Coalescence of geodesics and the BKS midpoint problem in planar first-passage percolation

We consider first-passage percolation on $\mathbb Z^2$ with independent and identically distributed weights whose common distribution is absolutely continuous with a finite exponential moment. Under the assumption that the limit shape has more than 32 extreme points, we prove that geodesics with nearby starting and ending points have significant overlap, coalescing on all but small portions near their endpoints. The statement is quantified, with power-law dependence of the involved quantities on the length of the geodesics. The result leads to a quantitative resolution of the Benjamini--Kalai--Schramm midpoint problem. It is shown that the probability that the geodesic between two given points passes through a given edge is smaller than a power of the distance between the points and the edge. We further prove that the limit shape assumption is satisfied for a specific family of distributions. Lastly, related to the 1965 Hammersley--Welsh highways and byways problem, we prove that the expected fraction of the square $\{-n,\dots ,n\}^2$ which is covered by infinite geodesics starting at the origin is at most an inverse power of $n$. This result is obtained without explicit limit shape assumptions.


Introduction
The model of first-passage percolation was introduced by Hammersley and Welsh [24] in 1965 to study the spread of a fluid in a porous medium. Since then, it has been studied extensively in the probability and the statistical physics literature. We refer to [29] for a general background and to [5] for a review of more recent results.
We study first-passage percolation on the square lattice (Z 2 , E(Z 2 )) with independent and identically distributed (IID) random environment. The model is specified by a weight distribution G, a probability measure on the non-negative reals. Each edge e ∈ E(Z 2 ) is assigned a random passage time t e with distribution G, independently between edges. Then, each finite path p in Z 2 is assigned the passage time (1.1) A random metric T on Z 2 is defined by setting the passage time between u, v ∈ Z 2 to where the infimum ranges over all paths connecting u and v. Any path achieving the infimum is termed a geodesic between u and v. A unique geodesic exists when G is atomless (in particular, under our assumption (ABS) below) and will be denoted γ(u, v). The focus of first-passage percolation is the study of the large-scale properties of the random metric T and its geodesics.
1.1. Results. We proceed to describe our main results. Background and further discussion is provided in Section 1.4. Throughout we assume that G possesses an exponential moment, and also that G is absolutely continuous.
(ABS) The first-order behavior of the metric T is governed by the following result. Define the metric ball of radius t by Limit Shape Theorem (Cox and Durrett [13]). For any distribution G satisfying (EXP) and (ABS), there exists a deterministic convex set B G such that for all > 0, The set B G is called the limit shape corresponding to G. The theorem holds under weaker assumptions but the above generality suffices for the purposes here.
1.1.1. Coalescence of geodesics. Our first result concerns the coalescence of geodesics with nearby starting and ending points. It is shown that, with high probability, such geodesics overlap almost entirely, differing only in short segments near their endpoints. The statement is quantified, obtaining power-law dependence of the involved quantities on the length of the geodesics.
We define Sides(B G ) as the number of sides of B G : if B G is a polygon then Sides(B G ) is its number of edges while if B G is not a polygon then Sides(B G ) := +∞ (equivalently, Sides(B G ) is the number of extreme points of B G ). Our proof requires a lower bound on Sides(B G ). This assumption is weaker than the condition that the limit shape be strictly convex, a condition which is believed, but not proved, to follow from assumption (ABS) (see [5,Question 11]). Theorem 1.3 below identifies an explicit class of distributions G satisfying (EXP), (ABS) and the required lower bound on Sides(B G ) so that our result holds unconditionally for this class.
We point out that the coalescence set of two geodesics is necessarily a path. This follows from the fact that there is a unique geodesic between every pair of points. See Figure 1 for simulation results showing the phenomenon of coalescence.
1.1.2. The influence of edges. The passage time of the geodesic between given endpoints is naturally a function of the weights assigned to all edges. To what extent is this passage time influenced by the weight assigned to a specific edge? This notion is formalized here by the probability that the geodesic passes through that edge. It is clear that the influence of edges near the endpoints cannot be uniformly small, but it is not clear whether the influence must diminish for edges far from the endpoints. This issue was highlighted by Benjamini-Kalai-Schramm [11] in their seminal study of the variance of the passage time, where the following problem, later termed the BKS midpoint problem, was posed: Consider the geodesic between 0 and v. Does the probability that it passes at distance 1 from v/2 tend to zero as v → ∞? A proof that this probability tends to zero as a power of v (and analogous estimates for other edges) would simplify the argument of [11].
The BKS midpoint problem on the square lattice was resolved positively by Damron-Hanson [15] under the assumption that the limit shape boundary is differentiable and then resolved unconditionally by Ahlberg-Hoffman [1]. While both resolutions apply to the more general setup of ergodic edge weights (rather than simply IID), they also share the drawback that no quantitative decay rate for the probability is obtained. As a consequence of our quantitative control on the coalescence of geodesics, we are able to prove power-law decay rates for the influence of edges. These apply, in particular, for the "midpoint edges", yielding a quantitative resolution of the BKS midpoint problem. Theorem 1.2. Suppose G satisfies (EXP), (ABS) and Sides(B G ) > 40. There exists C > 0 (depending only on G) such that for all u, v, z ∈ Z 2 ,

2)
where D u,v z := min{ u − z , v − z }. A variant of the result may also be obtained under the weaker assumption Sides(B G ) > 32, see (3.11).
It is clear that one cannot have a decay rate in (1.2) which is uniform in z at a given distance from u and v and is faster than a power law, since for any integer 0 < k ≤ 1 2 u − v the geodesic γ(u, v) must pass through at least one vertex z with D u,v z = k.
1.1.3. Many sides to the limit shape. The following theorem identifies a wide class of weight distributions for which the limit shape has many sides (so that the assumptions of Theorem 1.1 and Theorem 1.2 are satisfied). Theorem 1.3. Let X be a random variable supported on [0, 1] with Var(X) = σ 2 > 0. There exists 0 (σ) > 0, depending only on σ, such that the following holds for all 0 < < 0 (σ). Let G be the distribution of 1 + X. Then the limit shape corresponding to G satisfies Sides(B G ) ≥ log(1/ ) log log (1/ ) . Remark 1.4. The proof of Theorem 1. 3 gives not only that there are many sides, but also that there are many sides close to the (1, 0) direction. More precisely, we prove that there are many extreme points between the direction (1, 0) and (1, √ ).
1.2. Attractive geodesics. The main technical proposition underlying the proof of our coalescence result is presented in this section (Proposition 1.5 below). Roughly, it shows that if two geodesics spend significant amount of time near each other then they intersect. Its proof does not rely on the planar geometry and may be adapted also to geodesics in Z d for d > 2. Planarity is used when deducing Theorem 1.1 from the proposition, in order The pictures depict the geodesics in independent samples of the environment. Theorem 1.1 states that nearby geodesics coalesce with high probability. The geodesics in the fourth simulation did not coalesce and, moreover, were far from each other for most of the way. This is compatible with our results as Proposition 1.5 shows that geodesics which stay close to each other for a significant amount of time have a very high probability to coalesce.
to verify that two geodesics with nearby starting and ending points will spend a significant amount of time near each other, with high probability. We wish to make precise the idea that a geodesic γ is attractive in the sense that any geodesic which spends significant amount of time near γ must share an edge with γ. Our formalization of this idea is in (1.3); it requires the following definitions. Denote For a finite path p in Z 2 : • Write X(p) for the interval whose endpoints are the x-coordinates of the endpoints of p. Precisely, if p has endpoints (t 1 , s 1 ) and (t 2 , s 2 ), with t 1 ≤ t 2 , then X(p) : we refer to the points (x, f p (x)) as pioneer points of p. • Given r > 0, the r-tube of (the pioneer points of ) p is the set • Given an interval J = [a, b] with integer a, b ∈ X(p) and a second path q in Z 2 , we say that q is r-close to p on J if the following conditions hold: (1) q has a vertex u ∈ Tube r (p) ∩ S a and a vertex v ∈ Tube r (p) ∩ S b .
(2) In the sub-path of q between u and v, the number of edges with both endpoints in Tube r (p) ∩ S J is at least 1 2 |J|. Figure 2. Illustration of the event that the path q is r-close to the path p on the interval J. The blue region depicts Tube r (p) ∩ S J . This event will be used in the attractive geodesics proposition, Proposition 1.5, where p will have length of order L and J will have length of order m.
The proposition below states that a geodesic is attractive with high probability, provided that it satisfies the following technical requirement of bounded slope: For ρ, m > 0, we say that a finite path p in Z 2 has (ρ, m)-bounded slope if for all In words, the slope of p between its pioneer points is bounded above by ρ for every pair of pioneer points with horizontal separation at least m. This requirement is discussed further following the statement of the proposition. There exist C ρ , c ρ , α ρ > 0, depending only on G and ρ, such that the following holds.
Consider the geodesic γ := γ((0, 0), (L, s)) for integer L > 0 and s. Let (I i ) N −1 i=0 be intervals of the form I i = [a i , a i+1 ] where 0 = a 0 < a 1 < · · · < a N = L are integers and with m ≤ |I i | ≤ 2m for some m. Define the event Att r,ξ := every geodesic which is r-close to γ on at least and the parameters satisfy and max{r, log 2 L} ≤ α ρ m r . (1.5) In our application, the parameters r, N, m will be chosen as suitable powers of L, so that, in particular, assumption (1.5) is satisfied.
To deduce from the proposition that geodesics are typically attractive, we need to prove that they typically have (ρ, m)-bounded slope. This is handled by the next result, for which we require the following limit shape assumption, the limit shape B G is not a dilation of the 1 ball.
(N 1 ) For horizontal geodesics (in the sense of (1.8) below), the assumption may be waived.
We can relax the restriction (1.6) (to at most 90 − δ degree slope) with extra limit shape assumptions. The proof under condition 1.8 (without assumption (N 1 )) uses only the convexity and the lattice symmetries of the limit shape.
We make several remarks regarding the results of this section. First, the notion of attractive geodesic is not invariant to rotations of Z 2 , as the x-axis plays a special role in the definition of r-closeness (the interval J in its definition should be thought of as a subset of the x-axis). We can thus define a notion of "vertically attractive geodesic" by exchanging the role of the x and y axes in our definitions, and our statements will apply just as well for this notion. This fact is especially relevant for the 45-degree assumption (1.6) as one sees that if this assumption is not satisfied by γ, then it will be satisfied once the x and y axes are exchanged. In this sense (1.6) is not a serious restriction. In the proof of Theorem 1.1, thanks to this symmetry of the lattice, we can assume without loss of generality that the geodesic satisfies the 45-degree assumption. Proposition 1.6 ensures that the geodesic does not make "big jumps" with high probability so that, in particular, any sub-path of the geodesic does not make "big jumps". In the proof of Theorem 1.1, we apply Proposition 1.6 to the whole geodesic, and then apply Proposition 1.5 to suitable sub-paths of the geodesic near its endpoints in order to prove that these sub-paths are typically attractive for suitable choices of r and m.
A second related observation is that if one first rotates the Z 2 lattice by 45 degrees, thus making the line x = y into the new x-axis, one obtains yet another notion of attractive geodesic (where the interval J in the definition of r-closeness should be thought of as a subset of the line x = y in the original coordinate system). The proofs of our propositions apply also in this rotated coordinate system. It is then worthwhile to note that if the limit shape in the original coordinate system was a dilation of the 1 ball then after the rotation the limit shape will be a dilation of the ∞ ball, allowing to apply Proposition 1.6. In this sense, a version of our results holds without need to verify assumption (N 1 ).

1.3.
Overview of the proofs. We briefly explain here how our main theorems are proved. Theorem 1.1 shows that geodesics which start near (0, 0) and end near a point y ∈ Z 2 will coalesce with high probability. We prove it using a trapping strategy, showing that all such geodesics stay, with high probability, between two reference coalescing geodesics γ + and γ − , thereby forcing the coalescence event by the planar geometry (see Figure 3). The reference geodesic γ + (γ − ) starts and ends at a suitably chosen distance h above (below) (0, 0) and y. For the reference geodesics to form a trap, we need to ensure that they stay ordered, meaning that γ + is always above γ − (in a suitable sense), and that they stay away from the neighborhoods of (0, 0) and y. We prove that these properties are satisfied with high probability, when h is somewhat large, using our assumptions on the limit shape (see Proposition 3.1). The coalescence of the reference geodesics is proved using the attractive geodesics proposition, applied to sub-geodesics of γ − of length L y located at the extremities of γ − : To this end, first, Proposition 1.6 is used to verify that the reference geodesics have bounded slope with high probability (assuming WLOG that γ − satisfies the '45-degree slope' condition as in the first remark after Proposition 1.6). Second, the planar geometry, translation invariance of the lattice and the fact that the reference geodesics remain ordered are used to prove that for r h, using Markov's inequality, the reference geodesics are r-close to each other above each segment with high probability. Theorem 1.2 shows that the probability of the event E u,v z that a vertex z lies on the geodesic between the vertices u and v is small, when z is separated from u and v. The theorem is deduced from the coalescence result, Theorem 1.1, using an averaging trick (see Figure 4) as used in the later proofs of the BKS-type concentration bound by Damron-Hanson-Sosoe [16]. Translation invariance of the lattice shows that the probability of E u,v z equals the probability of E u+w,v+w z+w for every w. This gives, in particular, that where Λ is a discrete square of side length . For w ∈ Λ , with suitably small, Theorem 1.1 shows that all geodesics of the form γ(u + w, z + w) coalesce and all geodesics of the form γ(z + w, v + w) coalesce, with high probability. When this happens, then all geodesics of the form γ(u + w, v + w) for which E u+w,v+w z+w occurs (with w ∈ Λ ) must coincide on the box z + Λ . Therefore, on this event, the quantity inside the expectation in (1.9) does not exceed the order 1 (with high probability, since geodesics only spend order time in z + Λ ), yielding the required bound.
The main idea in the proof of Theorem 1.3 is to take advantage of the fact that in some directions there are more deterministic paths of a given length. For example, there is a unique path of length 2n from (0, 0) to (2n, 0) while there are 2n n paths of length 2n from (0, 0) to (n, n). We also use the fact that the weight distribution is a small perturbation of a constant in order to argue that the geodesics are close to being shortest paths in the graph Z 2 . Using this, we prove that two directions which are not very close cannot be on the same flat edge of the limit shape.
The proof of the attractive geodesics proposition in Section 2 starts with a main steps part (Section 2.1) which may serve as an overview of it. Among the ingredients in the proof is a Mermin-Wagner style argument which is explained in Section 2.3. 1.4.1. Coalescence of geodesics. Theorem 1.1 proves that geodesics of length n whose starting and ending points are at distance h = n 1/8− coalesce with high probability. We briefly review here the literature on similar results.
The most progress has been achieved for "exactly-solvable models": Directed last-passage percolation models (in two dimensions) for which exact formulas have been found for the basic statistics. In these models the exponent 2/3 was shown to govern the coalescence (as is also predicted for first-passage percolation). The first result is due to Wütrich [54] who proved that with high probability geodesics will not coalesce at distance h = n 2/3+ . This was later improved by Pimentel [40] to show that the geodesics will not coalesce with uniformly positive probability when h = Cn 2/3 for any C > 0. Basu, Sarkar and Sly [9] proved that 2/3 is the right exponent by showing that geodesics at distance Cn 2/3 will coalesce with probability tending to 1 as C ↓ 0. Zhang [55], and independently Balázs, Busani and Seppäläinen [6] and Seppäläinen and Shen [45] improved the quantitative bounds and estimated the probability of coalescence up to constants as C → 0 and as C → ∞. See also [46,8,47,48,10] for the study of infinite geodesics in exactly-solvable models.
In the first-passage percolation setting there are no quantitative and unconditional coalescence results such as Theorem 1.1 (taking into account Theorem 1.3). In fact, the only quantitative result we are aware of is that of Alexander [3] who obtained statements with precise exponents, but under strong assumptions which are currently proved only in the exactly-solvable models (in particular, they are not known to hold for any weight distribution in the first-passage percolation setting). A non-quantitative coalescence result was proved by Licea and Newman [31,35] for infinite geodesics (semi-infinite paths for which every finite sub-path is a geodesic) in two dimensions. They showed that for almost all directions θ ∈ [0, 2π), any two infinite geodesics with asymptotic direction θ must coalesce. Their results were strengthened by Damron-Hanson [14] and Ahlberg-Hoffman [1]: For θ ∈ [0, 2π) denote by v θ the unique point in ∂B G in the direction θ. Damron-Hanson proved that in two dimensions, for any direction θ ∈ [0, 2π) such that the limit shape is differentiable at v θ , there exist no disjoint infinite geodesics with θ as a direction. Ahlberg-Hoffman developed an ergodic theory of random coalescing geodesics in dimension 2. They proved that the properties of coalescence described in [14] are not only valid for some geodesics but are in some sense valid for a dense set of geodesics. The results of [14,1] are also non-quantitative, but have the advantage of applying to the more general setup of ergodic edge weights.
An interesting direction for extending Theorem 1.1 is to prove a quantitative coalescence result for infinite geodesics. To this end, one would naturally need a coalescence result in which the distance to coalescence does not depend on the overall length of the geodesics. The following is an example of such a statement: Let γ n,h be the geodesic from (0, 0) to (n, h). Prove that for some C > 0 depending on G, universal α 1 , α 2 > 0 and all h, n > 0, (1.10) The obstacles in adapting our argument to prove (1.10) are to control the vertical fluctuations of the geodesics (as in Proposition 1.6) and to create suitable 'traps' for the geodesics as in Section 3. We believe these may be overcome by relying on stronger limit shape assumptions than those in Theorem 1.1 but have not pursued this extension here.
1.4.2. The influence of edges. As previously mentioned, the works of Damron-Hanson [14] and Ahlberg-Hoffman [1] provided a non-quantitative resolution of the BKS midpoint problem. In addition, the aforementioned work of Alexander [3] provided quantitative bounds for the BKS problem under strong assumptions which are currently known to hold only in the exactly-solvable models.
The exponent 1/16 in Theorem 1.2 follows from optimizing between the different parameters in our proof. The correct exponent is expected to be 2/3, the same as the exponent predicted to govern the transversal fluctuations of geodesics. Theorem 1.2 can be seen as a bound on the "first-order influence of edges" in the sense that it bounds the probability that a single given edge is in the geodesic. It is natural to ask also about "higher-order influences" in the sense of asking about the probability that several edges are simultaneously in the geodesic. In this direction, we offer the following natural problem: Does the correlation between the choices of first and last edges in the geodesic from (0, 0) to (n, 0) tends to zero as n → ∞?
1.4.3. Limit shape properties. Theorem 1.3 proves that the limit shape has many sides for a particular class of distributions. We are only aware of few related results in the literature, as we now describe.
Damron-Hochman [17], relying on results of Marchand [32] and Kesten [29], construct an atomic distribution whose limit shape has an infinite number of sides.
Basdevant-Gouéré-Théret [7] determined the first-order behavior of the limit shape corresponding to the weight distribution Bernoulli(1− ) as tends to 0. Using their result one can show that the number of extreme points of the limit shape corresponding to Bernoulli(1 − ) tends to infinity as ↓ 0.
1.4.4. Related models. Analogs of our main results (Theorem 1.1 and Theorem 1.2) continue to hold for point-to-line geodesics (i.e., geodesics from nearby starting points to the same line will coalesce with high probability), requiring only notational modifications in the proof.
The proofs of our main results should also extend, with minimal changes, to directed firstand last-passage percolation models in a planar geometry (under the added condition that the slope between the starting and ending points of the geodesics under study is bounded away from the maximal and minimal allowed slopes). In fact, the proofs should simplify in this setting, since if a directed geodesic starts and ends above another directed geodesic then they must deterministically preserve their order throughout.
Lastly, one may also consider first-passage percolation in the "slab" S [0,n] for some integer n > 0 (this may be thought of as first-passage percolation on Z 2 in which the weights of the edges not fully contained in S [0,n] are set to infinity). Analogs of our main results can also be proved in this setting, with minimal modifications to our proofs, for geodesics connecting the sides of the slab (i.e., starting at (0, y 1 ) and ending at (n, y 2 ) for some y 1 , y 2 ). Similarly to the directed models, geodesics connecting the sides of the slab deterministically preserve their ordering in this setting. The lower bound on the required number of sides of the limit shape in this setting stems solely from the analog of Proposition 1.6, so the coalescence result should hold without any limit shape assumptions in the case of horizontal geodesics (i.e., geodesics satisfying (1.8)).
1.4.5. Higher dimensions. In dimension d ≥ 3, Proposition 1.5 remains true with minimal modifications of the proof, as long as condition 1.5 is suitably modified. Among the requirements entering it is that the "cost" r to connect two geodesics at distance r is smaller than the increase generated by the Mermin-Wagner style argument (Section 2.3)) that is of order m/r d−1 . However, planarity is crucially used in the proof of Theorem 1.1 to keep the ordering between the geodesics (i.e., to show that if a geodesic has its endpoints above those of another geodesic then it will very likely remain above the other geodesic throughout). Ordering, in turn, is used to ensure (using Markov's inequality) that geodesics with nearby starting and ending points spend significant time near each other with high probability (so that Proposition 1.5 is applicable). As the ordering is lost in dimensions d ≥ 3, we do not know how to apply Proposition 1.5 in order to deduce coalescence.
1.4.6. The assumptions. The assumption (EXP) is used to ensure that the probability that the passage time between u, v ∈ Z 2 is larger than ρ u − v , for some constant ρ > 0, is exponentially small in u − v (see Claim 2.9). Weaker decay rates may also suffice in our arguments.
The second assumption (ABS) is mostly used in Claim 2.14 as part of the proof of our Mermin-Wagner style result. It may be possible to push our arguments to a class of non absolutely-continuous distributions but we have not attempted to do so.
The assumed lower bound on the number of sides of the limit shape is required in order to have sufficient control on the geometry of geodesics to produce the "traps" used in the proof of Theorem 1.1. In particular, we want to ensure that a geodesic that has endpoints far above another geodesic is unlikely to go below that other geodesic.

1.5.
Reader's guide. The rest of the paper is organized as follows. In the next section we prove Proposition 1.5, which is the main technical ingredient in the proof of the coalescence result of Theorem 1.1. The latter theorem, as well as our estimate on the influence of edges, Theorem 1.2, are proved in Section 3. In Section 4 we study the geometry of geodesics and prove Proposition 1.6 and Proposition 3.1 using our assumptions on the limit shape. These propositions are used in Section 3 to control the amount of time that a geodesic spends "going in a wrong direction". Finally, in Section 5 we establish the lower bound on the number of sides of the limit shape given in Theorem 1.3. The latter proof is independent of the rest of the paper.

Proof of the attractive geodesics proposition
In this section we prove Proposition 1.5. We assume that L is sufficiently large for the arguments (as a function of the distribution G and the parameter ρ) as the constants in the proposition may be adjusted to fit smaller L. We also assume throughout that G satisfies (EXP) and (ABS) and we continue with the notation of the proposition. for integer L > 0 and s. Call an interval J ⊂ [0, L] with integer endpoints attractive if every geodesic which is r-close to γ on J necessarily has an edge in common with γ. The following containment of events is immediate, We thus focus on giving a lower bound for the probability of the left-hand side event. As the first step we develop a sufficient condition for an interval to be attractive.
The following basic bound, controlling the passage time and length of geodesics, will be helpful.
Let Ω basic be the event that for all where we write |p| for the number of edges in a path p. Then The lemma is proved in Section 2.2. The notation ρ 1 , ρ 2 is reserved throughout our argument to the constants from the lemma.
To make use of the lemma for the geodesic γ, we first note that when γ has (ρ, m)bounded slope then |f γ (L)| ≤ ρL. Thus, as we've assumed that L is large as a function of ρ, the subgeodesic of γ between (0, 0) and (L, f γ (L)) satisfies the estimates (2.2).
Let P := {p : p is a path in Z 2 with P(γ = p) > 0}, so that, in particular, each p ∈ P is a path from (0, 0) to (L, s) and γ ∈ P almost surely.
Let J = [a, b] ⊂ [0, L] be an interval with integer endpoints and p ∈ P. Write for the passage time from the pioneer point of p above a to the pioneer point of p above b (using the geodesic between these two points, which may differ from p). A central role in our analysis is played by the following notion of the restricted passage timeT p (J), defined as the minimal passage time among (simple) paths q satisfying (1) q is edge-disjoint from p.
(2) One endpoint of q is in Tube r (p) ∩ S a and the other is in Tube The number of edges of q with both endpoints in Tube r (p) ∩ S J is at least 1 2 |J|. Let us prove this. Assume Ω basic and assume that γ has (ρ, m)-bounded slope. We show that the existence of a geodesic γ which is r-close to γ on J and is edge-disjoint from γ implies that Ω(J) does not occur. First, it follows from the properties of γ that it contains a subgeodesic γ connecting some (a, y 1 ) ∈ Tube r (γ) ∩ S a to some (b, y 2 ) ∈ Tube r (γ) ∩ S b which satisfies properties (1),(2),(3) above with q = γ and p = γ. Moreover, We claim that γ also satisfies property (4) so that it belongs to Q γ (J). This follows from Ω basic , as the endpoints of γ are in [−L 2 , L 2 ] by our upper bound (1.5) on r (with α ρ ≤ 1, say) and since γ has (ρ, m)-bounded slope. Second, since γ is a geodesic, its passage time must be at most that of the path going along the geodesic from (a, y 1 ) to (a, f γ (a)), then along γ from (a, f γ (a)) to (b, f γ (b)) and finally along the geodesic from (b, f γ (b)) to (b, y 2 ). On Ω basic , the latter path has passage time at most T γ (J) + 2ρ 1 max{r, log 2 L}. SinceT γ (J) ≤ T (γ ), we conclude that Ω(J) does not hold.
With the sufficient condition (2.4) in hand, and taking into account the containment (2.1) and Lemma 2.1, we see that Proposition 1.5 follows from the following statement: There exist C ρ , c ρ > 0, depending only on G and ρ, such that The next sections present the proof of this estimate, which relies on the following ingredients: (1) An upper bound for the passage time T γ (I i ) of many of the intervals (I i ). The main observation here is that Talagrand's concentration inequality self-improves when applied to sub-geodesics of γ due to the concavity of the square root function.
for the expected passage time between u and v and the deviation from the expectation. We may note that E is a (deterministic) metric on Z 2 , since T is a (random) metric on Z 2 . Talagrand's concentration inequality provides the following control on D(u, v).
There exist C, c > 0, depending only on G, such that the following holds. Let Then The lemma is proved in Section 2.2. The observation made in this section, stated in (2.9) below, is that (2.7) may be improved 'on average' for sub-geodesics of γ due to the concavity of the square root function.
be the expected passage time between the pioneer points of p above the endpoints of J. We emphasize that even when we shall apply this definition with p = γ, the right-hand side is calculated using (2.6), i.e., without taking into account the information that γ is a geodesic (so that E γ (J) is a random variable, different from the deterministic quantity E[T γ (J)]). Define p[J] to be the subpath of p between the points (a, f p (a)) and (b, f p (b)), and define where the time of a path was defined in (1.1). Note that T γ (J) = T (γ[J]) almost surely as γ is a geodesic. The quantity D γ (J) is a measure of the deviation of the passage time of the sub-geodesic of γ between the pioneer points at a and b.
It is straightforward to check that the following statements hold almost surely, We conclude that on Ω Tal ∩ {γ has (ρ, m)-bounded slope}, The assertion (2.10) is harnessed in the following way. It is straightforward that if (2.10) holds then for each τ satisfying we have that either Ω 1,τ or Ω 2,τ occurs, with (2.13) We will use this conclusion with τ = m r , noting that (2.11) is satisfied due to our assumption (1.5) (choosing α ρ sufficiently small). For a path p ∈ P, it will be convenient to denote by Ω 1,τ (p) and Ω 2,τ (p) the events appearing in (2.12) and (2.13), respectively, in which all occurrences of D γ (I i ) are replaced by D p (I i ).
2.1.3. The restricted passage time is long with non-negligible probability. In this section we provide lower bounds for the probability that a restricted passage timeT p (I i ) is long and further discuss the independence properties of the restricted passage times. Lemma 2.3. There exists c ρ > 0, depending only on G and ρ, such that the following holds. For each path p ∈ P having (ρ, m)-bounded slope, each 0 ≤ i ≤ N − 1 and each The proof of Lemma (2.3), relying on Talagrand's concentration inequality, is given in Section 2.2. The proof of (2.15) additionally uses a "Mermin-Wagner style argument" developed in Section 2.3. On an intuitive level, the argument yields that the distribution of T p (I i ) "contains a Gaussian component with variance of order m r " (see Lemma 2.15 for the precise result). This implies the following statement.
Lemma 2.4. Let p ∈ P having (ρ, m)-bounded slope and let J ⊂ [0, L] be an interval with integer endpoints satisfying m ≤ |J| ≤ 2m. There exist C ρ , c ρ > 0, depending only on G and ρ, such that for each 0 ≤ α ≤ c ρ √ mr and each real a, The argument leading from Lemma 2.4 to the inequality 2.15 is explained in Section 2.2. We remark that the "standard deviation lower bound m r " is obtained as the ratio between the length of J and the square root of the volume of the r-tube of p above J (using in the process that paths in Q p (J) must spend a significant fraction of their time in the r-tube). This is of the same nature as the relation χ ≥ 1−(d−1)ξ 2 on Z d , between the fluctuation exponent χ and transversal exponent ξ, obtained by Wehr-Aizenman [52, Section 6] and Newman-Piza [37,Theorem 5]. Our arguments may also be used to obtain such a relation.
We also remark that it would have been helpful to know the natural fact that T p (I i ) ≥ E p (I i ) occurs with probability bounded away from zero uniformly in L, m, r. Such a fact would both simplify and lead to better probability lower bounds in Lemma 2.3.
Recall that our goal is to prove the probability bound (2.5). This requires showing that several of the restricted passage timesT γ (I i ) are simultaneously large. To this end, the following independence property is handy: Set (2.16) Indeed, recall thatT γ (I i ) is the minimal passage time among the paths in Q γ (I i ), and that the paths q ∈ Q γ (I i ) have endpoints with x coordinates a i and a i+1 and satisfy is measurable with respect to the weights of the edges with both endpoints in S [a i −(ρ 3 −1)m, a i+1 +(ρ 3 −1)m] , from which (2.16) follows as edges have independent weights. Lemma 2.3 and the independence property (2.16) will be used in the following way. For a path p ∈ P having (ρ, m)-bounded slope and a vector of The following bounds the probability of E p,d . Proposition 2.5. If α ρ is sufficiently small (as a function of G and ρ) then (2.18) The proof uses the following special case of Chernoff's bound (see, e.g., [23, equation (7)]). Let X 1 , . . . , X k be independent random variables taking values in {0, 1} and write µ := k i=1 E(X i ) for the expectation of their sum. Then Proof of Proposition 2.5. If the first condition in (2.18) holds then there exists a subset I ⊂ {0, 1 . . . , N − 1} such that d i ≤ 6 m r for i ∈ I, |i 1 − i 2 | ≥ 2ρ 3 for all distinct i 1 , i 2 ∈ I and |I| ≥ N 2 2ρ 3 . The variables (T p (I i )) i∈I are then independent by (2.16). Thus, by (2.19), r log L by (2.15) and where we use that µ 1 ≥ 2ξN when α ρ is chosen sufficiently small (here c ρ is the constant from (2.15)).
Similarly, if the second condition in (2.18) holds then there exists The variables (T p (I i )) i∈I are again independent by (2.16). Thus, by (2.19), where by (2.14) (checking that m r ≥ 5ρ 1 max{r, log 2 L} by (1.5) when α ρ is sufficiently small), and where we use that µ 2 ≥ 2ξN when α ρ is sufficiently small (here c ρ is the constant from (2.14)).

Monotonic events.
Recall that Harris' inequality (generalized to dependent variables by the FKG inequality) states that two increasing events in independent random variables are non-negatively correlated [25]. In this section we explain the use that we make of this inequality in our context.
Let p ∈ P. Observe that the event {γ = p} is decreasing in the weights (t e ) e∈p and increasing in the weights (t e ) e / ∈p . Thus, by Harris' inequality, it is non-negatively correlated with every event sharing the same monotonicity properties. We employ the following variant of this observation. Lemma 2.6. Let p ∈ P. Let E be an event which is decreasing in the weights (t e ) e / ∈p (for every fixed value of (t e ) e∈p ). Then the following inequality of conditional probabilities holds almost surely, (2.20) Proof. The IID structure of the environment implies that the (t e ) e / ∈p remain independent after conditioning on (t e ) e∈p . As {γ = p} is an increasing event in (t e ) e / ∈p while E is decreasing in these variables, we may apply Harris' inequality in the conditional probability space to obtain (2.20).

2.1.5.
Putting all the ingredients together. In this section we explain how the results of the previous sections are combined to prove (2.5), from which the attractive geodesic proposition, Proposition 1.5, follows. We assume throughout that the constant α ρ in (1.5) is taken sufficiently small for the arguments.
Write P ρ,m := {p ∈ P : p has (ρ, m)-bounded slope}. Also set D p := (D p (I i )) 0≤i≤N −1 (recalling the definition of D p (J) from (2.8)). First, with the second equality following by comparing the definition (2.3) of Ω(J), the definition (2.8) of D p (J) and the definition (2.17) of E p,d , and with the inequality following from Lemma 2.2. Second, for each p ∈ P ρ,m , with the first equality following from the discussion after (2.10) and the second equality following from the definition of Ω 1,τ (p) and Ω 2,τ (p) following (2.13), making use of the intersection with the event {γ = p}. Third, we condition on the passage time of the edges on the path p and observe that D p and hence also the Ω i,τ (p) events are measurable with respect to this conditioning. Thus, where the first inequality follows from Lemma 2.6 (as E p,Dp is decreasing in (t e ) e / ∈p , for each fixed value of (t e ) e∈p ) and the second inequality follows from Proposition 2.5 (using the IID structure of the environment, as E p,d is independent of (t e ) e∈p while D p is measurable with respect to these variables), noting that the Ω i,τ (p) events exactly ensure that condition (2.18) holds.
Putting the previous displayed equations together, we finally conclude that This concludes the proof of (2.5), and hence of Proposition 1.5, once we note that ξN ≥ c ρ log 2 L by (1.5), for some c ρ > 0 depending only on G and ρ.

Basic lemmas.
In this section we prove that the events Ω basic (Lemma 2.1) and Ω Tal (Lemma 2.2) occur with high probability, and also prove Lemma 2.3 using Lemma 2.4. Lemma 2.2 is deduced from Talagrand's concentration inequality.
Theorem 2.7 (Talagrand's inequality [50]). There exist C, c > 0, depending only on G, such that for all u, v ∈ Z 2 we have that Proof of Lemma 2.2. We have The result follows from Theorem 2.7 and a union bound.
To prove Lemma 2.1, we need the following two claims.
Claim 2.9. There exist c, ρ 1 > 0, depending only on G, such that for every u, v ∈ Z 2 and n ≥ ρ Proof of Lemma 2.1. We have The result follows from Claim 2.8, Claim 2.9 and a union bound.
We proceed to prove the claims.
Proof of Claim 2.9. Let p be a deterministic path between u and v such that |p| = u − v 1 . For instance, one can choose the path that first goes straight in the vertical direction and then straight in the horizontal direction. For each α > 0, Markov's inequality and the independence of the edge weights yield that every n ≥ ρ 1 u − v 1 , The claim follows by choosing α to be the constant from our assumption (EXP) and choosing ρ 1 sufficiently large.
To prove Claim 2.8 we need the following result due to Kesten, a corollary of Proposition (5.8) in [29].
Let us now prove Lemma 2.3. The following preliminary claim shows thatT p (J) is unlikely to be much smaller than T p (J).
Claim 2.11 (Connection cost). There exist C, c > 0, depending only on G, such that the following holds. For each path p ∈ P and each interval J = [a, b] ⊂ [0, L] with integer endpoints, Proof. By the triangle inequality, The claim thus follows from Claim 2.9, applied with n = ρ 1 max{r, log 2 L}, and a union bound (using 1.5 with α ρ ≤ 1).
Proof of Lemma 2.3, inequality (2.15), using Lemma 2.4. Let J = [a, b] ⊂ [0, L] be an interval with integer endpoints satisfying m ≤ |J| ≤ 2m. Set X := T p (J) − E p (J). As in the previous proof, taking into account Claim 2.11 and the fact that m r ≥ 4ρ 1 max{r, log 2 L} by (1.5) with small α ρ , we see that it suffices to prove that for some c ρ > 0. Denote also ζ := P X ≤ −7 m r .
As in the previous proof, (2.23) In addition, Lemma 2.4 with α = 14 shows that

Perturbing the weights (a Mermin-Wagner style argument).
In this section we prove Lemma 2.4. Our basic tool is a "Mermin-Wagner style argument"; by this terminology, we mean the idea of perturbing a distribution (in our case, the edge passage time distribution) in a way which, on the one hand, significantly alters the observable of interest (in our case, the restricted passage time) and, on the other hand, can be usefully compared with the original distribution. This basic (and somewhat vague) approach has been key in many proofs of the Mermin-Wagner theorem in statistical physics, including [18,33,19,39,41,34], [22,Theorem 9.2] and [38, Section 2.6]), whence the name, but has also been used in other contexts, e.g. in [42,12,30,21]. Our treatment here draws inspiration from [39,41,34,30] and has the benefit of providing Gaussian lower bounds on the tail probabilities.

2.3.1.
The basic probabilistic estimate. The following statement is the basic "Mermin-Wagner style estimate" that we will use. Given a subset S we write S n for its n-fold Cartesian product and given a probability measure ν we write ν n for its n-fold product measure. In our application, the measure ν will be the distribution G of the edge passage time.
Claim 2.13. Let τ = (τ 1 , . . . , τ n ) ∈ [0, ∞) n and let X = (X 1 , . . . , X n ) be a vector of independent standard Gaussian random variables. Then, for every measurable A ⊂ R n , Proof. Let f : R n → [0, ∞) be the density of X, i.e., Observe that the density of X ± τ is f (x ∓ τ ) and that Thus, on the one hand, While, on the other hand, by the Cauchy-Schwarz inequality, The claim follows by combining the previous two displayed equations.
The second step is to define the bijections g ± ν,σ . For the rest of the section fix an absolutelycontinuous probability measure ν on R.
Therefore, Claim (2.13) and the fact that h −1 (ν) = ν Gauss imply proving (2.27). Lastly, we proceed to define the sets (B δ ) and establish (2.25). We use the following real analysis statement, which follows from the absolute continuity of ν.
Claim 2.14. For δ > 0, define the set Second, as h is increasing, for any x ∈ D M we have for all x ≤ y ≤ x + 1, ).
An analogous statement holds with x − 1 ≤ y ≤ x. Thus, for each M > 0 there exists δ > 0 for which D M ⊆ A δ . This finishes the proof of (2.28) using (2.29).

2.3.2.
Application to the restricted passage timeT p (J). Throughout this section we fix a path p ∈ P having (ρ, m)-bounded slope and an interval J ⊂ [0, L] with integer endpoints satisfying m ≤ |J| ≤ 2m. Our goal is to use Lemma 2.12 to prove (a generalization of) Lemma 2.4. As previously mentioned, on an intuitive level, the result may be thought of as saying that the distribution ofT p (J) "contains a Gaussian component with variance of order m r ". The following is our precise statement. Lemma 2.15. There exist C ρ , c ρ > 0, depending only on G and ρ, such that for each We note that Lemma 2.4 is the special case b = ∞ of this result. The rest of the section is devoted to the proof of Lemma 2.15.
We aim to use Lemma 2.12 to change the weight environment (t e ) e∈E(Z 2 ) . Since we are only interested in the effect of this change on the restricted passage timeT p (J), we restrict attention to a suitable finite set of edges Σ in Z 2 which contains the edges of all paths q ∈ Q p (J) as well as all edges in the set E p (J) below.
Define the set of edges Let δ 0 > 0 be a small constant, chosen as a function only of G and ρ following Claim 2.16 below. We apply Lemma 2.12 with ν = G and with (τ e ) e∈Σ given by We take 0 ≤ α ≤ 1 4 δ 0 √ mr so that 0 ≤ τ e ≤ 1 for all e. Note that The lemma provides us with two bijections, T + G,τ : We next show that Ω δ 0 is very likely when δ 0 is sufficiently small.
Henceforth we fix δ 0 to the value δ 1 of the last claim. We proceed to deduce Lemma 2.15.
On the one hand, by (2.27), ). On the other hand, by (2.33), if (t e ) ∈ Ω δ 0 and a ≤T p (J) ≤ b thenT p (J) + ≥ a + α m r and Lemma 2.15 follows by combining the last two displayed equations with (2.31) and Claim 2.16.

Coalescence of geodesics in Z 2 and the BKS midpoint problem
In this section we deduce our main results, Theorem 1.1 and Theorem 1.2, from the attractive geodesics proposition, Proposition 1.5. We will also need the following proposition which shows that typically a geodesic does not "go in the wrong direction for a long time". This proposition will allow us to "trap" geodesics. For x = (x 1 , x 2 ) ∈ R 2 we write x := ( x 1 , x 2 ) ∈ Z 2 where for t ∈ R, t denotes the largest integer smaller than t. Identifying R 2 with C we have Re iθ = R cos θ , R sin θ .
This proposition is proved in Section 4. Throughout this section and the next ones we will denote by C, c generic positive constants which may depend only on the edge weight distribution G and the variables in the subscript (e.g., c may depend on in addition to G), whose value may change from one appearance to the next, with the value of C increasing and the value of c decreasing. Similarly, labeled constants such as C 0 or c (which may also depend on G and additionally on their subscript variables) do not change their value throughout the section where there are defined.
3.1. Coalescence of geodesics in Z 2 . In this section, we prove Theorem 1.1 using Propositions 1.5, 1.6 and 3.1.
Let θ u ∈ (π/4, π/2) depending only on G be as in Proposition 3.1 applied to k = 3. Let κ be the smallest integer (depending only on θ u ) such that Set w − = (− , −κ ), z − = y + ( , −κ ), w + = (− , κ ) and z + = y + ( , κ ). Let γ − (respectively γ + ) be the geodesic between w − and z − (respectively w + and z + ). Our goal is to prove that the geodesics γ − and γ + coalesce with high probability and that all geodesics starting in Λ and ending in y + Λ are trapped between γ − and γ + and forced to coalesce (see figure 5). Denote by T the event where the following holds • The geodesic γ + stays above Λ and y + Λ , that is, γ + does not intersect the set • Any geodesic that starts at a point u ∈ Λ and ends at a point v ∈ (y + Λ ) does not circle γ + or γ − , that is, it does not intersect the following set On the event T , the geodesics starting in Λ and ending in y + Λ are "trapped" between γ + and γ − . Let us prove that the event T occurs with high probability. Let us first prove that with high probability, the geodesics γ − and γ + stay respectively below and above Λ and y + Λ . Set 0 := 1/3(1/16−1/17). In particular, we have + 0 < 1/16− 0 . Thanks to Proposition 3.1 and the invariance under translation, under the assumption Sides(B G ) > 32, we have P Re iθ + w + ∈ γ + for some R ≥ y 1/16+ 0 and |θ| ≥ θ u ≤ C exp − y c 0 . Let θ 1 be the angle in absolute value between the axis x = − and the line D that joins (− , κ ) and (− , ) (see figure 5). It is easy to check that Let us assume that γ + does not stay above Λ . Then γ + intersects the set {− , . . . , } × (−∞, ]. It follows that γ + intersects D at least twice (the first time corresponds to its starting point (− , κ )). Let us denote by x the last point of intersection of γ + with D. Thanks to Proposition 3.1 since the angle −π/2 + θ 1 < −θ u , we have with probability at least 1 − C exp(−c y c 0 ). Let γ be the subpath between w + and x. The path γ is a geodesic between w + and x. Since γ goes intersects the set {− , . . . , } × (−∞, ], it follows that |γ| ≥ (κ − 1) ≥ κ − 1 2 y 1/8− .
Recall that 1 16 By inequality (3.1), it yields that for y large enough (depending on 0 , κ and ρ 1 ), we have By Claim 2.8, this event occurs with a probability at most C exp(−c log 2 y ). It follows that γ + stays above Λ with probability at least 1 − C exp(−c log 2 y ). By similar arguments, we conclude with high probability at least 1 − C exp(−c log 2 y ), the geodesic γ + stays above Λ + y and the geodesic γ − stays below Λ and Λ + y. Lastly, we need to prove that any geodesic γ starting at a point u ∈ Λ and ending at a point v ∈ (y + Λ ) cannot exit the trap. The only option for γ to exit the trap is to leave the slab and intersect the set V . It is easy to check that the direction of the geodesic between u and v is contained in [θ 0 − 0 /2, θ 0 , 0 /2] for y large enough. Let w be the last point of intersection of γ with the line x = − . Thanks to Proposition 3.1, we have that w − u ≤ n 1/16+ 0 with probability at least 1 − C exp(− y c 0 ). It follows from a similar use of Claim 2.8 as above, that with probability at least 1 − C exp(−c log 2 y ), the geodesic γ does not go through a vertex in V . By union bound, it follows that any geodesic γ from Λ to y + Λ cannot exit the trap with probability at least 1 − C exp(−c log 2 y ). Finally, we have and on the event T , when γ − and γ + coalesce, any geodesic γ from Λ to y + Λ will also coalesce : We turn to show that γ − and γ + coalesce with high probability. Let ρ be as in Proposition 1.6 and ρ 2 be as in Lemma 2.1 (depending only on the limit shape). Let δ > 0 depending on . Set where α ρ is the constant in Proposition 1.5. Let (I k ) N −1 k=0 be intervals of the form I k = [a k , a k+1 ] where a k = − + km. We denote by γ − [a 0 , a N ] the subpath of γ − between its starting point and the point (a N , f γ − (a N )). Let us prove that the geodesics γ − and γ + are r-close on most of the intervals (I k ) N −1 k=0 . We recall that the definition of r-closeness interval was defined before Proposition 1.5. We first need to prove that with high probability γ + stays above γ − . By translation invariance in law of the environment, it yields that , then one of the geodesic has to circle around the other: the event E − ∪ E + occurs where

It yields that
where we used Cauchy-Schwarz inequality in the last line. Thanks to Claim 2.8, the quantity E (f γ − (x) − f γ + (x)) 2 is at most polynomial in y . Hence, there exists a positive constant C 0 such that for every x ∈ {− , . . . , a N } By Markov's inequality, we have By union bound, it follows that Finally, we can control the total number of r-close intervals using again Markov's inequality By Proposition 1.6, thanks to our choice of ρ, we have Note that if a path has (ρ, m)-bounded slope, it is also true for any subpath. Besides, by Proposition 1.5, we have with ξ := cρ √ r log y . Choose t = 4 y −δ/8 log 3 y .
Let K be large enough depending on G such that With this choice of parameters, we have for large enough y ≥ K The latter inequality implies that there are strictly less intervals where the geodesics γ + and γ − are not r-close than intervals that are not attractive for γ − . It yields that there must exist at least one attractive interval among the (I k ) N −1 k=0 where the geodesics γ + and γ − are r-close. On this interval, the geodesics γ − and γ + intersect. Combining inequalities (3.5), (3.6) and (3.7), it yields that for y ≥ K Let w and z be respectively the first intersection of γ − with x = − + N m and x = + y 1 − N m. Note that both γ 0 and γ 1 also intersect w and z and coincide between these two points. We can upper bound the symmetric difference by the length of the subpaths of γ 0 and γ 1 from their endpoints to w and z . With probability at least 1 − Ce −c log 2 y , by inequality (3.6), we have w − w − ≤ (1 + ρ)(N m + ) and z − z − ≤ (1 + ρ)(N m + ). It follows that for y large enough (depending only on G) Similarly, we have Finally, combining the two previous inequalities, on the event Ω basic (with L = y ), we have By combining the previous inequality with inequalities (3.4), (3.6), (3.8), (3.9) and Lemma 2.1, there exists a constant C depending only on G such that P ∃u, z ∈ Λ y 1/8− ∃v, w ∈ (y + Λ y 1/8− ) γ(u, v) γ(z, w) > When the endpoints of the geodesics are getting closer ( increasing), we need more sides on the limit shape for the trap to be efficient. Indeed, it is easier to circle the other geodesic when the endpoints are getting closer. For our later application of this theorem to prove the quantified version of BKS, we will need to use it for = 1/16. We state here another version of the theorem that will be sufficient for this application.
If we assume that Sides(B G ) > 40, we can chose ≤ 1/16 and 0 = 1/64. We apply Proposition 3.1 for k = 4. The inequality (3.2) becomes 1 32 and inequality (3.3) will hold for y large enough depending only on G.
3.2. BKS midpoint problem. In this section, we prove Theorem 1.2 using the quantitative coalescence result of Theorem 3.2.
Proof of Theorem 1.2. We assume that Sides(B G ) > 40. Let ∈ (0, 1/16]. Let u, v, z ∈ Z 2 . Without loss of generality let us assume that u − z ≤ v − z . Set We will use here an averaging trick by considering all geodesics from u + Λ to v + Λ . Set E 0 , E 1 be the following coalescence events Thanks to Theorem 3.2, we have (3.10) Let γ 1 and γ 2 be two geodesics with starting points in u + Λ and ending points in v + Λ . On the event E 0 ∩ E 1 , if γ 1 ∩ (z + Λ ) = ∅ and γ 2 ∩ (z + Λ ) = ∅, then γ 1 and γ 2 must intersect before and after intersecting z + Λ . Hence, By translation invariance, we have By Lemma 2.1 (applied for L = y ), we have for y large enough (depending on G) Combining the two previous inequalities together with (3.10), it follows that By taking = 1/16, we get

The result follows.
Under the weaker assumption Sides(B G ) > 32, thanks to Theorem 3.2, we can prove that for every > 0, there exists C > 0 (depending on G and ) such that for all u, v, z ∈ Z 2 , (3.11)

From the limit shape to the geometry of geodesics
In this section, we assume some properties on the limit shape and derive properties of the geodesic in the limiting norm. From these properties, we can control the asymptotic behavior of geodesics. In particular, we show that typically a geodesic does not "go in the wrong direction for long" (Proposition 3.1). We also prove Proposition 1.6. 4.1. Geometry of the geodesics in the limiting norm. In this section, we prove a characterisation of being in the same flat edge of the limit shape and deduce some properties of the geodesics in the limiting norm. The following theorem states that under some mild assumptions on the distribution G, one can prove that asymptotically when n is large, the random variable T (0, nx) behaves like n · µ(x) where µ(x) is a deterministic constant depending only on the distribution G and the point x. More precisely, we have the following theorem.
Time constant. Let G be a distribution such that E[t e ] < ∞. There exists a deterministic function µ on R 2 depending on G such that ∀x ∈ Z d lim n→∞ T (0, nx) n = µ(x) a.s. and in L 1 .
The constant µ(x) is the so-called time constant.
This constant may be interpreted as an inverse speed in the direction x. Kesten proved in [29] that µ is a norm if and only if G({0}) < p c (d). Under our assumption (ABS), the function µ is a norm. In particular, one can prove that B G is the unit ball for the norm µ.
We quantify how far a path that "goes in the wrong direction" is from being a µ-geodesic. The following claim is a useful property of the time constant. It will be used in the proof of Proposition 1.6.
Proof. Set x = (a, b) ∈ R 2 . Without loss of generality assume |a| ≥ |b|. By the triangular inequality, symmetry and homogeneity of µ The result follows.
To lighten notation, in this section we shorten B G to B. We let B(θ) be the unique x such that xe iθ is on the boundary of B, where as usual we identify R 2 with C. The unit ball of the norm µ is B and therefore µ(Re iθ ) = R/B(θ).
We say that directions θ 1 and θ 2 are on the same flat edge of the limit shape if the interior of B does not intersect the line connecting B(θ 1 )e iθ 1 and B(θ 2 )e iθ 2 (this is the line connecting the two points on the boundary of the limit shape that are at angles θ 1 and θ 2 ). We say that θ is a vertex direction if there are two distinct lines passing through B(θ)e iθ such that the interior of B does not intersect any of them. (1) The directions θ 1 and θ 2 are on the same flat edge of the limit shape.
(2) For all R 1 , R 2 ≥ 0 we have Note that by the triangle inequality and homogeneity, the right hand side of (4.1) is always larger than the left hand side of (4.1).
Proof. Statement (3) clearly follows from (2). We start by showing that (2) follows from (1). Suppose θ 1 and θ 2 are on the same flat edge. Substituting µ(R 1 e iθ 1 ) = R 1 /B(θ 1 ) and µ(R 2 e iθ 2 ) = R 2 /B(θ 1 ) we obtain This is a linear combination of the points B(θ 1 )e iθ 1 and B(θ 2 )e iθ 2 with coefficients that sum up to 1. Thus, the point in the left hand side of (4.2) is on the flat edge containing θ 1 and θ 2 and its norm has to be 1. This proves the equality in (4.1). The proof that (1) follows from (3) is similar. Indeed, if (4.1) holds for some R 1 and R 2 then the point in the left hand side of (4.2) is on the boundary of the limit shape. Since the point is on the line containing B(θ 1 )e iθ 1 and B(θ 2 )e iθ 2 , the angles θ 1 and θ 2 have to be on the same flat edge of the limit shape.
The following claim is a quantitative version of Claim 4.2. Let 0 ≤ θ 1 < θ 2 < π/2 be two angles not on the same flat edge or such that θ 1 is a vertex direction. The following claim proves that any path in R 2 from (0, 0) to R 1 e iθ 1 that contains the point R 2 e iθ 2 is very far from being a µ-geodesic. Claim 4.3. Let 0 ≤ θ 1 < θ 2 < π/2 such that either θ 1 and θ 2 are not on the same flat edge of the limit shape or θ 1 is a vertex direction. There exists a constant c 4 depending on the edge distribution, θ 1 and θ 2 such that for all R 1 , R 2 and Proof.
The assumption on θ 1 and θ 2 exactly ensures that there exists a line l with a negative or infinite slope, that passes through B(θ 1 )e iθ 1 , is disjoint from the interior of B and does not contain B(θ 2 )e iθ 2 . Let r 2 > 0 be the unique radius for which r 2 e iθ 2 is on l. Note that r 2 > B(θ 2 ) and therefore µ(r 2 e iθ 2 ) > 1. Thus, letting R 1 := R 2 B(θ 1 )/r 2 we have Next, suppose that R 1 = R 1 . In this case, by substituting the definition of R 1 we get that the point is on the line l. See Figure 6. It follows that the norm of this point is at least 1 and therefore Note that (4.6) clearly holds also in the case that R 1 = R 1 . Combining (4.6) and (4.5) we obtain This finishes the proof of (4.3). The inequality (4.4) follows from (4.3). Indeed, for all x, y 1 , y 2 such that |y 1 | ≤ |y 2 | we have that µ(x, y 1 ) ≤ µ(x, y 2 ). Thus, µ( The following claim relates the number of sides in the limit shape to the number of vertex directions in [0, π/4]. Claim 4.4. Suppose that the limit shape is not a polygon with 8(k + 1) sides or less. Let θ 0 ∈ [0, π/4]. Then, there are directions θ 0 < θ 1 < · · · < θ k < π/2 such that for every 1 ≤ j ≤ k, θ j is a vertex direction and θ 0 and θ 1 are not on the same flat edge.
Proof. If the limit shape is not a polygon then this claim clearly follows. Indeed, in this case one can find infinitely many directions in [π/4, π/2) such that no two of them are on the same flat edge of the limit shape. Next, suppose that the limit shape is a polygon with more than 8(k + 1) sides. By symmetry, the number of vertices directions is a multiple of 4. Hence, there is at least 8k + 12 vertices directions. Let m denote the number of vertices directions in (π/4, π/2). By symmetry, there are at most 8m + 8 vertices directions in total. It follows that m ≥ k + 1. Let θ 1 < · · · < θ m be the vertices directions in (π/4, π/2). Set θ i = θ i+1 for i ≤ k. Since there exists a vertex direction between θ 0 and θ 1 , θ 0 and θ 1 are not on the same flat edge. This concludes the proof.

Geometry of geodesics in FPP.
In this section we prove Proposition 3.1. We deduce from the results on the geometry of the geodesics in the limiting norm, results on the geometry of geodesics in FPP under some assumption on the limit shape. The following result due to Alexander (Theorem 3.2 in [2]), that bounds the "non-random fluctuations" of the passage time, enables to make the connection between geodesics in the limiting norm and geodesics in FPP.
Theorem 4.5. If the edge distribution satisfies E[e αte ] < ∞ for some α > 0. Then, for every x ∈ Z 2 we have that The following lemma allows us to control the behavior of geodesics using properties of the limit shape. This lemma will be the key ingredient to prove Proposition 3.1.
Lemma 4.6. Let 0 ≤ θ 1 < θ 2 < π/2 satisfy either that θ 1 and θ 2 are not on the same flat edge of the limit shape or that θ 1 is a vertex direction. Let > 0 and let γ be the geodesic from (0, 0) to R 1 e iθ 1 . Then, P R 2 e iθ 2 ∈ γ or R 2 e −iθ 2 ∈ γ for some where the constant c may depend on , θ 1 and θ 2 .
We postpone the proof of Lemma 4.6 for now and first prove Proposition 3.1. Proof of Proposition 3.1. We let θ j be the vertices directions from Claim 4.4 and fix some θ k+1 ∈ (θ k , π/2). For all 0 ≤ j ≤ k + 1 we let γ j (R) and γ j (R) be the geodesics from (0, 0) to Re iθ j and from (0, 0) to Re −iθ j respectively. For 0 ≤ j ≤ k define the events It follows from Lemma 4.6 and a union bound that P(A j ) ≤ C exp(−n c ). Next, for 1 ≤ j ≤ k + 1 define By Claim 2.8 we have that P(B j ) ≤ C exp(−n c ) for all j ≤ k + 1. Next, we claim that To this end, for 1 ≤ j ≤ k + 1 define the sets Suppose that the event on the left hand side of (4.7) holds and let z ∈ γ be a point of the form z = Re iθ with R ≥ n 2 −k−1 +2 and |θ| > θ k+1 . Let p j be the last intersection of γ with A j before reaching (n, 0). Note that it is possible that some of these points are identical.
Clearly, on the path γ from the origin to (n, 0) we first visit z then p k+1 , p k ,. . . , and lastly p 1 . Without loss of generality assume that p j = R j e iθ j . Let us assume that for every j we have R j−1 ≥ n 2 −j+1 + . In particular, we have R 1 ≥ n 1/2+ then A 0 holds. Otherwise, if R k ≤ n 2 −k−1 + , then we have |γ| ≥ R ≥ n R k and the event B k occurs. If R k ≥ n 2 −k−1 + , there exists j ≥ 1 such that R j ≥ n 2 −j + and R j−1 ≤ n 2 −j+1 + . Fix such j. If R j−1 ≤ n 2 −j+1 then B j−1 holds. If n 2 −j+1 ≤ R j−1 ≤ n 2 −j + then A j−1 holds. This proves inequality (4.7) and concludes the proof.
Next, we turn to prove Lemma 4.6.
Proof of Lemma 4.6. Fix some . There exists C depending on µ such that µ(R 1 e iθ 1 ) − µ( R 1 e iθ 1 ) ≤ C and therefore by Theorem 4.5 Thus, by Talagrand's inequality By the same argument we also have and Since , on the complement of these bad events, we see that all error terms are negligible compared to R 2 and therefore one can switch all the µ terms in Claim 4.3 with the corresponding passage times and change c 4 slightly. Thus, T 0, R 2 e iθ 2 + T ( R 2 e iθ 2 , R 1 e iθ 1 ) ≥ c 5 R 2 + T 0, R 1 e iθ 1 , with probability at least 1−C exp(−R c 1 ). Of course, on this event we cannot have R 2 e iθ 2 ∈ γ and therefore Using a union bound over all points of the form R 2 e iθ 2 for some R 1/2+ 1 ≤ R 2 ≤ R 2 1 and the fact that with very high probability |γ| ≤ R 2 1 and therefore it is unlikely that R 2 e iθ 2 ∈ γ for some R 2 ≥ R 2 1 we get By the same arguments we also have P R 2 e −iθ 2 ∈ γ for some This finishes the proof of the lemma.

4.3.
Controlling vertical jumps in the geodesic. The aim of this section is to prove that with high probability the geodesic between (0, 0) and (L, s) with |s| ≤ L does not make large vertical jumps (Proposition 1.6). Let x ∈ R 2 and θ 1 < θ 2 . Let Cone(x, θ 1 , θ 2 ) denote the cone centered at x between the angle θ 1 and θ 2 , that is We will need in the proof the following claim. It follows easily from the definition of a cone.
Denote by int(B G ) be the directions in the interior of a flat edge: Note that if θ / ∈ int(B G ), then it corresponds to an extreme point of the limit shape.
Proof of Proposition 1.6. We assume here (N 1 ). Let n ≥ 1, |s| ≤ n, let δ > 0 and m ≥ n 1/2+δ . Without loss of generality, we can assume that s ≥ 0. Let γ be the geodesic between (0, 0) and (n, s). Set Under the assumption (N 1 ), we have θ 0 > 0. If not, by symmetry we get , π, 3π 2 and it implies that the limit shape is a dilation of the 1 ball. Set We define [b(θ), u(θ)] to be the flat edge of θ for θ ∈ int(B G ). We set u(θ) = b(θ) = θ for θ / ∈ int(B G ). Note that by symmetry, we always have u(θ) − b(θ) ≤ π/2. For x, y ∈ Z 2 , we denote by E x,y the set of unreachable points for the geodesic between x and y (with high probability the geodesic from x to y does not go through these points). Write y = x + re iθ , set Note that this definition is not symmetric in x and y. Consider the following event E = ∀x, y ∈ [−n 2 , n 2 ] 2 ∩ Z 2 such that x − y 2 ≥ m, the geodesic between x and y does not intersect E x,y .
On this event, the geodesics behave well, they don't deviate too much from the optimal direction. Note that in the case of a flat edge, there is a cone of optimal directions. It is easy to check thanks to Lemma 4.6 that for n large enough depending on δ and , by union bound Next, we show that on the event E ∩ Ω basic the geodesic γ is (ρ, m)-bounded. Note that on the event Ω basic , for n large enough the geodesic remain inside the box [−n 2 , n 2 ] 2 . Let x = (a, b) and y = (a + d, b + c) with 0 ≤ a + d ≤ n and d ≥ m such that the geodesic γ goes through x and y in this order. We aim to bound |c| on the event E. Write (n, s) = x + R x e iθx with R x ≥ 0 and − π 2 ≤ θ x ≤ π 2 .
Thanks to Claim 4.7, we cannot have both x ∈ Cone(y, b(θ y ) − , u(θ y ) + ) and y ∈ Cone(x, b(θ x ) − , u(θ x ) + ). It follows that this case cannot occur on the event E ∩ Ω basic .
Hence, the geodesic γ has (2, m)-bounded slope on the event Ω Tal ∩ Ω basic . The result follows from Lemmas 2.1 and 2.2.

The number of sides of the limit shape
In this section we prove Theorem 1.3. Recall that X is a random variable supported on [0, 1] with Var(X) = σ 2 and let E := E[X]. Let G be the distribution of 1 + X and let B and µ be the limit shape and limiting norm corresponding to G . The limiting norm of a distribution is defined in Theorem 4.1. Recall the definitions and notations at the beginning of Section 4. We start with the following proposition.
We turn to prove Theorem 1.3 using the proposition.
As long as is sufficiently small we have that < δ 1 ≤ · · · ≤ δ k ≤ √ and that δ i and δ i+1 satisfy the assumptions of Proposition 5.1 for all 1 ≤ i ≤ k. This gives that the directions (1, δ i ) and (1, δ i+1 ) are not on the same flat edge of B and therefore B has at least k − 1 vertices between the angles arg(1, ) and arg(1, √ ). It follows from the symmetries of the lattice that B has at least 8(k − 1) vertices. Indeed, by the reflection symmetry around the y = x line, there are 2(k − 1) vertices in the first quadrant and by the 90-degrees rotation symmetry there is at least this number of vertices in all other quadrants. The theorem follows by taking sufficiently small.
We start with a proof of Lemma 5.2. To this end we need the following claim. Proof of Lemma 5.2. Let δ 0 := 1/M 0 where M 0 is the constant from Claim 5.4. Consider the sequence of points x i := (i/δ, i) ∈ Z 2 for 0 ≤ i ≤ δn. Let p i be the path from x i−1 to x i that goes one step up and then 1/δ steps to the right and let q i be the path from x i−1 to x i that goes 1/δ steps to the right and then one step up. Finally, defineT i := min(T (p i ), T (q i )). By the triangle inequality we have This finishes the proof of (5.3). Finally, by (5.2) we have that E T n ≤ (1 + δ)(1 + /2) − c σ √ δ n.
The lemma follows from the last inequality.