Geodesics in First Passage Percolation

Transcription

1 Geodesics in First Passage Percolation Christopher Hoffman arxiv:math/ v1 [math.pr] 5 Aug 2005 Abstract We consider a wide class of ergodic first passage percolation processes on Z 2 and prove that there exist at least four one-sided geodesics a.s. We also show that coexistence is possible with positive probability in a four color Richardson s growth model. This improves earlier results of Häggström and Pemantle [9], Garet and Marchand [7] and Hoffman [11] who proved that first passage percolation has at least two geodesics and that coexistence is possible in a two color Richardson s growth model. 1 Introduction 1.1 First passage percolation First passage percolation is a process introduced by Hammersley and Welsh as a time dependent model for the passage of a fluid through a porous medium which has provided a large number of problems of probabilistic interest with excellent physical motivation [10]. Study of this model led to the development of the ergodic theory of subadditive processes by Kingman [15]. It also has links to mathematical biology through Richardson s growth model [9]. A good overview of first passage percolation is contained in [13]. Let µ be a stationary measure on [0, EdgesZd and let ω be a realization of µ. For any x and y we define τx,y, the passage time from x to y, by τx,y = inf ωv i,v i+1 where the sum is taken over all of the edges in the path and the inf is taken over all paths connecting x to y. The time minimizing path from x to y is called a geodesic. An infinite path v 1,v 2,... is called a geodesic if for all 0 < i < j j 1 τv i,v j = ωv k,v k+1. Key words: first passage percolation, Richardson s growth model 2000 Mathematics Subject Classifications: 60K35 82B43 k=i 1

2 In this paper we prove that for a very general class of first passage percolation processes that there exist at least four disjoint infinite geodesics a.s. For notational reasons it will often be convenient to think of τ as a function defined on R 2 R 2 by setting τx + u,y + v = τx,y for any x,y Z 2 and any u,v [ 1 2, The most basic result from first passage percolation is the shape theorem. Define Rt = {v : τ0,v t}. The shape theorem says that there is a nonempty set R such that modulo the bound- converges to R a.s. ary Rt t Theorem 1.1. [3] Let µ be stationary and ergodic, where the distribution on any edge has finite d + ǫ moment for ǫ > 0. There exists a closed set R which is nonempty, convex, and symmetric about reflection through the coordinate axis such that for every ǫ > 0 there exists a T such that P 1 ǫr < Rt < 1 + ǫr for all t > T > 1 ǫ. t This theorem is an example of a subadditive ergodic theorem. In general, little is known about the shape of R other than it is convex and symmetric. Cox and Durrett have shown that there are nontrivial product measures such that the boundary of R contains a flat piece yet it is neither a square nor a diamond [6]. However for any nonempty, convex, and symmetric set R there exist a stationary measure µ such that the shape for µ is R [8]. Kesten proved that the variance of a n = τ0,n,0 is On [13]. Benjamini, Kalai, and Schramm proved for a class of measures that the variance of a n is On/log n [2]. It is conjectured that the variance of a n is On 2/3. Recent work on last passage percolation supports this conjecture and also suggests what the limiting distribution of a n /n 1/3 might be [1], [5], [12]. Another widely studied aspect of first passage percolation are geodesics. We let Gx,y be the geodesic connecting x and y. Define Γx = y Z d{e Gx,y}. We refer to this as the tree of infection of x. We define KΓx to be the number of topological ends in Γx. This is also the number of infinite self avoiding paths in Γx that start at x. Newman has conjectured that for a large class of µ, KΓ0 = a.s. [14] Häggström and Pemantle proved that if d = 2, µ is i.i.d. and ωe has exponential distribution then with positive probability KΓ0 > 1. In independent work Garet and Marchand [7] and Hoffman [11] extended this result in two directions. Their results apply to a wide class of ergodic measures µ on any d 2. Newman has proved that if µ is i.i.d. and R has certain properties then KΓ0 = a.s. [14]. Although these conditions are plausible there are no known measures µ 2

3 with S that satisfy these conditions. In this paper we prove an analogous theorem but with a much weaker condition on R. Unfortunately even this weaker condition, that R is not a polygon, hasn t been verified for any version of i.i.d. first passage percolation. Now we will introduce some more notation which will let us list the conditions that we place on µ for the rest of this paper. We say that µ has unique passage times for all x and y z Pτx,y τx,z = 1. Now we are ready to define the class of measures that we will work with. We say that µ is good if 1. µ is ergodic, 2. µ has unique passage times, 3. the distribution of µ on any edge has finite d + ǫ moment for some ǫ > 0 4. R is bounded. Throughout the rest of the paper we will assume that µ is good. Unfortunately there is no general necessary and sufficient condition to determine when the shape R is bounded. See [8] for examples. 1.2 Spatial Growth Models Richardson s growth model, a simple competition model between diseases, was introduced by Häggström and Pemantle [9]. The rules for this model are as follows. Each vertex z Z 2 at each time t 0 is either infected by one of k diseases z t {1,...,k} or is uninfected z t = 0. Initially for each disease there is one vertex which is infected by that disease. All other vertices are initially uninfected. Once a vertex is infected one of the diseases it stays infected by that disease for all time and is not infected by any disease. All of the diseases spread from sites they have already infected to neighboring uninfected sites at some rate. We now explain the relationship between first passage percolation and Richardson s growth models. For any ω [0, EdgesZd with unique passage times and any x 1,...,x k Z d we can project ω to ω x1,...,x k {0,1,...,k} Zd [0, by { i if τxi,z t and τx ω x1,...,x k z,t = i,z < τx j,z for all i j; 0 else. If µ has unique passage times then µ projects onto a measure on {0,1,...,k} Zd [0,. It is clear that the models start with a single vertex in states 1 through k. Vertices in states i > 0 remain in their states forever, while vertices in state 0 which are adjacent to a vertex in state i can switch to state i. We think of the vertices in states i > 0 as infected with one of k infections while the vertices in state 0 are considered uninfected. For this model it is most common to choose µ to be i.i.d. with an exponential distribution on each edge. This makes the spatial growth process Markovian. 3

4 As each z Z d eventually changes to some state i > 0 and then stays in that state for the rest of time, we can define the limiting configuration ω x1,...,x k z = lim t ω x1,...,x k z,t We say that mutual unbounded growth or coexistence occurs if the limiting configuration has infinitely many z in state i for all i k. More precisely we define Cx 1,...,x k to be the event that {z : ω x1,...,x k z = 1} = = { ω x1,...,x k z = k} =. We refer to this event as coexistence or mutual unbounded growth. 1.3 Results Our results depend on the geometry of R. Let Sidesµ be the number of sides of R if R is a polygon and infinity if R is not a polygon. Note that by symmetry we have Sidesµ 4 for any good measure µ. Let Gx 1,...,x k be the event that there exist disjoint geodesics g i starting at x i. In this paper we prove the following theorem about general first passage percolation. Theorem 1.2. Let µ be good. For any ǫ > 0 and k Sidesµ there exists x 1,...x k such that PGx 1,...,x k > 1 ǫ. We also get the two closely related theorems. Theorem 1.3. If µ is i.i.d. and the distribution on any edge is exponential then for any k Sidesµ P KΓ0 k > 0. Theorem 1.4. Let µ be good. For any k Sidesµ/2 P KΓ0 k > 1. Theorem 1.3 extends a theorem of Häggström and Pemantle [9]. They proved that under the same hypothesis that P KΓ0 > 1 > 0. Garet and Marchand [7] and Hoffman [11] extended the results of Häggström and Pemantle to a general class of first passage percolation processes in any dimension. As an easy consequence of Theorem 1.4 we get Corollary 1.5. There exists a good measure µ such that P KΓ0 = = 1. Proof. This follows easily from Theorem 1.4 and [8] where it is proven that there is a good measure µ such that R is the unit disk. 4

5 Our main result on a multiple color Richardson s growth model is that with positive probability coexistence occurs. Theorem 1.6. If µ is good and k Sidesµ then for any ǫ > 0 there exist x 1,...,x k such that PCx 1,...,x k > 1 ǫ. Häggström and Pemantle [9] proved that if µ is i.i.d. with exponential distribution then PC0,0,0,1 > 0. Garet and Marchand [7] and Hoffman [11] proved that in any dimension mutual unbounded growth is possible when k = 2. Our result extends the previous results in two ways. First it shows that coexistence is possible with four colors. Second it shows that if the points x 1,...,x k are sufficiently far apart then the probability of coexistence approaches one. None of the three proofs that coexistence is possible in the two color Richardson s growth model were able to show that the probability of coexistence went to one as the initial sites x 1 and x 2 moved farther apart. Corollary 1.7. There exists a nontrivial i.i.d. measure µ and x 1,...,x 8 such that PCx 1,...,x 8 > 0. Proof. By [4] there exists a µ which is i.i.d. such that R is neither a square nor a diamond. As R is symmetric Sidesµ 8. Thus the result follows from Theorem Notation Much of the notation that we introduce is related to the shape R. For v R 2 \ 0,0 let T 1 v = sup{k : kv R}. Theorem 1.1 says lim 1 n τ0,nv = T v. 1 Also we have that T kv = kt v and T v = 1 for all v R. Let the set V consist of all v R such that there is a unique line L v which is tangent to R through v. For such a v let wv be a unit vector parallel to L v. Let L n,v be the line through nv in the direction of wv. For any k Sidesµ there exists points v 1,...,v k R such that v i V for all i and the lines L vi are distinct. The existence of a unique line tangent to R at the point v is useful because it is equivalent to T v + wvb 1 lim = 0. 2 b 0 b Let S R 2 we define the function B S x,y = inf z S n τx,z inf z S n τy,z. 5

6 Lemma 2.1. For any set S Z 2 and any x,y,r Z 2 1. B S x,y τx,y and 2. B S x,y + B S y,z = B S x,z. Proof. These properties follow easily from the subadditivity of τ and the definition of B S. These functions are useful in analyzing the growth model because of the following fact. Lemma 2.2. If there exists c > 0, and x 1,...,x k such that for infinitely many n then Proof. If for a fixed i and all j i PB Ln,vi x j,x i > 0 i j > 1 c PCx 1,...,x k > 1 c. B Ln,vi x j,x i > 0 then there exists z L n,vi such that τz,x i < τz,x j for all j i. Thus there is a z L n,vi such that ω x1,...,x k z = i. For a fixed i each z is in only one L n,vi, so if there exist infinitely many n such that for all i and j i B Ln,vi x j,x i > 0 then for every i there are infinitely many z such that ω x1,...,x k z = i. By assumption we have that there exist infinitely many n such that PB Ln,vi x j,x i > 0 for all i and j i > 1 c. Thus by the Borel-Cantelli lemma we have that with probability at least 1 c there exist infinitely many n such that for all i and j i B Ln,vi x j,x i > 0. In conjunction with the previous paragraph this proves the lemma. Lemma 2.3. PGx 1,...,x k > PCx 1,...,x k. 6

7 Proof. For any y {z : ω x1,...,x k z = i} the geodesic from x i to y lies entirely in {z : ω x1,...,x k z = i}. By compactness if {z : ω x1,...,x k z = i} = then there exists an infinite geodesic g i which is contained in the vertices {z : ω x1,...,x k z = i}. The following lemma is certainly not new but we include a sketch of the proof for completeness. We use the notation 1 A to represent the indicator function of A. Lemma 2.4. For all v R 2 \0 and ǫ > 0 there exists δ > 0 such that for all sufficiently large M and all events A with PA < δ we have that In particular for any v Eτx,x + Mv1 A < Mǫ. lim M 1 M Eτ0,Mv = T v. Proof. Let τ x,y be the distance from x to y by travelling parallel to the x axis and then parallel to the y axis. We work on an ergodic component for the shift maps. By the ergodic theorem for any ǫ > 0 there exists δ such that for all v and M sufficiently large such that if PA < δ then The lemma follows because Eτ x,x + Mv1 A < Mǫ. τx,x + Mv τ x,x + Mv. 3 Outline We start by outlining a possible method to prove that there are infinitely many geodesics starting at the origin. Then we show the portion of this plan that we can not prove. Finally we show how to adapt this method to get the results in this paper. It is easy to construct geodesics beginning at 0. We can take any sequence W 1,W 2,... of disjoint subsets of Z 2 and consider G0,W n, the geodesic from 0 to W n. Using compactness it is easy to show that there exists a subsequence n k such that G0,W nk converges to an infinite geodesic. 7

8 If we take two sequences of sets W n and W n we can construct a geodesic for each sequence. It is difficult to determine whether or not the two sequences produce the same or different geodesics. The tool that we use to distinguish the geodesics are Busemann functions. Every geodesic generates a Busemann function as follows. For any x,y Z 2 and infinite geodesic G = v 0,v 1,v 2,... we can define To see the limit exists first note that B ω Gx,y = B G x,y = lim n τx,v n τy,v n. B G x,y = lim τx,v n τy,v n n = lim τx,v n τv 0,v n + τv 0,v n τy,v n n = lim τx,v n τv 0,v n + lim τv 0,v n τy,v n. n n As G is a geodesic the two sequences in the right hand side of the last line are bounded and monotonic so they converge. Thus B G x,y is well defined. Two distinct geodesics may generate the same Busemann function but distinct Busemann functions mean that there exist distinct geodesics. To construct a geodesic we pick a,b Z 2 and we set W n to be W n = {w Z 2 : w a,b n}. If we could show that for every z Z 2 that G n z, the geodesic from z to W n, converges then it is possible to show that and lim M lim M 1 B0,bM, am = 0. 3 M 1 B0,aM, bm = M inf T v. 4 v a,b=a 2 +b 2 Thus for any a,b which is not a scalar multiple of a,b we would be able to show that 1 lim M M B0,b M, a M 0. 5 Thus for any a,b and a,b which are not scalar multiples we get distinct geodesics. In this way it would be possible to construct an infinite sequence of distinct geodesics. We are unable to show that the geodesics Gz,W n converge. But for some a,b Z 2 we can establish versions of 3 and 4. These are Lemmas 4.3 and 4.2. These lemmas form the heart of our proof. 4 Proofs Although it is convenient to write τx,y for x,y R 2, the distribution of τx,y is equal to the distribution of τx + z,y + z only if z Z 2. For z Z 2 the distribution 8

9 of τx,y may not be equal to the distribution of τx + z,y + z which will make the notation more complicated. But the distributions are close enough so that this lack of shift invariance for noninteger translations will not cause any significant problems. To deal we this lack of translation invariance we let Lemma 4.1. For all a and b Ia,b = I v a,b = sup x L a,v E inf τx,y y L b,v Ia,b = I0,b a. 6 Moreover, for any m such that m v > 1, any n,d, and any r L d,v Id,n < E inf τr mv,y. 7 y L n,v Proof. If wv has rational slope then 6 follows easily from the shift invariance of µ. If wv has irrational slope then for any z Z 2 which is the nearest lattice point to a point in L a,v there is a u Z 2 which is the nearest lattice point to a point in L 0,v such that the line L b a,v translated by z u is arbitrarily close to L b,v. This creates a coupling between shortest paths between u and L b a,v and shortest paths between z and L b,v such that with high probability such that with high probability the paths are translates of each other. Thus by Lemma 2.4 for any x L a,v and any ǫ > 0 there is a point in t L 0,v such that E inf τx,y y L b,v As ǫ was arbitrary 8 implies E inf τt,y y L b a,v Ia,b I0,b a. < ǫ. 8 We get the other inequality in exactly the same way and 6 is true. Let z Z 2 be the nearest lattice point to r mv. Let u Z 2 be any lattice point nearest to a point in L d,v. We need to prove that E inf τz,y y L n,v E inf τu,y y L n,v. 9 As m v > 1 the translation of L n,v by u z lies on the opposite side of L n,v as do z and u. As µ is shift invariant we get that 9 is true. Lemma 4.2. For any v V, any ǫ > 0, for all sufficiently large M and all r R 2 the density of n such that P M1 ǫ < B Ln,v r,r + Mv < M1 + ǫ > 1 ǫ. 10 is at least 1 ǫ. 9

10 Proof. By Lemma 2.1 for any r,n,m and v B Ln,v r,r + Mv τr,r + Mv 11 and by Theorem 1.1 for any r,v and sufficiently large M P τr,r + Mv < M1 + ǫ > 1 ǫ. 12 Thus for sufficiently large M the upper bound on B Ln,v r,r + Mv is satisfied for all n with probability at least 1 ǫ. Now we bound the probability that B Ln,v r,r + Mv is too small. Let d be such that r L d,v. For any sufficiently large M and any n > d + M Id,n Id + M,n < E inf τr mv,y sup E inf τx,y 13 y L n,v x L d+m,v y L n,v < E inf τr mv,y E inf τr + Mv,y y L n,v y L n,v < E inf τr mv,y inf τr + Mv,y y L n,v y L n,v < E B Ln,v r mv,r + Mv 14 < E τr mv,r + Mv 15 < M1 + ǫ follows from 7 and the definition of Id + M, n, 14 follows from the definition of B Ln,v, 15 follows from Lemma 2.1, and 16 follows from Theorem 1.1. Let k be such that d + km n < d + k + 1M. For k and M by 6 Id, n = Id,n Id + M,n + Id + M,n Id + 2M,n + + Id + k 1M,n Id + km,n + Id + km,n = Id,n Id + M,n + Id,n M Id + M,n M + + Id,n k 1M Id + M,n k 1M + Id,n km 17 By Theorem 1.1 for any sufficiently large M and any n > d + M Id,n > k + 1M1 ǫ. 10

11 By 17 and 16, Id,n is the sum of k+1 terms bounded above by M1+ǫ. Without loss of generality ǫ < 1/4. Thus the number of j < k such that Id,n jm Id + M,n jm > M1 ǫ is at least k1 2 ǫ. As ǫ and n were arbitrary we get that for any ǫ > 0 the density of n such that Id,n Id + M,n > M1 ǫ 18 is at least 1 ǫ. Now we show that for any M,n,r and v such that 18 is satisfied we have that with high probability B Ln,v r,r + Mv is large. Let E be the event that B Ln,v r,r + Mv > M1 + ǫ. By Lemma 2.1 and Theorem 1.1 we can make PE arbitrarily small by making M sufficiently large. Then we get Id,n Id + M,n > M1 ǫ E B Ln,v r mv,r + Mv > M1 ǫ 19 E B Ln,v r mv,r + E B Ln,v r,r + Mv1 E C +E B Ln,v r,r + Mv1 E > M1 ǫ 20 E B Ln,v r,r + Mv1 E C > M1 2ǫ E B Ln,v r,r + Mv1 E E B Ln,v r,r + Mv1 E C > M1 2ǫ E B Ln,v r,r + Mv1 E C Eτr,r + Mv1 E 21 > M1 3ǫ follows from follows from Lemma follows from Lemma 2.1 and Theorem is due to Lemma 2.4. As the expected value of the function B Ln,v r,r + Mv1 E C is close to its maximum, M1 + ǫ, we get that with high probability the function is close to its maximum. Thus we get that for any ǫ > 0 and all sufficiently large M, the set of n such that P M1 ǫ < B Ln,v r,r + Mv > 1 ǫ 23 has density at least 1 ǫ. Putting together 11, 12 and 23 proves the lemma. 11

12 Lemma 4.3. For any v V, ǫ > 0, any M with sufficiently large absolute value and any r R 2 the density of n such that P BLn,vr,r + Mwv < ǫ M > 1 ǫ is at least 1 ǫ. Proof. First we prove the upper bound in the case that M is positive. Fix ǫ > 0. Since R has a unique tangent line at v by 2 we can find b > 0 such that T v + bwv1 + ǫb < 1 + 2ǫb. 24 By Lemma 4.2 for any ǫ,b > 0 and all sufficiently large M the density of n such that P B Ln,v r Mv,r M1 ǫb > 1 ǫ 25 is at least 1 ǫ. By Lemma 1.1 for any ǫ,b > 0 and all sufficiently large M P τr Mv,r + Mbwv MT v + bwv1 + ǫb > 1 ǫ. 26 Choose M large enough such that both 25 and 26 are satisfied. Thus with probability at least 1 2ǫ the density of n such that the following inequalities are satisfied is at least 1 ǫ. τr Mv,r + Mbwv τr Mv,r + Mbwv τr Mv,r + Mbwv B Ln,v r Mv,r B Ln,v r Mv,r + Mbwv B Ln,v r Mv,r + B Ln,v r,r + bmwv B Ln,v r,r + bmwv MT v + bwv1 + ǫb M1 ǫb B Ln,v r,r + bmwv 27 M1 + 2ǫb M1 ǫb B Ln,v r,r + bmwv 28 3bMǫ B Ln,v r,r + bmwv. The first two lines follow deterministically from Lemma is true with probability at least 1 2ǫ. This follows from 26 and follows from 24. Thus we have that for any sufficiently large M the density of n such that P B Ln,v r,r + bmwv 3bMǫ > 1 2ǫ is at least 1 ǫ. By replacing wv with wv and interchanging r and r + bmwv we get that for any sufficiently large M the density of n such that P B Ln,v r,r + bmwv 3bMǫ > 1 2ǫ is at least 1 ǫ. The case that M is negative follows in the same manner by replacing wv with wv. As ǫ was arbitrary the lemma follows. 12

13 Lemma 4.4. Let v V. For all y R 2 let s = sv,y and t = tv,y be such that v + swv = y + tv. 29 Then for all ǫ > 0 and all sufficiently large M the density of n with P B Ln,v My,Mv > Mt ǫ > 1 ǫ 30 is at least 1 ǫ. Proof. Fix ǫ > 0. If and then B Ln,v My,My + tv > Mt ǫ 31 B Ln,v Mv + swv,mv > ǫm 32 B Ln,v My,Mv = B Ln,v My,Mv + swv + B Ln,v Mv + swv,mv = B Ln,v My,My + tv + B Ln,v Mv + swv,mv > Mt ǫ ǫm > Mt 2ǫ. The first line follows from Lemma 2.1, the second from 29, and the third from 31 and 32. By Lemma 4.2 for any sufficiently large M the density of n such that 31 is satisfied with probability at least 1 ǫ is at least 1 ǫ. If s 0 then by Lemma 4.3 for any sufficiently large M the density of n such that 32 is satisfied with probability at least 1 ǫ is at least 1 ǫ. If s = 0 then Mv + swv = Mv and 32 is satisfied for all M and n. As ǫ is arbitrary the lemma is true. Proof of Theorem 1.6. By the definition of Sidesµ for any k Sidesµ we can find v 1,...v k such that v i V for all i and the lines L vi are all distinct. The fact that all v i R and that the tangent lines are distinct implies that tv i,v j > 0 for any i j. By multiple applications of Lemma 4.4 there exists c > 0 such that for all ǫ > 0 there exists M such that the density of n with P B Ln,vi Mv j,mv i > cm i j > 1 ǫ. 33 is at least 1 ǫ. We then choose x i to be the point in Z 2 nearest to Mv i. Thus by Lemma 2.2 and 33 we have coexistence with probability at least 1 ǫ. Proof of Theorem 1.2. This follows from Lemma 2.3 and Theorem 1.6. For the following proofs we will use the following notation. For x,y R 2 we use the notation x,y = x 2 + y 2 and Ballc,r = {a R 2 : c a < r}. Let x,v,y V have distinct tangent lines. Let A = Ax,v,y R be the open arc of R from x to y that contains v. For any x,v,y V we consider the event G0,L n,v MR MA

14 Lemma 4.5. Let x,v,y V have distinct tangent lines and let ǫ > 0. For any M sufficiently large we have that the density of n such that is at least 1 ǫ. P34 is satisfied > 1 ǫ Proof. First we claim that for fixed M,n,ǫ and z R \ A that if and Mz G0,L n,v then τ0,l n,v = τ0,mz + τmz,l n,v τ0,mz τ0,mv ǫm 35 = τ0,mz + τmz,l n,v τmv,l n,v + τmv,l n,v τ0,mv ǫm + B Ln,v Mz,Mv + τmv,l n,v τ0,mv + τmv,l n,v + B Ln,v Mz,Mv ǫm τ0,l n,v + B Ln,v Mz,Mv ǫm. The first line is true because Mz G0,L n,v. The third line is true because of 35 and the definition of B Ln,v. The last line is true because of the subadditivity of τ. Thus B Ln,v Mz,Mv ǫm. Fix {y i } k i=1, y i R \ A for all i, such that for every z R \ A there exists y i with y i z < ǫ/10t 1,0. Next we note that if B Ln,v My i,mv 10ǫM and τmy i,mz ǫm then B Ln,v Mz,Mv = B Ln,v Mz,My i + B Ln,v My i,mv 10ǫM τmy i,mz > 2ǫM. Thus to bound the probability that there exists z R\A such that Mz G0,L n,v we need only to bound the probabilities of 1. τ0,mz τ0,mv ǫm for all z R, 2. B Ln,v My i,mv 10ǫM for all y i, and 3. τmy i,mz ǫm for all y i and Mz BallMy i,ǫm/t 1,0. For sufficiently large M the first and third events happen with probability 1 ǫ by Theorem 1.1. By the argument in Lemma 4.4 we have that the density of n such that the second event happens with probability at least 1 ǫ is at least 1 ǫ. 14

15 Proof of Theorem 1.4. Let v 1,... v k V have distinct tangent lines. By Lemma 4.5 we see that for i = 1,...,k/2 there exists M and infinitely many n such that the geodesics G0,L n,v2i are pairwise disjoint in M R. They all intersect at 0 so for infinitely many n the geodesics are pairwise disjoint in the complement of MR. Thus KΓ0 k. To prove Theorem 1.3 we consider the event GMv,L v,n MR BallMv,ǫM. 36 Lemma 4.6. Let ǫ > 0 and for any M sufficiently large we have that the density of n such that P36 is satisfied > 1 ǫ is at least 1 ǫ. Proof. If there exists z MR \ BallMv,ǫM and z GMv,L nv then there exists z GMv,L nv such that z Z = MR \ BallMv,ǫM. Choose {y i } k i=1 such that for any z Z there exists y i such that Suppose the following events happen: z y i < ǫ/100t 1,0. 1. B Ln,v Mv,My i < ǫmt 1,0/10 for all i, 2. τmv,mz > ǫmt 1,0/3 for all z such that z v ǫ and 3. τmy i,mz < ǫmt 1,0/10 for all i and z such that z y i < ǫm/100t 1,0. Then we claim that 36 is satisfied. To see this we note that by 1 and 3 B Ln,v Mv,Mz = B Ln,v Mv,My i + B Ln,v My i,mz for all z Z. If Mz GMv,L n,v then < ǫmt 1,0/10 + τmy i,mz < ǫmt 1,0/5. 37 B Ln,v Mv,Mz = τmv,mz. Thus by condition 2 if z Z and Mz GMv,L n,v then B Ln,v Mv,Mz = τmv,mz > ǫmt 1,0/3 which contradicts 37 and establishes the claim. Thus to prove the lemma we need to show that the density of n such that the probability of all of the events in 1, 2 and 3 occurring is greater than 1 ǫ. By the argument in Lemma 4.4 for sufficiently large M with probability at least 1 ǫ/3 the density of n such that 1 occurs is at least 1 ǫ. By Theorem 1.1 the probabilities of the last two events can be made greater that 1 ǫ/3. 15

16 For the final proof we will be dealing with multiple realizations of first passage percolation. To deal with this we will use the notation τ ω x,y B ω S x,y and Gω x,y to represent the quantities τx,y B S x,y and Gx,y in ω. Proof of Theorem 1.3. Given k by Theorem 1.2 we can choose M and x 1,...,x k MR such that with positive probability there exist disjoint geodesics G i starting at each x i. By Lemma 4.4 we have that there exists a measurable choice of geodesics G i and vertices x i such that for any i j B Gi x j,x i > There exist finite paths G i MR and an event E of positive probability that satisfy the following condition. For each ω E and i, the paths G i and G i agree in MR. Let y i Z 2 be the first vertex in G i after G i exits MR for the last time. We can find a i > 0 and restrict to a smaller event of positive probability where a i < τx i,y i < a i We can pick some large K and further restrict our event as follows. Let z,z Z 2 \MR be such that there exist x,x Z 2 MR and z z = x x = 1. We require that for any such z and z that there exists a path from z to z that lies entirely outside of MR and has passage time less than or equal to K. For K large enough the resulting event Ê will have positive probability. We now create a new event E by taking any ω Ê and altering the passage times in MR. We will do this in a way such that E has positive probability and the inequality KΓ0 k is satisfied for all ω E. First we choose paths Ĝi MR that connect 0 to y i such that Ĝi G j = for all i j. This is possible by Lemma 4.6. A configuration ω E if 1. there exists an ω Ê such that ωe = ω e for all edges e with both endpoints in MR C and a i < τ ω 0,y i < a i + 2, for every z Z 2 \ MR there exists i such that and 3. for every z Z 2 \ MR and every i G0,z MR = Ĝi, Gy i,z MR C. Note that the first and last conditions imply that for every z Z 2 \ MR and every i τ ω y i,z τ ω y i,z. 41 Also note that the first condition implies if v G i \ MR then τ ω y i,v = τ ω y i,v

17 Fix ω E and v G i \ MR. We claim that G ω 0,v = Ĝi G ω y i,v. By condition 2 we know that G ω 0,v must pass through some y l. Then we calculate τ ω 0,y i + τ ω y i,v < a i τ ω y i,v < τ ω x i,y i τ ω y i,v < τ ω x i,v The first inequality follows from 40 and 42. The second inequality follows from 39. The third is true because x i,y i and v are all on G i. Next we calculate τ ω 0,y j + τ ω y j,v > a j + τ ω y j,v > τ ω x j,y j 1 + τ ω y j,v > τ ω x j,v 1 > τ ω x i,v 1 + B Gi x j,x i > τ ω x i,v The first inequality follows from 40 and 41. The second inequality follows from 39. The third from the subadditivity of τ, the fourth from the definition of B Gi and the final from 38. Combining 43 and 44 we get that for every v G i the geodesic G ω 0,v passes through y i. As this holds true for every i we have that P KΓ0 k PE. The conditions on E can be satisfied by picking the passage times through the edges in Ĝi to be in some appropriate interval to satisfy 40 and by choosing the edges not in Ĝi to have passage times larger than 10K maxa i. Thus we see that P KΓ0 k PE > 0. Acknowledgements I would like to thank Itai Benjamini for introducing me to this problem and Oded Schramm for noticing a error in an earlier version of this paper. References [1] J. Baik, P. Deift and K. Johansson. On the distribution of the length of the longest increasing subsequence of random permutations, J. Amer. Math. Soc , [2] I. Benjamini, G. Kalai, and O. Schramm. First Passage Percolation Has Sublinear Distance Variance. Ann. Probab , no. 4,

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