Multiple geodesics with the same direction

The directed last-passage percolation (LPP) model with independent exponential times is considered. We complete the study of asymptotic directions of infinite geodesics, started by Ferrari and Pimentel \cite{FP}. In particular, using a recent result of \cite{CH2} and a local modification argument, we prove there is no (random) direction with more than two geodesics with probability 1.


Introduction
Commonly, geodesics are known as the generalization of straight lines in Euclidean geometry to curved spaces. For example, the geodesics on a sphere are parts of great circles (arcs). In a random environment, geodesics can be defined as paths optimizing a deterministic procedure. In this paper, the random environment is given by a LPP model in which geodesics are paths with maximal time. Precisely, let us consider i.i.d. random variables ω(z), z ∈ N 2 , exponentially distributed with parameter 1. The last-passage time to z is defined by where the above maximum is taken over all directed paths from the origin to z. See Martin [8] for a quite complete survey. Almost surely, the maximum G(z) is reached by only one path: this will be the geodesic to z. Thus, the collection of all these geodesics provides a random tree rooted at the origin and spanning all N 2 . It is called the geodesic tree and is denoted by . Of course, changing the random environment or the procedure affects the geodesics and the infinite graph they generate. One of the first issues about such an infinite graph concerns its number of topological ends, i.e. the number of infinite self-avoiding geodesics from any fixed vertex. Alexander [1] proved the Minimal Spanning Forest is one-ended. The same statement holds for the Directed Spanning Forest: see Coupier and Tran [4].
In first-passage percolation (FPP), results differ according to the nature-discrete or continuous -of the random environment. Hoffman [6] proved that for a large class of ergodic FPP processes on the lattice Z 2 , the number of infinite self-avoiding geodesics is a.s. greater than four. It can even 517 DOI: 10.1214/ECP.v16-1656 be infinite in some cases. When the lattice Z 2 is replaced with the location of an homogeneous Poisson point process on R 2 , Howard and Newman talk about euclidean FPP models. In [7], they got accurate estimates on the fluctuations of geodesics leading to conclusions not yet obtained in the lattice context; quoting them, a.s. every infinite geodesic has an asymptotic direction and every direction has an infinite geodesic.
Since then, this strategy has been successfully applied to the Radial Spanning Tree by Baccelli and Bordenave [2], and to the LPP model by Ferrari and Pimentel [5]. These last results are recalled in Theorem 0.
In this paper, we finish the study of directions of infinite geodesics in the LPP model. Our results strengthen and complete Ferrari and Pimentel's ones. Indeed, Ferrari and Pimentel [5], and Martin [9] had already remarked the almost sure existence of random directions with more than one geodesic. We prove (Part 1. of Theorem 1) the random set of directions with two geodesics is a.s. dense in [0; π 2 ] and countable. Furthermore, with probability 1, there is no direction with more than two geodesics (Part 2. of Theorem 1). Besides, Ferrari and Pimentel have proved that there is a.s. only one infinite geodesic in each deterministic direction belonging to a subset of [0; π 2 ] of full Lebesgue measure. In Part 3. of Theorem 1, we prove this holds for any direction. In a recent work [3], it has been shown that the probability for the three sites (0, 2), (1, 1) and (2, 0) to be crossed by infinite geodesics is equal to 6 − 8 log 2. Theorem 2 states such a phenomenon associated to the sites (0, n), (1, n − 1), . . . , (n, 0) occurs with positive probability. Our results are based on a local modification of the geodesic tree. First, remark that any given asymptotic property of the geodesic tree depends on times ω(z), |z| ≤ m only through the lastpassage times G(z), |z| = m (where m ∈ N and | · | denotes the L 1 -norm). Hence, Proposition 3 allows to replace the geodesic tree on the set {|z| ≤ m} with any deterministic directed tree, without changing its structure on {|z| ≥ m}. To our knowledge, no result similar to those of Theorem 1 has been established for the models previously mentionned in this section (except for Part 3. whenever an isotropy property holds). Actually, the LPP model takes the advantage to be deeply linked to a particle system, namely the Totally Asymmetric Simple Exclusion Process (TASEP). This coupling, due to Rost [10] and described in [3], allows to transfer results about the TASEP to the LPP model. The reader can refer to Theorem 1 of [5], Theorems 1 and 2 of [3]. Parts 2. and 3. of Theorem 1 derive from this coupling too.
The paper is organized as follows. Section 2 contains the definition of the directed LPP model and the results. The local modification argument is detailled in Section 3. Finally, Theorems 1 and 2 are proved in Section 4.

Results
Let IP be the Borel probability measure on Ω = [0, ∞) N 2 induced by the family {ω(z), z ∈ N 2 } of i.i.d. random variables exponentially distributed with parameter 1. A directed path γ from (0, 0) to z is a finite sequence of sites (z 0 , z 1 , . . . , z k ) with z 0 = (0, 0), z k = z and z i+1 − z i = (1, 0) or (0, 1), for 0 ≤ i ≤ k − 1. The quantity z ′ ∈γ ω(z ′ ) represents the time to reach z via γ. Its maximum over the set Γ(z) of all directed paths from (0, 0) to z is called the last-passage time to z and is denoted by G(z): Each path of Γ(z) goes by either z − (1, 0) or z − (0, 1). So, the function G satisfies the recurrence relation: . Since IP is a product measure over N 2 of non-atomic laws, the maximum G(z) is a.s. reached by only one (directed) path, called the geodesic to z. Let us denote it by γ z . The collection of all the geodesics γ z , z ∈ N 2 , can be interpreted as a tree rooted at the origin, spanning all the positive quadrant N 2 and whose edge set is made up of pairs {z ′ , z} with z − z ′ = (1, 0) or (0, 1) and z ′ ∈ γ z . This random tree is called the geodesic tree and is denoted by . See Figure 1 for an illustration. An infinite geodesic is merely a infinite branch of the geodesic tree . Precisely, this is a semiinfinite directed path (z n ) n∈N such that, for all n, the geodesic γ z n is exactly (z 0 , . . . , z n ). The existence of infinite geodesics different from the horizontal and vertical axes has been stated by Ferrari and Pimentel. Their results (Propositions 7 and 8 of [5]) are summarized below. Let us recall that an infinite geodesic (where · denotes the euclidean norm).

IP-a.s. each infinite geodesic has a direction in
, there exists at least one infinite geodesic with direction α. 3. There exists a (deterministic) set D ⊂ [0; π 2 ] of full Lebesgue measure such that, for all α ∈ D, IP-a.s. there exists at most one infinite geodesic with direction α.
The first two parts of Theorem 0 are based on a clever and efficient technic developed by Howard and Newman [7]. Let f be a positive function on R + . A tree R is said f -straight if for all but finitely many vertices z of R, the subtree of R coming from z is included in the semi-infinite cone rooted at z with direction [(0, 0); z) and angle f ( z ). Howard and Newman proved (Proposition then the two assertions corresponding to Parts 1. and 2. of Theorem 0 are satisfied. Thenceforth, an upperbound for the fluctuations of (finite) geodesics (Lemma 10 of [5]) allows to prove that the geodesic tree is a.s. f -straight for a suitable function f . Besides, Part 3. of Theorem 0 derives from an appropriate use of Fubini's theorem.
Our main result completes the study of directions of infinite geodesics of . 3. For all α ∈ [0; π 2 ], IP-a.s. there is exactly one geodesic with direction α. Let us first remark that multiplying the ω(z)'s by a same factor λ does not alter the graph structure of . As a consequence, Theorem 1 (and also Theorem 2 below) remains true when replacing the parameter 1 of the exponential distribution of the ω(z)'s with any positive real number. Parts 2. and 3. of Theorem 1 rely on the local modification argument of the geodesic tree (see the next section) and on asymptotic results of the LPP model, namely Theorem 1 of [5] and Theorem 2 of [3]. Conversely, the density result of the set of directions with two geodesics is only based on geodesic arguments and Theorem 0. Let us add this set is necessarily random.
This section ends with a coexistence result. Let n be a positive integer. Let z be the subtree of rooted at z; z is the collection of geodesics passing by z at which the common part from the origin to z is deleted. We will say there is n-coexistence if the vertex sets of the n subtrees (0,n−1) , (1,n−2) , . . . , (n−1,0) are simultaneously unbounded. The 2-coexistence always occurs since the vertical axis {(0, x), x ≥ 1} and the horizontal axis {(x, 0), x ≥ 1} are respectively included in the vertex sets of (0,1) and (1,0) . When n = 3, it has been proved (Theorem 1 of [3]) that 3-coexistence occurs with probability 6 − 8 log 2.
Theorem 2. For all n, the n-coexistence occurs with positive probability.

Local modification of the geodesic tree
Let us denote by |·| the L 1 -norm: |(x, y)| = x + y. In this section, we focus on events depending on times ω(z), for |z| < m, only through the geodesic tree they generate. Let us start with specifying such events. Let m ∈ N * . A directed tree on {|z| ≤ m} is a graph whose vertex set is {|z| ≤ m} and whose edge set contains Now, we can state the main result of this section. Let us consider an event S, with positive probability, describing some asymptotic properties of the geodesic tree , as for instance "having more than two infinite geodesics with the same direction". Then, Proposition 3 allows to locally change -in order to get additional properties -so that the altered event still occurs with positive probability.
This means that whatever the value of the vector G m , the tree m has a positive probability to satisfy the T m -dependent event B. Moreover, IP(S) > 0 forces the conditional expectation IE[1I S | G m ] to be nonzero. Combining with (2), we get the product To conclude, it remains to adduce the independence of events B and S conditionally to G m . Indeed, conditionally to G m , the event S only depends on times on {|z| > m} while B depends on times on {|z| < m}.
Before proving Lemma 4, let us introduce some notations. Let X m ∈ R m+1 + be the vector of times of the diagonal {|z| = m} , 0), ω(m − 1, 1), . . . , ω(0, m)) and let Y m ∈ R m+1 + such that G m = Y m + X m . In the sequel, we will always use bold letters for elements of R m+1 + . The random vectors G m , X m and Y m respectively induce the probability measures IP G m , IP X m and IP Y m on R m+1 + , which is endowed with its Borel σ-algebra m+1 . By hypothesis on the ω(z)'s, the random vectors X m and Y m are independent. Moreover, IP X m is absolutely continuous with respect to the Lebesgue measure λ m+1 on R m+1 + . Then, so does for

Proof (of Lemma 4) Consider two events
where f m denotes the density of IP X m with respect to λ m+1 , and with z = (z 1 , . . . , z m+1 ).
Since the density f m is componentwise decreasing, A ′ ǫ can be written as The sequence (A ′ ǫ ) ǫ>0 increases as ǫ ց 0. Hence, About the above equalities, the second one comes from the fact that the density of the exponential distribution is positive at each point of R + . The third one derives from (4). So, let ǫ > 0 small enough such that IP G m (A ′ ǫ ) is positive. A second use of (4) gives Now, let B be a T m -dependent event. Then we write Thanks to the independence between the random vectors X m and Y m , and the fact that the event B belongs to σ(ω(z), |z| < m), the last integral becomes : Let y ∈ [0; ǫ] m+1 and x ∈ R m+1 + . The event A ′ ǫ has been constructed in order to f m (x) is larger than ǫ whenever x + y belongs to A ′ ǫ . Hence, Combining the previous inequalities, we get Finally, Lemma 4 follows from (5), (6) and Lemma 5 stated below.

Proof of Theorem 1
Let us start with proving Part 2. of Theorem 1. The idea of the proof can be summarized as follows. From three infinite geodesics the local modification argument allows to assume that they respectively go by sites (0, 2), (1, 1) and (2, 0). The subtree (1,1) is the only site of {|z| = m} whose geodesic goes by (1, 1) .
In other words, B(m, k) means the intersection between the vertex set of (1,1) and the diagonal {|z| = m} is reduced to (k, m − k). This event only describes the graph structure of m ; it is T m -dependent (various directed trees of T m satisfy B(m, k)). Then, Proposition 3 applies and gives IP (B(m, k) ∩ S(m, k)) > 0 .
Let us denote by C(1, 1) the vertex set of (1,1) . On B(m, k) ∩ S(m, k), the set C(1, 1) is unbounded since it contains all the sites of the middle infinite geodesic of S(m, k). It is also trapped between two geodesics with the same direction. That forces C(1, 1) to have a null density with respect to N 2 : lim n→∞ 1 n 2 Card C(1, 1) ∩ [0, n] 2 = 0 . Now, such a situation never happens by Theorem 2 of [3]. This contradicts (8).

Proof of Theorem 2
Let n be a positive integer. Consider n different angles θ 1 < . . . < θ n in [0; π 2 ]. We denote by γ 1 , . . . , γ n the (different) infinite geodesics with direction respectively θ 1 , . . . , θ n . Their existence is almost surely ensured by the third part of Theorem 1. Thus, there exists an integer m such that, with positive probability, the γ i 's intersect the diagonal {|z| = m} on n different sites. In the same way, one can find n sites z 1 , . . . , z n in {|z| = m} such that, with positive probability, for any 1 ≤ i ≤ n, γ i goes by z i . Let us denote by S(m; z 1 , . . . , z n ) this event. Thus remark that each site z i belongs to the quadrant (i − 1, n − i) + N 2 . So, z i can be reached by a directed path coming from (i − 1, n − i). Now, let us consider the event B(m; z 1 , . . . , z n ) for which, for any 1 ≤ i ≤ n, the geodesic to z i goes through (i − 1, n − i). This is a T m -dependent event.
On the other hand, on B(m; z 1 , . . . , z n ) ∩ S(m; z 1 , . . . , z n ), an infinite geodesic emanates from each site of the diagonal {|z| = n − 1}. This proves that n-coexistence occurs with positive probability.