The time constant vanishes only on the percolation cone in directed first passage percolation

We consider the directed first passage percolation model on ${\bf Z}^2$. In this model, we assign independently to each edge $e$ a passage time $t(e)$ with a common distribution $F$. We denote by $\vec{T}({\bf 0}, (r,\theta))$ the passage time from the origin to $(r, \theta)$ by a northeast path for $(r, \theta)\in {\bf R}^+\times [0,\pi/2]$. It is known that $\vec{T}({\bf 0}, (r, \theta))/r$ converges to a time constant $\vec{\mu}_F (\theta)$. Let $\vec{p}_c$ denote the critical probability for oriented percolation. In this paper, we show that the time constant has a phase transition divided by $\vec{p}_c$, as follows: (1) If $F(0)<\vec{p}_c$, then $\vec{\mu}_F(\theta)>0$ for all $0\leq \theta\leq \pi/2$. (2) If $F(0) = \vec{p}_c$, then $\vec{\mu}_F(\theta)>0$ if and only if $\theta\neq \pi/4$. (3) If $F(0)=p>\vec{p}_c$, then there exists a percolation cone between $\theta_p^-$ and $\theta_p^+$ for $0\leq \theta^-_p<\theta^+_p \leq \pi/2$ such that $\vec{\mu} (\theta)>0$ if and only if $\theta\not\in [\theta_p^-, \theta^+_p]$. Furthermore, all the moments of $\vec{T}({\bf 0}, (r, \theta))$ converge whenever $\theta\in [\theta_p^-, \theta^+_p]$. As applications, we describe the shape of the directed growth model on the distribution of $F$. We give a phase transition for the shape divided by $\vec{p}_c$.

1 Introduction of the model and results.
In this directed first passage percolation model, we consider the vertices of the Z 2 lattice and the edges of the vertices with the Euclidean distance 1.We assign independently to each edge a non-negative passage time t(e) with a common distribution F .More formally, we consider the following probability space.As the sample space, we take Ω = e∈Z 2 [0, ∞), whose points are called configurations.Let P = e∈Z 2 µ e be the corresponding product measure on Ω, where µ e is the measure on [0, ∞) with distribution F .The expectation with respect to P is denoted by E(•).For any two vertices u and v in Z 2 , a path γ from u to v is an alternating sequence (v 0 , e 1 , v 1 , ..., v i , e i+1 , v i+1 , ..., v n−1 , e n , v n ) of vertices v i and edges e i between v i and v i+1 in Z 2 , with v 0 = u and v n = v.For a vertex u, its northeast edges from u are denoted by u = (u 1 , u 2 ) to (u 1 + 1, u 2 ) or to (u 1 , u 2 + 1).Given a path (v 0 , e 1 , v 1 , ..., v i , e i+1 , v i+1 , ..., v n−1 , e n , v n ), if each edge e i is a northeast edge from v i , the path is called northeast, or directed.For short, we denote northeast edges or northeast paths by NE edges or NE paths.
Given a path γ, we define its passage time as For any two vertices u and v, we define the passage time from u to v by where the infimum is over all possible paths from u to v. We also define where the infimum is over all possible NE paths from u to v. If there does not exist a NE path from u and v, we simply define A NE path γ from u to v with T (γ) = T (u, v) is called an optimal path of T (u, v).We need to point out that the optimal path may not be unique.If we focus on a special configuration ω, we may write T (u, v)(ω), instead of T (u, v).
In addition to vertices on Z 2 , we may also consider points on R 2 .In particular, we often use the polar coordinates {(r, θ)} = R + × [0, π/2], where r and θ represent the radius and the angle between the radius and the X-axis, respectively.We may extend the definition of passage time over R + × [0, π/2].If u = (r, θ) in R + × [0, π/2], we define T (0, u) = T (0, u ′ ), where u ′ is the nearest neighbor of u in Z 2 .Possible indetermination can be eliminated by choosing an order on the vertices of Z 2 and taking the smallest nearest neighbor for this order.Similarly, T (u, v) can be defined for any u, v ∈ R 2 .Moreover, with this extension, for any points u and v in R 2 , we may consider a path of Z 2 from u to v.
We call µ F (θ) a time constant.Furthermore, by the same subadditive argument, for a non-zero vector (r, θ) ∈ R + × [0, π/2], lim We also call µ F (θ) a time constant.By the subadditive argument again, we know (see Proposition 2.1 (iv) in Martin (2004)) that µ F (θ) is finite and convex in θ. (1.2) In general, we require that t(e) has a finite first moment or m-th moment.However, we sometimes require the following stronger tail assumption: E exp(ητ (e)) < ∞ for η > 0. (1.3) Recall the undirected first passage percolation model for {T (u, v)}.Kesten (1986) showed that there is a phase transition divided by critical probability, p c , of bond percolation for a time constant.More precisely, he showed that time constant µ F (θ) vanishes if and only if F (0) ≥ p c .Therefore, F (0) > p c , F (0) = p c , and F (0) < p c are called the supercritical, the critical, and the subcritical phases, respectively.It is natural to examine a similar situation for the directed first passage percolation model.In this paper, our focus is that there is also a phase transition for µ F (θ) divided by critical probability, p c , of directed bond percolation.We will demonstrate for the supercritical and critical phases, which are quite different from the undirected first passage percolation model (see Kesten andZhang (1997), andZhang (1995)).We will also examine the subcritical phase, which is similar to the undirected model (see Kesten (1986)).
1.1.Supercritical phase.We now focus on the supercritical phase: F (0) > p c .Before introducing our results, we would like to introduce a few basic oriented percolation results.If we rotate our lattice counterclockwise by 45 • and extend each edge by a factor of √ 2, the new graph is denoted by L with oriented edges from (m, n) to (m + 1, n + 1) and to (m − 1, n + 1).Each edge is independently open or closed with probability p or 1 − p.An oriented path from u to v is defined as a sequence The path has the vertices v i = (x i , y i ) and v i+1 = (x i+1 , y i+1 ) for 0 ≤ i ≤ m − 1 such that y i+1 = y i + 1 and v i and v i+1 are connected by an oriented edge.An oriented path is open if each of its edges is open.For two vertices u and v in L, we say u → v if there is an oriented open path from u to v.For A ⊂ (−∞, ∞), we denote a random subset by The right edge for this set is defined by By a subadditive argument (see section 3 (7) in Durrett (1984)), there exists a non-random constant α p such that where α p > 0 if p > p c , and α p = 0 if p = p c , and α p = −∞ if p < p c .Now we rotate the lattice back to Z 2 .If p ≥ p c , the percolation cone is the cone between two polar equations θ = θ ∓ p in the first quadrant, where (see Marchand (2002)) Note that if p = p c , then the percolation cone shrinks to the positive diagonal line.In fact, for any point (r, θ) with θ ∈ [θ − p , θ + p ], it can be shown (see Lemma 3 in Yukich and Zhang ( 2006)) that P[∃ a NE zero-path from the origin to (r, θ)] > C. (1.5) In this paper, C and C i are always positive constants that may depend on F , but not on t, r, k, or n.Their values are not significant and may change from appearance to appearance.With these definitions, we have the following theorem regarding the passage time on the percolation cone: In contrast to the passage time on the percolation cone, we have another theorem: , then for all r, there exist δ = δ(F, θ) > 0 and Together with Theorems 1 and 2, we have the following corollary: We would like to discuss µ F (θ) as a function of F .Recall that in the general first passage percolation model, Yukich and Zhang (2006) showed that the time constant is not third differentiable in the direction of θ ± p .We find out that the same proof together with can be carried out to show the same result for directed first passage percolation.Here we state the following result but omit the proof.We denote by µ F (θ, p) the time constant for F (0) = p.If t(e) only takes two values 0 or 1 and F (0) > p c , then Except in these two directions, we believe that there is no other singularity.
Remark 2. Note that µ F (θ) can also be considered as a function of θ.By the convexity in (1.2), we can show that µ F (θ) is continuous in θ.We believe that θ ∓ p are also the singularities for µ F (θ) in θ.
1.2.Critical phase.We focus on the critical phase: F (0) = p c .Now, as we mentioned, the percolation cone shrinks to the positive diagonal line.Similar to the supercritical phase, we can show the following theorem: Theorem 4. If F (0) = p c and θ = π/4, then there exist δ = δ(F, θ) > 0 and The time constant at θ = π/4 has double behaviors: supercritical and subcritical behaviors.First, we show that it has a supercritical behavior: (1.7) In addition, if F (0) = p, then lim p→ pc µ F (π/4) = 0. (1.8) Remark 3. Cox and Kesten used a circuit method (1981) to show the following result, which is a stronger result than (1.8).If However, their method cannot be applied for the directed model, since a path may not be directed after using a piece of circuit.Therefore, we might need a new method to solve this problem.
Together with Theorems 4 and 5, we have the following corollary: To pursue the convergent rate, we need to use the isoperimetric inequality by Talagrand (1995).Denote by S the sets of all NE paths from the origin to (r, θ) with the minimum passage time.Let where |A| is the number of vertices in A for some vertex set A. Since we only focus on NE paths, α ≤ Cr.
Denote by M a median of T (0, (r, θ)).By Theorem 8.3.1 (see Talagrand (1995)), if (1.3) holds, then there exist constants By this isoperimetric inequality together with a simple computation, we can show the following argument.For all r > 0 and 1 ≤ x ≤ √ r, if (1.3) holds, then With this concentration inequality, we can use Alexander's result (1996) to show the following.For all r, if (1.3) holds, there exists C = C(F, θ) such that for all 0 < r r µ(θ) ≤ E T (0, (r, θ)) ≤ r µ F (θ) + C √ r log r. (1.9) With (1.9) and Theorem 5, if (1.3) holds and F (0) = p c , then there exists C = C(F ) such that E T (0, (r, π/4)) ≤ C √ r log r. (1.10) Remark 4. The upper bound might not be tight at the right side of (1.10).In fact, we believe the following conjecture in a much tight upper bound: (1.11) Note that (1.11) holds for the undirected first passage time (see Chayes, Chayes, and Durrett (1986)).In contrast, the lower bound is more complicated.It might depend on how F (x) ↓ F (0) = p c as x ↓ 0. When the right derivative of F (0) is large enough, we believe the following conjecture occurs as the same as the undirected model (see Zhang (1995)): Conjecture 4.There exists F with F (0) = p c such that E T (0, (r, π/4)) ≤ C.
(1.12)However, when the right derivative of F (0) is small, we believe that it has a subcritical behavior similar to the behavior of undirected passage time.More precisely, lim r→∞ E T (0, (r, π/4)) = ∞. (1.13) In fact, we may simply ask the same questions when t(e) only takes 0 and 1 with F (0) = p c .
Conjecture 5.If t(e) only takes 0 and 1 with F (0) = p c , show that (1.14) Note that (1.14) is indeed true (see Chayes, Chayes, and Durrett (1986)) for the undirected critical model.Furthermore, Kesten and Zhang (1997) showed a central limit theorem for the passage time in the undirected critical model.Here, we partially verify (1.14) for the directed critical model: Theorem 7. If t(e) only takes two values 0 and 1 with F (0) = p c , then Remark 5.As we mentioned above, we know the continuity of µ F (θ) in θ.We believe that there is a power law when θ → π/4.More precisely, we assume that F (0) = p c and t(e) only takes values 0 and 1. Conjecture 6. µ F (θ) ≈ |θ − π/4| α for some 0 < α < 1.
1.3.Subcritical phase.Finally, we focus on the subcritical phase: F (0) = p < p c .On this phase, we show the following theorem: Theorem 8.If F (0) < p c , then for all r and 0 ≤ θ ≤ π/2, there exist δ = δ(F ) and By Theorem 8, there exists C = C(F ) such that for all r and θ E[ T (0, (r, θ))] ≥ Cr. (1.15) With (1.15) and (1.1), we have the following corollary: Remark 6.We would like to focus on a special case in the subcritical phase.In fact, Hammersley and Welsh (1965) considered t(e) + a for some real number a.They used F ⊕ a(x) = F (x − a) to denote the distribution.Clearly, if a > 0, each edge takes at least a time a, so F ⊕ a(0) = 0. Therefore, it is in a subcritical phase.Durrett and Liggett (1981) consider the case that where T ′ (u, v) is passage time from u to v with passage time t(e) on edge e.Thus, the directed first passage percolation model on F ⊕ a(x) is equivalent to the supercritical phase discussed before.
1.4.Shape of the growth model.We may discuss the shape theorem for this directed first passage percolation.Define the shape as (1.17) By (1.1) and (1.17), we know that With (1.17), we define the directed growth shape as With these definitions, Martin (2004) then C is a convex compact set, and for any ǫ > 0, The result in (1.19) is called shape theorem.In the subcritical case, for all 0 ≤ θ ≤ π/2, by Corollary 8, C is a convex compact set such that the shape theorem holds.We denote the shape between two angles by . Furthermore, we denote by ρ(θ) the boundary point (ρ(θ), θ) of C.
In the supercritical case, by Corollary 3 and (1.19), for any small 0 < δ, such that the shape theorem holds: and eventually with probability 1.On the other hand, by Corollary 3 again, equal the percolation cone. (1.22) In the critical case, for any small 0 < δ, by Corollary 6 and (1.19), such that the shape theorem holds: and eventually with probability 1.On the other hand, by Corollary 6 again, C(π/4, π/4) and lim t C t (π/4, π/4) t equal the positive diagonal line. (1.25) In particular, in both the supercritical and critical phases, by Theorems 1 and 2, and Theorem 5, the continuity of In the subcritical case, by Corollary 9, the shape theorem in (1.19) holds.We can describe the phases of the shapes as Fig. 1.
Remark 7. Since the shape is convex, by (1.25), on the critical and super critical phases, the slope of the line passing through (ρ(θ 1 ), θ 1 ) and (ρ(θ 2 ), θ 2 ) cannot be more than tan(θ − p ) sq q q q q q q q q q q q q q r s q q q q q q q q q q q q q q r E T Critical phase: sq q q q q q q q q q q q q q q s s q q q q q q q q q q q q q q q s E T Subcritical phase:F (0) < p c s q q q q q q q q q q q q q q q q q sq q q q q q q q q q q q q q q q q Figure 1: The graph shows shape C in subcritical, critical, and supercritical phases.In the supercritical phase, the right figure, the shape is the percolation cone between two angles θ ± p , and the other two parts of the shape are finite.In the critical phase, the middle figure, the percolation cone shrinks to the positive diagonal line.The other two parts of the shape are finite.In the subcritical phase, the left figure, the shape is finite.
We may relate the directed first passage percolation to the following directed growth model.At time 1, a cell A 1 consists of the unit square with the center at the origin.Each square has four edges: the north, the east, the south, and the west edges.Two squares are connected if they have a common edge.Suppose that at time n we have connected n unit squares, denoted by A n .Let ∂A n be the boundary of A n .A square is a boundary square of A n if one of its edges belongs to ∂A n .We collect all the north and the east edges in ∂A n from the boundary squares.We denote these edges by the northeast edges of A n .At time n + 1, a new square is added to A n such that it connected to the northeast edges of A n .The location of the new square is chosen with a probability proportional to the northeast edges of A n .Now we consider F has an exponential distribution with rate 1.Instead of associating the passage time to the edges, we may associate the passage time to vertices.By the same discussion, we can define the directed growth shape C t and show the shape theorem of (1.19).By a similar computation (see page 131 in Kesten (1986)), we can show that shapes A n and C tn have the same distribution with Thus, the shape theorem for C t implies that n −1/2 A n has also an asymptotic shape.
Unlike the undirected model, the oriented percolation model in higher dimensions has been more limited.For example, we cannot define the right edge r n for the oriented percolation model when d > 2. However, if one could develop a similar argument of the percolation cone as we defined in section 1.1, our techniques for first passage percolation would apply for higher dimensions.Finally, we conclude this section with the following strictly monotone conjecture: 2 Preliminaries.
2.1.Renormalization method.We introduce the method of renormalization in Kesten and Zhang (1990).We define, for a large integer M and w = (w 1 , w 2 ) ∈ Z 2 , the squares by We denote these M-squares by {B M (w) : (2.1) For each M-square B M (w), there are eight M-square neighbors.We say they are adjacent to B M (w).We denote B M (w) and its eight M-square neighbors by BM (w).BM(w) is called a 3M-square.Since γ is connected, γ M has to be connected through the square connections.
If B M (w) ∩ γ = ∅ and BM (w) does not contain the origin, note that γ has to cross BM (w) \ B M (w) to reach to B M (w), so BM (w) contains at least M vertices of γ in its interior.We collect all such 3M-squares { BM (w)} such that their center M-squares contain at least a vertex of γ.We call these 3M-squares center 3M-squares of γ.With these definitions, the following lemma (see Zhang, page 22 (2008)) can be calculated directly.
Lemma 1.For a connected path γ, if |γ M | = k, then there are at least k/15 disjoint center 3M-squares of γ.

Results for oriented percolation.
We assign either open or closed to each edge with probability p or 1 − p independently from the other edges.For two sets A and B, if there exists a NE open path from u ∈ A to v ∈ B, we write A → B as the event.
First, we focus on the subcritical phase: p < p c .Let Durrett (section 7, (6) (1984)) showed the following lemma: Lemma 2. If p < p c , then there exist C i for i = 1, 2 such that Now we focus on the critical and supercritical phases: p ≥ p c .Given two points u = (u 1 , u 2 ) and v = (v 1 , v 2 ), we define the slope between them by With these definitions, Zhang (Lemma 3 ( 2008)) showed the following lemma: 2.3.Analysis for the shape C t .Now we would like to introduce a few geometric properties for C t .In the remainder of section 2.3, we only consider t(e) when it takes value 0 or 1 with F (0) = p.If t(e) = 0 or 1, e is said to be open or closed.
Given a set Γ ⊂ R 2 , we let Γ ′ be all vertices on Z 2 contained in Γ.It is easy to see that As we defined in the last section, C t is finite, and so is C ′ t .A set A is said to be directly connected in Z 2 if any two vertices of A are connected by a NE path in A.
Given a finite directly connected set Γ of Z 2 , we define its vertex boundary as follows.For each v ∈ Γ, v ∈ Γ is said to be a boundary vertex of Γ if there exists u ∈ Γ but u is adjacent to v by either a north or an east edge.We denote by ∂Γ all boundary vertices of Γ.We also let ∂ o Γ be all vertices not in Γ, but adjacent to ∂Γ by north or east edges.∂ e Γ is denoted by these NE edges between ∂Γ and ∂ o Γ.
We define the event as With these definitions, Zhang (see propositions 1-3 in Zhang (2006)) proved the following lemmas for undirected first passage percolation.The proofs can be carried out by changing paths to directed paths, so we omit the proofs.In fact, these lemmas are easily understood by drawing a few figures.
Lemma 4. C ′ t is directly connected.
Lemma 6.The event of {C ′ t = Γ} only depends on the zeros and ones of the edges on 2.4.Monotone property for the time constant.Finally, we would like to introduce a monotone lemma for the time constant.Comparing two distributions F 1 and F 2 , we have the following lemma: Proof.Smythe and Wierman (1978) proved the same result in their Theorem 7.12 for undirected first passage percolation.The same proof can be carried out to show Lemma 7. 2 3 Subcritical phase.
In section 3, we assume that F (0) < p c .Since F is right-continuous, we take ǫ > 0 small such that F (ǫ) < p c . (3.1) We say that an edge is open if t(e) ≤ ǫ, otherwise e is said to be closed.With (3.1), we know that Now we work on a NE path from the origin to (r, θ).As before, we use γ M to denote the squares of γ.If a square in γ M contains a closed edge, we call the square a bad square.Otherwise, it is a good square.If there is a path γ from the origin to (r, θ) such that it has less than Cr closed edges for some small C, then there is less than C|γ M | bad squares in γ M .Now we account the choices of these squares.Note that γ is connected, and so is γ M .We assume that |γ M | = k.
By Lemma 1 in section 2, we know there are at least |γ M |/15 center 3M-squares of γ.Thus, for C < 1/(30), there are at least disjoint 3M-squares such that their center squares contain an edge of γ and all the M-squares in these 3M-squares are good.We also call these 3M-squares good.By a standard method (see (4.24) in Grimmett (1999)), there are at most 7 2k choices for all possible choices of γ M .When γ M is fixed, we select these good 3M-squares.There are at most 2 k choices for these good 3M-squares.
For each good 3M-square BM (w), there exists a NE open path crossing the 3M-square from a vertex at the boundary of B M (w) to another vertex at the boundary of BM (w).There are at most 4M choices for the starting vertex, and the path contains at least M edges.For a fixed BM (w), we denote by E w the event that there exists a NE open path from B M (w) to the boundary of BM (w).By Lemma 2, there are (3.4) Note that E w and E u are independent with the same distribution for fixed w and u if B w (M) and B u (M) are two center squares of two different 3M-squares.With these observations and (2.1), if we take M large, for small C > 0, there exist C i = C i (F ) for i = 1, 2 in (3.4) and C j = C j (F, M, C) for j = 3, 4 such that P[∃ a NE path γ from the origin with |γ| ≥ r and with less than Cr closed edges] (3.5) Proof of Theorem 8. On { T (0, (r, θ)) ≤ ǫ 2 (r − 1)}, there exists a NE path γ from the origin with |γ| ≥ r − 1 and T (γ) ≤ ǫ 2 (r − 1) for some ǫ > 0. Note that if |γ| ≥ r − 1 and T (γ) ≤ ǫ 2 (r − 1), then γ contains less than ǫ(r − 1) closed edges.So, if we take C in (3.5) such that C = ǫ for some small ǫ, by (3.5), there exist constants (3.6)Therefore, for some δ > 0, there exist C i (F, δ) for i = 1, 2 such that (3.7) Thus, Theorem 8 follows from (3.7). 2 4 Outside the percolation cone.
The proofs for theorems outside the percolation cone also need the method of renormalization.We assume that in Theorems 2 and 4 Thus, we say an edge is open or closed if t(e) = 0 or t(e) > 0. With (4.1), we have Similar to the proof in the last section, we work on a path γ from the origin to (r, θ) and denote the M-squares of γ by γ M for a large M. If a square in γ M contains an edge e with t(e) > 0, we say the square is a bad square.Otherwise, it is a good square.
If there is a path γ from the origin to (r, θ) with less than Cr closed edges for some small C, then there is less than C|γ M | bad squares in γ M .Now we account the number of the choices for these squares.Note that γ is connected, and so is γ M .We assume that As we proved in section 3, there are at most 7 2k choices for all possible γ M .When γ M is fixed, we select these bad squares.There are at most choices for these bad squares.We list all the bad squares as for l ≤ Ck.For each S i , path γ will meet the boundary of S i at v ′ i and then use less than 2M edges to meet v ′′ i , another boundary point of S i .We denote the path from the origin to Note that bad edges are only contained in bad squares, so γ i does not contain a bad edge.In other words, Let E i be the event.Since there exists a NE path with less than 2M edges from v ′ i to v ′′ i , Also, there are at most (4M) 2 choices for v ′ i and v ′′ i when S i is fixed.For a fixed S i and fixed {∃ open γ i }} and {  Thus by Lemma 3, (4.9), (4.10), and (4.7), there exist If we substitute (4.11) into (4.8),there exist Therefore, if we take M large and then C = C(M, C 2 ) small, there exist C i = C i (F, θ, C) for i = 3, 4 such that P[∃ a NE γ from 0 to (r, θ) with less than Cr closed edges] ≤ C 3 exp(−C 4 r).

In summary, if θ < θ −
p , for all r, there exist C = C(F, θ) and C i = C i (F, θ, C) for i = 1, 2 such that P[∃ a NE γ from 0 to (r, θ) with less than Cr closed edges] ≤ C 1 exp(−C 2 r). (4.12) With (4.12), we show Theorems 2 and 4: Proofs of Theorems 2 and 4. Suppose that there exists a NE path γ from the origin to (r, θ) with T (γ) ≤ C 1 r.
By (4.12), we may choose the C in (4.12) such that (4.13) For each closed edge e, we know that t(e) > 0. For ǫ > 0, we take δ > 0 small such that For each closed edge, if it satisfies t(e) ≤ δ, we say it is a bad edge.Thus, choices for these closed edges.If these closed edges are fixed, as we mentioned above, each edge has a probability less than ǫ such that it is also bad.In addition, we also have another 2 2r choices to select these bad edges from these closed edges.Therefore, if we take ǫ = ǫ(F, δ, C) small, then there exist C i = C i (F, δ, C) for i = 2, 3 such that P[∃ a NE γ from 0 to (r, θ) with more than Cr closed edges, these closed edges contain more than Cr/2 bad edges] for θ < θ − p , then there is a NE path γ from 0 to (r, θ) with a passage time less than C 1 r.Therefore, by (4.13) and (4.14), P[ T (0, (r, θ)) ≤ C 1 r] ≤ P[∃ a NE γ from 0 to (r, θ) with more than Cr closed edges, these closed edges contain less than Cr/2 bad edges, If there is a NE path from 0 to (r, θ) with less than Cr/2 bad edges among these Cr closed edges, note that each good edge costs at least passage time δ, so the passage time of the path is more than δCr/2.Thus, if we select C 1 such that By (4.15) and (4.16), for (4.17) When θ > θ + p , by symmetry, we still have (4.17).Therefore, Theorems 2 and 4 follow.2 5 Inside the percolation cone.
In section 5, we assume that . Edge e is called an open or a closed edge if t(e) = 0 or t(e) > 0, respectively.We define τ (e) = 0 if t(e) = 0, or τ (e) = 1 if t(e) > 0. We also denote by the passage time T τ (u, v) corresponding to τ (e).Let (5.1) We also assume that (r, θ) ∈ Z 2 without loss generality.If (r, θ) ∈ B τ (t), then T τ (0, (r, θ)) ≤ t. (5.2) Note that B τ (t) will eventually cover all the vertices in R × [0, π/2] as t → ∞, so there exists a t such that (r, θ) ∈ B τ (t).Let σ be the smallest t such that (r, θ) ∈ B τ (t).We will estimate σ to show that there exist C i = C i (F ) for i = 1, 2 such that for all large k, (5.3) where Γ, containing the origin, takes all possible vertex sets in the first quadrant.We also remark that for Γ 1 and Γ 2 , In other words, all Γ in the above sum do not contain (r, θ).Thus, by Lemma 5, there is no NE open path from ∂ o (Γ) to (r, θ), without using edges of Γ ∪ ∂ e Γ.Otherwise, σ < k, which is contrary to the assumption that σ ≥ k.For a fixed Γ, we denote by E k (Γ) the above event that there is no NE open path without using edges of Γ ∪ ∂ e Γ from ∂ o (Γ) to (r, θ).Note that E k (Γ) only depends on configurations of edges outside Γ ∪ ∂ e Γ, so by Lemma 6, for any fixed Γ with Γ ∩ (r, θ) = ∅, E k (Γ) and {B τ (k − 2) = Γ} are independent. (5.5) Note that if there is a NE open path from the origin to (r, θ), then there exists a NE open path outside Γ ∪ ∂ e Γ from ∂ o Γ to (r, θ).By (1.5), there exists 0 < δ < 1 such that for a fixed Γ, (5.6) With these observations, Note that for a fixed Γ, by Lemma 5 again, Therefore, (5.7) Thus, (5.3) follows if we iterate (5.7).We show Theorem 1 by (5.3).In fact, if t(e) is bounded from above by a constant, then Theorem 1 is implied by (5.3) directly.However, if we restrict ourselves only on a moment condition, the proof is complicated, as follows: Proof of Theorem 1.On {σ = k}, there exists an optimal path γ k in τ (e) from the origin to (r, θ) with only k edges {e i } such that τ (e i ) = 1.For each configuration on {σ = k}, we use a unique way to select an optimal path in τ (e) with these k edges.We still denote the path by γ k .For configuration ω, and path γ k (ω), let e 1 , e 2 , • • • , e k ⊂ γ k with τ (e i ) = 1.Note that on {σ = k}, only t(e i ) > 0, but the others are zero-edges, so where β is a fixed NE path, the second sum takes over all possible NE paths β that are from the origin to (r, θ), and the third sum takes over all possible k edges e i for i = 1, 2, • • • , k on path β.With these decompositions, the events for different paths β and different selections of e i in β.
On the event in (5.8) for a fixed β and these fixed e i , 1 ≤ i ≤ k, we denote by event E j if t(e j ) = max{t(e 1 ), • • • , t(e k )} and t(e i ) < t(e j ) for i = 1, • • • , j − 1.
Proof of Theorem 5. First we show (1.7) in Theorem 5. We may take h small such that F has two different situations at F (0): . Let us assume that case (b) holds.For all ǫ ≤ h, we construct another distribution: By this definition, for each n, G n (x n ) > p c .
In other words, there is a jump point at h.We focus on case (a) (i).We take ǫ > 0 small such that Then we construct another distribution: As we defined, t(e) is the random variable with distribution F .Let g ǫ (e) be the random variable with distribution G ǫ .In addition, t(e) and g ǫ (e) are signed values independently edge by edge as we defined.The key step is to couple these two random variables together.Define g ǫ (e) as follows: If t(e) = 0, then g ǫ (e) = 0.
Finally, we focus on case (a) without other assumptions.As we mentioned, t(e) is not a constant.Thus, there exists h 1 > h such that F (h 1 ) > F (0).We construct With this definition, H ≤ F, and H(0) = p c , and H(x) has a jump point at h 1 .By the analysis of case (a) (i), we have Proof of Theorem 7. In this proof, we assume that t(e) only takes 0 (open) and 1 (closed) with probability p c and 1 − p c , respectively.Let L r be the line y = −x + r inside the first quadrant.Note that L 0 is just the origin.Bezuidenhout and Grimmett (1991) showed that for fixed L r 1 and for 0 < δ < 1, there exists r 2 = r 2 (r 1 ) such that (r, θ)) = lim r→∞ 1 r E T (0, (r, θ)) = inf r 1 r E T (0, (r, θ)) = µ F (θ) a.s. and in L 1 .(1.1) sr, θ)) = µ F (r, θ) a.s. and in L 1 .
[e is closed and bad] = P[0 < t(e) ≤ δ] ≤ ǫ.Now, on {∃ a NE γ from 0 to (r, θ) with more than Cr closed edges}, we estimate the event that there are at least Cr/2 bad edges in γ.By (4.7), |γ| ≤ 2r.Now we fix path γ.Since each vertex in γ can be adjacent only from a north or an east edge, there are at most 2 2r choices for γ.If γ is fixed, there are at most

P
[L r 1 → L r 2 ] ≥ δ.(6.18)If L r 1 → L r 2 , then NE path from L r 1 to L r 2 has to use at least one edge with passage time 1.Let I(L r 1 , L r 2 ) be the indicator of the event that there is no NE open path from L r 1 to L r 2 .For large r, let r 1 ≤ r 2 ≤ • • • ≤ r m ≤ r be a sequence such that for i < m − 1,P[L r i → L r i+1 ] ≥ δ. (6.19)Note that any NE path from the origin to (r, π/4) has to cross the strip between L i andL i+1 for i = 1, • • • , m − 1, so E T (0, (r, π/4)) ≥ E m i=1 I(L r 1 , L r 2 ) = δm.(6.20)By (6.18), we have m → ∞ as r → ∞.Therefore, by (6