First Passage percolation on a hyperbolic graph admits bi-infinite geodesics

Given an infinite connected graph, a way to randomly perturb its metric is to assign random i.i.d. lengths to the edges. An open question attributed to Furstenberg is whether there exists a two-sided infinite geodesic in first passage percolation on Z^2, and more generally on Z^n for n>1. Although the answer is generally conjectured to be negative, we give a positive answer for graphs satisfying some negative curvature assumption. Assuming only strict positivity and finite expectation of the random lengths, we prove that if a graph X has bounded degree and contains a Morse geodesic (e.g. is non-elementary Gromov hyperbolic), then almost surely, there exists a bi-infinite geodesic in first passage percolation on X.


Introduction
First passage percolation is a model of random perturbation of a given geometry. In this paper, we shall restrict to the simplest model, where random i.i.d lengths are assigned to the edges of a fixed graph. We refer to [ADH15,GK12,Ke86] for background and references.
Let us briefly recall how FPP is defined. We consider a connected nonoriented graph X, whose set of vertices (resp. edges) is denoted by V (resp. E).
For every function ω : E → (0, ∞), we equip V with the weighted graph metric d ω , where each edge e has weight ω(e). In other words, for every v 1 , v 2 ∈ V, d ω (v 1 , v 2 ) is defined as the infimum over all path γ = (e 1 , . . . , e m ) joining v 1 to v 2 of |γ| ω := m i=1 ω(e i ). Observe that the simplicial metric on V corresponds to the case where ω is constant equal to 1, we shall simply denote it by d. We will now consider a probability measure on the set of all weight functions ω. We let ν be a probability measure supported on [0, ∞). Our model consists in choosing independently at random the weights ω(e) according to ν. More formally, we equip the space Ω = [0, ∞) E with the product probability that we denote by P.
A famous open problem in percolation theory is whether with positive probability, first passage percolation on Z 2 admits a bi-infinite geodesic. In his Saint-Flour course from 84', Kesten attributes this question to Furstenberg (see [Ke86]). Licea and Newman [LN96] made partial progress on this problem, which is still open and mentioned that the conjecture that there are no such geodesics arose independently in the physics community studying spin glass. Wehr and Woo [WW98] proved absence of two sided infinite geodesic in a half plane, assuming the lengths distribution is continuously distributed with a finite mean.
For Riemannian manifolds, the existence of bi-infinite geodesics is influenced by the curvature of the space. It is well-known that complete simply connected non-positively curved Riemannian manifolds (such as the Euclidean space R n or the hyperbolic space H n ) admit bi-infinite geodesics. Therefore, simply connected manifolds without two-sided geodesics must have positively curved regions. It is easy to come up with examples of complete Riemannian surfaces with bubble-like structures that create short cuts avoiding larger and larger balls around some origin. See [Ba95] for background.
To help the reader's intuition, let us roughly describe a similar example in the graph setting. Starting with the standard Cayley graph of Z 2 , it is not difficult to choose edges lengths among the two possible values 1/10 and 1, such that the resulting weighted graph has no bi-infinite geodesics. To do so, consider a sequence of squares C n centered at the origin, whose size grows faster than any exponential sequence (e.g. like n n ). Then attribute length 1/10 to the edges along C n for all n, and 1 to all other edges. This creates large "bubbles" with relatively small neck in the graph (which in a sense can be interpreted as large positively curved regions). One easily checks for all n, for every pair of points at large enough distance from the origin, any geodesic between them never enters C n (as it is more efficient to go around the shorter edges of its boundary, than traveling inside it). Random triangulations admit bubbles and indeed no two sided infinite geodesics. Geodesics go via "mountain passes" these are random analogous of the Morse geodesics defined below. See [CL15] for the study of FPP on random triangulations.
The Euclidean plane being flat, its discrete counterpart Z 2 (and more generally Z d for d ≥ 2) is in some sense at criticality for the question of existence of bi-infinite geodesics in FPP. Therefore, one should expect that in presence of negative curvature, FPP a.s. exhibits bi-infinite geodesics. For instance, this should apply to FPP on Cayley graphs of groups acting properly cocompactly by isometries on the hyperbolic space H d . We will see that this is indeed the case.
Let us first introduce some notation. Let X be a simple graph, with no double edges. Recall that a path γ = (e 1 , . . . , e n ) between two vertices x, y is a sequence of consecutive edges joining x to y. We denote (x = γ(0), . . . , γ(n) = y) the set of vertices such that for all 0 ≤ i < n, γ(i) and γ(i + 1) are joined by the edge e i+1 . For all i < j, we shall also denote by γ([i, j]) the subpath (e i+1 , . . . , e j ) joining γ(i) to γ(j). Similarly, we define infinite paths indexed by N (resp. bi-infinite paths indexed by Z).
Definition 1.1. Let X be an infinite connected graph, and let C ≥ 1 and K ≥ 0. A path γ of length n between two vertices x and y is called a (C, K)-quasi-geodesic finite path if for all 0 < i < j ≤ n, Similarly, we define (C, K)-quasi-geodesic infinite (or bi-infinite) paths. An infinite (or a bi-infinite) path will simply be called a quasi-geodesic if it is (C, K)-quasi-geodesic for some constants C and K. Definition 1.2. A bi-infinite path γ in X is called a Morse quasi-geodesic (resp. Morse geodesic) if it is a quasi-geodesic (resp. a geodesic) and if it satisfies the so-called Morse property: for all C ≥ 1 and K > 0, there exists R such that every (C, K)-quasi-geodesic joining two points of γ remains inside the R-neighborhood of γ.
It is well-known and easy to deduce from its definition that in a weighted graph with bounded degree, whose weights are bounded away from 0, a Morse geodesic always lies at at bounded distance from a bi-infinite geodesic. Theorem 1.3. Let X be an infinite connected graph with bounded degree, that contains a Morse quasi-geodesic γ. Assume Eω e < ∞ and ν({0}) = 0. Then for a.e. ω, X ω admits a bi-infinite geodesic. Moreover for a.e. ω there exists a finite subset A ⊂ X such that for every sequence of pairs of vertices (x n , y n ) going to infinity on opposite sides of γ, the ω-geodesic between x n and y n crosses A.
That is, if the underling graph admits a two sided infinite Morse geodesic, the random FPP metric will have a two sided infinite geodesic a.s.
Very recently Ahlberg and Hoffman [AH15] made substantial progress on the structure of geodesic rays for FPP on Z 2 solving the midpoint problem (from [BKS03]) which is related to the non existence of bi-infinite geodesic. They showed that the probability a shortest path between (−n, 0) and (n, 0) will go via (0, 0) is going to 0 with n. Their upperbound on the probability is going very slowly to 0, and is likely far from the truth. The theorem above shows that the probability of the midpoint event is not going to 0 in the distance, along a Morse geodesic. This can be useful in proving linear variance.
We briefly recall the definition of a hyperbolic graph. A geodesic triangle in a graph X consists of a triplet of vertices x 0 , x 1 , x 2 ∈ V, and of geodesic paths γ 0 , γ 1 , γ 2 such that γ i joins x i+1 to x i+2 where i ∈ Z/3Z. Given δ ≥ 0, a geodesic triangle is called δ-thin if for every i ∈ Z/3Z, every vertex v i on γ i lies at distance at most δ from either γ i+1 or γ i+2 . Said informally, a geodesic triangle is δ-thin if every side is contained in the δ-neighborhood of the other two sides. A graph is hyperbolic if there is δ < ∞ so that all geodesic triangles are δ-thin. It is well-known [Gr87] that in a hyperbolic graph, any bi-infinite quasi-geodesic is Morse. In particular, we deduce the following Corollary 1.4. Let X be a hyperbolic graph with bounded degree containing at least one bi-infinite geodesic. Assume Eω e < ∞ and ν({0}) = 0. Then for a.e. ω, X ω admits a bi-infinite geodesic. Moreover for every bi-infinite quasi-geodesic γ, for a.e. ω there exists a finite subset A ⊂ X such that for every sequence of pairs of vertices (x n , y n ) going to infinity on opposite sides of γ, the ω-geodesic between x n and y n crosses A.
Note that the case where ν is supported in an interval [a, b] ⊂ (0, ∞) is essentially obvious. Indeed, for all ω, the weighed graph X ω is bi-Lipschitz equivalent to X. We deduce that a Morse quasi-geodesic in X remains a Morse quasi-geodesic in X ω (adapting the definition to weighted graphs), and therefore lies at bounded distance from an actual bi-infinite geodesic.
We finish this introduction mentioning that Theorem 1.3 applies to a wide class of Cayley graphs, including Cayley graphs of relatively hyperbolic groups, Mapping Class groups, and so on.

Preliminarily lemmas
We start with a useful characterization of Morse quasi-geodesics, that one may take as a definition. We deduce the following criterion, which we shall use in the sequel. Proof. Assume by contradiction that there exists a constant C > 0, and for every n, an integer R ≥ n, vertices x, x ′ , y, y ′ and a path γ of length ≤ Cd(x, y) such that , and γ avoids the R-neighborhood of γ 0 . By choosing n large enough, we can assume that R > D. Applying Proposition 2.1 to the path obtained by concatenating γ with geodesics from x ′ to x and from y ′ to y yields the desired contradiction.
The hypothesis Eω e < ∞ is used (only) in the following trivial lemma.
Lemma 2.3. Let X be a connected graph, and let γ be a self avoiding path. Assume 0 < b = Eω e < ∞. Then for a.e. ω, there exists r 0 = r 0 (ω) such that for all i ≤ 0 ≤ j, Proof. This immediately follows from the law of large number, using that the edges length distributions are i.i.d.

Proof.
The assumption implies that for all λ > 0, there exists δ > 0 such that ν([0, 0 + δ]) < λ. Let γ be a path of length n. Assume that |γ| ω ≤ ε|γ|, and let N be the number of edges of γ with ω-length ≥ δ. It follows that δN ≤ εn, so we deduce that N ≤ εn/δ. This imposes that at least (1 − ε/δ)n edges of γ have ω-length ≤ δ. Recall that by Stirling's formula, given some 0 < α < 1, the number of ways to choose αn edges in a path of length n is Thus the probability that γ has ω-length at most εn is less than a universal constant times In other words, we have proved that for all λ > 0, there exists ε > 0 such that which is equivalent to the statement of the lemma. Proof. Let q be an upper bound on the degree of X, and let n ≥ 1 be some integer. Every path of length n lying at distance at most n from o is such that d(o, γ(0)) ≤ 2n, hence such a path is determined by a vertex in the ball B(o, 2n), whose size is at most q 2n + 1, and a path of length n originated from this vertex. Therefore the number of such paths is at most (q + 1) 3n .
On the other hand, we deduce from the previous lemma that for c > 0 small enough, the probability that there exists some path γ of length n, and at distance at most n from o, and satisfying |γ| ω ≤ c|γ| is less than 1/(q + 1) 4n . Hence the lemma follows by Borel-Cantelli.

Proof of Theorem 1.3
We let γ 0 be some Morse quasi-geodesic. First of all, we do not loose generality by assuming that our Morse quasi-geodesic γ 0 is a bi-infinite geodesic of the graph X. We let o = γ 0 (0) be some vertex. We consider two sequences of vertices (x n ) and (y n ) on γ 0 which go to infinity in opposite directions.
We let Ω ′ ⊂ Ω be a measurable subset of full measure such that the conclusions of Lemmas 2.3 and 2.5 hold. For all n and for all ω, we pick measurably an ω-geodesic γ n ω between x n and y n . Note that Lemmas 2.3 and 2.5 imply that such a geodesic exists: by Lemma 2.3, we have that d ω (x n , y n ) is finite, and by Lemma 2.5, paths of length ≥ M have ω-length going to infinity as M → ∞.
If we can prove the second part of Theorem 1.3, i.e. that for all ω ∈ Ω ′ , there exists a constant R ω > 0 such that for all n, d(γ n ω , o) ≤ R ω , then the first part of Theorem 1.3 follows by a straightforward compactness argument 2 . So we shall assume by contradiction that for some ω ∈ Ω ′ , there exists a sequence R n going to infinity such that γ n ω avoids B(o, 100R n ).
Clearly, for all p ≤ k ≤ q, d(γ n ω (k), γ 0 ) ≥ R n . Moreover, recall that γ n ω avoids B(o, 100R n ). Hence if x and y are points on γ 0 such that d(γ n ω (p), x) = d(γ n ω (q), y) = R n , we deduce by triangular inequality that d(x, o) ≥ 99R n and d(y, o) ≥ 99R n . But since x and y lie on both sides of o, this implies that d(x, y) ≥ 198R n (because γ 0 is a geodesic). Now by triangular inequality, we conclude that End of the proof of Theorem 1. 3 We now let i and j be integers such that d(γ n ω (p), γ 0 (i)) = d(γ n ω (q), γ 0 (j)) = R n . Note that by triangular inequality, By Lemmas 2.5 and 2.2, we have On the other hand, since γ n ω is an ω-geodesic between x n and y n , we have where the second inequality follows from Lemma 2.3. Gathering these inequalities, we obtain (for n large enough) which yields a contradiction since φ(R n ) → ∞ as n → ∞. This ends the proof of Theorem 1.3.

Remarks and questions
• (Cayley graphs) We do not know a single example of an infinite Cayley graph, for which FPP a.s. admits no bi-infinite geodesics (to fix the ideas, assume the edge length distribution is supported on the interval [1, 2]).
• (Adding dependence) Given a hyperbolic Cayley graph, rather than considering independent edges lengths it is natural to consider other group invariant distributions. Under which natural conditions (mixing?) on this distribution do bi-infinite geodesics a.s. exists? • (Poisson Voronoi and other random models) A variant of random metric perturbation is obtained via Poisson Voronoi tiling of a measure metric space. It seems likely that our method of proof applies to the hyperbolic Poisson Voronoi tiling, see [BPP14]. Recently other versions of random hyperbolic triangulations were constructed, [AR13] [C14]. Since those are not obtained by perturbing an underling hyperbolic space, our proof does not apply to this setting. • (Variance along Morse geodesics) We conjecture that under a suitable moment condition on the edge-length distribution, the variance of the random distance grows linearly along the Morse quasi-geodesic, unlike in Euclidean lattices [BKS03]. For lengths which are bounded away from zero and infinity, Morse's property ensures that geodesics remain at uniformly bounded distance from γ, hence reducing the problem to "filiform graphs", i.e. graphs quasi-isometric to Z. A (very) special class of filiform graphs is dealt with in [A15] where a linear variance growth is proven.