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Coexistence in two type first passage percolation model: expansion of a paper of O. Garet and R. Marchand
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Coexistence in two type first passage percolation model: expansion of a paper of O. Garet and R. Marchand
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COEXISTENCE IN TWO TYPE FIRST PASSAGE PERCOLATION MODEL: EXPANSION OF A PAPER OF O. GARET AND R. MARCHAND Kai Fan A thesis submitted for the degree of Master of Arts in Applied Math University of Southern California August 2019 1 Thanks My master thesis started from May 2017. Thanks for my thesis advisor Prof. Alexander who spent at least one hour per week to guide my works polishing my math skills and broaden my knowledge in various areas. Also thanks for Prof. Lototsky and Prof. Arratia spend their time to review my works and give valuable advice. A special thanks for Prof. Baxan- dale. His precious silver made me an honest man. Thanks for my parents, they gave me nancial support and encouraged me to choose what I am interested in. 2 Contents 1 Explanations of the Notations 4 2 Reminder on the directional asymptotic speed results. 6 3 Coexistence result 12 4 Competition model and optimal paths. 19 5 Mutual unbounded growth and existence of two disjoint paths for diuse passage times. 24 6 References 39 3 Abstract: This paper is based on `Coexistence in Two Type First Passage Percola- tion Model' by Garet and Regine Marchand. The paper studies two-type of coexistence model through rst-passage of percolation on Z d or on the in- nite cluster in Bernoulli percolation. This is somewhat an annotated version of that paper. We have expanded out many of the details and added com- ments and graphical illustrations; other signicant portions have been little changed. Also, we complete some proofs incorporated a correction they pro- vide through personal communication to make the reference more readable. 1 Explanations of the Notations Structure of Z d and E d : First we dene Z d . In general, Z d means a set of points and each points has integer coordinate. In this paper, we dened Z d to be a graph with this set of points and edges. Each pair of neighbour points are connected with an edge and the Euclidean length is equal to 1. Meanwhile this set of edges is denoted byE d . A (simple) path is a sequence = (x 1 ;x 2 ;:::;x n ;x n+1 ), where x i and x i+1 are neighbor to each other with at least one edge e i that can take nite passage time to pass through, also each edge as we dened above has Euclidean distance equal 1. The number n of edges in is called the length of and is denoted byj j. The travel time: We need rst introduce Bernoulli percolation. For each edge, we dene it is either open with probability p or closed with prob- ability 1p and each edge is independent, which constructs a Bernoulli percolation onE d . Here closed edge means it will need innite passage time to pass though that edge. We used e to represent the passage time to pass through a certain edge e, which has non-negative value. And is dened to bef e g e2E d. Then we construct a sample space =f0; 1g E d , which is the union of all edges onE d . 0 means the edges are closed and 1 means the edges are open. Denote p c (d) as the critical threshold for Bernoulli percolation on the edges E d of Z d . Here, p c (d) is a threshold merely depending on certain dimension. And for dierent dimensions, it has dierent real value. Also p< 1, then we have two equivalent formulations: all e =1 for closed edges or all e <1 but can not use closed edges. On =f0; 1g E d , consider the measureP p : on =f0; 1g E d P p = (p 1 + (1p) 0 ) E d : 4 This a measure dened onE d and for each edge it has Bernoulli distribution. Therefore, the whole conguration consists Bernoulli percolation. The chemical distance: Now we consider what variables we need to use to describe the distance and passage time between two points. From the denitions we give above. There are two variables that we can use to satisfy our requirements: the environment of edges that shows either the edges are open or closed, denoted as !. D(x;y) between x and y inZ d only depends on the Bernoulli percolation structure ! and is dened as follows: D(x;y)(!)=inf j j, which is the inmum ofj j over the set of all for which edges on are open. Otherwise, we set by convention D(x;y) = +1. The clusters of an environment ! are the connected components of the graph induced onZ d by the open edges in!. Asp>p c (d), there almost surely exists one and only one innite cluster, denoted by C 1 . On S = (R + ) E d , dene probability measure S on S = (R + ) E d , and S is stationary and ergodic. Basic on the properties of S , our probability space will then be = E S . A point in will be denoted (!;), with ! corresponding to the environment, which is 0 or 1 we mentioned before. The nal probability is on = E S writing asP =P p S . For (!;)2 , and (x;y)2Z d Z d , we dene the travel time from x to y: d(x;y)(!;) = inf (d( )) = inf P e2 e e is the travel time for a single edge on geodesic line. And the inmum is taken on the set of paths whose extremities are x and y and that are open in the environment!. Of coursed(x;y) = +1 if and only ifD(x;y) = +1. Finite and semi-innite geodesic: A path from x to y which realizes the distanced(x;y) is called a nite geodesic. An innite path = (x i ) i0 is called a semi-innite geodesic if (x n ;x n+1 ;:::;x p ) is a nite geodesic for every np. Innite geodesics: innite self-avoiding nearest-neighbor path is an innite geodesic (for a given edge-weight realization) if every nite subpath of is a nite geodesic. This paper is mainly focus on nite and semi-innite geodesic. We use the properties of nite geodesic line in proving coexistence in section 5. The innite geodesics is also a research area related to existence of geodesic rays but not highly related to our paper. For the details please see Homan, C. (2005) 5 Moreover, we suppose that S satises the following integrability and dependence conditions: m = sup e2E d Z e dS ()<1; (1) also this can be understood as nite expection of e . 9> 1;9A;B > 0 such that:8E d S ( P e2 i jj) A jj (2) For instance, if S is the product measure E d , assumption (2) follows from the Marcinkiewicz{Zygmund inequality when the passage time of an edge has a moment of order strictly greater than 2. For more details, see Theorem 3.7.8 in Stout (1974). For dierent rst-passage percolation models S can generalize dierent forms. Here we give three forms we mainly discuss: (a) The case of classical i.i.d. rst-passage percolation: take p = 1, i.e., all the edges ofZ d are open, andS = E d , where is a probability measure onR + (b) The case of classical i.i.d. rst-passage percolation, but allowing the passage times to take the value1 with positive probability: takep c (d)<p< 1, a probability measure v onR + , and setS = E d . This is equivanlent to consider p=1 andS = ~ E d , where ~ is a probability measure onR + [f1g that charges1 with probability 1p. (c) The case of stationary rst-passage percolation, as considered by Boivin (1990): take p = 1 andS a stationary probability measure. 2 Reminder on the directional asymptotic speed results. In the following content, we need to get the asymptotic results concerning travel time from the origin to points that tend to innity, it is natural to conditionP p on the event that 0 is in the innite cluster: P p (:) =P p (:j02C 1 ) and P = P p S . 6 For B2B( E ), with Bf0$1g andP p (B) > 0, we will also dene the probability measure P B by 8C2B( ), P B (C) = P(C\(B S )) Pp(B) f0$1g is dened asf!2f0; 1g E d : 0$1g, which is an event that9 conguration such that 0 can be passed to innity through open edges. Picture 2.1, we give explanation of the innite cluster, here we dene 0 is in innite cluster in this picture and therefore at least one path can go to innite. In this picture, three paths goes to innity(black lines). Each edge on the black lines is open, which guarantees it is in innite cluster. Dierent from B =f0$1g, Bf0$1g is a event to represent paths go from 0 to innity with some other conditions. For example, B can be existence of 2 disjoint paths to innity rather than just one path to innity. Directional asymptotic speed results: In classical rst-passage percola- tion, for eachu2Z d nf0g, the travel timed(0;nu) goes to innity asn goes to innity. Here, as all points inZ d are not necessarily accessible from 0, we must introduce the following denitions: 7 For any set X, and u2Z d , we rst dene u on the translation operator u on X E d by the relation: 8!2X8e2E d ( u !) e =! ue where u e denotes the natural action of Z d on E d : if e = fa;bg, then ue =fa +u;b +ug. The translation operator u is used to shift the orgin from 0 to u. Denition 2.1. For each u2Z d nf0g and B2B( E ), let T B u (!) = inffn 1; nu !2Bg, The picture 2.2 gives us an explanation of notation T B u (!). As we dened before, it is multiple ofu from a innite cluster in whichB occurs to the next conguration in whichB occurs. And letBf0$1g, nu !2B means some of those start points is in the innite cluster. For example, geodesic lines start from 0;u; 2u; 3u; 4u;:::, but only 0; 3u; 7u; 12u can connect to innity with some certain conditions(some other may connect to innite but not the conditions for B may not satised). Then T B u (!) for start point 3u is 4(from 3u to 7u), which is a shift from 3u to 7u and T B 3;u (!) is 3 + 4 + 5 = 12. And also dene the associated random translation operator on = E S 8 B u (!;) = ( T B u (!) u (!); T B u (!) u ()) This translation operator gives a tool translate a innite cluster cong- uration in which B occurs to another conguration in which B also occurs. This is equivalent to moving the starting point 0 to another starting point. And this helps us to get a composed version: T B n;u (!) = P n1 k=0 T B u (( B u ) k !): T B u andT B n;u only depends on the environment! not on the passage times , whereas the operator B u acts on the whole conguration (!;). The new conguration B u (!;) is the initial conguration viewed from the rst point of the form nu to be in the innite cluster in which B is occurs. In classical rst passage percolation, we need to use a famous directional asymptotic speed result: if (d(e)) e2E d are i.i.d. non-negative integrable ran- dom variables, then for every x2Z d , there exists (x) 0 such that a.s. lim n!1 d(0;nx) n = lim n!1 Ed(0;nx) n =(x). The proof for this result is a classic application of the sub-additive ergodic theorem. There are several version with dierent hypotheses. And the one with Kingman's original assumptions can be found in Liggett, T. (1985). The result for this case has been extended in full details in a previous work of Garet and Marchand (2004) to rst-passage percolation on an innite Bernoulli cluster, and we will fully discuss this in section 5. The aim of this section is to introduce some explanations and modied proofs to satisfy our requirements. Lemma 2.2. B u is a P-preserving transformation, is ergodic for P and E P T B u = 1 Pp(B) and T B n;u n ! 1 Pp(B) , P a.s. The lemma 2.2 gives us a tool to prove coexistence result in section 4. It provides a description of the frequency of eventB whereBf0$1g, this means there are some starting points nu having congurations in which B occurs and the distant between n th and (n + 1) th points divided by n has a limit number. And for P-preserving transformation, it starts from one point in innite cluster where B occurs to shift to another point in the innite cluster where B occurs by translating by T B u (( B u ) k !)u. Here k is the times that start point shifts. For example, in picture 2.2 the point 0 shifts to 3u, 7u, 12u by take k = 1; 2; 3 respectively. 9 Proof of Lemma 2.2. Garet and Marchand (1999) gives the fully explanation of why this dynamical system is ergodic. From this result, we can use classical ergodic theorem to get E P T B u = 1 Pp(B) because B u is preserving transforma- tion. And for the limit statement, since B u is a P-preserving transformation, we can use pointwise ergodic theorem, lim n!1 P n1 k=0 T B u (( B u ) k !) n converges to the time average and equal to 1 Pp(B) . Lemma 2.3. Let B2B( E ), with Bf0$1g. And P p (B) > 0. For u2Z d nf0g, there exists a constant f B u 0 such that d(0;T B n;u (!)u)(!;) n !f B u , P B a.s. Moreover, this convergence also holds in L 1 ( P B ) and f B u E P B d(0;T B u u) < 1 The lemma 2.3 gives us a converge result. This can be understood as the passage time from 0 to n th u in which B occurs is about nf B u . u u 3u 3u 5u 5u 7u 7u 9u 9u 11u 11u 13u 13u 0 0 2u 2u 4u 4u 6u 6u 8u 8u 10u 10u 12u 12u 14u 14u Picture 5.3 A B M n Picture 2.3 The picture 2.3 gives us an explanation to understand lemma 2.2 and 2.3. In this case n = 3, soT B 3;u u = 12 based on our denitions. The passage time of dash line isd(0;T B n;u u) and gives us a converge result of T B n;u n ! 1 Pp(B) , P a.s. 10 Proof of Lemma 2.3. These results are proved with full details when B = f0$1g in Garet and Marchand (2004). Since the proof is essentially the same, we omit it. Now, for eachu2Z d nf0g, we dene the asymptotic speed in the direction u by (u) =P p f0$1gf A u For the choice A =f0$1g. We also dene (0) = 0. Corollary 2.4. Let B2B( E ), with Bf0$1g and P p (B) > 0. For u2Z d nf0g, we have: d(0;T B n;u (!)u)(!;) n ! (u) Pp(B) , P B a.s., d(0;T B n;u (!)u)(!;) T B n;u (!) !(u), P B a.s. Proof. By the denition (!;)2 = E S , then deneT B n;u (!)=inffn 1; nu !2Bg = inffn 1;! nu 2Bg From Garet and Marchand (2004), sinceBA =f0$1g, we can get: d(0;T A n;u (!)u)(!;) n !f A u for each u2Z d nf0g, we have asympototic speed in direction u such that: (u) =P p (0$1)f A u , and (0) = 0 From Lemma 2.3, we can get: d(0;T A n;u (!)u)(!;) T A n;u (!) !(u), P B a.s. Which has same limit to the previous lemma. And sinceB is a subset event of A, then ( d(0;T A n;u (u)) T A n;u ) n0 is a subsequence of ( d(0;T B n;u (u)) T B n;u ) n0 , then they converge to a same limit, we can get corollary 2.4. is a seminorm when it is applied to classical i.i.d rst passage percola- tion. What's more, in classical i.i.d. rst passage percolation with passage time law , satises the properties of norm as soon as (0) < p c (d). S can also imply norm properties. For this case, the condition forS is a prod- uct measure such that Z d , p (0) < p c (d). For the details, see Garet and Marchand (2004). 11 3 Coexistence result In the previous sections, we have already discuss the rst-passage percolation model and some basic properties. Now, we are going to introduce coexistence and its properties. Considering any pair of starting points inZ d ,x;y respec- tively, the invasion from every pair of starting points (x ory) moves through the open edges inZ d to reach as far as it can. When one of them rst reaches a point, the other source cannot invade that point. One invaded region can not encircle the other one when both sources invade innite subsets. Now we provide mathematical expressions of coexistence. For the rst- passage percolation model as we discussed in previous sections, consider some pair of sources x and y, which are dierent distinct points, inZ d . if fz2Z d ;d(x;z)<d(y;z)g andfz2Z d ;d(x;z)>d(y;z)g are both innite sets, then we call it coexistence. The rst set shows that the passage time fromx toz is less than passage time fromy toz, which means z satisfying this inequality is invaded by source x. And likewise the second set showsz is invaded by the other sourcey. Also we deneCoex(x;y) is an event that the coexistence occurs for sources x andy. The main goal of this paper is to nd some conditions forP(Coex(x,y))> 0. We also give a special cases that coexistence never happen: Suppose d = 2, S = E d , and p (0) > p c (2) = 1 2 . There almost surely exists an innite cluster of open edges with passage time zero. And ford = 2, x;y2 Z 2 , the supercritical innite cluster almost surely contains a circuit that surrounds x and y and disconnects them from innity. This means the passage time between any two points on the circuit is 0. Also, supercritical innite cluster almost surely contains a circuit that surrounds x and y and disconnects them from innity as d = 2 (Harris(1960), Grimmett (1999)). If x and y reach the circuit at the same time, since the passage time between any two points on the circuit is 0;x and y will reach the points outsides the circuit simultaneously. Also if x reaches the circuit rst, then it will reach any point outside the circuit before y. 12 Circuit x y In this picture, the passage time between any two points on the circuit is 0. If x reach the circuit rst then it will reach any point outside the circuit before y. The following Theorem 3.1 gives us a bound condition for coexistence. Theorem 3.1. Let d 2, p > p c (d), S a stationary ergodic probability measure on (R + ) E d satisfying (1) and (2), and be the related seminorm describing the directional asymptotic speeds. Let B2B( E ), with Bf0$1g and P p (B) > 0, and y2 Z d . We have ifEd(0;T B 1;y y)< 2(y) Pp(B) , then P B (Coex(0;T B 1;y y))> 0. Moreover, if x2 Z d is such that (x) > 0, then y = rx satises the previous condition provided that r is large enough. A special case is that when p = 1, which corresponds to classical rst- passage percolation, we can take B = E , and then T B 1;y y is simply equal to y. Before we move on to the prove for Theorem 3.1, we need to introduce a fundamental argument (H aggstr om and Pemantle (1998)) to illustrate The- orem 3.1. This is called symmetry argument. We use i.i.d case S = E d withp = 1 to show symmetry argument, but stationary ergodic cases will be more general in real proof. 13 M n in this picture is some sequence of points go to innity dened inZ d . Assume A is 8 steps across and 3 steps up. B is 4 steps across and 3 steps up, which is closer to M n than A. The dash line is another path from B to M n that takes longer passage time toM n thanA. The picture assumes when M n go to innity, M n are innitely often closer from B than from A with a probability bounded away from 1/2. The mathematical expression is written as: lim n!1 supP(d(B;M n )<d(A;M n ))> 1 2 To illustrate this, we separately draw three lines. Assume those two solid lines represent the passage time fromB toM n is shorter thanA toM n . And also assume the dash line represents the passage time fromB toM n could be longer than A to M n even if M n is closer to B than A. The main idea here is the dash line occurs with smaller probability than the probability of solid lines occur for some sequence of M n . And we need to prove the limit term above is hold for innitely many n. This is a fundamental idea to construct S 0 and S 1 in step 1 of Theorem 3.1. Likewise, for a single start point, we have another situation like the fol- 14 lowing picture: Similar to M n , we assume J n and I n are sequence of points in Z d and d(0;J n ), d(0;I n ) are passage time to J n (respectively I n ). And assume I n is 8 steps across and 3 steps up from 0, J n is 3 steps across and 3 steps up from 0. From our assumption of M n , J n , and I n , we can deduce the eventfd(0;I n ) < d(0;J n )g happens with the same probability as the event fd(B;M n )<d(A;M n )g. Dene sequence events A n =fd(0;J n )>d(0;I n )g, therefore for some > 1=2, we need to showP(A n ) holds for innitely manyn. And also for symmetric condition, we dene eventsB n =fd(0;I n )> d(0;J n )g, if we can show for some > 1=2, P(A n ) holds for innitely many n as well, then we can get Theorem 3.1. However, the dierence of passage time between those passage time is not clear enough. We need a more quantitative information to know how close for the start points to the end points. Proof of Theorem 3.1. Step 1: In the previous content, we only consider the invasion from 0 tony. And it can actually invadeny as well. Hereny is a symmetric point of ny. 15 Picture 5.3 A B M n Picture 2.3 0 Picture 3.1 In picture 3.1, let 0 and T B 1;y y are two points that have paths connected to T B n;y y and T B n;y y. B occurs in those 4 points when the origin is shifted there. This picture gives an illustration of dierence of passage time from T B 1;y y to T B n;y y and 0 to T B n;y y. And the dierence of passage time from T B n;y y to 0 and T B n;y y to T B 1;y y. To prove Theorem 3.1, we rst choosing y2Z d nf0g such that: E P B d(0;T B 1;y y)< 2(y) P p (B) (3) And here we dene: S 0 = lim sup jjZjj 1 !+1 fd(0;z)<d(T B 1;y y;z)< +1g, S 1 = lim sup jjZjj 1 !+1 f+1>d(0;z)>d(T B 1;y y;z)g S 0 represents pointz is reached earlier by start point 0 andS 1 is reached by T B 1;y y through shorter time. And when both S 0 and S 1 happen, the coex- istence comes into being, which is Coex(0;T B 1;y y) =fS 0 \S 1 g. Intuitively, one expects that the dierence betweend(0;z) andd(T B 1;y y;z) will be the greatest if z2Ry, and we will eectively consider such z. From picture 3.1, we also expect that d(T B 1;y y;z)d(0;z) is positive if z2 R y, and d(T B 1;y y;z)d(0;z) is negative if z2 R + y. For the convenience of the reader, we also denote, forn2Z + andx2Z d , ~ T n;x =T B n;x x. Dene ~ T 0;x = 0, and for x 0, X n =d(0; ~ T n;y )d( ~ T 1;y ; ~ T n;y ) X 0 n =d( ~ T 1;y ; ~ T n;y )d(0; ~ T n;y ) 16 By the triangle inequality, one hasjX n jd(0; ~ T 1;y ) andjX 0 n jd(0; ~ T 1;y ). Note that for !62 S 1 , X n (!) 0 if n is large enough. And for !62 S 0 , X 0 n (!) 0 if n is large enough. It follows that for !62S 0 T S 1 , X n (!) +X 0 n1 (!)d(0; ~ T 1;y )(!). For large n, we can dene: Q n = P n k=1 (X k +X 0 k1 ),Z n = Qn n andZ = lim supZ n ;n! +1. This step means we expectX n (!) andX 0 n1 (!) both to be positive, and if there is no coexistence, then for !62S 0 T S 1 , the sum of those two will be bounded by d(0; ~ T 1;y )(!). And we will use Fatous's lemma to prove this is not consistent. Step 2: by constructZ n , we can use Fatous's lemma to get lower bound for expectation ofZ. Using transfer operator, we can change the start points to what we need. The previous remark implies that: 8!62S 0 \ S 1 ;Z(!)d(0; ~ T 1;y )(!) (4) By Lemma 2.3, d(0; ~ T 1;y ) is integrable underP B . SincejZ n jd(0; ~ T 1;y ), it follows(for instance, by Fatou's lemma)that E P B Z =E P B lim supZ n lim supE P B Z n , as n! +1 Sinced( ~ T 1;y ; ~ T n;y ) =d(0; ~ T n1;y ) B y , it follows from the invariance of P B under B y that E P B X n =E P B d(0; ~ T n;y )E P B d(0; ~ T n1;y ) Then it follows that E P B (X 1 +X 2 +::: +X n ) = E P B d(0; ~ T n;y ). Similarly, as d( ~ T 1;y ; ~ T n;y ) = d(0; ~ T n+1;y ) B y , E P B X 0 n =E P B (d( ~ T 1;y ; ~ T n;y )d(0; ~ T n;y )) =E P B (d(0; ~ T n+1;y )E P B d(0; ~ T n;y )) 17 And E P B (X 0 0 +X 0 1 +::: +X 0 n1 ) =E P B d(0; ~ T n;y ) =E P B d(0; ~ T n;y ), using for the last equality the fact that P B is invariant under ( B y ) n and the fact that a distance is symmetric. Then, E P B Z n = 2E P B d(0; ~ Tn;y ) n . Since, via Corollary 2.4, E P B d(0; ~ Tn;y ) n con- verges to (y) Pp(B) , it follows that E P B Z 2(y) P p (B) (5) This is deduce from: E P B Z =E P B lim supZ n lim supE P B Z n limE P B Z n = lim 2E P B d(0; ~ T n;y ) n = 2(y) P p (B) Putting together (3), (4) and (5), we see that P B (S 0 T S 1 ) = 0, or equiv- alently, P B ((S 0 T S 1 ) c ) = 1. This would yield to a contradition. This con- cludes the proof of the rst assertion. If x2Z d and (x) > 0, we can use Corollary 2.4. E P B d(0;T B n;y (!)x)(!;) n ! (x) Pp(B) , P B a.s., then when we choosey =nx asn is large enough, (y) Pp(B) < 2(y) Pp(B) , and we get the desired result. Theorem 3.2. Under the same assumptions as in Theorem 3.1, suppose, moreover, that is not identically null. Then, we have the following: (a) Forx2Z d with(x)6= 0, there is an innite set of values forn2Z + such thatP(Coex(0;nx))> 0 (b)P(9x;y2Z d ;Coex(x;y)) = 1. The position of species will in uence the result of coexistence. Theorem 3.2 gives us the result to come back deterministic sources. It can be under- stood by thinking the larger the distance between the two sources is, the higher the probability of coexistence will be. This a beginning of coexistence by using tools in section 2. 18 Proof Theorem 3.2. Let x2 Z d be such that (x) > 0 and N 2 Z + . Let A =f0$1g. By Lemma 2.3 and Theorem 2.4, Ed(0;T A 1;rx rx) n tends to (x) Pp(A) , so for all large integers rN, we have Ed(0;T A 1;rx rx) n < 2(x) Pp(A) . By Theorem 3.1, one has P(S 0 \S 1 )> 0 The second point is a consequence of the ergodicity assumption. 4 Competition model and optimal paths. In order to ensure uniqueness and/or existence of optimal paths, we rst give assumption. And this assumption is also what we used in the previous sections. ASSUMPTIONS. We consider rst-passage percolation onZ d , withd 2. The open edges are given by a Bernoulli percolation on the edges E d of Z d with parameter p2 (p c (d); 1] on E =f0; 1g E d P p = (p 1 + (1p) 0 ) E d . For each edge, it is either open with probability p or closed with prob- ability 1p and each edge is independent, which constructs a Bernoulli percolation onE d . The passage times of the edges are given by a probability measureS on s = (R + ) E d such thatS is stationary and ergodic. Finally, we consider the product measureP =P p S v on E S . This meansP p andS v are independent from each other. We also need two distinct initial source s 1 , s 2 inZ d . Then we can construct paths starting from those two points to satisfy our requirements. Based on this assumption we can dene the following two-type rst- passage percolation model. Denition 4.1. Under the previous assumptions, we set the following: A 1 (s 1 ;s 2 ) =fx2Z d ;d(s 1 ;x)<d(s 2 ;x)g, A 2 (s 1 ;s 2 ) =fx2Z d ;d(s 2 ;x)<d(s 1 ;x)g. Here we say A i (s 1 ;s 2 ) is the set of sites inZ d that are nally infected by type i . The time of infection of x2Z d is t(x) = minfd(s i ;x); 1 i 2g. We say that x is nally infected if smallest passage time taken by the paths either from s 1 or s 2 to x is nite. If this time is innite, there are no paths on which all edges are open from s i to x, where 1i 2. 19 Based on this denition, we can get three situations: d(s 1 ;x)<d(s 2 ;x), d(s 1 ;x)>d(s 2 ;x)andd(s 1 ;x) =d(s 2 ;x). From the conditions of coexistence we gave in section 3, if one species has already invaded a point, another specie stop invading that point. But when d(s 1 ;x) =d(s 2 ;x), s 1 and s 2 reach a point x simultaneously, we can not dene an infection type. Also forA 1 (s 1 ;s 2 ) andA 2 (s 1 ;s 2 ) both innite, we say the two infections mutually grow unboundedly. In this case, either species could not surround the other one, which leads to Coex(s 1 ;s 2 ) Lemma 4.2. If x2Z d is such that d(s 1 ;x) is reached on at least one nite path and such that d(s 1 ;x) < d(s 2 ;x), then for every y in an optimal path realizing d(s 1 ;x), we have d(s 1 ;y)<d(s 2 ;y). This Lemma is easy to understand directly using triangle inequality. And all points in an optimal path realizing d(s 1 ;x) is in species s 1 . Proof of Lemma 4.2. Let (s 1 ;x) be an optimal path froms 1 tox. And also lety be a point inZ d such thaty2 (s 1 ;x) andd(s 2 ;y)d(s 1 ;y). Then by the triangle inequality, d(s 2 ;x)d(s 2 ;y) +d(y;x)d(s 1 ;y) +d(y;x) Also since y is on (s 1 ;x), then we have d(s 2 ;x)d(s 1 ;x). However, we have already assumed that d(s 1 ;x)<d(s 2 ;x). This is a contraction. Assumption for uniqueness of optimal paths. If is a nite subset ofE d , denote byF c the -algebra generated by (! e ; e );e62 . We suppose 8 nite subset ofE d , for all random variable A which areF Cmeasurable,S ( e =AjF c) = 0: (6) Assumption (6) is also called as non-atom assumption. It says there are no atoms in the condition distribution, even if is equal to all bonds except e. This assumption imples the uniqueness of geodesic path, which is widely used in section 5. Lemma 4.3. Under the additional assumption (6), we have the following: (i) If and 0 are paths that dier in at least one edge, then P(d( ) = P e2 e =d( 0 ) = P e2 0 e <1) = 0: 20 Thus, the optimal paths, when they exist, are unique. (ii) For every 2R, if x;x 0 ;y;y 0 are distinct points inZ d , P(9finite path such that d(x;y) =d( ); 9finite path 0 such that d(x 0 ;y 0 ) =d( 0 ) and d(x;y)d(x 0 ;y 0 ) =) = 0 s 1 x picture 4.1 The picture 4.1 gives the explanation for lemma 4.3. We assume all edges on the path from s 1 tox are xed excepte. For example, suppose the passage time of (in the picture, this is the upper path froms 1 tox) is 14 except e and the passage time of 0 is 17. Therefore, P(d( ) =d( 0 )<1jF c) = 0 is equal toS v ( e = 3jF c) = 0. Proof of Lemma 4.3. (i) : suppose and 0 are two paths that dier at least one edge. Lete be edge either on or on 0 but not on the parts of path that have same edges. To make clarications, d( ) and d( 0 ) are passage times for and 0 . Also let =feg, we know the edge e is either on or on 0 . And since those two events are equivalent, we only choosee on 0 . Therefore, d( 0 ne) are passage times for path 0 except edge e. Then we can get the eventfd( ) =d( 0 )g is equal to the eventfd( 0 ne) + e =d( )g. Therefore, P(d( ) =d( 0 )) =P(d( ) =d( 0 ne) + e ) 21 And this is equal toP(d( ) =d( 0 ne) + e ; e =AjF c), where A is random variable on c satisfying A =d( )d( 0 ne) and [f 0 neg c . This is also equal toS ( e =AjF c). Therefore, by choosing =feg we can get: S ( e =AjF c) = 0 ,P(d( ) =d( 0 ne) + e ; e =AjF c) = 0 ,P(d( ) =d( 0 )jF c) = 0 And sinceP(d( ) =d( 0 )jF c) = 0 a.s.,P(d( ) =d( 0 )) = 0. Then we get the desired result. (ii): Similar to (i),82R d . Choosinge on 0 , andA =d( 0 )d( 0 ne)+ then we have: S ( e =AjF c) = 0 ,P(d( ) =d( 0 ne) + e ; e =AjF c) = 0 ,P(d( ) =d( 0 ne) + e jF c) = 0 ,P(d( ) =d( 0 ) +jF c) = 0 ,P(d( ) =d( 0 ) +) = 0 From assumption (6), we have already knownS ( e =AjF c) = 0. Therefore, P(9finite path such that d(x;y) =d( );9finite path 0 such that d(x 0 ;y 0 ) =d( 0 ) and d(x;y)d(x 0 ;y 0 ) =) = 0 Consider the following extra assumption onP limd(0;x) = +1; asjjxjj 1 ! +1;P a:s: (7) (7) tells us whenx is far enough, the passage time from 0 tox can also be large enough. From this assumption we can deduce that for each x;y2Z d with d(x;y)< +1, there always exists at least one path from x to y with d(x;y) = d( ). Assumption (7) is ensured by the shape theorem, which we will fuly explain the relationship between those two in section 5. Here we only brie y introduce the shape theorem. Suppose B(t) is the union of set containing points that the paths reaching those points take passage time less 22 than t. This means when x is inside B(t), d(0;x) t and d(0;x) > t for x outside B(t). Then B(t) t has an upper bound and a lower bound. The main idea is use this basic shape theorem to deduce assumption (7). We will give proof in detail in section 5. 9A;B > 038E d ;S (2 S ; X e2 i Bjj) A jj : (8) jj in Assumption (8) is the size of , the number of the points inside . Suppose, for instance, that the functional associated toP is a norm and that one of the three following conditions fullled, then we can get the shape theorem based on dierent backgrounds: Assumption (8) holds for some >d 2 + 2d 1. p = 1 and the passage times of bonds have a moment of order >d. p = 1,S is a product measure and the passage times of bonds have a second moment. Kesten (1986) rst assumed the second moment assumption and based on this assumption Lemma 3.5 in his paper proved the shape theorem for i.i.d. case. Assumption (8) with > d is the one used by Boivin (1990) for the shape theorem in stationary rst passage percolation. Both Kesten (1986) and Boivin (1990) considered only for all edges open. Garet and Marchand (2004) extended to the case that the edges can be closed, considering only p> p c (Z). The Assumption (8) with >d 2 + 2d 1 is the one Garet and Marchand (2004), Lemma 3.7 used. Also, ifS is the product measure E d , assumption Assumption (8) fol- lows from the Marcinkiewicz-Zygmund(J. Marcinkiewicz and A. Zygmund(1937)) inequality as soon as the passage time of an edge has a moment of order strictly greater than 2. Here we give a brief proof for Assumption (8): We rst assumeA,B are positive numbers and2 S . Then we can get: S ( P e2 e Bjj) E(( P e2 e) 2 ) (Bjj) 2 (from Markov inequality with moment order of 2). Then use Marcinkiewicz-Zygmund inequality such that: 23 E(( P e2 e ) 2 ) (Bjj) 2 C()jj 1 jjEj e j 2 (Bjj) 2 C()Ej e j 2 B 2 jj Here C() is a constant depending on . Through dening a constant A = C()Ejej 2 B 2 we can get Assumption (8). 5 Mutual unbounded growth and existence of two disjoint paths for diuse passage times. In this section, we will prove the possibility of coexistence in two-type rst- passage percolation for diuse passage times. As for semi-innite geodesics we discuss before in the rst section, we will talk about the existence of two semi-innite geodesics in the corresponding one-type rst-passage percola- tion. The relation between coexistence and semi-innite geodesics is that some branches in each invaded region are semi-innite geodesics. Suppose two sources starting from s 1 and s 2 achieving coexistence, and letfx i g is a set of points invaded bys 1 (respectivelyfy i g is a set of points invaded bys 2 ). Then there is at least one path in each invaded area is innite geodesic path. Lemma 5.1 is the most important part of section 5. It gives us a result that the position of of two initial sources are not related to the existence of coexistence. Even if we change the position of those sources, it still have same positive probability to grow mutual unbounded. The method we use to proof Lemma 5.1 is based on separating the con- guration into two parts. The rst part is constructing a nite box and two initial sources are inside the box. The second part is conguration outside the nite box. Through choosing the position of the initial sources and keeping the conguration outside the box unchanged, we get the desired result. Lemma 5.1. Consider Z d , with d 2 and p2 (p c (d); 1]. Choose a station- ary ergodic probability measure S v on S = (R + ) E satisfying the nonatomic Assumption (6) and Assumption (9): 8finite subset of E d ;8e2 ;80a<b;S ( e 2 [a;b]jF c)> 0 a:s: (9) 24 Then if s 1 ;s 2 and s 0 1 ;s 0 2 are two pairs of distinct points inZ d , P(Coex(s 1 ;s 2 ))> 0()P(Coex(s 0 1 ;s 0 2 ))> 0. Assumption (9) is also called nite energy property. It assumes if we keep and the conditions outside xed, the passage time of e can be any positive value. This is also a vital tool in proving step 3 of Lemma 5.1. The next Theorem 5.2 comes from the combination of Theorem 3.2 and Lemma 5.1. Theorem 5.2. ConsiderZ d , withd 2 andp2 (p c (d); 1]. Choose a station- ary ergodic probability measure S on S = (R + E d ) satisfying the integrabil- ity Assumptions (1), (2). Suppose the semi-norm based on the probability measure S is not identically null. Also, suppose S satises the nonatomic Assumption (6) and nite energy property (9). Then we have 8x6=y2Z d ,P(Coex(x;y))> 0. Theorem 5.3. (Garet and Marchand (2004), Theorem 3.2) Assume that (H ) holds for some > 1. For each u2Z d nf0g, we set (u) =P p (0$1)f u , where f u is given by Lemma 2.3 and (0) = 0. Then lim n!1 d(0;Tn;uu) Tn;uu =(u), P a:s: Garet and Marchand (2004) considers 0 < p 1. Here we also give Kesten, H. (1986), which is the version based on p = 1: If (d(e)) e2E d are i.i.d. non-negative integrable random variables with F (0) < p c (d), then for every x 2 Z d , there exists (x) > 0 ((0) = 0) such that a.s. lim n!1 d(0;nx) n = lim n!1 Ed(0;nx) n =(x). In the Theorem 5.2, (1) and (2) are integrability and dependence condi- tions ofS . We also need to have the semi-norm based on the probability measure S , which is to describe the directional asymptotic speeds in the section 2, is not identically null. If is identically null, and d(0;nx) is the passage time from 0 to nx, then we have: lim n!1 d(0;nx) n = lim n!1 Ed(0;nx) n = 0 25 When is identically null, there is positive probability of 0 passage time. Kesten, H. (1986) We only consider is not identically null in order to deduce Property (7) from the shape theorem. And this is after Theorem 5.4. From the denition of , we can deduce the following properties: For x;y2Q d , c2Q: 1. (x +y)(x) +(y) 2. (cx) =jcj(x) 3. is invariant under symmetries ofZ d that x the origin. 4. is uniformly continuous and Lipschitz on bounded subsets of Q d , so it has a unique continuous extension toR d . The rst subadditivity helps us to show the existence of the time constant. This is fromEd(0;x +y)Ed(0;x) +Ed(x;x +y). And we use translation invariance to transferEd(x;x +y) toEd(0;y), then we get(x +y)(x) + (y). The time constant describes the rst order approximation of the random ball B(t) as t goes to innity. Here we introduce B(t) andB : For each t 0, B(t) =fy2R d :d(0;y)tg Since the passage time d(0;y) is only dened on Z d for points y. Here we extendy toR d : y2y 0 + [0; 1) d , wherey 0 is the unique vertex inZ d , then we choose d(0;y 0 ) to be the passage time from 0 to y. B(t) grows linearly in t and, when normalized, it converges to a deterministic subsetB ofR d called the limit shape. The setB s not universal and depends on the distribution of the passage times. Garet and Marchand (2004) gives denition ofB (x;r) and the Hausdor distanceD(A;B). Before we introduce the shape theorem, we need to give those defnition rst. For x2R d and r 0, we dene B (x;r) =fy2R d ;(xy)r)g The Hausdor distance between two non empty compact subsets A and B ofR d is dened by 26 D(A;B) = inffr 0;AB +B (0;r) and BA +B (0;r)g Theorem 5.4. (Garet and Marchand (2004), Theorem 5.3) Assume Assumption (8) holds for some >d 2 + 2d 1 and that is a norm. Then, lim t!+1 D( Bt t ;B (0; 1)) = 0, P a:s: This theorem is for 0 < p 1 and here we also give another theorem based on p = 1 (Kesten, H. (1993)): Suppose (x) is not identically null and letM be the set of Borel prob- ability measures on [0; +1). Also let t i , i = 1;:::2d, be independent copies of e such that E minft d 1 ;:::;t d 2d g <1 and F (0) < p c (d), where p c (d) is the threshold for bond percolation in Z d . Then for each 2M, there exists a deterministic, convex, compact setB inR d such that for each > 0, P((1)B B(t) t (1 +)B )=1 as t!1. This means B(t) t has an upper bound and a lower bound. The Assumption (7) is also ensured by the shape theorem. Let us recall Assumption (7): limd(0;x) = +1, asjjxjj 1 ! +1,P a.s. This tells us when point x goes to innite, the passage time for infection to reach x will go to innite as well. Here we give a brief proof to show how to deduce Assumption (7) from Theorem 5.4. From shape theorem, we know lim t!1 B(t) t =B and B(t) (1 +)tB for all large t. Denefx n g n2Z+ be a xed sequence of points on Z d such that as p > q,jjx p jj >jjx q jj. Let t > 0. Also, letfg m g be a set of all points (1 +)tB (upper bound), thenB(t) (1 +)tB can be expressed as: when jjx n jj>maxfjjg m jjg,d(0;x n )>t. And we can always nd a pointx n outside (1 +)tB such thatjjx n jj>maxfjjg m jjg. Then according to the denition of innite limit, takejjx N jj maxfjjg m jjg. Then ifjjx n jj>jjx N jj, we getd(0;x n )>d(0;x N )t. Therefore, lim jjxnjj!1 d(0;x n ) = 1. Theorem 5.2 is based on Assumption (6) and (7), which separately tells us no ties occur and the existence of geodesic lines. Therefore, d(x;y) is achieved for each pair of points (x;y) inZ d . 27 First recall the denition of A 1 (s 1 ;s 2 ) and A 2 (s 1 ;s 2 ) in section 4, which denes the site `closer' to s 1 (respectively s 2 ). From Lemma 4.2, we know A 1 (s 1 ;s 2 ) and A 2 (s 1 ;s 2 ) are connected. Also Assumption (6) and (7) guar- antee existence of optimal path and no ties occur. We dene t(x) be the passage time from one of the sources to x such that this source rst reaches x. For example, if s 1 reaches x rst, then by existence of geodesic lines t(x) is equal to d(s 1 ;x). The path (x) that achieved d(s 1 ;x) is unique. When coexistence occurs, we have two infected region separately from s 1 and s 2 and each infected region contains at least one innite geodesic line. Then we have the infected region is actually the union of A 1 (s 1 ;s 2 ) (the infected region bys 1 ) andA 2 (s 1 ;s 2 ) (the infected region bys 2 ). Therefore, under the Assumption (6) and Property (7), for each infected point (the point reached by either one of sources), we can always ensure the point can be only infected by only one source rather than simultaneously infected by two sources. When we consider the whole infected region, each point in this region is infected by either s 1 ors 2 . Thus the whole infected region is spanned by nite paths (The union of ( (x)) x2Z d) starting from s 1 or s 2 . Still under the same as- sumptions (existence and no ties occur), we can also extend to the denition of infected region starting with sources more than two. Lemma 5.5 and Theorem 5.6 say that the mutual unbounded growth in the two-type rst passage percolation model and the existence of two distinct semi-innite geodesics for xed starting sources in the corresponding rst- passage percolation model are equivalent. In Lemma 5.5 and Theorem 5.6, we dened `infection tree' to describe the union of ( (x)) x2Z d ;t(x)<1 . Here we dene `rooted in 0' means the starting source is from 0 rather than s 1 or s 2 . Since the union of ( (x)) are rooted in both s 1 and s 2 , there are two `infection trees' starting from s 1 and s 2 in this region. Lemma 5.5. Under the same assumption as in Lemma 5.1. plus the extra assumption (7),9s 1 ;s 2 2Z d such thatP(Coex(s 1 ;s 2 ))> 0()P(there exist two edge-disjoint semi-innite geodesics in the infection tree rooted in 0)> 0 Combining these results with Theorem 3.2, we obtain the following: Theorem 5.6. Under the same assumptions as in Theorem 5.2, plus the extra assumption (7), P(there exist two edge-disjoint semi-innite geodesics in the infection tree rooted in 0)> 0. 28 The following two examples give us two specic cases. Those cases give us two dierent laws included in our fundamental assumption in section 5. Those assumptions include non-atomic Assumption (6) and Assumption (7) (existence of geodesic lines). Under the conditions in those cases we can directly use the results of Lemma 5.5 and Theorem 5.6. Example 1. Choose rst-passage percolation on Z d , d 2, with a family (t(e)) e2E d of i.i.d.non-negative random variables and we assume assumption (6) and (7), which guarantee for each pair of points inZ d , there exists a path between them achieving geodesic and no ties occur. Also we use exponential law, which is used in Richardson's model as well. Then: (a) When the competition model starts with two sources, mutual un- bounded growth occurs with positive probability for every pair of distinct sources inZ d . (b) When we consider only one source for rst-passage percolation model, the event that the innite geodesic lines inZ d has two edge-disjoint innite branches occurs with positive probability. For dimension 2, those results have already been proved by H aggstr om and Pemantle (1998). And their research extend those results to higher dimensions and applying to more general distributions for passage times. Example 2. For rst-passage percolation on Z d , d 2, with a family (t(e)) e2E d of i.i.d. nonnegative random variables that has no ties between nite passage time, which means ties can only occur when some edges are closed(0<p< 1), and has 0 in its support, for instance, tpU [0;1] + (1 p) 1 , with p c (d)<p< 1. Then: (a) When considering the two-type competition model, coexistence occurs with positive probability for every pair of distinct sources inZ d , which is from Theorem 5.2. (b) When considering the rst-passage percolation model with only one source, the event that the innite geodesic lines of the innite open cluster has two edge-disjoint innite branches occurs with positive probability, which is from Theorem 5.6 REMARK. Here we also provide some possible extension. 1. Based on the results of Deijfen and H aggstr om (2003), fertile nite sets could be considered as initial sources rather than points. We rst need to give denition of fertile nite sets : dene two nite nonempty disjoint 29 setsS 1 andS 2 inZ d and if there exist two innite paths 1 and 2 such that 1 (resp. 2 ) starts from points in S 1 (resp. in S 2 ) and 1 , 2 have no same points, then (S 1 ;S 2 ) are called fertile nite sets. Since this denition only change argument around initial sources, we can still use Assumption (10). Let's say S 1 , S 2 and S 0 1 , S 0 2 are two pairs of fertile nite sets inZ d , then: P(Coex(S 1 ;S 2 ))> 0()P(Coex(S 0 1 ;S 0 2 ))> 0. 2. From above we only discuss rst-passage percolation on two sources. Considering N infections from N initial sources s 1 ;s 2 ;:::;s N , each of them is trying to invade the points on Z d . s 1 ;s 2 ;:::;s N are dened as the sources onZ d . Also, the coexistence eventCoex(s 1 ;s 2 ;:::;s N ) is dened as the event that there exists an innite set that is the union of N innite infected re- gion starting from s 1 ;s 2 ;:::;s N . Theorem 5.1 may fail in some special cases when applies to multitype Coexistence. For example, when one source is surrounded by other sources, the one surrounded can not go anywhere else outside the surrounding sources. Theorem 5.2 can not be used in multitype rst-passage percolation case as well. This is because Theorem 5.2 is based on Theorem 3.1. And the proof of Theorem 3.1 is valid only if the sources and infected points lie in a line. We cannot restrict the points in a line when the number of sources are more than two. Therefore, we can not directly use Theorem 5.2 to prove coexistence of N infections. In the picture 5.1,s 1 ,s 2 ,s 3 ,s 4 ands 5 are ve points ofZ d . s 1 is surrounded by the rest of four points such that any path starting from s 1 to innity must pass through at least one of those four points. Therefore, in this case multitype Coexistence could not occurs. Also since Lemma 5.1 and Lemma 5.5 are quite similar, we will give fully proof of Lemma 5.1 and only sketch proof for Lemma 5.5. 30 Proof of Lemma 5.1. Since there no dierence by rstly choosings 1 ,s 2 ors 0 1 , s 0 2 . We can just prove for one side and the proof of another side is trivial. Here we give the proof from left side to right side, which is P(Coex(s 0 1 ;s 0 2 )) > 0. Chooses 1 ;s 2 ands 0 1 ;s 0 2 two pairs of distinct points inZ d and denote by an hypercubic box inZ d such that s 1 ;s 2 and s 0 1 ;s 0 2 are inside . We also dene @ =fx2 ;9y62 ;jjxyjj 1 = 1g. By enlarging if necessary, we can assume that s 1 ;s 2 ;s 0 1 ;s 0 2 are at a distance at least 3 from @. For an edge e2E d , we say that e2 if and only if its two extremities are in and at least one is not in @. For a point (!;) in = E S =f0; 1g E d (R + ) E d , dene: (! ; ) =f(! e ; e );e2 g and (! c; c) =f(! e ; e );e2E d n g. For two points x;y that are in c [@, we dene d c(x;y)(!) as the inmum among all the paths from x to y whose edges are not in , of P e2 e . Remember that the box has been chosen large enough to contains 1 ;s 2 . Fors2 ;x2 c andr2@, let us denote by r (s;x) the set of paths from s to x such that r is the last points of the path which is in . Since @ is nite, there exists at least one r2@ such that d(s;x) = inf 2r (s;x) d( ). 31 Picture 5.1 Picture 5.2 S S 1 ' S 1 S 2 S 2 r 2 Picture 5.3 x In the Picture 5.2, rst we assumed Assumption (6) and Assumption (9). We dene s is inZ d and is an hypercubic box in Z d such that s is inside .fr i g is a set of points on the boundary of such that there at least one bound connected those points are open. Thereforefr i g are the last points for those paths from s to exist . Suppose r 1 ;:::r 4 be those last points. We also dene the set of paths from s through r 1 to x (respectively through r 2 , r 3 , r 4 ) to be I 1 (respectively I 2 , I 3 , I 4 ). And suppose at least one path in I 2 achieves d(s;x), Then by assumption (6), there is an unique path achieving d(s;x). Then we just let R s (x) be r 2 . But if there are no paths achieving d(s;x), and suppose both I 2 and I 3 have the same smallest inmum but do not achieve d(s;x). Then we use lexicographic order to choose R s (x)=r 2 as r 2 is before r 3 in lexicographic order. Note that d(s;x) =d(s;R s (x)) +d(R s (x);x) =d(s;R s (x)) +d c(R s (x);x). As @ is nite, there exists two distinct points r 1 ;r 2 2@ such that P(A 1 (s 1 ;s 2 )\fx2 c : R s1 (x) =r 1 g is infinite A 2 (s 1 ;s 2 )\fx2 c : R s2 (x) =r 2 g is infinite)> 0: (10) Let r 1 ;r 2 be two such points. Now we introduce the following events: I 1 =There is an innite set of vertices V 1 in c such that8 y2 V 1 , d c(y;r 1 ) +d(r 1 ;s 1 )<d c(y;r 2 ) +d(r 2 ;s 2 ) I 2 =There is an innite set of vertices V 2 in c such that8y2 V 2 , d c(y;r 2 ) +d(r 2 ;s 2 )<d c(y;r 1 ) +d(r 1 ;s 1 ) 32 C=There are two innite sets of vertices V 1 and V 2 in c such that sup y2V 1(d c(y;r 1 )d c(y;r 2 ))< inf y2V 2(d c(y;r 1 )d c(y;r 2 )) The picture 5.3 will also brie y explain the main idea of proof. The con- gurations outside are xed and the congurations inside are changeable, which shows in the Picture 5.3, the dash lines connected start from s 1 and s 2 are changeable tos 0 1 ands 0 2 , and this guaranteesI 1 ,I 2 , andC are still oc- curring. After the changes inside , r 1 and r 2 are still R s 1 (y 1 i ) and R s 2 (y 2 j ). This means the paths from s 0 1 (respectively s 0 2 ) to the innite sets (y 1 i ) i1 (respectively(y 2 j ) j1 ) are geodesic or the inmum over those paths through r 1 (respectively r 2 ) is equal to overall inmum d(s 1 ;y 1 i ). The picture 5.3 gives the explanation of step 3 in proving changing original source. That is with the xed congurations outside , the step 3 proves we can change the conguration inside and start from dierent original sources. 33 S 1 S 1 ' S 2 S 2 ' Picture 5.3 In Picture 5.3, we assume Assumption (6) and (9), but not Assumption (7). Dene is a nite box onZ d s 1 ;s 2 are start points inside on Z d . We suppose there are innite sets V 1 = (y 1 i ) i1 , V 2 = (y 2 j ) j1 , those are clusters of innite points consisted by y 1 i and y 2 j respectively onZ d andr 1 ,r 2 are two points on the boundary. Also s 1 is closer toV 1 than s 2 (respectively s 2 is closer to V 2 than s 1 ). Here we let r 1 = R s1 (y 1 i ) and r 2 = R s2 (y 2 j ), which has three probabilities: (1) (Geodesic) If at least one path froms 1 toy 1 i throughr 1 achieved(s 1 ;y 1 i ), and the path from s 2 to y 2 j through r 2 achieves d(s 2 ;y 2 j ), then from previous denitions r 1 = R s1 (y 1 i ) and r 2 =R s2 (y 2 j ) (2)(Non-geodesic) If all paths do not achieve d(s 1 ;y 1 i ) (respectively d(s 2 ;y 2 j )), but the in- mum over those paths through r 1 (respectively r 2 ) is equal to overall inmum d(s 1 ;y 1 i ) (respectively d(s 2 ;y 2 j )), then we choose the point on the boundary that has smallest in- mum to be desired path and the last point for this path to exit is dened to be R s1 (y 1 i )(respectively R s2 (y 2 j )) (3) (Non-geodesic)Under conditions of Assumption (2), if more then one path whose pas- sage time has same smallest inmum (it has ties), then we use lexicographic order to get R s1 (y 1 i ) (respectively R s2 (y 2 j )). Noticing that the path through r 1 (respectively r 2 ) means r 1 (respectively r 2 ) is the last point on the boundary to be passed through. Note that the eventC is inF c butI 1 \I 2 depends on the whole cong- uration. Step 1. Let us prove thatP(C)> 0: We need to rst prove that the event in (10) is included in I 1 \I 2 . On the event in (10), We set V 1 = A 1 (s 1 ;s 2 )\fx 2 Z d ;R s 1 (x) = r 1 g and V 2 = A 2 (s 1 ;s 2 )\fx2Z d ; R s 2 (x) = r 2 g, so both are innite. Let y2 V 1 . On the event in (10), R s 1 (y) =r 1 , so 34 d(s 1 ;r 1 ) +d(r 1 ;y) =d(s 1 ;y) <d(s 2 ;y) d(s 2 ;r 2 ) +d(r 2 ;y) d(s 2 ;r 2 ) +d c(r 2 ;y): This step is changingd(r 2 ;y) tod c(r 2 ;y), which only depends on outside conditions. As r 1 is that last point of to be in , we have d(r 1 ;y) =d c(r 1 ;y), so d c(y;r 1 ) +d(r 1 ;s 1 )<d c(y;r 2 ) +d(r 2 ;s 2 ): (11) This shows the event in (10) is included in I 1 , and, proceeding similarly, it is also included in I 2 . So (10) implies: P(I 1 \I 2 )> 0 (12) On I 1 , we need to choose and to order one of the innite sets of the denition of the event: for any such innite set, we order its elements in the increasing order, and then among all such ordered sets, we choose the smallest one for the lexicographic order and we denote it (y 1 i ) i1 . We can do the same to dene (y 2 i ) i1 onI 2 . Now, we build a setG =G(! c; c) of good congurations (! ; ) inside , depending on the conguration outside . On C, we proceed in the same manner: order each component of any good (V 1 ;V 2 ) in the increasing order to obtain ((V 1 i ) i1 ; (V 2 i ) i1 ), then among all these couples, choose the smallest one ((y 1 i ) i1 ; (y 2 i ) i1 ) for the lexicographic order. We set, on I 1 \I 2 and on C, m 1 =m 1 (! c; c) = sup i1 (d c(y 1 i ;r 1 )d c(y 1 i ;r 2 )) m 2 =m 2 (! c; c) = inf j1 (d c(y 2 j ;r 1 )d c(y 2 j ;r 2 )) and consider the quantities: d c(y;r 1 )d c(y;r 2 ) (13) 35 d(s 2 ;r 2 )d(s 1 ;r 1 ) (14) Here m 1 and m 2 areF c-measurables. Then from denitions of m 1 , m 2 , and (11), we get m 1 (14) if we choose y 1 i and (14) m 2 if we choose y 2 i . This means the dierence of passage time for the two paths outside depends on inside dierence. More precisely, whend(s 2 ;r 2 )d(s 1 ;r 1 )m 2 , all y 2 i is closer to s 2 than s 1 . Note that CI 1 \I 2 \fm 1 <m 2 g, to understand this relationship we can divided the congurations into the parts on and c . So, with Lemma 4:3:(ii), P(I 1 \I 2 \fm 1 m 2 g) =E(P(I 1 \I 2 \fm 1 m 2 gjF c)); E(P(d(r 2 ;s 2 )d(r 1 ;s 1 ) =m 1 jF c)) = 0 (From Assumption (6)) [ So from (12), we haveP(C)P(I 1 \I 2 \fm 1 <m 2 g) =P(I 1 \I 2 )> 0. Step 2. From the condition of nite energy property, we can give the bounds for each path 0 i . It is also easy to see, if m 1 (! c; c) < m 2 (! c; c), that we can nd a 1 ;a 2 ;b 1 ;b 2 2R + such that b 1 m 1 <b 2 m 2 , a 2 m 2 <a 1 m 1 , b 1 a 1 = 1 and b 2 a 2 = 1. Dene also M =M(! c; c) = maxfa 1 m 1 ;b 2 m 2 g + maxfd c(r 1 ;z) :z2@;d c(r 1 ;z)<1g + maxfd c(r 2 ;z) :z2@;d c(r 2 ;z)<1g; First, since has chosen large enough, it is possible to draw with the edges in , a path 0 1 that links s 0 1 to r 1 and a path 0 2 that links s 0 2 to r 2 such that 0 1 and 0 2 have no vertex and no edge in common. Like in picture 5.3. Denote byj 0 1 j (resp.j 0 2 j) the number of edges in 0 1 (resp. 0 2 ). We dene now G as the set of (! ; ) that satisfy the following conditions: 36 (i) 8e2 0 1 ; ! e = 1 and a 2 m 2 =j 0 1 j< e <a 1 m 1 =j 0 1 j; (ii) 8e2 0 2 ; ! e = 1 and b 1 m 1 =j 0 2 j< e <b 2 m 2 =j 0 2 j; (iii) if p< 1; then8e2 n ( 0 1 [ 0 2 );! e = 0; if p = 1; then8e2 n ( 0 1 [ 0 2 ); e >M: Here we want to getm 1 d( 0 2 )d( 0 1 )m 2 , which means after changing the congurations inside , d( 0 2 )d( 0 1 ) is still restricted by m 1 and m 2 . Through (i) we can geta 2 m 2 d( 0 1 )a 1 m 1 . And getb 1 m 1 d( 2 )b 2 m 2 through (ii). Under the nite energy assumptions (9), on the eventfm 1 < m 2 g, we haveP(G(! c; cjF c )> 0) a.s., so (12) implies: P(G\C) =E(1 C P(GjF c))> 0 (15) Here we need to use conditional expectation to solve P(G\C) because C is completely outside but G is partially conditional onF c Step 3. This step gives us that we can x the conguration outside but changing the congurations and original resources inside . Let us prove that on the event C\GCoex(s 0 1 ;s 0 2 ), Suppose then that (!;)2C\G. We have in the conguration (!;): a 2 m 2 <d( 0 1 )<a 1 m 1 thanks to condition (i) in the denition of G. b 1 m 1 <d( 0 2 )<b 2 m 2 thanks to condition (ii) in the denition of G. Thus, by dierence and by the choice of a 1 ;b 1 ;a 2 ;b 2 , we have m 1 =m 1 (b 1 a 1 )<d( 0 2 )d( 0 1 )<m 2 (b 2 a 2 ) =m 2 : (16) Moreover, as soon as a path from s 0 1 to r 1 diers from 0 1 by at least one edge, it must use an edgee in n ( 0 1 [ 0 2 ), and this edge is either closed or such that e >M thanks to conditions (iii) in the denition of G. Thus, d( )>Ma 1 m 1 >d( 0 1 ). Consequently, 0 1 is the optimal path from s 0 1 to r 1 . On the other hand, every path from s 0 2 to r 1 has to use an edge e in n ( 0 1 [ 0 2 ), and then by the same argument, d( ) > M a 1 m 1 > d( 0 1 ), and then d(s 0 2 ;r 1 ) > d(s 0 1 ;r 1 ). Consequently, r 1 is nally infected by s 0 1 and 37 d(s 0 1 ;r 1 ) is reached for the optimal path 0 1 ; in the same manner,r 2 is nally infected by s 0 2 and d(s 0 2 ;r 2 ) is reached for the optimal path 0 2 . Let us now prove that for each i 1, R s 0 1 (x 1 i ) = r 1 . Let i 1 and note z = R s 0 1 (x 1 i ). If z6= r 1 , then the path from s 0 1 to x 1 i is not always geodesic, and we would have d(s 0 1 ;x 01 i ) =d(s 0 1 ;z) +d c(z;x 1 i )d(s 0 1 ;r 1 ) +d c(r 1 ;x 1 i ) and then d(s 0 1 ;z)d(s 0 1 ;r 1 ) +d c(r 1 ;x 1 i )d c(z;x 1 i )d(s 0 i ;r 1 ) +d c(r 1 ;z). The last inequality is just the triangle inequality for d c. But each path from s 0 1 to z must then contain at least one edge e in n ( 0 1 [ 0 2 ), and so such that e >M or ! e = 0, and then by denition of M, we must have d(s 0 1 ;z)Ma 1 m 1 +d c(r 1 ;z)>d(s 0 1 ;r 1 ) +d (r 1 ;z), which contradicts the previous inequality. In the same manner, we can prove the following: (a) For each i 1;R s 0 2 (x 0 i ) =r 2 ; (b) For each j 1;R s 0 2 (x 2 j ) =r 2 ; (c) For each j 1;R s 0 1 (xj 2 ) =r 1 : Let us now prove that for each i 1, we haved(s 0 1 ;x 1 i )<d(s 0 2 ;x 1 i ). We have d(s 0 1 ;x 1 i ) =d(s 0 1 ;r 1 ) +d c(r 1 ;x 1 i ) =d( 0 1 ) +d c(r 2 ;x 1 i ) + (d c(r 1 ;x 1 i )d( c(r 2 ;x 1 i ))) d( 0 1 ) +d c(r 2 ;x 0 i ) +m 1 <d( 0 2 )m 1 +d c(r 2 ;x 1 i ) +m 1 <d(s 0 2 ;r 2 ) +d c(r 2 ;x 1 i ) =d(s 0 2 ;x 1 i ) because R s 0 2 (x 1 i ) = r 2 . Similarly, we can prove that for each j 1, we have d(s 0 1 ;x 2 j ) > d(s 0 2 ;x 2 j ). We have thus proved the desired inclusion C 1 \C 2 \ fm 1 <m 2 g\GCoex(s 0 1 ;s 0 2 ). Now, (12) ensures thatP(Coex(s 0 1 ;s 0 2 ))> 0, which ends the proof. 38 Proof of Lemma 5.5. Let us give the line of the proof for the direct implica- tion. By Lemma 4.3, we can assume thats 1 = 0 ands 2 = 1. Under assump- tions (6) and (7), A 1 (0; 1) and A 2 (0; 1), with edges the ones in[ x2Z d (x), are both connected trees, denoted, respectively, by T (0) and T (1). Thus, if A 1 (0; 1) and A 2 (0; 1) are innite, by a classical compactness argument, one can nd, from 02 A 1 (0; 1), a semi-innite geodesic which is completely in A 1 (0; 1), and from 12A 2 (0; 1), a semi-innite geodesic which is completely in A 2 (0; 1). Following the proof of Lemma 5.1, inequality (10) is now replaced by P(T (0) contains a innite branch 1 starting from 0 and whose last point in @ is r 1 , T (1) contains a innite branch 2 starting from 1 and whose last point in @ is r 2 )> 0: (17) The proof follows exactly the same lines as the previous one: just replace C 1 and C 2 by C 1 =fThere is a simple path (x 1 i ) i1 in c such thatjjx 1 1 r 1 jj 1 = 1 and 8i 1; c(r 1 ;x 1 i ) = (r 1 ;x 1 1 ;:::;x 1 i1 ;x 1 i )g, C 2 =fThere is a simple path (x 2 i ) j1 in c such thatjjx 2 1 r 2 jj 1 = 1 and 8j 1; c(r 2 ;x 2 i ) = (r 2 ;x 2 1 ;:::;x 2 i1 ;x 2 i )g, The modication argument is the same, the only dierence is to choose two paths 0 1 and 0 2 starting both from 0, reaching, respectively, r 1 and r 2 , with no edge and no point in common except 0. The passage times are then modied exactly in the same manner. The converse implication can also be proved by an analogous modication argument. 6 References [1] AUFFINGER, A., DAMRON, M. and HANSON, J. (2016). 50 years of rst passage percolation. University Lecture Series, 68. American Math- ematical Society, Providence, RI, 2017. v+161 pp. MR3729447 [2] BARSKY, D. J., GRIMMETT, G. R. and NEWMAN, C. M. (1991). Percolation in half-spaces: Equality of critical densities and continuity of 39 the percolation probability. Probab. Theory Related Fields 90 (1991), no. 1, 111{148. MR1124831 [3] COX, J. T. and DURRETT, R. (1981). Some limit theorems for perco- lation processes with necessary and sucient conditions. Ann. Probab. 9 (1981), no. 4, 583{603. MR0549555 [4] DEIJFEN, M. and H AGGSTR OM, O. (2003). A Stochastic Model for Competing Growth onR d . Markov Process. Related Fields 10 (2004), no. 2, 217{248. MR1970474 [5] GARET, O. and MARCHAND, R. (2004). Asymptotic shape for the chemical distance and rst-passage percolation in random environment. ESAIM Probab. Stat. 8 (2004), 169{199. MR0901151 [6] GRIMMETT, G. (1999). Percolation, 2nd ed. Grundlehren der Mathe- matischen Wissenschaften [Fundamental Principles of Mathematical Sci- ences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. MR1675079 [7] GARET, O. and MARCHAND, R. (2005). Coexistence in Two-Type First-Passage Percolation Models. Ann. Appl. Probab. 15 (2005), no. 1A, 298{330. MR2462555 [8] HOFFMAN, C. (2005). Coexistence for Richardson type competing spa- tial growth models. Ann. Appl. Probab. 15 739{747. MR2114988 [9] HOFFMAN, C(2005). Geodesics in rst passage percolation. Ann. Appl. Probab. 18 (2008), no. 5, 1944{1969. MR2462555 [10] HOWARD, C.D. (2004).Models of rst-passage percolation. In Prob- ability on Discrete Structures 125{173. Encyclopaedia Math. Sci. 110. Springer, Berlin. MR2023652 [11] KESTEN, H. (1986). Aspects of rst passage percolation. Ecole d'Ete de Probabilites de Saint-Flour XIV|1984. Lecture Notes in Math. 1180 125{264. Springer, Berlin. MR0876084 [12] KESTEN, H. (1993) On the speed of convergence in rst-passage per- colation. Ann. Appl. Probab., 3(2):296{338, 1993. MR1221154 40 [13] MARCINKIEWICZ, J. and ZYGMUND, A. (1937). Sur les fon- cions independantes. Fund. Math., 28:60{90, 1937. Reprinted in J ozef Marcinkiewicz, Collected papers, edited by Antoni Zygmund, Panstwowe Wydawnictwo Naukowe, Warsaw, 1964, pp. 233{259. MR1546069 [14] MARCHAND, R. (2018). Personal communication. 41
Abstract (if available)
Abstract
This paper is based on ‘Coexistence in Two Type First Passage Percolation Model’ by Garet and Regine Marchand. The paper studies two-type of coexistence model through first-passage of percolation on ℤᵈ or on the infinite cluster in Bernoulli percolation. This is somewhat an annotated version of that paper. We have expanded out many of the details and added comments and graphical illustrations
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Coexistence in two type first passage percolation model: expansion of a paper of O. Garet and R. Marchand
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