) ≥ E T F + (x, y). Choosing x = 0 and y = ne 2 , Taking limits,μ(e 2 ) ≥ µ + (e 2 ), and this proves the theorem.
Remark 6.1. The same argument applies in the case of a columnar defect, i.e., when edges along the vertical axis are assigned weights according to F 0 while remaining edges are assigned weights according to F − = F + . Assuming that E Y ± and E Y 0 are finite, this shows thatμ(e 2 ) = µ − (e 2 ) = µ + (e 2 ) as long as Proof of Theorem 1.6. There are distributions such that in the second-coordinate directionμ < min{µ − , µ + }, as seen in the following example. Let F + be an arbitrary non-degenerate distribution with µ + = 1, and let F − be the degenerate distribution δ 1 . From [Kes86,Theorem 6.4], we can find y < µ + = 1 such that F + (y) > 0. Now choose any integer K > 2y/(1 − y) and note that (K + 2)y < K .
where T iKe 2 is the operator that translates the point iKe 2 to the origin. In words, the path γ i follows the edges in the translate T iKe 2 K of K (from (0, iK) to (0, (i + 1)K)) if they all have weight at most y and follows the e 2 -axis otherwise.
We can now compute the passage time of γ up to its intersection with (0, jK) as where N is the random number of occurrences of translates of A: This is an upper bound for the minimal passage time, so As j → ∞ the left side converges toμ(e 2 ) almost surely. The law of large numbers implies that the right side converges almost surely to 1 + P(A) (K+2)y−K K , which by (6.1) is strictly less than 1.
This does not yet prove Theorem 1.6, since F − is degenerate, so we do a limiting argument. Let F

Randomly introduced columnar defects
We finish by returning to the effect of columnar defects. We will show that for every ε > 0, if we introduce a defect independently with probability ε for each column {n} × Z, then the time constant will change in the vertical direction.
Together with the characterization of [dBK93, Theorem 2.9(b)], Theorem 7.1 gives a weak criterion for randomly introduced columnar defects to result in a strictly smaller time constant: Let λ = inf supp(F ) and let p c denote the critical probability for oriented bond percolation on Z 2 . If F satisfies F (0) < p c and F (λ) < p c , then µ F 0 < µ F for any F 0 = F which is more variable than F .
Proof. Let F and F 0 be given, and fix δ > 0. First note that the argument used to prove Theorem 1.5 also shows that, for every η ∈ {0, 1} Z , lim inf n→∞ T η (0, ne 2 )/n ≥ µ F 0 P-almost surely. So, it suffices to prove the remaining upper bound.
Let T η,K denote the restriction of T η to paths contained in the cylinder given by |x| ≤ K. The limit lim n→∞ 1 n T η,K (0, ne 2 ) exists almost surely by the Subadditive Ergodic Theorem. Moreover, there is K = K(δ) such that for every η with η x = 1 for all |x| ≤ K, the limit is bounded by µ F 0 (e 2 ) + δ (see e.g. [Ahl13a, Proposition 8]). In particular, lim sup n→∞ T η (0, ne 2 ) n ≤ µ F 0 (e 2 ) + δ with P-probability 1 . (7.1) By Borel-Cantelli, {η x+m = 1 for all |x| ≤ K} occurs for some m ≥ 0 for P ε -almost every η. Let M < ∞ denote the least positive integer m for which it does. By subadditivity so division by n and taking limits we obtain from (7.1) that Since δ > 0 was arbitrary, this concludes the proof.
Remark 7.2. One may define various versions of the above defected model. For example, we could take all edge-weights for horizontal edges to be distributed as F as well (with defects only present on vertical edges) and the limit in this case would be the time constant for the lattice with horizontal weights assigned from F and vertical ones from F 0 .

A Proof sketch for Proposition 2.2
We finish with an outline of the proof of Proposition 2.2. The first inequality appeared in [Ahl13b] and the proof is a modification of the strategy of Grimmett and Kesten [GK84,Kes86], so we omit it. The proof of the second inequality is more involved and we sketch the proof here. The original proof of Proposition 2.2 in [Ahl13b,Ahl13a] was for x ∈ Z 2 , and not for x ∈ R 2 . However, the latter easily follows from the former: let z x ∈ Z 2 be such that and we can apply the integer case explained below to the right side.
The goal of the rest of the appendix is to outline the proof of (2.6) using a regeneration argument from [Ahl13b,Ahl13a]. Given z ∈ Z 2 and r > 0, let C(z, r) denote the infinite cylinder a∈R {x ∈ R 2 : |x − az| ≤ r}. We will assume throughout that E Y + < ∞ and assign i.i.d. passage times from F + to all edges in (Z 2 , E).
When clearly understood from the context, the reference to z and r will be dropped. Note that {A n (z, r)} n≥1 are i.i.d., so {ρ j − ρ j−1 } j≥1 are independent geometrically distributed with success probability P(A 0 (z, r)) and {T C(z,r) (H ρ j−1 z 1 , H ρ j z 1 )} j≥1 are i.i.d. This sequence will serve as the increments in a renewal sequence. We note the following: Proposition A.1. If E Y + < ∞, then E T C(z,r) (H ρ 0 z 1 , H ρ 1 z 1 ) 2 < ∞ for all large r.
The proof is straightforward and involves bounding T C(z,r) by the minimum of passage times of a large number of disjoint paths of length (ρ 1 − ρ 0 ) z 1 .
Outline of proof. The main tools are Chebychev's inequality and Wald's equation. Fix ε > 0 and choose r large enough that T C(z,r) (H 0 , H z 1 ) has finite variance (see Proposition A.1). Next choose m large for 1 m E T C(z,r) (H 0 , H m z 1 ) to be close to µ C(z,r) (see (A.1)). Set y = mz.
T C(z,r) (H 0 , H ρ j y 1 ) j≥1 is not a renewal sequence, but by the choice of cross-sections defining the ρ i 's, it follows that T C(z,r) (H 0 , H ρ j y 1 ) is well-approximated by the sum of T C(z,r) (H ρ i−1 y 1 , H ρ i y 1 ) for i = 1, 2, . . . , j (where ρ 0 = 0). Note that these terms are i.i.d. by construction, and have finite variance. We will therefore use optimal stopping to approximate T C(z,r) (H 0 , H n z 1 ) by a stopped sum. In particular, T C (H ρ i−1 y 1 , H ρ i y 1 ) − µ τ (y, r) > x + P ν(n)µ τ (y, r) − nµ + (y) > x + small error .
We apply Chebychev's inequality to the second term in the right side for the upper bound M/x. For the first term, we use both Chebychev's inequality and Wald's lemma for a similar bound M/x, and this implies the proposition for q = 1. For larger q, one strengthens the bound by considering disjoint cylinders of radius r aligned next to each other.
Step 4. Large deviations for the half-plane. To derive the estimate for the half-plane from Proposition A.2 we need to compare travel times on a cylinder with those in the half-plane. To circumvent the fact that the finite chunk C r of C(z, r) in between H 0 and H n z 1 may intersect the left half-plane we will shift space slightly. Namely, shifting C r by the vector (r + 1, r + 1) completely includes it in the first quadrant, and P T + (H 2r+2 ,H n z 1 +2r+2 ) > x ≤ P T C(z,r) (0, nz) > x , whereH n+2r+2 = H n+2r+2 ∩[n, ∞) 2 . We now combine this with (A.2) and Proposition A.2: for every ε > 0, q ≥ 1 and z ∈ H + , there exists M 2 = M 2 (ε, q, z) such that for every n ∈ N and x ≥ n P T + (2e 1 , 2e 1 + nz) − nµ + (z) > 2εx z 1 ≤ M 2 P(Y > x/M 2 ) + M 2 x q . (A. 3) The first term on the right is a bound on a sum of terms of the form T + (x, y), where x and y are at bounded distance. It is an error from 'surgery' needed to connect 2e 1 to H 2r+2 , andH n z 1 +2r+2 to 2e 1 + nz. (A.3) is close to the statement we aim to prove, and it remains only to remove the dependence of M 2 on the direction z. The argument is similar to the step from radial convergence of passage times to the shape theorem (as in the proof of (1.2)). One first obtains a control on large passage times of the type P T + (2e 2 , 2e 2 + z) > M 3 x ≤ M 3 P(Y > x/M 3 ) + M 3 x q for all z ∈ H + and x ≥ z 1 and then argues as in (1.2).