The time constant and critical probabilities in percolation models

We consider a first-passage percolation model on a Delaunay triangulation of the plane. In this model each edge is independently equipped with a nonnegative random variable, with distribution function F, which is interpreted as the time it takes to traverse the edge. Vahidi-Asl and Wierman have shown that, under a suitable moment condition on F, the minimum time taken to reach a point x from the origin 0 is asymptotically \mu(F)|x|, where \mu(F) is a nonnegative finite constant. However the exact value of the time constant \mu(F) still a fundamental problem in percolation theory. Here we prove that if F(0)<1-p_c^* then \mu(F)>0, where p_c^* is a critical probability for bond percolation on the dual graph (Voronoi tessellation).


Introduction
First-passage percolation theory on periodic graphs was presented by Hammersley and Welsh [4] to model the spread of a fluid through a porous medium.In this paper we continue a study of planar first-passage percolation models on random graphs, initiated by Vahidi-Asl and Wierman [9], as follows.Let P denote the set of points realized in a two-dimensional homogeneous Poisson point process with intensity 1.To each v ∈ P corresponds an open polygonal region C v = C v (P), the Voronoi tile at v, consisting of the set of points of R 2 which are closer to v than to any other v ′ ∈ P. Given x ∈ R 2 we denote by v x the almost surely unique point in P such that x ∈ C vx .The collection {C v : v ∈ P} is called the Voronoi Tiling of the plane based on P.
The Delaunay Triangulation D is the graph where the vertex set D v equals P and the edge set D e consists of non-oriented pairs (v, v ′ ) such that C v and C v ′ share a onedimensional edge (Figure 1).One can see that almost surely each Voronoi tile is a convex and bounded polygon, and the graph D is a triangulation of the plane [7].The Voronoi Tessellation V is the graph where the vertex set V v is the set of vertices of the Voronoi tiles and the edge set V e is the set of edges of the Voronoi tiles.The edges of V are segments of the perpendicular bisectors of the edges of D. This establishes duality of D and V as planar graphs: V = D * .To each edge e ∈ D e is independently assigned a nonnegative random variable τ e from a common distribution F, which is also independent of the Poisson point process that generates P. From now on we denote (Ω, F , P) the probability space induced by the Poisson point process P and the passage times (τ e ) e∈De .The passage time t(γ) of a path γ in the Delaunay Triangulation is the sum of the passage times of the edges in γ.The first-passage time between two vertices v and v ′ is defined by where C(v, v ′ ) the set of all paths connecting v to v ′ .Given x, y ∈ R 2 we define T (x, y) := T (v x , v y ).
To state the main result of this work we require some definitions involving a bond percolation model on the Voronoi Tessellation V.Such a model is constructed by choosing each edge of V to be open independently with probability p.An open path is a path composed of open edges.We denote P * p the law induced by the Poisson point process and the random state (open or not) of an edge.Given a planar graph G and A, B ⊆ R 2 we say that a self-avoiding path γ y] denotes the line segment connecting x to y).For L > 0 let A L be the event that there exists an open path γ In this case we also say that γ crosses the rectangle [0 and consider the percolation threshold, We have that p * c ∈ (0, 1), which follows by standard arguments in percolation theory.For more in percolation thresholds on Voronoi tilings we refer to [1,2,11].
To show the importance of Theorem 1 we recall two fundamental results proved by Vahidi-Asl and Wierman [9,10].Consider the growth process and let τ 1 , τ 2 , τ 3 be independent random variables with distribution F.
Further, if E min j=1,2,3 and µ(F) > 0 then for all ǫ > 0 P-a.s.there exists t 0 > 0 such that for all t > t 0 where We note here that the asymptotic shape is an Euclidean ball due to the statistical invariance of the Poisson point process.Unfortunately the exact value of the time constant µ(F), as a functional of F, still a basic problem in first-passage percolation theory.Our result provides a sufficient condition on F to ensure µ(F) > 0.
Proof of Corollary 1. Together with the Borel-Cantelli Lemma, Theorem 1 and (4) imply which is the desired result.
For FPP models on the Z 2 lattice Kesten (1986) have shown that F(0) < 1/2 = p c (Z 2 ) (the critical probability for bond percolation on Z 2 ) is a sufficient condition to get (2) by using a stronger version of the BK-inequality.Here we follow a different method and apply a simple renormalization argument to obtain a similar result.We expect that our condition to get (2) is equivalent to where θ(p) is the probability that bond percolation on D occurs with density p, since it is conjectured that p c + p * c = 1 (duality) for many planar graphs.In fact, by combining Corollary 1 with (6) we have: Proof of Corollary 1.To see this assume we have a first-passage percolation model on D with Then P-a.s.there exists an infinite cluster W ⊆ D composed by edges e with τ e = 0. Denote by T (0, W) the first-passage time from 0 to W. Then for all t > T (0, W) we have that B 0 (t) is an unbounded set.By ( 6) (since such a distribution satisfies (3) and ( 5)), this implies that µ(F) = µ(p) = 0 if 1 − p > p c .On the other hand, by Corollary 1, µ(p) > 0 if 1 − p < 1 − p * c , and so (2) must hold.
Other passage times have been considered in the literature such as T (0, H n ), where H n is the hyperplane consisting of points x = (x 1 , x 2 ) so that x 1 = n, and T (0, ∂[−n, n] 2 ).The arguments in this article can be used to prove the analog of Theorem 1 when T (0, n) is replaced by T (0, H n ) or T (0, ∂[−n, n] 2 ).For site versions of FPP models the method works as well if we change the condition on F to F(0) < 1 − pc , where now pc is the critical probability for site percolation.Similarly to Corollary 2, in this case one can also obtain the inequality 1/2 ≤ pc .For more details we refer to [8].

Renormalization
For the moment we assume that F is Bernoulli with parameter p.Let L ≥ 1 be a parameter whose value will be specified later.Let z = (z 1 , z 2 ) ∈ Z 2 and we denote by ∂A its boundary.For each z ∈ Z 2 and r ∈ {j/2 : j ∈ N} consider the box

Divide B
L/2 z into thirty-six sub-boxes with the same size and declare that B L/2 z is a full box if all these thirty-six sub-boxes contain at least one point of P. Let , where I E denotes the indicator function of the event E.
Proof of Lemma 1.First notice that By the definition of a two-dimensional homogeneous Poisson point process, Now, let X e * := τ e , where e * is the edge in V e (the Voronoi tessellation) dual to e. Then {X e * ; e * ∈ V e } defines a bond percolation model on V with law P * p .Consider the rectangles ] .We denote by A i L the event A L (recall the definition of p * c ) but now translate to the rectangle R i L , and by F L the event that an open circuit σ * in V which surrounds B L/2 0 and lies inside B 3L/2 0 does not exist.Thus one can easily see that Notice that if there exists an open circuit σ * in V which surrounds B L/2 0 and lies inside B 3L/2 0 , then every path γ in C L has an edge crossing with σ * and thus t(γ) ≥ 1.Therefore, ) Since p > p * c , by using ( 8), ( 9), (10) and the definition of p * c , we get Lemma 1.
To obtain some sort of independence between the random variables Y L z we shall study some geometrical aspects of Voronoi tilings.Given A ⊆ R 2 , let I P (A) be the sub-graph of D composed of vertices v 1 in D v and edges (v 2 , v 3 ) in D e so that C v i ∩ A = ∅ for all i = 1, 2, 3. ).
Proof of Lemma 2. By the definition of the Delaunay Triangulation, Lemma 2 holds if we prove that To prove this we claim that If (12) does not hold then there exist Although, x 1 and x 2 belong to C v (P) and so ) Now suppose (11) does not hold.Without lost of generality, we may assume that there exists ) c , which is a contradiction with (12) and (13).
For each l ≥ 1, we say that the collection of random variables {Y z : z ∈ Z 2 } is ldependent if {Y z : z ∈ A} and {Y z : z ∈ B} are independent whenever l < d ∞ (A, B) := min{|z − z ′ | ∞ : z ∈ A and z ′ ∈ B} .
By repeating this argument inductively (on the number of good boxes which are crossed by γ) one can get (14).Lemma 5 follows directly from (14).

Figure 1 .
Figure 1.The Delaunay Triangulation and the Voronoi Tessellation.

Lemma 2 .z
Let L > 0 and z ∈ Z 2 .Assume that P and P ′ are two configurations of points so that P ∩ B ′ is a full box with respect to P, for all z ′ ∈ C z .ThenI P (B 3L/2 z ) = I P ′ (B 3L/2 z