Percolation and first-passage percolation on oriented graphs

We give the first properties of independent Bernoulli percolation, for oriented graphs on the set of vertices $\Z^d$ that are translation-invariant and may contain loops. We exhibit some examples showing that the critical probability for the existence of an infinite cluster may be direction-dependent. Then, we prove that the phase transition in a given direction is sharp, and study the links between percolation and first-passage percolation on these oriented graphs.

In percolation and directed percolation on the cubic lattice Z d , infinite clusters do not have the same geometry. In the unoriented setting, as soon as the opening parameter p exceeds the critical value p c (d) for the existence of an infinite cluster, one can build an infinite path in any given direction. On the contrary, in the oriented setting, clusters starting from the origin only live in the first quadrant; more precisely, when the opening parameter p exceeds the critical value − → p c (d), a deterministic cone gives the directions in which infinite paths are found.
Between these two models, it seems natural to ask what may happen for percolation for oriented graphs, on the set of vertices Z d , whose connections do not forbid any direction, or in other words, for oriented graphs that contain loops.
In the present paper, we first exhibit one example of such an oriented graph, where every direction is permitted, but such that we observe two phase transitions: if p is small, there there exists no infinite path, then when p increases there is a phase where infinite paths exist but not in any direction (as in classical supercritical oriented percolation), and finally, when p is large enough, infinite paths can grow in any direction (as in classical supercritical unoriented percolation).
Then, coming back to the general framework, we give some properties of percolation on oriented graphs on Z d that give an echo to some standard results for unoriented percolation, with a particular attention to the links between oriented percolation and first-passage oriented percolation.

The framework and one example
We deal here with an oriented graph whose vertices are the elements of Z d , and whose edges are the couples (x, y) such that y − x belongs to a given finite set denoted by Dir. Hence, if E denotes the set of edges, one has For a given parameter p ∈ (0, 1), we endow the set Ω = {0, 1} E with the Bernoulli product P p = Ber(p) ⊗E : under this probability measure, the edges are independently open (state 1) with probability p or closed (state 0) with probability 1 − p, and we are interested in the connectivity properties of the random graph G(ω) whose edges are the ones that are open in ω.
For x ∈ Z d , we denote by C + (x) the set of points that can be reached from x by a path in the random graph G, i.e. the points y such that there exists a sequence (x 0 , . . . , x n ) with x 0 = x, x n = y and (x i , x i+1 ) ∈ E for each i ∈ {0, . . . , n − 1}.
For u ∈ R d \{0}, we define The field (D u (x)) x∈Z d is stationary and ergodic. We set The quantity D u (x) measures the extension of the oriented open cluster issued from x in direction u and p c (u) is the critical parameter for the existence of an oriented open cluster that is unbounded in direction u. Note however that D u (x) = +∞ does not imply the existence of infinitely many points of C + (x) close to the half-line R + u. In other words, the only allowed communications are the following: for all x, x ′ , y ∈ Z, • (x, y) → (x, y − 1) Let us denote by (e 1 , e 2 ) the canonical basis for R 2 : with this set of edges, we give an advantage to direction e 2 when compared to direction −e 2 .
We first observe that for M large enough, there exist values for the opening parameter p such that there is percolation in direction e 2 but not in direction −e 2 : Theorem 1.1. Denote by − → p c (2) the critical value for classical oriented percolation on Z 2 + .
• For the last inequality, we build a dynamic independent directed percolation from bloc events with length M/2 that partition the horizontal lines. Remember that M ≥ 5. The probability that a given point (x, y) in the segment(  [6] (see also Grimmett [7]), that stochastically dominates an independent directed bond percolation on Z 2 , with parameter 1 − (1 − p) M/2 . Then, percolation in direction e 2 is possible as soon as where the last inequality comes from Liggett's bound [10]: − → p c (2) ≤ 2/3. The desired result follows.

A sharp percolation transition
We now come back to our general framework. Let Ψ : Z d → R be a subadditive function, i.e. such that for any x, y ∈ Z d , Ψ(x + y) ≤ Ψ(x) + Ψ(y). We define The graph (Z d , E) being translation-invariant, the distribution of r Ψ (x) does not depend on x.
If A, B and S are subsets of Z d , the event A S → B means that there exists a path (x 0 , . . . , x n ) with x 0 ∈ A, x n ∈ B, x i ∈ S for i ∈ {1, . . . , n − 1} and the bonds ( For p ∈ [0, 1] and 0 ∈ S ⊂ Z d , we define Note that in the above definition, the set S may be infinite. Then, we have the following result:  (1) For p <p c (Ψ), there exists c = c(Ψ, p) > 0 such that for each n ≥ 1, (2) For p >p c (Ψ), In particular, (1) and (2) imply thatp c (Ψ) = p c (Ψ).
Note that Ψ u (x) = u, x is linear and thus subadditive, and, for this map, Proof. • At first, let us prove that (1) and (2) • Proof of (1): it is very similar to Duminil-Copin-Tassion [3,4,5]. Since it is short, we give it to stay self-contained.
Let p <p c (Ψ). By the very definition ofp c (Ψ), we can find S ⊂ Z d that contains the origin and such that ϕ p (S) < 1 and sup S Ψ < +∞. Fix a positive integer L ≥ sup S∪Dir Ψ. We set , an open path starting from 0 and escaping from Λ kL eventually leaves S. Then, Note that We thus obtain • Proof of (2). In Duminil-Copin-Tassion, the idea is to use the Russo inequality. It is a bit more tricky here, because the events {0 ↔ ∂Λ n }, which correspond to the exit of finite boxes in Duminil-Copin-Tassion, now depend on infinitely many bonds. The proof is cut into three lemmas.
We begin with a lemma on a general graph.
We denote by T Y the σ-field generated by the events {x → Y }, for x ∈ V . We denote by B Y the random subset of V composed by the points that are not linked to Y .
Remember that e ∈ E is said to be pivotal for an event Then, for any e ∈ E, Proof of Lemma 2.2. Let us denote by Γ the set of oriented paths in Y c from a point in Y c to a point in Y . Then the subsets ∩ γ∈A ∩ e∈γ {ω e = 1}, for A ⊂ Γ, form a π-system that generates T Y , so it is enough to prove that for each A ⊂ Γ, one has The quantities that appear on each side of (3) are the limit of analogous quantities for a sequence of finite subgraphs of G. So, by dominated convergence, it is sufficient to prove (3) for a finite graph. From now on, we assume that G is finite.
Decomposing on the (finite number of) possible values of B Y , we thus only have to prove that for any subset S of vertices such that Fix a set S such that X ⊂ S ⊂ Y c . Let us denote by Note that on the event B Y = S, as X ⊂ S, pivotal edges for X → Y are necessarily in E 2 and that when e / ∈ E 2 , both members vanish. The event B Y = S is measurable with respect to the states of the edges in E 2 ∪ E 3 , and implies that all edges in E 2 are closed. Thus both members vanish if A ⊂ E 3 . Denote by A 3 the set of possible configurations of edges in E 3 that correspond to B Y = S. Finally, we thus have to prove that for any S such that But now, by independence, We come back to the case of a graph on Z d .
Proof of Lemma 2.3. The event {0 → Λ c n } depends on infinitely many bonds, so one can not directly apply the Russo formula. However, since {0 → Λ c n } is an increasing event, the following inequality is preserved (see for example Grimmett [7], page 43): Now consider the random set S n of points from which Λ c n can not be reached. Note that {0 → Λ c n } = {0 ∈ S n }. For each S ⊂ Z d and (x, y) ∈ E, we define the random variable Integrating the result of Lemma 2.2, we have for each e ∈ E: which gives the desired inequality.
Then, for any a and b in I with a ≤ b, Proof of Lemma 2.4. Let a, b ∈ I with a < b. We fix ε > 0 and define on [a, b]: It is sufficient to prove that F ε is non-decreasing for each ε > 0. Indeed, it will imply that which gives the lemma when ε tends to 0. Let Assume by contradiction that B = ∅ and define c = inf B. Let (x n ) be a sequence in B that tends to c. By the previous observation, the inequality F ε (x n ) ≥ F ε (c) holds for n large enough. Since As F ε is the sum of a function which is upper semi-continuous from the left and of a continuous function, it is still upper semi-continuous from the left. So and by definition of c, F ε (t) ≥ F (a) for each t ∈]a, c[, so F ε (c) ≥ F ε (a). This brings a contradiction.
Let p ∈ [p ′ , 1) and h ∈ (0, 1 − p): With Lemma 2.3 (note that as p >p c (Ψ), c n ≥ 1), we obtain that We can now apply Lemma 2.4 on [p ′ , 1[: as f n is non-increasing, g • f n is nondecreasing, so it is clearly upper semi-continuous from the left: for any p > p ′ .
Finally, we obtain (2) by letting n go to infinity.

Percolation and first-passage percolation on the (unoriented) edges of Z d .
Consider first Z d endowed with the set E d of edges between nearest neighbors. In the first-passage percolation model, iid non negative and integrable random variables (t e ) e∈E d are associated to edges. Let us denote by ν their common law. We refer the reader to the recent review paper on first passage percolation by Damron et al [1]. For each path γ in the graph (Z d , E d ), we define t(γ) = e∈γ t e , and ∀x, y ∈ Z d , t(x, y) = inf γ:x→y that can been seen as a random pseudo-distance on Z d . Using Kingman's subadditive ergodic theorem allows to define where the limits hold almost surely and in L 1 . The functional µ ν is homogeneous and subadditive, and can be extended to a symmetric semi-norm on R d . With some extra integrability assumption, we obtain the analytic form of the asymptotic shape theorem: The subadditivity and the symmetries of the lattice imply quite simply that µ ν is a norm if and only if it µ ν ((1, 0, . . . , 0)) > 0 is strictly positive. Moreover, it has long been known (see for example Cox-Durrett [2] or Kesten [9]) that µ ν is a norm if and only ν({0}) < p c (Z d ), where p c (Z d ) is the critical percolation parameter for independent percolation on the edges of Z d .
Our idea here is to find, in oriented percolation on (Z d , E), an analogous characterization of directions of percolation in terms of the semi-norm for an associated oriented first-passage percolation on (Z d , E). Things are necessarily more intricate, since we saw that for oriented percolation the critical probability may depend on the direction. (Z d , E). We suppose that to each oriented bond e ∈ E is associated a random variable t e , the (t e )'s being i.i.d. integrable non-negative random variables, with ν as common distribution; we denote by p the probability p = P(t e = 0) = ν({0}).

Oriented percolation and first-passage percolation on
In this section, we assume that the semi-group of Z d generated by Dir is the whole set Z d . Then, the graph (Z d , E) is transitive.
As in the classical setting, we can define the passage time of an oriented path as in (4), use Kingman's subadditive ergodic theorem to define the associated functional µ ν as in (5), which is now positively homogeneous and subadditive but not necessarily symmetric. By sudadditivity, Thus µ ν can be extended in the usual way to a non-symmetric semi-norm on R d . Finally, we get, under some extra integrability assumption, the analytic form of the asymptotic shape theorem as in (6).
Our hope is to characterize the directions of percolations in (Z d , E) when edges are open with probability p, i.e. the u ∈ R d such that with the help of the semi-norm µ ν for some law ν for the passage times of the edges. Since the only relevant parameter here is ν({0}) = p, we take from now on we denote by µ p the associated semi-norm on R d and we set which is a closed and convex set, but not necessarily bounded. We thus need some basics in the theory of convex sets.

Convex sets.
As A p is closed and convex, we can associate to A p two nonempty closed convex cones: • The recession cone 1 of A p is • The barrier cone of A p is The polar cone of a closed non-empty convex cone C is defined by The map C → C • is an involutive map in the set of closed non-empty convex cones.
Note also that C ∩ C • = {0}. Here, 0 + (A p ) is the polar cone associated to Bar(A p ) (see Rockafellar [12] Corollary 14. 2.1 p 123). In other words, characterizing the directions x ∈ R d such that µ p (x) = 0 is equivalent to characterizing the directions y ∈ R d such that b p (y) > 0.

3.4.
Results. Let us define, for p ∈ [0, 1], Note that BG(p) is non-increasing in p. The set BG(p) collects the directions in which the growth of the cluster issued from 0 is bounded. It is thus natural to make the following conjecture: For the moment, we only manage to prove the following result: This result will be a direct consequence of corollaries 3.4 and 3.7.
As in the classical setting, we can describe the asymptotic behavior of the pointto-hyperplane passage times with µ p . For u ∈ R d \{0} and n ≥ 0, set Proof. As in the unoriented case, it will follow from the analytic form of the shape theorem. However, the existence of directions for which µ p vanishes requires some attention.
• Let L > b p (u). There exists x ∈ R d with u, x = 1 and µ p (x) ≤ L. For n ≥ 1, denote by x n one vertex in H n (u) which is the closest to nx.
Since t(0, H n (u)) ≤ t(0, x n ), we have lim Letting L go to b p (u), we obtain that lim t(0, H n (u)) n ≤ b p (u).
• If u ∈ Bar(A p ), then b p (u) = 0 and the desired convergence is clear.
• If u ∈ int(Bar(A p )), there exists ε > 0 such that the open ball centered in u with radius ε is included in Bar(A p ); moreover, b p (u) > 0. By contradiction, assume that there exists ℓ ∈ (0, b p (u)) such that Then, one can build an infinite increasing sequence integers (n k ) and sites (x k ) such that t(0, x k ) ≤ ℓn k and u, x k = n k + O(1). By a compactness argument, we can assume that x k By the asymptotic shape theorem, t(0,x k ) x k tends to µ p (x), and we get the inequality Assume that u, x = 0, then µ p (x) = 0, so x ∈ 0 + (A p ). But Bar(A p ) is the polar cone of 0 + (A p ): by definition of ε, u + εx/2 ∈ Bar(A p ), so 0 ≥ u + εx/2, x = ε/2, which is a contradiction. So assume that u, x = 0: we can definex = x u,x and then u,x = 1 and µ p (x) ≤ ℓ, which contradicts the definition of b p (u). x, u = +∞, for each n ≥ 1, one can find x n ∈ C + (0), with x n , u ≥ n. Then, x n ∈ H n (u) and t(0, H n (u)) = 0. We then apply Theorem 3.3.
Proof. The idea is close to the one used by Grimmett and Kesten [8] to obtain large deviations inequalities for first-passage percolation: along an optimal path from 0 to H n (u), we expect to find a number proportional to n/N of disjoint streches whose increase in the u-direction is at least N . However, as p < p c (u), the first point of Theorem 2.1 ensures that P p sup y∈C+(0) y, u ≥ N decreases exponentially fast with N , so with high probability, streches whose increase in the u-direction is at least N have to use edges with passage time 1, and should globally contribute to an amount of time κn for some small κ > 0. A renormalisation argument allows to make all this accurate. However, we did not manage to implement the renormalisation argument under the assumption p < p c (u), and we rather work under the stronger assumption 1. We can assume without loss that u 1 < 1. Then, we can find δ ∈ R 2 \{0} with u, δ = 0 such that v = u + δ and w = u − δ satisfy p < min(p c (v), p c (w)), v 1 < 1 and w 1 < 1. By construction, u = v+w 2 . We define the following set We can easily check that T is bounded and thus is a triangle: for any x ∈ T , (7) −10 ≤ x, u ≤ 1, −11 ≤ x, δ ≤ 11, and As sup{ x, v : x ∈ [−1, 1] 2 } = v 1 , we also check that [−1, 1] 2 ⊂ T , and we set θ = max( v 1 , w 1 ) < 1.
2. For an integer N ≥ 4, we partition Z 2 into boxes (B N (k)) k∈Z 2 = (2N k + {−N, . . . , N − 1} 2 ) k∈Z 2 . We set B N = B N (0), and T N is the image of T by the dilatation with ratio N . Note that B N is included inside T N . We then define naturally the translated triangles T N (k) = 2N k + T N .
Consider now a path γ from 0 to H n (u). As in Grimmett-Kesten [8], we now associate to this path a squeleton Γ = (i 0 , i 1 , . . . , i ℓ ) of distinct N -boxes and a sequence (b 0 , b 1 , . . . , b ℓ ) of sites such that, except for the last point, b k ∈ B N (i k ) ⊂ T N (i k ), in the following manner. Set i 0 = 0 and b 0 = 0. Suppose i 0 , . . . , i n , b 0 , . . . , b n have been defined.
• If the last point γ last of γ belongs to T N (i n ), then we end the process and set ℓ = n. • Otherwise, let b n+1 be the first point of the path that is outside T N (i n ) and define i n+1 as the only index such that b n+1 ∈ B N (i n+1 ). We also a crossing type for b n+1 : we say that the crossing type of i n is up; -otherwise b n+1 − 2N i n , u < −10N and we say that the crossing type of i n is down.
We then remove the loops from this sequence, and we obtain, by relabeling the coordinates of the remaining N -boxes if necessary, the squeleton Γ = (i k ) 0≤l≤ℓ of the path γ, see Grimmett-Kesten [8] for details. We denote by I up (γ) and I down (γ) the number of crossings of the squeleton that are of the respective types up and down. Note that I up + I down (γ) = ℓ. Let us now establish rough bounds for I up and I down by using the following decomposition We have the following estimates: • γ last and b ℓ are both in T N (i ℓ ), so with (7), γ last − b ℓ , u ≤ 11N .
• For any k ∈ {1, . . . , ℓ}, let a k be the last point of the path γ before b k to be in T N (i k−1 ). As a k ∈ T N (i k−1 ), with (7) we have a k − 2N i k−1 , u ≤ N .
As γ is a path from 0 to H n (u), this leads, for any fixed N ≥ K, to From this, we first deduce that for every n large enough, 3. For k ∈ Z 2 , we say that the box B N (k) is good if for each x ∈ B N (k), max sup and that B N (k) is bad otherwise. As p < min(p c (v), p c (w)), by the first point of Theorem 2.1, there exists α > 0 such that for every n ≥ 1, y, v ≥ n ≤ e −αn and P p sup Thus, for every N ≥ 1, P p (B N is bad) ≤ 2(2N + 1) 2 e −α(1−θ)N and lim N →+∞ Let us also denote by I G up (γ) the number of boxes that belong to the squeleton, whose associated crossing is of type up, and that are also good. Assume that i k is the index of a crossing of type up, and for instance that b k+1 − 2N i k , v > N . Then, as b k ∈ B N (i k ), with (7), If moreover the box B N (i k ) is good, then sup y∈C+(b k ) y, v < N (1 − θ), and this implies that the portion of the path γ between b k and b k+1 uses at least one edge with passage time 1; and the same is true if the crossing is up because b k+1 − 2N i k , w > N . So (10) t(γ) ≥ I G up (γ). From now on, we will denote it as I G up (Γ, ε) the number of good boxes associated to the couple formed by a squeleton and a sequence of up/down status for its crossings. 4. Remember that C is defined in (7). We now fix the last parameters: choose α > 0 and ρ ∈ (0, 1) such that β = 8(C + 1) 2 (ρe −α + (1 − ρ)) 3/4 < 1.