ASYMPTOTICS FOR FIRST-PASSAGE TIMES ON DELAUNAY ...
arXiv:math/0510605v1 [math.PR] 27 Oct 2005
ASYMPTOTICS FOR FIRST-PASSAGE TIMES ON DELAUNAY TRIANGULATIONS LEANDRO P. R. PIMENTEL
Abstract. In this paper we study planar first-passage percolation (FPP) models, originally defined in the context of Zd lattice by Hammersley and Welsh [6], on random Delaunay triangulations. The setup is as follows: to each edge e attach a positive random ¯ ) between two vertexes v and v ¯ is defined as the variable τe ; P the first-passage time T (v, v ¯ . By using subadditivity, Vahidiinfimum of e∈γ τe over all paths γ connecting v to v Asl and Wierman [19] showed that the rescaled first-passage time converges to a constant, called the time constant. We show a sufficient condition to ensure that the time constant is strictly positive and derive some upper bounds for fluctuations. Our proofs are based on renormalization ideas and on the method of bounded differences.
1. Introduction Let P ⊆ R2 denote the set of points realized in a two-dimensional homogeneous Poisson point process with intensity 1. To each v ∈ P corresponds a polygonal region Cv , named ¯ | for all v ¯ ∈ P. the Voronoi tile at v, consisting of points x ∈ R2 such that |x − v| ≤ |x − v The family composed by Voronoi tiles is called the Voronoi tiling of the plane based on P. For a concise introduction in the subject we refer to Moller [16]. The Delaunay Triangulation D = (Dv , De ) is the graph where the vertex set Dv := P ¯ ) such that Cv and Cv′ share a oneand the edge set De consists of non-oriented pairs (v, v dimensional boundary (Figure 1). One can see that (with probability one) each Voronoi tile is a convex and bounded polygon, and the graph D is a triangulation of the plane. 0.75
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Figure 1. The Voronoi Tiling and Delaunay Triangulation. 2000 Mathematics Subject Classification. Primary: 60K35; Secondary: 82B. 1
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LEANDRO P. R. PIMENTEL
The Voronoi Tessellation V = (Vv , Ve ) is the graph where the vertex set Vv consists of vertexes of the Voronoi tiles and the edge set Ve is the set of edges of the Voronoi tiles. The edges e∗ of V are segments of the perpendicular bisectors of the edges e of D. This establishes duality of D and V as planar graphs.
To each edge e ∈ De is independently assigned a nonnegative random variable τe from a common distribution F, which is also independent of the Poisson point process that generates P. We assume that both P and {τe : e ∈ De } are functions of a configuration ω ∈ Ω and denote by P its joint law. The expectation and the variance are denoted by E and by V, respectively. Let t(γ) :=
X
τe ,
e∈γ
be the passage time along the path γ. The first-passage time between two vertexes v and v′ is defined by T (v, v′) := inf{t(γ); γ ∈ C(v, v′ )} where C(v, v′) denotes the set of all self-avoiding paths connecting v to v′ . A geodesic connecting v to v′ is a path ρ(v, v′) ∈ C(v, v′ ) which attains the minimum: t(ρ(v, v′ )) = T (v, v′)
(a sufficient condition on F to ensure the existence of geodesics is given by Corollary 3.2). For each x ∈ R2 we denote v(x) the almost-surely unique point v ∈ P such that x ∈ Cv . For x, y ∈ R2 let T (x, y) := T (v(x), v(y)) and ρ(x, y) := ρ(v(x), v(y)) . The set of points reached from x by time t is defined by Bx (t) := {y ∈ R2 : y ∈ Cv where v ∈ Dv and T (v(x), v) ≤ t} , We remark that others FPP models (euclidean FPP) were introduced by Howard and Newman [7], where the underline graph is the complete graph with vertex set P and to ¯ ) is attached the passage time τ(v,¯v) := |v − v ¯ |α (α > 0 is a fixed each edge e = (v, v parameter). We recall fundamental results in the subjects, proved by Vahidi-Asl and Wierman [19, 20]. Let ET (0, n) µ(F) := inf ∈ [0, ∞] n>0 n where 0 := (0, 0) and n := (n, 0). µ(F) is usually refereed the time constant. Assume that τ1 , τ2 , τ3 are independent random variables with common distribution F: if E( min {τj }) < ∞ j=1,2,3
(1.1)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
3
then µ(F) < ∞ and for all unit vectors ~x ∈ S 1 (|~x| = 1) P-a.s.
ET (0, n) T (0, n~x) = lim = µ(F) . n→∞ n→∞ n n lim
(1.2)
In addition, if E( min {τj }2 ) < ∞ j=1,2,3
holds and µ(F) > 0 then for all ǫ > 0 P-a.s. there exists t0 > 0 such that for all t > t0 (1 − ǫ)tD(1/µ) ⊆ B0 (t) ⊆ (1 + ǫ)tD(1/µ) ,
where D(r) := {x ∈ R2 : |x| ≤ r}.
Two natural questions arise from (1.2). • When is µ(F) > 0? • What is the right order of T (0, n) − µn?
It is expected that F(0) < pc := inf{p > 0; θ(p) = 1} , where θ(p) is the probability that bond percolation on D occurs with density p, is a sufficient and necessary condition to have µ(F) > 0 (Kesten [12]). Heuristics arguments indicate that FPP models belongs to the KPZ universality class, which brings the conjecture that the right order is nχ , where χ = 1/3 (Kadar, Parisi and Zhang [10], Krug and Spohn [14]). However the only models for which this has been proved are certain growth models related to random permutations (Baik, Deift and Johansson [2], Johansson [9]). For lattice FPP Kesten [13] showed that χ ≤ 1/2 (see also Alexander [1]), and for euclidean FPP Howard and Newman [8] showed the same upper bound. In this work we prove some results related to the questions above. To state them we require some definitions involving a bond percolation model. Let A and B be two subsets of R2 . Let [x, y] denote the line segment connecting x to y. We say that a self-avoiding path γ ∗ = (v1∗ , ..., vk∗ ) in V is a path connecting A to B ∗ if [v1∗ , v2∗] ∩ A 6= ∅ and [vk−1 , vk∗ ] ∩ B 6= ∅. The bond percolation model on the Voronoi Tessellation V, with parameter p and probability law P∗p , is constructed by choosing each edge e∗ in Ve to be open independently with probability p. An open path is a path composed by open edges. For each R > 0 denote by AR the event that there exists an open path γ = (vj∗ )1≤j≤h in V, connecting {0} × [0, R] to {3R} × [0, R], and with vj∗ ∈ [0, 3R] × [0, R] for all j = 2, . . . , h − 1. Define the function η ∗ (p) := lim inf P∗p (AR ), R→∞
and consider the percolation threshold, p∗c := inf{p > 0 : η ∗ (p) = 1} < 1 (the inequality in (1.3) will follow from Proposition 3.1).
(1.3)
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LEANDRO P. R. PIMENTEL
For each κ > 1/2 we set ν = ν(κ) := (4κ − 2)/7. For the remainder of the paper we will use the symbols aj , cj , bj for j = 1, 2, . . . to represent strictly positive constants, whose value may change from appearance to appearance but will not depend on n, r, s or t in the notation we will follow here. Theorem 1. If F(0) < 1 − p∗c and (1.1) holds then 0 < µ(F) < ∞ . If (1.1) is strengthened to aτ
E(e ) =
Z
(1.4)
eat dF(t) < ∞ for some a > 0
(1.5)
then: • For all ǫ > 0 there exists n0 > 0 such that for all n > n0 V(T (0, n)) ≤ n1+ǫ ;
(1.6)
• For all κ > 1/2 there exist constants cj > 0 such that for all n ≥ 1 and r ∈ [0, c1 nκ ] ν
and
P(|T (0, n) − ET (0, n)| ≥ rnκ ) ≤ e−c2 r ,
(1.7)
µn ≤ ET (0, n) ≤ µn + c3 nκ (log n)1/ν .
(1.8)
We note that our condition to get µ(F) > 0 should be equivalent to F(0) < pc , since it is expected that pc + p∗c = 1 (duality) for many planar graphs1. From Theorem 1, the proof of which is given in Section 4, we get an upper bound for the fluctuations of T (0, n) about µn and an improved shape theorem. Corollary 1.1. If F(0) < 1 − p∗c and (1.5) holds then for all κ > 1/2 there exist constants cj > 0 such that for all r ∈ [c3 (log n)1/ν , c1 nκ ] c2 ν
P(|T (0, n) − µn| ≥ rnκ ) ≤ e− 2 r .
(1.9)
Proof of Corollary 1.1. (1.9) follows immediately from (1.7) and (1.8). Corollary 1.2. If F(0) < 1 − p∗c and (1.5) holds then for all κ > 1/2, P-a.s. there exists t0 > 0 such that for all t > t0 (t − tκ )D(1/µ) ⊆ B0 (t) ⊆ (t + tκ )D(1/µ) . 1It
follows from (1.4) that pc + p∗c ≥ 1 holds (Pimentel [17])
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
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Proof of Corollary 1.2. Notice that, for large |x|, if
|T (0, x) − µ|x|| > |x|κ
then
|T (0, z) − µ|z|| > |z|κ /4 or max T (x, z) > |z|κ /4 . |x−z|≤1
(1.10)
Under (1.5), one can easily see that P-a.s. the event in the right hand side of (1.10) occurs for finitely many z ∈ Z2 . From (1.9) (and the Borel-Cantelli lemma) it follows that P-a.s. the event in the left-hand side of (1.10) occurs for finitely many z ∈ Z2 . Therefore, P-a.s. there exist M > 0 such that for all x ∈ R2 with |x| > M |T (0, x) − µ|x|| ≤ |x|κ ,
which yields Corollary 1.2 (further details are left to the reader). Overview. In Section 2 we will study some geometrical aspects of Voronoi tilings and of self-avoiding paths on Delaunay triangulations, which will play an important role in control the asymptotic behavior of first-passage times. In Section 3 we explore the duality between bond percolation models on Delaunay triangulations and on Voronoi tessellations to relate the size of F(0) and the value of µ(F). The fluctuations of the first-passage time about its asymptotic value are considered in Section 4. 2. Preliminaries The main tool we handle to control the fluctuations of the first-passage time about its asymptotic value is the method of bounded increments, which was also considered by Kesten [13] in the lattice FPP context, and also by Howard and Newman [8] in the euclidean FPP context, to study the same question. This method represents T (0, n) − ET (0, n) as a sum of martingales increments and, after estimating these increments, applies standard bounds for martingales with bounded increments. One of the great difficulties here is that, as distinct from the lattice context, a local increment in a configuration ω is felt not only by the travel times but also by the (random) graph, which has a long range dependence. For this reason we shall define a truncation of the Poisson process. For z ∈ Z2 , r > 0 and s ∈ {j/2 : j ∈ N} let
2 Bs,r z := rz + [−sr, sr] .
2
2
(2.11)
Order the points of Z in some arbitrary fashion, say Z := {u1 , u2 , . . . }. Let δ > 0 be a fixed parameter which value will be specified later 2. Let n ≥ 1 and for each k ≥ 1 let δ
Bnk := Bu1/2,n . k
2Its
value will depend on ǫ and κ in Theorem 1
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LEANDRO P. R. PIMENTEL
Let |A| denote the number of elements belonging to the set A. Define the point process Pn := Pn (P) (whose distribution will also depend on δ) as follows: • If |Bnk ∩ P| = 0 then set Bnk ∩ Pn := {vk } where vk is a random point uniformly distributed in Bnk ; 2δ 2δ • If |Bnk ∩ P| > 4n2δ then set Bnk ∩ Pn := {vk1 , . . . , vk4n } where (vk1 , . . . , vk4n ) is a sequence of 4n2δ points uniformly chosen in Bnk ∩ P. We make the convention P∞ = P and denote by Dn the Delaunay Triangulation based on Pn . 2.1. Renormalization and full boxes. The geometry of Voronoi tilings is study through renormalization ideas. In a few words, it consists in defining a larger box to be a good box if some property is satisfied in a neighborhood of that box. In the course of the proofs we shall utilize different definitions of good boxes but all of them will carry the notion of full box. 1/2,r
Formally, we divide a square box B = Bz into thirty-six sub boxes of the same length, say B1 , . . . , B36 . We stipulate B is a full box, with respect to the point configuration Pn , if all those thirty-six sub boxes have at least one point belonging to Pn (Figure 2). In other words, [B = ∪36 j=1 Bj is a full box ] := [Bj ∩ Pn 6= ∅ ∀ j = 1, . . . , 36 ] .
(2.12)
Figure 2. Renormalization: a full box 1/2,r
1/2,r
We say that Λ := (Bz1 , . . . , Bzk ) is a circuit of boxes if (z1 , . . . , zk ) is a circuit in Z2 . Let λ be the closed polygonal path composed by the line segments connecting rzj to rzj+1, where j = 1, . . . , k − 1, together with [zk , z1 ]. To each circuit Λ we associate four subsets of the plane: Λout will denote the (topological) interior of the unbounded 1/2,r component of R2 \ ∪kj=1 Bzj , while Λin will denote the interior of the bounded component 1/2,r of R2 \ ∪kj=1 Bzj ; λout will denote the interior of the unbounded component of R2 \λ, while λin will denote the interior of the bounded component of R2 \λ.
ASYMPTOTICS FOR FIRST-PASSAGE TIMES 1/2,r k )j=1
Lemma 2.1. Assume that Λ := (Bzj to Pn .
7
is a circuit composed by full boxes with respect
If Cv ∩ Λin 6= ∅ ⇒ Cv ⊆ λin .
(2.13)
If Cv ∩ Λout 6= ∅ ⇒ Cv ⊆ λout .
(2.14)
Proof of Lemma 2.1. By convexity of Voronoi tiles, if (2.13) does not hold then there exist x1 ∈ ∂Λin ∩ Cv and x2 ∈ ∂λin ∩ Cv (∂A denotes the boundary of the set A) and thus r/2 ≤ |x1 − x2 | . 1/2,r k )j=1
is a full box, there exist v1 , v2 ∈ Pn so that √ √ |v1 − x1 | ≤ 2r/6 and |v2 − x2 | ≤ 2r/6 .
Since every box in (Bzj
Although, x1 and x2 belong to Cv and so |v − x1 | ≤ |v1 − x1 | and |v − x2 | ≤ |v2 − x2 | . Thus,
√ r/2 ≤ |x1 − x2 | ≤ |x1 − v| + |x2 − v| ≤ 2r/3 , √ which leads to a contradiction since 2/3 < 1/2. By an analogous argument, one can prove (2.14). For each A ⊆ R2 let GPn (A) be the sub-graph of Dn composed by edges e = (v1 , v2 ) so that Cvi ∩ A 6= ∅ for i = 1, 2. We denote by τ (A) the sequence of passage times associated to edges in GPn (A).
Given a random variable X and a measurable event F we denote by X | F the random variable X condition on the event F . 1/2,r
Lemma 2.2. Let FΛ be the event that Λ := (Bzj )kj=1 is a circuit composed by full boxes with respect to Pn . Assume that X is a random variable which only depend on (GPn (Λin ), τ (Λin )) and that Y is a random variable which only depend on (GPn (Λout ), τ (Λout )). Then X | FΛ is independent of Y | FΛ . Proof of Lemma 2.2. It follows from Lemma 2.1.
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LEANDRO P. R. PIMENTEL
2.2. Greedy lattice animals and site percolation. In this section we introduce two models which will also play an important rule in the study of Voronoi tilings. The first one, named Greedy lattice animals and introduced by Cox, Gandolfi, Griffin and Kesten [3]), consists of the following: a lattice animal is a connected subset of Z2 containing the origin. Let Φr and Φr denote the set of animals with at most r sites and with at least r sites, respectively. Let X := {Xz : z ∈ Z2 } be a collection of i.i.d. non-negative random variables and define X Ms (X) := max{ Xz : A ∈ Φs } . (2.15) z∈A
A greedy lattice animal is a lattice animal which attains the maximum in the above definition. Lemma 2.3. If E(ecX0 ) < ∞ for some c > 0
(2.16)
holds then there exist b1 , b2 > 0 such that
P(Ms > b1 s) ≤ e−b2 s . Proof of Lemma 2.3. Under (2.15), E(Ms ) ≤ bs for some b > 0 (Cox, Gandolfi, Griffin, Kesten [3]). From Corollary 8.2.4 of Talagrand [18], if υs denotes a median of Ms then P(|Ms − υs | > u) ≤ 4 exp(−k0 min{s−1 u2 , u}) . for some constant c0 > 0. By integrating in u from 0 to ∞ both sides of the last inequality we get that, for some constant ¯b > 0, √ |EMs − υs | ≤ E|Ms − υs | ≤ ¯b s .
and so
P(|Ms − EMs | > u) ≤ exp(−c0 min{s−1 u2 , u}) ,
√ if u > 2¯b s. This implies that
P(Ms ≥ 2bs) ≤ P(|Ms − EMs | > bs) ≤ e−bs/2c0 ,
(2.17)
which yields Lemma 2.3. Now, for each n ≥ 1 consider Y n := {Yzn : z ∈ Z2 }, a collection of Bernoulli random variables. Let φ := inf inf2 P(Yzn = 1) . n≥1 z∈Z
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
9
Assume that this collection is l-dependent: {Yzn : z ∈ A} and {Yzn : z ∈ B} are independent whenever l ≤ d∞ (A, B) := min{|z − z′ |∞ : z ∈ A and z′ ∈ B} .
This defines a family of l-dependent site percolation models index by n ∈ [1, ∞] (a site z is called open if Yzn = 1). Here we are interested on the density of open sites in the set of lattice animals with at least s sites. For more details in the subject see the reference book of Grimmet [4]. For each animal A we denote by ΞkA the set of all B ⊆ A such that if z, ¯z ∈ B then d∞ (z, ¯z) ≥ k. Let X mk (A, Y n ) := max{ Yzn : B ∈ ΞA } z∈B
and mks (Y n ) := min{mk (A, Y n ) : A ∈ Φs } .
(2.18)
Lemma 2.4. For all l ≥ 1 there exists p0 ∈ (0, 1) such that if φ > p0 then for all k ≥ 1 there exist b3 , b4 > 0 such that for all n ∈ [1, ∞] and s ≥ 1, P(mks (Y n ) < b3 s) ≤ e−b4 s .
(2.19)
Proof of Lemma 2.4. By Theorem 0.0 of Ligget, Schomman, Stacey [15], the family {Y n : n ∈ [1, ∞]}
is stochastically dominated from below by a collection of i.i.d Bernoulli random variables with parameter ρ(φ) → 1 when φ → 1.
Therefore, Lemma 2.4 will follow if we prove that (2.19) holds for mks (Y ), where Y := {Yz : z ∈ Z2 } is a collection of i.i.d Bernoulli random variables with parameter ρ sufficiently close to 1. Since m1s (Y ) ≤ c(k)mks (Y ), for some constant c(k) (only depending on k), we may restrict our attention to k = 1. P For each animal A, z∈A Yz is a binomial random variable with parameter |A| and ρ. Since (the number of animals A with |A| = j is at most C j , for some constant C > 0) we have at most cj animals with |A| = j, we get that X P(ms (Y )1 < bs) = P(∃ A : Yz < bs and |A| ≥ s) z∈A
≤
X j≥s
c
j
bs X l=0
Clj ρl (1 − ρ)j−l ≤ (1 − ρ)−bs
X j≥s
[2c(1 − ρ)]j
(1 − ρ)−bs [2c(1 − ρ)]s exp{s log 2c + s(1 − b) log(1 − ρ)} = 1 − 2c(1 − ρ) 1 − 2c(1 − ρ) if b < 1 and ρ > 1 − 1/2c. Thus, if we fix ρ > 1 − 1/2c and =
b3 < 1 + log 2c[log(1 − ρ)]−1
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LEANDRO P. R. PIMENTEL
then we can find b4 = b4 (ρ) > 0 so that (2.19) holds for m1s (Y ). 2.3. Self-avoiding paths on Delaunay triangulations. Let Cr and C r denote the set of self-avoiding and finite paths γ in Dn , starting from v(0), with |γ| ≥ r and with |γ| ≤ r, respectively. Fix L > 0 and for each path γ in Dn let A(γ) ⊆ Z2 be the set of points z ∈ Z2 so that ∩ [v, v′] 6= ∅ for some v, v′ ∈ γ (Figure 3). Define
1/2,L Bz
grL(Pn ) := min{|A(γ)| : γ ∈ Cr (Pn )}
and GLr (Pn ) := max{|A(γ)| : γ ∈ C r (Pn )} .
Figure 3. γ and A(γ) Let x, y ∈ R2 and define the path γ(x, y) := (v1 , ..., vk ) in D(Pn ) as follows: set v1 := v(x); if v1 6= v(y) then let v2 be the (almost-surely) unique nearest neighbor of v1 such that the edge of Cv1 , that is perpendicular to [v1 , v2 ], crosses [x, y]; otherwise we set k = 1 and the construction is finished; given vl with l ≥ 1, if vl 6= v(y) then we set vl+1 to be the unique nearest neighbor of vl different from vl−1 such that the edge of Cvl , that is perpendicular to [vl , vl+1 ], crosses [x, y]; otherwise we set k := l and the construction is finished. We denote γr := γ(0, r1). Proposition 2.1. For each L > 1 there exist cj = cj (L) > 0 such that for all n ∈ [1, ∞], for all r > 0, P(grL (Pn ) < c1 r) ≤ e−c2 r , (2.20) and
P(GLr (Pn ) > c3 r) ≤ e−c4 r .
(2.21)
Proposition 2.2. There exists z0 > 0 such that, for all n ∈ [1, ∞], for all r > 0 and z > z0 P(|γr (Pn )| > zr) ≤ e−c1 zr . (2.22)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
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Proof of Proposition 2.1. First notice that to prove (2.20) we can restrict the attention to 1/2,1 L = 1. Let Xzn := |Bz ∩ Pn | and denote X n := {Xzn : z ∈ Z2 }. Consider the random variable Msn := Ms (X n ) defined in (2.15). By definition, if gr1 (Pn ) ≤ cr then there exists a path γ ∈ Cr (Pn ) with |A(γ)| ≤ cr. Thus X n Mcr ≥ |B1/2,1 ∩ Pn | ≥ |γ| ≥ r , z z∈A(γ)
which yields that n P(gr1(Pn ) ≤ cr) ≤ P(Mcr ≥ r) .
(2.23)
By the definition of Pn , for each z ∈ Z2 and then
|B1/2,1 ∩ Pn | ≤ |B1/2,1 ∩ P∞ | + 1 , z z Mrn ≤ Mr∞ + r .
Combining this with (2.23), one gets that ∞ P(gr1 (Pn ) ≤ cr) ≤ P(Mcr ≥ (1 − c)r) .
Since (2.16) holds for X ∞ , together with Lemma 2.3, this implies (2.20). The proof of (2.21) is founded on renormalization techniques as follows. Denote Cz the 1/2,L is good box, or circuit composed by sites z′ ∈ Z2 with |z − z′ |∞ = 1. We say that Bz 1/2,L n equivalently Yz (L) = 1, if B¯z is a full box with respect to Pn for all z¯ ∈ Cz (see (2.12)). We denote Y n (L) := {Yzn (L) : z ∈ Z2 }.
Lemma 2.5. {Yzn (L) : z ∈ Z2 } is a 3-dependent collection of Bernoulli random variables. Further, for all ǫ > 0 there exists L0 > 0 so that inf inf P(Yzn (L0 )) ≥ 1 − ǫ .
n≥1 z∈Z2
Proof of Lemma 2.5. Since Cz and Cz′ are disjoint if |z − z′ | ≥ 3, the first part of this lemma follows directly from the definition of Pn . 1/2,L
To prove the second part notice that if L/36 ≥ nδ then each sub-box of Bz length L/6 contains at least one box Bkn (used to construct Pn ). Thus
with
P(YzL (Pn ) = 1) = 1 .
Now, if L/36 < nδ and B is a box with length L/6 then P(B ∩ Pn 6= ∅) ≤ P(B ∩ P = 6 ∅) + P(Pn ∩ B 6= P ∩ B) 2
2δ
2 /36
≤ e−L /36 + e−bn ≤ e−L which yields the second part of Lemma 2.5.
2
+ e−b(L/36) ,
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LEANDRO P. R. PIMENTEL
Consider the random variable m3r (Y n (L)) defined in (2.18). Assume we have a path γ ∈ C r . Then its animal A(γ) contains a subset B ∈ Ξ3A with m3 (Y n (L), A) good boxes. 1/2,L 1/2,L By Lemma 2.1, if Bz is a good box in A(γ) then γ has an edge lying in ∪¯z∈Cz B¯z . 1/2,L 1/2,L On the other hand, if Bz1 and Bz2 are 3-distant then 1/2,L
Thus, we must have that
∪¯z∈Cz1 B¯z
1/2,L
∩ ∪¯z∈Cz2 B¯z
= ∅.
m3 (Y n (L), A) ≤ |γ| ≤ r ,
which implies that
P(GLr (Pn ) > cr) ≤ P(m3cr (Y n (L)) < r) . Together with Lemma 2.5 and Lemma (2.4) this proves (2.21) for large L0 , which finishes the proof of Proposition 2.1. 1/2,L
Proof of Proposition 2.2. For this proof we say that Bz is a good box, or equivalently n n n Zz (L) = 1, if it is a full box. Then Z (L) := {Zz (L) : z ∈ Z2 } is a collection of independent random variables. Let Hnj be the set of closed sites (Zzn (L) = 0) in Z2 which are connected to (j, 0), and ¯ n := {(0, j)} ∪ Hj ∪ ∂Hn . H j
j
¯ n. Also define Ins (Z n (L)) := ∪sj=1 H j By Lemma 2.1,
γr (Pn ) ⊆ ∪z∈Inr/L Bz1/2,L ,
and thus
|γr (Pn )| ≤
X
z∈In r/L
|Bz1/2,L ∩ Pn | .
(2.24)
Since P(ZzL (Pn ) = 0) = o(L) uniformly in n (by the same argument in Lemma 2.5), we can find L0 sufficiently large so that E(exp(|Hn0 |) < ∞. This yields that P(|Ins | > ys) ≤ ec1 ys ,
(2.25)
for y > y0 and c1 , y0 constants whose values do not depend on n. From now on we fix such L0 and write Msn := Ms (Z n (L0 )) (defined in (2.15)) and Ins := Ins (Z n (L0 )). By Lemma 2.3, for sufficiently small c > 0 one can find b1 > 0 so that n ∞ P(McL ≥ zr) ≤ P(McL ≥ (z − 1)r) ≤ e−b1 zr . 0 zr 0 zr
(2.26)
By (2.24), P(|γr (Pn )|
P(|γr (Pn )| ≥ zr) ≤ n ≥ zr , |Ir/L0 | ≤ cL0 z(r/L0 )) + P(|Inr/L0 | > cL0 z(r/L0 )) n P(McL ≥ zr) + P(Inr/L0 > cL0 z(r/L0 )) . 0 zr
≤
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
13
Together with (2.26) and (2.25), this yields (2.22). Remark: the connective function on disordered graphs. Problems related to selfavoiding paths are connected with various branches of applied mathematics such as long chain polymers, percolation and ferromagnetism (Kesten [11], Hammersley [5]). One fundamental problem is the asymptotic behavior of the connective function κr defined by the logarithm of the number of self-avoiding paths (on some graph G) starting at v and with r steps. For planar and periodic graphs subadditivity arguments yields that r −1 κr (v) converges, when r → ∞, to some value κ ∈ (0, ∞) (the connectivity constant) independent of the initial vertex v. In disordered planar graphs subadditivity is lost but, if the underline graph possess some statistical symmetries (ergodicity), we may believed that the rescaled connective function still converges to some constant. From Proposition 2.1 we obtain a linear upper bound for the connective function of the Delaunay triangulation, where we set v to be the closest vertex to the origin: there exist a constant c > 0 so that κr lim sup ≤ c. (2.27) r r→∞ To prove (2.27) recall we have associate to each self-avoiding path γ and ordered sequence A(γ) = (z1 , ..., zn ) of nearest neighbors integer sites, which maybe seen as animal in Z2 (Figure 3). The number of self-avoiding paths associated to an ordered sequence (z1 , ..., zn ) should be at at most |B1/2,1 ∩ P|! × · · · × |B1/2,1 ∩ P|! z1 zn (n! is factorial of n). Now, every self-avoiding path with r steps intersects at most Gr boxes, and thus Nr ≤ 4Gr max {|B1/2,1 ∩ P|! × · · · × |B1/2,1 z1 z|A| ∩ P|!} , |A|≤Gr
which implies that κr = log Nr ≤ (log 4)Gr + max { |A|≤Gr
|A| X i=1
log |B1/2,1 ∩ P|!} . zi
(2.28)
Denote Xz := log |B1/2,1 ∩ P|! , z 2 and let X := {Xz : z ∈ Z } and Ms := Ms (X) (see (2.15)). By (2.28), if a > 1 and c < 1/2 log 4 then
P(κr > ar) ≤ P(Gr > car) + P(Mcar > ar/2) .
(2.29)
14
LEANDRO P. R. PIMENTEL
Since (2.16) holds for X, by Lemma 2.3 there exist b1 , b2 > 0 such that if a > 1 and c < min{1/2 log 4, 1/2b1 } then 2 P(Mcar > ar/2) ≤ e−b2 car .
(2.30)
By Proposition 2.1 we can find c3 , c4 > 0 so that, if ca > c3 then P(Gr > car) ≤ e−c4 r . Combining this with (2.29) and (2.30) we obtain that for a > c3 / min{1/2 log 4, 1/2b1 } P(κr > ar) ≤ e−b2 car + e−c4 r .
Together with Borel-Cantelli lemma, this proves (2.27). 3. Density of open edges in percolation and the time constant In this section we prove a sufficient condition on the passage time distribution F to ensure that µ(F) > 0. We also show that the same condition ensures the existence of geodesics. These results will follow from the subsequent propositions. Proposition 3.1. There exist p0 ∈ (0, 1) so that if p ∈ (p0 , 1) then η ∗ (p) = lim inf P∗p (AR ) = 1 . R→∞
In particular,
p∗c
< 1.
Let tr := inf{
X e∈γ
τe : γ ∈ Cr (Pn )} .
(3.31)
Proposition 3.2. If F(0) < 1 − p∗c then there exist constants cj > 0 such that for all n sufficiently large, for all r ≥ 1 P(tr ≤ c1 r) ≤ e−c2 r . 1/2,L
Proof of Proposition 3.1. Now we say that Bz if the following holds:
is a good box, or equivalently Vz (L, p) = 1,
1/2,L
1/2,L
• B¯z is a full box, with respect to P, and |B¯z |¯ z − z|∞ ≤ 1 ; 1/2,L • Xe∗ = 1 for all e∗ ∈ Ve which intersects B¯z .
∩ P| ≤ 4L2 for all ¯z with
We denote B1 (z, L) and B2 (z, L) the event specified in the first and in the second item above, respectively. The law of large numbers implies that lim P(B1 (z, L)) = 1 .
L→∞
(3.32)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
15
1/2,L
Since B¯z is a full box for all z¯ with |¯ z − z|∞ ≤ 1, by Lemma 2.1, the set of edges 3/2,L 1/2,L ∗ is contained in B¯z . Let V , E and F be the number of e ∈ Ve intersecting B¯z 3/2,L vertexes, edges and faces of the Delaunay triangulation on B¯z , respectively. Then 3E = 2F and, by the Euler formula, V −E +F = 2. 3/2,L has the same order as of vertexes in the This yields that the number of edges in B¯z same box. Thus, for some constant b > 0, P(B1 (z, L) ∩ B2 (z, L)c ) ≤ bL2 (1 − p) . Together with (3.32), this yields that for all δ > 0 we can chose L sufficiently large, and then p = p(L, δ) sufficiently close to 1, to have that P(B1/2,L is a good box ) > 1 − δ . z
(3.33)
From Lemma 2.2 we have that V := {Vz (L, p) : z ∈ Z2 } is a collection of 3-dependent random variables. Notice that by a similar argument to construct γr (see subsection 2.3) and by using Lemma 2.1 one can get that [∃ a crossing composed by good boxes of [0, 3R/L] × [0, R/L]] ⊆ [∃ an open crossing, with respect to X, of [0, 3R] × [0, R]] .
Combining (3.33) with 3-dependence, one can find L0 large enough and then p0 sufficiently close to 1 so that the probability of the first event in the above inclusion goes to 1 as R goes to infinity, which finishes the proof of Proposition 3.1. Proof of Proposition 3.2. First we claim it suffices to prove Proposition 3.2 if P(τe = 1) = p = 1 − P(τe = 0). To see this, assume that F(0) < 1 − p∗c . Since F is right-continuous, we can chose ǫ > 0 small so that F(ǫ) < 1 − p∗c . Let τeǫ := I(τ≥ ǫ) and denote tǫr the infimun defined in (3.31) but considering the travel times τeǫ . Therefore
which yields the last claim.
P(τeǫ = 0) < 1 − p∗c and ǫtǫr ≤ tr ,
Now assume that P(τe = 1) = p = 1 − P(τe = 0). Let L > 0 and z ∈ Z2 and define that is a good box if the following holds:
1/2,L Bz
1/2,L
• For all z′ ∈ Z2 with |z − z′ | = 2, Bz′
is a full box;
16
LEANDRO P. R. PIMENTEL 1/2,L
• For all γ ∈ C(Bz
3/2,L
, ∂Bz
) we have t(γ) ≥ 1 .
Let Wzn (L) := I(B1/2,L is a good box ) , z and p(n, L) := inf2 P(Wzn (L) = 1) . z∈Z
Lemma 3.1. W n (L) := {Wzn (L) : z ∈ Z2 } is 5-dependent for all L > 0. Moreover, if P(τe = 0) < 1 − p∗c then for all ǫ > 0 there exists Lǫ > 0 (independent of n) so that for all n ≥ Lǫ 1 − ǫ < p(n, Lǫ ) ≤ 1 . Proof of Lemma 3.1. 5-dependence follows from Lemma 2.2. Now consider the event 1/2,L
HzL,n := [Bz′
is a full box (with respect to Pn ) ∀ |z − z′ | = 2]
and the event Then
GL,n := [t(γ) ≥ 1 ∀ γ ∈ C(Bz1/2,L , ∂Bz3/2,L )] . z c P(Wzn (L) = 0) ≤ P((HzL,n )c ) + P((GL,n z ) ).
(3.34)
As we have seen before, P((HzL,n )c ) → 0 when L → ∞ uniformly in n .
This yields Lemma 3.1 if we prove that if P(τe = 0) < 1 − p∗c then for all ǫ > 0 there exists Lǫ > 0 so that for all n ≥ Lǫ 1 − ǫ < P(GLz ǫ ,n ) ≤ 1 . (3.35) We first prove (3.35) for n = ∞. Let Xe∗ := τe , where e∗ is the edge in Ve (with point configuration P) dual to e. Then {Xe∗ : e∗ ∈ Ve } defines a bond percolation model on V with law P∗p . Let RL1 := [L/2, 3L/2] × [−3L/2, 3L/2] , RL2 := [−3L/2, 3L/2] × [L/2, 3L/2] , RL3 := [−3L/2, −L/2] × [−3L/2, 3L/2] , RL4 := [−3L/2, 3L/2] × [−3L/2, −L/2] .
Let AiL be the event that there exists an open crossing in V, as in the definition of the event AL , but now translate to the rectangle RLi . 1/2,L
Denote by FL the event that an open circuit σ ∗ in V which surrounds B0 3/2,L inside B0 does not exist. Thus one can see that ∩4i=1 AiL ⊆ (FL )c .
and lies
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
17 1/2,L
Notice also that, if there exists an open circuit σ ∗ in V which surrounds B0 and lies 3/2,L 1/2,L 3/2,L inside B0 then every path γ connecting B0 to ∂B0 has an edge crossing with σ ∗ , and thus t(γ) ≥ 1. Together with translation invariance, this yields P((GL,∞ )c ) ≤ P∗p (FL ) ≤ 4(1 − P∗p (AL )) . 0
(3.36)
Since p > p∗c , by using (3.36) and the definition of p∗c one can obtain (3.35) for n = ∞. Now,
2δ
P(P∞ 6= Pn in B5/2,L ) ≤ c1 L2 e−c2 n . z for some constant c1 > 0. Thus, for all n ≥ L 2δ
c L,∞ c P((GL,n ) ) + c1 L2 e−c2 L . z ) ) ≤ P((Gz
(3.37)
By (3.36) and (3.37), given ǫ > 0 we can find Lǫ so that for all n ≥ Lǫ P((GLz ǫ ,n )c ) ≤ ǫ ,
which finishes the proof of this lemma.
Now consider the random variable m5r (W n (L)) defined by (2.18). Combining Lemma 2.4 with Lemma 3.1, one obtains that there exists L0 > 0 and b1 , b2 > 0, whose value depends only on L0 , such that for all n ∈ [1, ∞], for all r > 0 P(m5r (W n (L0 )) ≤ b1 r) ≤ e−b2 r .
(3.38)
Let γ ∈ Cr and consider the animal A(γ) previously defined. Then A(γ) contains a 1/2,L 1/2,L subset B ∈ Ξ5A with m5 (A(γ), W n (L)) good boxes. If Bz1 and Bz2 are 5-distant good boxes in A(γ) then (by definition) for each j = 1, 2 there exist a piece of γ, say γj , 1/2,L 3/2,L connecting ∂Bzj to ∂Bzj and so that t(γj ) ≥ 1. Since these good boxes are 5-distant, by Lemma 2.1, γ1 and γ2 must be disjoints. This yields that t(γ) ≥ 2. By repeating this argument inductively, one gets that m5 (A(γ), W n (L0 )) ≤ t(γ) ≤ r . Therefore, P(tr ≤ cr) ≤ P(gr < br) + P(m5br (W n (L)) ≤ cr) . Together with (3.38) and Proposition 2.1, this yields Proposition 3.2. Corollary 3.1. If F(0) < 1 − p∗c then there exist constants cj > 0 such that for all n sufficiently large, for all r > 0 P(T (0, r) < c1 r) ≤ e−c2 r .
In particular, if F(0) < 1 − p∗c then µ(F) > 0.
(3.39)
18
LEANDRO P. R. PIMENTEL
Proof of Corollary 3.1. If we have a path γ connecting 0 to r then it intersects at least r 1/2,1 unit boxes Bz . Thus, if b > 0 then P(T (0, r) < c1 r) ≤ P(Gbr > r) + P(tbr ≤ c1 r) . Combining Proposition 3.2 with Proposition 2.1, one can get (3.1). The second part follows from (3.1), together with Borel-Cantelli Lemma. Corollary 3.2. If F(0) < 1 − p∗c then for all x, y ∈ R2 P-a.s. there exist at least one geodesic connecting x to y. Further, if F is continuous, then we have P-a.s. uniqueness. Proof of Corollary 3.2. Let ¯ ) : |¯ dr (v) := inf{T (v, v v| ≥ r} . By Proposition 2.1 and Proposition 3.2, we can find sufficiently small c > 0 so that P-a.s. 0 < c < lim inf r
dr (v) . r
In particular, dr (v) → ∞ when r → ∞.
¯ ∈ Pn then we can find r large enough so that dr (v) > T (v, v ¯ )+1. Therefore, if we fix v, v ¯ will be attained by a path in the finite This implies that the passage time between v and v ¯ and lying inside the ball centered at v and with radius collection of paths connecting v to v r. To complete the proof,P notice thatPif F is a continuous function then P-a.s. there are no finite path γ and γ¯ with e∈γ τe = e∈¯γ τe .
4. Martingales with bounded increments and the rate of convergence We formulate the abstract martingale estimate used by Howard and Newman [8]. Lemma 4.1. Let (Ω, F , P) be a probability space and let {Fk }k≥0 be an increasing family of σ-algebras of measurable sets. Let {Mk }k≥0 , M0 = 0, be a martingale with respect to the filtration {Fk }k≥0 and let {Uk }k≥0 be a collection of positive random variables that are F -measurable. Assume that the increments ∆k = Mk − Mk−1 satisfy |∆k | ≤ c, for some constant c ≥ 0,
(4.40)
E(∆2k | Fk−1) ≤ E(Uk | Fk−1).
(4.41)
and
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
19
Assume further that for some finite constants 0 < c1 , υ, x0 with x0 ≥ c2 , we have that for all x ≥ x0 , X P( Uk > x) ≤ c1 exp(xυ ). (4.42) k≥1
Then almost surely M = limk→∞ Mk exists. Moreover, there exists constants 0 < c2 , c3 , whose values do not depend on c and x0 , such that for all x ≤ xυ0 , √ P (|M| > x x0 ) ≤ c2 exp(−c3 x). (4.43) Proof of Lemma 4.1. See the proof of Lemma 5.2 in Howard and Newman [8] We construct the underline probability space as follows. For each k ≥ 1 let and denote
Ik := {(a, b, c) : a ≥ 1, b ≥ k and c ≥ 1 if b 6= k or c > a if b = k} , b,c ωk := (Nk , (Uk,a)a≥1 , (τk,a )(a,b,c)∈Ik )
where Nk is a Poisson random variable with intensity n2δ , (Uk,a )a≥1 is a sequence of inde1/2,nδ b,c )(a,b,c)∈Ik is a sequence of pendent points uniformly distributed in Bnk = Buk , and (τk,a independent random variables with distribution F. Let (Ωk , Pk , Ak ) be the probability space induced by ωk and set ω = (ωk )k≥1 and ∞ Y Pk . P= k=1
To determine the point process P, we put Nk points in the box Bnk given by Uk,1 , . . . , Uk,Nk . For each e ∈ De we know there exists k ≥ 1 and (a, b, c) ∈ Ik so that e = (Uk,a , Ub,c ) and b,c so we set τe := τk,a . We shall use the following σ-fields: F0 := {∅, Ω}
Fk := σ-field generated by ω1 , . . . , ωk . Let Tn := T (0, n). The martingale representation of Tn − ETn is ∞ X Tn − ETn = {E(Tn | Fk ) − E(Tn | Fk−1)} . k=1
This representation is valid because M0 := 0 and Ml :=
l X k=1
{E(Tn | Fk ) − E(Tn | Fk−1 )}
= E(Tn | Fl ) − ETn
20
LEANDRO P. R. PIMENTEL
defines a Fk -martingale that, under (1.5), converges a.s. (and also in L2 ), to Tn − ETn . The increments of Ml are
∆k = E(Tn | Fk ) − E(Tn | Fk−1) ,
and the main step is to estimate E(∆2k | Fk−1 ).
4.1. Successive approximations. In order to satisfies the prescription of Lemma 4.1 we shall modified the configuration ω. We define ω ¯ k := ω ¯ k (n, ωk ) as follows: we change the point configuration P ∩ Bnk to Pn ∩ Bnk ; the travel time configuration is changed by putting travel times 3 b,c b,c τ¯k,a := min{τk,a , 8b−1 log n} .
¯ (¯ We note that Proposition 3.2 still holds for tr (P, τe )D¯e ) (with constants not depending on n), since for large enough n, τ¯e ≥ min{τe , 1}.
We denote ω ¯ := (¯ ωk )k≥1 . When a variable, such as travel times, geodesics and increments, is a function of ω ¯ we will decorate it with a bar. For instance we write T¯r (ω) := Tr (¯ ω ) and ρ¯r := ρr (¯ ω ). Lemma 4.2. If F(0) < 1 − p∗c and (1.5) holds then, and
2δ
P(|Tn − T¯n | > x) ≤ e−c1 n + e−c2 x
(4.44)
E({Tn − T¯n }2 ) ≤ c3 .
(4.45)
Proof of Lemma 4.2. To prove (4.44) we first claim that (for sufficiently large c > 0).
P(¯ ρn 6⊆ [−cn, cn]2 ) ≤ e−c2 n
(4.46)
Lemma 4.3. Under (1.5), there exists x0 , c0 > 0 so that for all x > x0 P(T¯r > xr) ≤ e−c0 xr . Proof of Lemma 4.3. By Proposition 2.2, there exist c1 , z0 > 0 so that if cx > z0 then P(T¯r > xr) ≤ P(T¯r > xr , |¯ γr | ≤ cxr) + e−c1 cxr . (4.47) By independence, P(T¯r > xr , |¯ γr | ≤ cxr) ≤ 3b
> 0 is given by (1.5)
cxr X j=0
P(
X
e∈¯ γr
τ¯e > xr | |¯ γr | = j)P(|¯ γ | = j) ≤
ASYMPTOTICS FOR FIRST-PASSAGE TIMES cxr X j=0
P(
j X i=0
21
τ¯i > xr)P(|¯ γ | = j) ≤ e−xar (Eeaτ )crx ≤ e−axr/2
aτ
if c < a log Ee . Together with 4.47, this yields Lemma 4.3. To prove (4.46) notice that, P(¯ ρn 6⊆ [−cn, cn]2 ) ≤ P(|¯ ρn | > bn) + P(∃ γ ∈ C bn : γ 6⊆ [−cn, cn]2 ) .
(4.48)
Now, P(|¯ ρn | > bn) ≤ P(T¯n > an) + P(T¯n ≤ an and |¯ ρn | > bn) a = P(T¯n > an) + P(tbn ≤ bn) . b Together with Proposition 3.2 and Lemma 4.3, this yield that if we take a > x0 , b > 0 so that a/b < c1 (given by Proposition 3.2) then for some constant c2 > 0.
P(|¯ ρn | > bn) ≤ ec2 n ,
(4.49)
By Proposition 2.1, if we take c sufficiently large, P(∃ γ ∈ C bn : γ 6⊆ [−cn, cn]2 ) ≤ P(Gbn ≥ cn)
will also decays exponentially fast with n. Together with (4.48) and (4.49), this proves (4.46). Turning back to the proof of (4.44), assume that ρ¯n ⊆ [−cn, cn]2 and P = Pn . Thus |T¯n − Tn | ≤ |T¯n | + |Tn | ≤ 2t(¯ γ )n ) .
This yields,
P(|T¯n − Tn | > x) ≤ P(P = 6 Pn in [−c1 n, c1 n]2 )+ P(¯ ρn 6⊆ [−cn, cn]2 ) + P(
X
[−c1 n,c1
τe I(τe > 8a−1 log n) > x) .
(4.50)
n]2
Notice that, since the mean number of Poisson points inside Bkn is n2δ , by using simple large deviations results for Poisson process we have that for each b > 0 there exists a constant b1 = b1 (b) > 0 such that for all n ∈ [1, ∞) 2δ
P(P ∩ [−bn, bn]2 6= Pn ∩ [−bn, bn]2 ) ≤ e−b1 n .
(4.51)
Let B ⊆ R2 and denote EB (P) the number of edges (v, v′) in D so that v, v′ ∈ B. Since the number of edges is comparable with the number of vertexes (see the proof of
22
LEANDRO P. R. PIMENTEL
Proposition 3.1) we have that for each b > 0 there exists b1 = b1 (b) > 0 such that for all r>0 2 (4.52) P(E[−br,br]2 (P) > 2(br)2 ) ≤ e−b1 r . To estimate the last term in the right hand side of (4.50) notice that, by (4.52), X τe I(τe > 8a−1 log n) > x) ≤ P( P(
X
[−c1 n,c1 n]2
[−c1 n,c1 n]2
2
τe I(τe > 8a−1 log n) > x | E[−c1 n,c1n]2 ≤ b1 n2 ) + eb2 n .
By using a similar procedure presented in Kesten [13] (see (2.37) there) one can prove that X τe I(τe > 8a−1 log n) > x | E[−c1 n,c1n]2 ≤ b1 n2 ) ≤ e−c1 x . P( [−c1 n,c1 n]2
Together with (4.46), (4.50) and (4.51), this yields (4.44). By Lemma 4.3, we also have that P(|T¯n − Tn | > ny) ≤ P(T¯n > ny/2) + P(Tn > ny/2) ≤ e−byn ,
if y ≥ y0 . Thus
E({T¯n − Tn }2 ) = 2n2
Z
(4.53)
∞
P(|T¯n − Tn | > ny)ydy 0 Z y0 Z ∞ 2 ¯ = 2n [ P(|Tn − Tn | > ny)ydy + P(|T¯n − Tn | > ny)ydy] . 0
y0
By (4.44) and (4.50), the last term is bounded by a constant and this completes the proof of Lemma 4.2 4.2. Main estimate. We introduce the the following notation: if ω = (ωk ), σ = (σk ) ∈ Ω then we denote [ω, σ]k := (ω1 , . . . , ωk , σk+1 , σk+2 , . . . ) . We also set ∞ Y Pk . νk := j=k
Since
Z
we have ¯k = ∆
T¯n [ω, σ]k νk+1 (dσ) = Z
Z
T¯n [ω, σ]k νk+1 (dσ) −
T¯n [ω, σ]k νk (dσ) , Z
T¯n [ω, σ]k−1νk (dσ)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
=
Z
23
{T¯n [ω, σ]k − T¯n [ω, σ]k−1}νk (dσ) .
Let Fk be the event that a geodesic from 0 to n has a vertex in Bnk . Let Ik be the indicator function of Fk and denote I¯k (ω) := Ik (¯ ω ). Lemma 4.4. |T¯n [ω, σ]k − T¯n [ω, σ]k−1| ≤ c1 n2δ log n × max{I¯k [ω, σ]k , I¯k [ω, σ]k−1} . Proof of Lemma 4.4. By Lemma 2.1, if I¯k [ω, σ]k = 0 then ρ¯n [ω, σ]k is also a path in D[ω, σ]k−1. Thus Tn [ω, σ]k−1 ≤ t(ρn [ω, σ]k ) = Tn [ω, σ]k . Analogously, if I¯k [ω, σ]k−1 = 0 then
Tn [ω, σ]k ≤ t(ρn [ω, σ]k−1) = Tn [ω, σ]k−1 . Therefore, max{I¯k [ω, σ]k , I¯k [ω, σ]k−1} = 0 ⇒ |T¯n [ω, σ]k − T¯n [ω, σ]k−1| = 0 .
(4.54)
Now assume I¯k [ω, σ]k = 1. Let vf (resp. vl ) be the first (resp. the last) vertex v ∈ δ ρn [ω, σ]k so that Cv ∩ Bz13,n . Define k ψn := ρ(0, vf )γ(vf , vu )ρ(vf , n)
which is the concatenation of these three paths. δ
By Lemma 2.1 and the definition of Pn , which ensures that Bu13,n has enough points k 13,nδ c n to isolate Bk from GPn ((Buk ) ) (see sub-section 2.1), D[ω, σ]k and D[ω, σ]k−1 coincide 2δ . Thus, ψn is also a path in D[ω, σ]k−1 and so outside Bu13,n k Tn [ω, σ]k−1 ≤ t(ψn ) = T (0, vf )[ω, σ]k + t(γ(vf , vl ))[ω, σ]k−1 + T (vl , n)[ω, σ]k ≤ Tn [ω, σ]k + t(γ(vf , vl ))[ω, σ]k−1 .
Analogously, if I¯k [ω, σ]k−1 = 1, then
Tn [ω, σ]k ≤ t(ψn ) = T (0, vf )[ω, σ]k−1 + t(γ(vf , vl ))[ω, σ]k + T (vl , n)[ω, σ]k−1 ≤ Tn [ω, σ]k−1 + t(γ(vf , vl ))[ω, σ]k .
Therefore,
|T¯n [ω, σ]k − T¯n [ω, σ]k−1| ≤
max{t(γ(vf , vl ))[ω, σ]k , t(γ(vf , vl ))[ω, σ]k−1} max{I¯k [ω, σ]k−1, I¯k [ω, σ]k } . Lemma 2.1 and the definition of Pn also ensures that,
max{|γ(vf , vl ))[ω, σ]k |, |γ(vf , vl ))[ω, σ]k−1|} ≤ c1 n2δ ,
(4.55)
24
LEANDRO P. R. PIMENTEL
for some constant c1 > 0. Together with and (4.55), this implies Lemma 4.4.
τ¯e ≤ 8a−1 log n
Let Uk := 2(c1 )2 n4δ (log n)2 I¯k , where c1 is the constant given by Lemma 4.4. Lemma 4.5. If F(0) < 1 − p∗c and (1.5) holds then for all sufficiently large n ∞ X E( Uk ) ≤ n1+5δ .
(4.56)
k=1
Further, for all δ ∈ (0, 1/8) for all y ≥ n1+7δ , P(
∞ X k=1
Uk > y) ≤ e−c2 y
1/2
.
Proof of Lemma 4.5. By definition ∞ ∞ X X 4δ 2 Ik ≤ c1 n4δ (log n)2 sup |¯ ρn | . Uk = c1 n (log n) k=1
k=1
Thus P(
∞ X k=1
Ik > ny) ≤ P(sup |¯ ρn | > ny)
≤ P(T¯n ≤ cny) + P(tny < cny) . By Proposition 3.2, if we chose c > 0 sufficiently small, P(tny < cny) ≤ ec¯yn . Therefore, E(
∞ X k=1
Ik ) ≤ E(sup |¯ ρn |) = n
≤n
Z
0
∞
Z
P(T¯n ≥ cny)dy +
∞ 0
Z
P(sup |¯ ρn | > ny)dy ∞
P(tny < cny)dy
0
= E(T¯n ) + c˜n , which proves (4.56). From Proposition 3.2 and Lemma 4.3, if x > c0 n then P(sup |¯ ρn | > x) ≤ P(T¯n ≥ cx) + P(tx < cx) ≤ ebx . Together with (4.56), this yields (4.57).
(4.57)
(4.58)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
25
Proof of Theorem 1. (1.4) follows from Corollary 3.1. Now we prove (1.6): by Lemma 4.4, the Schwarz inequality and the Fubini theorem, Z Z 2 ¯ E(∆k | Fk−1) = { {T¯n [ω, σ]k − T¯n [ω, σ]k−1}νk (dσ)}2 Pk (dω) ≤
c21 n4δ (log n)2 ≤
Z Z { max{I¯k [ω, σ]k , I¯k [ω, σ]k−1}νk (dσ)}Pk (dω)
2c21 n4δ (log n)2
Z
I¯k [ω, σ]k−1νk (dσ) = E(Uk | Fk−1 ) .
(4.59)
From Lemma 4.5 and (4.59), for all n sufficiently large VT¯n =
∞ X k=1
¯ 2 ) ≤ E( E(∆ k
∞ X k=1
Uk ) ≤ n1+5δ .
(4.60)
Now, Lemma 4.2 and the Schwarz inequality implies that VTn − VT¯n = E({Tn − T¯n }{Tn + T¯n }) + E(T¯n − Tn )E(T¯n + Tn ) ≤ 2{E({Tn − T¯n }2 )E({Tn + T¯n }2 )}1/2 ≤ 2{c3 E({Tn + T¯n }2 )}1/2
(c3 is given by Lemma 4.2).
By Lemma 4.3, E({Tn + T¯n }2 ) is at most of order n2 . Together with (4.60), this proves (1.6). To prove (1.7), let c = c(n) := c21 n2δ log n and x0 := n1+7δ , where c1 is the constant appearing in Lemma 4.4. Then x0 ≥ c2 (for large n). By Lemma 4.4 and 4.59, ¯k ≤ c ∆
and
¯ 2 | Fk−1 ) ≤ E(Uk | Fk−1) . E(∆ k
Together with Lemma 4.5, this yields (4.40), (4.41) and (4.42) (with υ = 1/2). By applying Lemma 4.1, one obtain that for all large n and x ≤ n(1+7δ)/2 P(|T¯n − ET¯n | > xn(1+7δ)/2 ) ≤ c2 e−c3 x .
On the other side, by Lemma 4.2, if x > 1 and n is large, P(|Tn − ETn | > xn(1+7δ)/2 ) ≤ P(|Tn − T¯n | > xn(1+7δ)/2 /3) + P(|T¯n − ET¯n | > xn(1+7δ)/2 /3) .
(4.61)
26
LEANDRO P. R. PIMENTEL
Combining this with Lemma 4.2 and (4.61) one get that for some constants b2 , b3 > 0, for n large and x ≤ n(1+7δ)/2 , P(|Tn − ETn | > xn(1+7δ)/2 ) ≤ b2 e−b3 x
4δ/(1+7δ)
.
(4.62)
Therefore, given κ ∈ (1/2, 1) we set δ := (2κ − 1)/7 and obtain (1.7) from (4.62).
(1.8) will follow from the same argument presented by Howard and Newman [8] to deal with the same inequality, but in the euclidean FPP context (see the proof of (4.3) there). For this reason we just sketch this proof and leave further details for the reader. By Lemma 4.2 of Howard and Newman [8], (1.8) will follows if we prove that, for some constant c1 , ET (0, 2n) ≥ 2ET (0, n) − c1 nκ (log n)1/ν (4.63) To prove (4.63) take δ < κ/2 and pick x1 := n, x2 , . . . , xl(n) in the boundary of Dn so that every x ∈ Dn is within distance nδ of one of the xj , and l(n) ≤ cn1−δ for some ¯ n := 2n − Dn . ¯ j := 2n − xj (the radial reflection of xj about n) D constant c > 0. Let x
Let ρ2n be a geodesic connecting 0 to 2n. Then ρ2n exits Dn for the last time at some ¯ n at some some point y ¯ n . Let xj(n) ∈ {x1 , . . . , xn } point yn and after that it enters in D ¯ ∈ {¯ ¯ l } be the closest points to yn and y ¯ n respectively. and xj(n) x1 , . . . , x Thus,
T (0, 2n) ≥ T (0, yn ) + T (¯ yn , 2n) ≥ T (0, xj(n) + T (¯ xn , 2n) − [(T (0, xj(n) − T (0, yn )) + (T (¯ xj(n) , 2n) − T (¯ yn , 2n))] . By sub-additivity, T (0, xj(n) ) − T (0, yn ) ≤ T (xj(n) , yn ) ≤ max sup T (xj , y) j
y−xj ≤nδ
and ¯ n ) ≤ max sup T (¯ T (¯ xj(n) , 2n) − T (¯ yn , 2n) ≤ T (¯ xj(n) , y xj , y) . j
y−xj ≤nδ
Therefore, T (0, 2n) ≥ ¯ j ) − [max sup T (xj , y) + max sup T (¯ min T (0, xj ) + min T (0, x xj , y)] . j
j
j
j
y−xj ≤nδ
y−xj ≤nδ
By symmetry, this implies that ET (0, 2n) ≥ 2ET (0, n) − 2[E(max{ET (0, xj ) − T (0, xj )}) − E(max sup T (xj , y))] . j
j
(4.64)
y−xj ≤nδ
Together with Lemma 4.3 of Howard and Newman [8], (1.7) yields E(max{ET (0, xj ) − T (0, xj )}) ≤ c2 nκ (log n)1/ν . j
(4.65)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
27
By noting that the expected number of edges inside {y : y − xj ≤ nδ } is of order n2δ and by using Lemma 4.3, one can prove that E(max sup T (xj , y)) ≤ c3 n2δ ≤ c3 nκ . j
y−xj ≤nδ
Combining this with (4.64) and (4.65), one gets (4.63). Acknowledgment. Part of this work was develop during my doctoral studies at Impa and I would like to thank my adviser, Prof. Vladas Sidoravicius, for his dedication and encouragement during this period. I also thank the whole administrative staff of IMPA for their assistance and CNPQ for financing my doctoral studies, without which this work would have not been possible. This work also amends some mistaken passages of my thesis (Pimentel [17]) and, for this reason, I should also express gratitude for the assistance given by Prof. Thomas Mountford. References [1] Alexander, K. S. (1997) Approximations of subadditive functions and convergence rates in limitingshape results, Ann.Probab. 25 30-55. [2] Baik, J.; Deift, P.; Johansson, K. (1999) On the distribution of the longest increasing subsequence in a random permutation, J. Amer. Math. Soc. 12 1119-1178. [3] Cox., J. T.; Gandolfi, A.; Griffin, P. S.; Kesten, H. (1993) Greedy lattice animals I: upper bounds, Ann. Appl. Probab. 3, 1151-1169. [4] Grimmett, G. (1999) Percolation (second edition), Springer. [5] Hammersley, J.M. (1961) On the rate of convergence to the connective function of the hypercubical lattice,Quart. J. Math. Oxford, Ser. 2 12, 250-256. [6] Hammersley, J.M.; Welsh, D.J.A. (1965) First-passage percolation, sub-additive process, stochastic network and generalized renewal theory, Springer-Verlag, 61-110. [7] Howard, C. D.; Newman, C. M. (1997) Euclidean models of first-passage percolation, Probab. Theory Related Fields 108, 153-170. [8] Howard, C. D.; Newman, C. M. (2001) Geodesics and spanning trees for Euclidean first-passage percolation, Ann. Probab. 29, 577-623. [9] Johansson, K. (2000) Shape fluctuations and random matrices, Comm. Math. Phys. 209, 437-476. [10] Kardar, K.; Parisi, G.; Zhang Y. Z. (1986) Dynamic scaling of growing interfaces Phys. Rev. Lett. 56, 889-892. [11] Kesten, H. (1963) On the number of self-avoiding walks J. Math. Phys. 4, 960-969. [12] Kesten, H. (1986) Aspects of first-passage percolation, Lectures Notes in Math. 1180, Springer-Verlag, 125-264 [13] Kesten, H. (1993) On the speed of convergence in first-passage percolation Ann. Appl. Probab. 3, 296-338. [14] Krug, J.; Spohn, H. (1991) Solids far from Equilibrium, (C. Godreche, ed.) Cambridge Univ. Press, Cambridge. [15] Ligget, T.M.; Schonmann, R.H.; Stacey, A.M. (1997) Domination by product measures, Ann. Probab. 25 , 71-95. [16] Moller, J. (1991) Lectures on random Voronoi tessellations, Lectures Notes in Stat. 87, SpringerVerlag.
28
LEANDRO P. R. PIMENTEL
[17] Pimentel, L. P. R. (2004) Competing growth, interfaces and geodesics in first-passage percolation on Voronoi tilings. Phd Thesis, IMPA, Rio de Janeiro. [18] Talagrand, M. (1995), Concentration of measure and isoperimetric inequalities in product spaces Inst. Hautes Etudes Sci. Publ. Math. 81, 73-205. [19] Vahidi-Asl, M.Q.; Wierman, J.C. (1990) First-passage percolation on the Voronoi tessellation and Delaunay triangulation, Random Graphs 87 (M. Karonske, J. Jaworski and A. Rucinski, eds.), 1990, 341-359. [20] Vahidi-Asl, M.C.; Wierman, J.C. (1992) A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation, Random Graphs 89 (A. Frieze and T. Luczak, eds.), Wiley, 247-262. ´ ´matiques, Ecole Institut de Mathe Polytechinique F´ ed´ erale de Lausanne, CH-1015 Lausanne, Switzerland. E-mail address: [email protected] URL: http://ima.epfl.ch/∼lpimente/
ASYMPTOTICS FOR FIRST-PASSAGE TIMES ON DELAUNAY TRIANGULATIONS LEANDRO P. R. PIMENTEL
Abstract. In this paper we study planar first-passage percolation (FPP) models, originally defined in the context of Zd lattice by Hammersley and Welsh [6], on random Delaunay triangulations. The setup is as follows: to each edge e attach a positive random ¯ ) between two vertexes v and v ¯ is defined as the variable τe ; P the first-passage time T (v, v ¯ . By using subadditivity, Vahidiinfimum of e∈γ τe over all paths γ connecting v to v Asl and Wierman [19] showed that the rescaled first-passage time converges to a constant, called the time constant. We show a sufficient condition to ensure that the time constant is strictly positive and derive some upper bounds for fluctuations. Our proofs are based on renormalization ideas and on the method of bounded differences.
1. Introduction Let P ⊆ R2 denote the set of points realized in a two-dimensional homogeneous Poisson point process with intensity 1. To each v ∈ P corresponds a polygonal region Cv , named ¯ | for all v ¯ ∈ P. the Voronoi tile at v, consisting of points x ∈ R2 such that |x − v| ≤ |x − v The family composed by Voronoi tiles is called the Voronoi tiling of the plane based on P. For a concise introduction in the subject we refer to Moller [16]. The Delaunay Triangulation D = (Dv , De ) is the graph where the vertex set Dv := P ¯ ) such that Cv and Cv′ share a oneand the edge set De consists of non-oriented pairs (v, v dimensional boundary (Figure 1). One can see that (with probability one) each Voronoi tile is a convex and bounded polygon, and the graph D is a triangulation of the plane. 0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3 0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
Figure 1. The Voronoi Tiling and Delaunay Triangulation. 2000 Mathematics Subject Classification. Primary: 60K35; Secondary: 82B. 1
2
LEANDRO P. R. PIMENTEL
The Voronoi Tessellation V = (Vv , Ve ) is the graph where the vertex set Vv consists of vertexes of the Voronoi tiles and the edge set Ve is the set of edges of the Voronoi tiles. The edges e∗ of V are segments of the perpendicular bisectors of the edges e of D. This establishes duality of D and V as planar graphs.
To each edge e ∈ De is independently assigned a nonnegative random variable τe from a common distribution F, which is also independent of the Poisson point process that generates P. We assume that both P and {τe : e ∈ De } are functions of a configuration ω ∈ Ω and denote by P its joint law. The expectation and the variance are denoted by E and by V, respectively. Let t(γ) :=
X
τe ,
e∈γ
be the passage time along the path γ. The first-passage time between two vertexes v and v′ is defined by T (v, v′) := inf{t(γ); γ ∈ C(v, v′ )} where C(v, v′) denotes the set of all self-avoiding paths connecting v to v′ . A geodesic connecting v to v′ is a path ρ(v, v′) ∈ C(v, v′ ) which attains the minimum: t(ρ(v, v′ )) = T (v, v′)
(a sufficient condition on F to ensure the existence of geodesics is given by Corollary 3.2). For each x ∈ R2 we denote v(x) the almost-surely unique point v ∈ P such that x ∈ Cv . For x, y ∈ R2 let T (x, y) := T (v(x), v(y)) and ρ(x, y) := ρ(v(x), v(y)) . The set of points reached from x by time t is defined by Bx (t) := {y ∈ R2 : y ∈ Cv where v ∈ Dv and T (v(x), v) ≤ t} , We remark that others FPP models (euclidean FPP) were introduced by Howard and Newman [7], where the underline graph is the complete graph with vertex set P and to ¯ ) is attached the passage time τ(v,¯v) := |v − v ¯ |α (α > 0 is a fixed each edge e = (v, v parameter). We recall fundamental results in the subjects, proved by Vahidi-Asl and Wierman [19, 20]. Let ET (0, n) µ(F) := inf ∈ [0, ∞] n>0 n where 0 := (0, 0) and n := (n, 0). µ(F) is usually refereed the time constant. Assume that τ1 , τ2 , τ3 are independent random variables with common distribution F: if E( min {τj }) < ∞ j=1,2,3
(1.1)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
3
then µ(F) < ∞ and for all unit vectors ~x ∈ S 1 (|~x| = 1) P-a.s.
ET (0, n) T (0, n~x) = lim = µ(F) . n→∞ n→∞ n n lim
(1.2)
In addition, if E( min {τj }2 ) < ∞ j=1,2,3
holds and µ(F) > 0 then for all ǫ > 0 P-a.s. there exists t0 > 0 such that for all t > t0 (1 − ǫ)tD(1/µ) ⊆ B0 (t) ⊆ (1 + ǫ)tD(1/µ) ,
where D(r) := {x ∈ R2 : |x| ≤ r}.
Two natural questions arise from (1.2). • When is µ(F) > 0? • What is the right order of T (0, n) − µn?
It is expected that F(0) < pc := inf{p > 0; θ(p) = 1} , where θ(p) is the probability that bond percolation on D occurs with density p, is a sufficient and necessary condition to have µ(F) > 0 (Kesten [12]). Heuristics arguments indicate that FPP models belongs to the KPZ universality class, which brings the conjecture that the right order is nχ , where χ = 1/3 (Kadar, Parisi and Zhang [10], Krug and Spohn [14]). However the only models for which this has been proved are certain growth models related to random permutations (Baik, Deift and Johansson [2], Johansson [9]). For lattice FPP Kesten [13] showed that χ ≤ 1/2 (see also Alexander [1]), and for euclidean FPP Howard and Newman [8] showed the same upper bound. In this work we prove some results related to the questions above. To state them we require some definitions involving a bond percolation model. Let A and B be two subsets of R2 . Let [x, y] denote the line segment connecting x to y. We say that a self-avoiding path γ ∗ = (v1∗ , ..., vk∗ ) in V is a path connecting A to B ∗ if [v1∗ , v2∗] ∩ A 6= ∅ and [vk−1 , vk∗ ] ∩ B 6= ∅. The bond percolation model on the Voronoi Tessellation V, with parameter p and probability law P∗p , is constructed by choosing each edge e∗ in Ve to be open independently with probability p. An open path is a path composed by open edges. For each R > 0 denote by AR the event that there exists an open path γ = (vj∗ )1≤j≤h in V, connecting {0} × [0, R] to {3R} × [0, R], and with vj∗ ∈ [0, 3R] × [0, R] for all j = 2, . . . , h − 1. Define the function η ∗ (p) := lim inf P∗p (AR ), R→∞
and consider the percolation threshold, p∗c := inf{p > 0 : η ∗ (p) = 1} < 1 (the inequality in (1.3) will follow from Proposition 3.1).
(1.3)
4
LEANDRO P. R. PIMENTEL
For each κ > 1/2 we set ν = ν(κ) := (4κ − 2)/7. For the remainder of the paper we will use the symbols aj , cj , bj for j = 1, 2, . . . to represent strictly positive constants, whose value may change from appearance to appearance but will not depend on n, r, s or t in the notation we will follow here. Theorem 1. If F(0) < 1 − p∗c and (1.1) holds then 0 < µ(F) < ∞ . If (1.1) is strengthened to aτ
E(e ) =
Z
(1.4)
eat dF(t) < ∞ for some a > 0
(1.5)
then: • For all ǫ > 0 there exists n0 > 0 such that for all n > n0 V(T (0, n)) ≤ n1+ǫ ;
(1.6)
• For all κ > 1/2 there exist constants cj > 0 such that for all n ≥ 1 and r ∈ [0, c1 nκ ] ν
and
P(|T (0, n) − ET (0, n)| ≥ rnκ ) ≤ e−c2 r ,
(1.7)
µn ≤ ET (0, n) ≤ µn + c3 nκ (log n)1/ν .
(1.8)
We note that our condition to get µ(F) > 0 should be equivalent to F(0) < pc , since it is expected that pc + p∗c = 1 (duality) for many planar graphs1. From Theorem 1, the proof of which is given in Section 4, we get an upper bound for the fluctuations of T (0, n) about µn and an improved shape theorem. Corollary 1.1. If F(0) < 1 − p∗c and (1.5) holds then for all κ > 1/2 there exist constants cj > 0 such that for all r ∈ [c3 (log n)1/ν , c1 nκ ] c2 ν
P(|T (0, n) − µn| ≥ rnκ ) ≤ e− 2 r .
(1.9)
Proof of Corollary 1.1. (1.9) follows immediately from (1.7) and (1.8). Corollary 1.2. If F(0) < 1 − p∗c and (1.5) holds then for all κ > 1/2, P-a.s. there exists t0 > 0 such that for all t > t0 (t − tκ )D(1/µ) ⊆ B0 (t) ⊆ (t + tκ )D(1/µ) . 1It
follows from (1.4) that pc + p∗c ≥ 1 holds (Pimentel [17])
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
5
Proof of Corollary 1.2. Notice that, for large |x|, if
|T (0, x) − µ|x|| > |x|κ
then
|T (0, z) − µ|z|| > |z|κ /4 or max T (x, z) > |z|κ /4 . |x−z|≤1
(1.10)
Under (1.5), one can easily see that P-a.s. the event in the right hand side of (1.10) occurs for finitely many z ∈ Z2 . From (1.9) (and the Borel-Cantelli lemma) it follows that P-a.s. the event in the left-hand side of (1.10) occurs for finitely many z ∈ Z2 . Therefore, P-a.s. there exist M > 0 such that for all x ∈ R2 with |x| > M |T (0, x) − µ|x|| ≤ |x|κ ,
which yields Corollary 1.2 (further details are left to the reader). Overview. In Section 2 we will study some geometrical aspects of Voronoi tilings and of self-avoiding paths on Delaunay triangulations, which will play an important role in control the asymptotic behavior of first-passage times. In Section 3 we explore the duality between bond percolation models on Delaunay triangulations and on Voronoi tessellations to relate the size of F(0) and the value of µ(F). The fluctuations of the first-passage time about its asymptotic value are considered in Section 4. 2. Preliminaries The main tool we handle to control the fluctuations of the first-passage time about its asymptotic value is the method of bounded increments, which was also considered by Kesten [13] in the lattice FPP context, and also by Howard and Newman [8] in the euclidean FPP context, to study the same question. This method represents T (0, n) − ET (0, n) as a sum of martingales increments and, after estimating these increments, applies standard bounds for martingales with bounded increments. One of the great difficulties here is that, as distinct from the lattice context, a local increment in a configuration ω is felt not only by the travel times but also by the (random) graph, which has a long range dependence. For this reason we shall define a truncation of the Poisson process. For z ∈ Z2 , r > 0 and s ∈ {j/2 : j ∈ N} let
2 Bs,r z := rz + [−sr, sr] .
2
2
(2.11)
Order the points of Z in some arbitrary fashion, say Z := {u1 , u2 , . . . }. Let δ > 0 be a fixed parameter which value will be specified later 2. Let n ≥ 1 and for each k ≥ 1 let δ
Bnk := Bu1/2,n . k
2Its
value will depend on ǫ and κ in Theorem 1
6
LEANDRO P. R. PIMENTEL
Let |A| denote the number of elements belonging to the set A. Define the point process Pn := Pn (P) (whose distribution will also depend on δ) as follows: • If |Bnk ∩ P| = 0 then set Bnk ∩ Pn := {vk } where vk is a random point uniformly distributed in Bnk ; 2δ 2δ • If |Bnk ∩ P| > 4n2δ then set Bnk ∩ Pn := {vk1 , . . . , vk4n } where (vk1 , . . . , vk4n ) is a sequence of 4n2δ points uniformly chosen in Bnk ∩ P. We make the convention P∞ = P and denote by Dn the Delaunay Triangulation based on Pn . 2.1. Renormalization and full boxes. The geometry of Voronoi tilings is study through renormalization ideas. In a few words, it consists in defining a larger box to be a good box if some property is satisfied in a neighborhood of that box. In the course of the proofs we shall utilize different definitions of good boxes but all of them will carry the notion of full box. 1/2,r
Formally, we divide a square box B = Bz into thirty-six sub boxes of the same length, say B1 , . . . , B36 . We stipulate B is a full box, with respect to the point configuration Pn , if all those thirty-six sub boxes have at least one point belonging to Pn (Figure 2). In other words, [B = ∪36 j=1 Bj is a full box ] := [Bj ∩ Pn 6= ∅ ∀ j = 1, . . . , 36 ] .
(2.12)
Figure 2. Renormalization: a full box 1/2,r
1/2,r
We say that Λ := (Bz1 , . . . , Bzk ) is a circuit of boxes if (z1 , . . . , zk ) is a circuit in Z2 . Let λ be the closed polygonal path composed by the line segments connecting rzj to rzj+1, where j = 1, . . . , k − 1, together with [zk , z1 ]. To each circuit Λ we associate four subsets of the plane: Λout will denote the (topological) interior of the unbounded 1/2,r component of R2 \ ∪kj=1 Bzj , while Λin will denote the interior of the bounded component 1/2,r of R2 \ ∪kj=1 Bzj ; λout will denote the interior of the unbounded component of R2 \λ, while λin will denote the interior of the bounded component of R2 \λ.
ASYMPTOTICS FOR FIRST-PASSAGE TIMES 1/2,r k )j=1
Lemma 2.1. Assume that Λ := (Bzj to Pn .
7
is a circuit composed by full boxes with respect
If Cv ∩ Λin 6= ∅ ⇒ Cv ⊆ λin .
(2.13)
If Cv ∩ Λout 6= ∅ ⇒ Cv ⊆ λout .
(2.14)
Proof of Lemma 2.1. By convexity of Voronoi tiles, if (2.13) does not hold then there exist x1 ∈ ∂Λin ∩ Cv and x2 ∈ ∂λin ∩ Cv (∂A denotes the boundary of the set A) and thus r/2 ≤ |x1 − x2 | . 1/2,r k )j=1
is a full box, there exist v1 , v2 ∈ Pn so that √ √ |v1 − x1 | ≤ 2r/6 and |v2 − x2 | ≤ 2r/6 .
Since every box in (Bzj
Although, x1 and x2 belong to Cv and so |v − x1 | ≤ |v1 − x1 | and |v − x2 | ≤ |v2 − x2 | . Thus,
√ r/2 ≤ |x1 − x2 | ≤ |x1 − v| + |x2 − v| ≤ 2r/3 , √ which leads to a contradiction since 2/3 < 1/2. By an analogous argument, one can prove (2.14). For each A ⊆ R2 let GPn (A) be the sub-graph of Dn composed by edges e = (v1 , v2 ) so that Cvi ∩ A 6= ∅ for i = 1, 2. We denote by τ (A) the sequence of passage times associated to edges in GPn (A).
Given a random variable X and a measurable event F we denote by X | F the random variable X condition on the event F . 1/2,r
Lemma 2.2. Let FΛ be the event that Λ := (Bzj )kj=1 is a circuit composed by full boxes with respect to Pn . Assume that X is a random variable which only depend on (GPn (Λin ), τ (Λin )) and that Y is a random variable which only depend on (GPn (Λout ), τ (Λout )). Then X | FΛ is independent of Y | FΛ . Proof of Lemma 2.2. It follows from Lemma 2.1.
8
LEANDRO P. R. PIMENTEL
2.2. Greedy lattice animals and site percolation. In this section we introduce two models which will also play an important rule in the study of Voronoi tilings. The first one, named Greedy lattice animals and introduced by Cox, Gandolfi, Griffin and Kesten [3]), consists of the following: a lattice animal is a connected subset of Z2 containing the origin. Let Φr and Φr denote the set of animals with at most r sites and with at least r sites, respectively. Let X := {Xz : z ∈ Z2 } be a collection of i.i.d. non-negative random variables and define X Ms (X) := max{ Xz : A ∈ Φs } . (2.15) z∈A
A greedy lattice animal is a lattice animal which attains the maximum in the above definition. Lemma 2.3. If E(ecX0 ) < ∞ for some c > 0
(2.16)
holds then there exist b1 , b2 > 0 such that
P(Ms > b1 s) ≤ e−b2 s . Proof of Lemma 2.3. Under (2.15), E(Ms ) ≤ bs for some b > 0 (Cox, Gandolfi, Griffin, Kesten [3]). From Corollary 8.2.4 of Talagrand [18], if υs denotes a median of Ms then P(|Ms − υs | > u) ≤ 4 exp(−k0 min{s−1 u2 , u}) . for some constant c0 > 0. By integrating in u from 0 to ∞ both sides of the last inequality we get that, for some constant ¯b > 0, √ |EMs − υs | ≤ E|Ms − υs | ≤ ¯b s .
and so
P(|Ms − EMs | > u) ≤ exp(−c0 min{s−1 u2 , u}) ,
√ if u > 2¯b s. This implies that
P(Ms ≥ 2bs) ≤ P(|Ms − EMs | > bs) ≤ e−bs/2c0 ,
(2.17)
which yields Lemma 2.3. Now, for each n ≥ 1 consider Y n := {Yzn : z ∈ Z2 }, a collection of Bernoulli random variables. Let φ := inf inf2 P(Yzn = 1) . n≥1 z∈Z
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
9
Assume that this collection is l-dependent: {Yzn : z ∈ A} and {Yzn : z ∈ B} are independent whenever l ≤ d∞ (A, B) := min{|z − z′ |∞ : z ∈ A and z′ ∈ B} .
This defines a family of l-dependent site percolation models index by n ∈ [1, ∞] (a site z is called open if Yzn = 1). Here we are interested on the density of open sites in the set of lattice animals with at least s sites. For more details in the subject see the reference book of Grimmet [4]. For each animal A we denote by ΞkA the set of all B ⊆ A such that if z, ¯z ∈ B then d∞ (z, ¯z) ≥ k. Let X mk (A, Y n ) := max{ Yzn : B ∈ ΞA } z∈B
and mks (Y n ) := min{mk (A, Y n ) : A ∈ Φs } .
(2.18)
Lemma 2.4. For all l ≥ 1 there exists p0 ∈ (0, 1) such that if φ > p0 then for all k ≥ 1 there exist b3 , b4 > 0 such that for all n ∈ [1, ∞] and s ≥ 1, P(mks (Y n ) < b3 s) ≤ e−b4 s .
(2.19)
Proof of Lemma 2.4. By Theorem 0.0 of Ligget, Schomman, Stacey [15], the family {Y n : n ∈ [1, ∞]}
is stochastically dominated from below by a collection of i.i.d Bernoulli random variables with parameter ρ(φ) → 1 when φ → 1.
Therefore, Lemma 2.4 will follow if we prove that (2.19) holds for mks (Y ), where Y := {Yz : z ∈ Z2 } is a collection of i.i.d Bernoulli random variables with parameter ρ sufficiently close to 1. Since m1s (Y ) ≤ c(k)mks (Y ), for some constant c(k) (only depending on k), we may restrict our attention to k = 1. P For each animal A, z∈A Yz is a binomial random variable with parameter |A| and ρ. Since (the number of animals A with |A| = j is at most C j , for some constant C > 0) we have at most cj animals with |A| = j, we get that X P(ms (Y )1 < bs) = P(∃ A : Yz < bs and |A| ≥ s) z∈A
≤
X j≥s
c
j
bs X l=0
Clj ρl (1 − ρ)j−l ≤ (1 − ρ)−bs
X j≥s
[2c(1 − ρ)]j
(1 − ρ)−bs [2c(1 − ρ)]s exp{s log 2c + s(1 − b) log(1 − ρ)} = 1 − 2c(1 − ρ) 1 − 2c(1 − ρ) if b < 1 and ρ > 1 − 1/2c. Thus, if we fix ρ > 1 − 1/2c and =
b3 < 1 + log 2c[log(1 − ρ)]−1
10
LEANDRO P. R. PIMENTEL
then we can find b4 = b4 (ρ) > 0 so that (2.19) holds for m1s (Y ). 2.3. Self-avoiding paths on Delaunay triangulations. Let Cr and C r denote the set of self-avoiding and finite paths γ in Dn , starting from v(0), with |γ| ≥ r and with |γ| ≤ r, respectively. Fix L > 0 and for each path γ in Dn let A(γ) ⊆ Z2 be the set of points z ∈ Z2 so that ∩ [v, v′] 6= ∅ for some v, v′ ∈ γ (Figure 3). Define
1/2,L Bz
grL(Pn ) := min{|A(γ)| : γ ∈ Cr (Pn )}
and GLr (Pn ) := max{|A(γ)| : γ ∈ C r (Pn )} .
Figure 3. γ and A(γ) Let x, y ∈ R2 and define the path γ(x, y) := (v1 , ..., vk ) in D(Pn ) as follows: set v1 := v(x); if v1 6= v(y) then let v2 be the (almost-surely) unique nearest neighbor of v1 such that the edge of Cv1 , that is perpendicular to [v1 , v2 ], crosses [x, y]; otherwise we set k = 1 and the construction is finished; given vl with l ≥ 1, if vl 6= v(y) then we set vl+1 to be the unique nearest neighbor of vl different from vl−1 such that the edge of Cvl , that is perpendicular to [vl , vl+1 ], crosses [x, y]; otherwise we set k := l and the construction is finished. We denote γr := γ(0, r1). Proposition 2.1. For each L > 1 there exist cj = cj (L) > 0 such that for all n ∈ [1, ∞], for all r > 0, P(grL (Pn ) < c1 r) ≤ e−c2 r , (2.20) and
P(GLr (Pn ) > c3 r) ≤ e−c4 r .
(2.21)
Proposition 2.2. There exists z0 > 0 such that, for all n ∈ [1, ∞], for all r > 0 and z > z0 P(|γr (Pn )| > zr) ≤ e−c1 zr . (2.22)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
11
Proof of Proposition 2.1. First notice that to prove (2.20) we can restrict the attention to 1/2,1 L = 1. Let Xzn := |Bz ∩ Pn | and denote X n := {Xzn : z ∈ Z2 }. Consider the random variable Msn := Ms (X n ) defined in (2.15). By definition, if gr1 (Pn ) ≤ cr then there exists a path γ ∈ Cr (Pn ) with |A(γ)| ≤ cr. Thus X n Mcr ≥ |B1/2,1 ∩ Pn | ≥ |γ| ≥ r , z z∈A(γ)
which yields that n P(gr1(Pn ) ≤ cr) ≤ P(Mcr ≥ r) .
(2.23)
By the definition of Pn , for each z ∈ Z2 and then
|B1/2,1 ∩ Pn | ≤ |B1/2,1 ∩ P∞ | + 1 , z z Mrn ≤ Mr∞ + r .
Combining this with (2.23), one gets that ∞ P(gr1 (Pn ) ≤ cr) ≤ P(Mcr ≥ (1 − c)r) .
Since (2.16) holds for X ∞ , together with Lemma 2.3, this implies (2.20). The proof of (2.21) is founded on renormalization techniques as follows. Denote Cz the 1/2,L is good box, or circuit composed by sites z′ ∈ Z2 with |z − z′ |∞ = 1. We say that Bz 1/2,L n equivalently Yz (L) = 1, if B¯z is a full box with respect to Pn for all z¯ ∈ Cz (see (2.12)). We denote Y n (L) := {Yzn (L) : z ∈ Z2 }.
Lemma 2.5. {Yzn (L) : z ∈ Z2 } is a 3-dependent collection of Bernoulli random variables. Further, for all ǫ > 0 there exists L0 > 0 so that inf inf P(Yzn (L0 )) ≥ 1 − ǫ .
n≥1 z∈Z2
Proof of Lemma 2.5. Since Cz and Cz′ are disjoint if |z − z′ | ≥ 3, the first part of this lemma follows directly from the definition of Pn . 1/2,L
To prove the second part notice that if L/36 ≥ nδ then each sub-box of Bz length L/6 contains at least one box Bkn (used to construct Pn ). Thus
with
P(YzL (Pn ) = 1) = 1 .
Now, if L/36 < nδ and B is a box with length L/6 then P(B ∩ Pn 6= ∅) ≤ P(B ∩ P = 6 ∅) + P(Pn ∩ B 6= P ∩ B) 2
2δ
2 /36
≤ e−L /36 + e−bn ≤ e−L which yields the second part of Lemma 2.5.
2
+ e−b(L/36) ,
12
LEANDRO P. R. PIMENTEL
Consider the random variable m3r (Y n (L)) defined in (2.18). Assume we have a path γ ∈ C r . Then its animal A(γ) contains a subset B ∈ Ξ3A with m3 (Y n (L), A) good boxes. 1/2,L 1/2,L By Lemma 2.1, if Bz is a good box in A(γ) then γ has an edge lying in ∪¯z∈Cz B¯z . 1/2,L 1/2,L On the other hand, if Bz1 and Bz2 are 3-distant then 1/2,L
Thus, we must have that
∪¯z∈Cz1 B¯z
1/2,L
∩ ∪¯z∈Cz2 B¯z
= ∅.
m3 (Y n (L), A) ≤ |γ| ≤ r ,
which implies that
P(GLr (Pn ) > cr) ≤ P(m3cr (Y n (L)) < r) . Together with Lemma 2.5 and Lemma (2.4) this proves (2.21) for large L0 , which finishes the proof of Proposition 2.1. 1/2,L
Proof of Proposition 2.2. For this proof we say that Bz is a good box, or equivalently n n n Zz (L) = 1, if it is a full box. Then Z (L) := {Zz (L) : z ∈ Z2 } is a collection of independent random variables. Let Hnj be the set of closed sites (Zzn (L) = 0) in Z2 which are connected to (j, 0), and ¯ n := {(0, j)} ∪ Hj ∪ ∂Hn . H j
j
¯ n. Also define Ins (Z n (L)) := ∪sj=1 H j By Lemma 2.1,
γr (Pn ) ⊆ ∪z∈Inr/L Bz1/2,L ,
and thus
|γr (Pn )| ≤
X
z∈In r/L
|Bz1/2,L ∩ Pn | .
(2.24)
Since P(ZzL (Pn ) = 0) = o(L) uniformly in n (by the same argument in Lemma 2.5), we can find L0 sufficiently large so that E(exp(|Hn0 |) < ∞. This yields that P(|Ins | > ys) ≤ ec1 ys ,
(2.25)
for y > y0 and c1 , y0 constants whose values do not depend on n. From now on we fix such L0 and write Msn := Ms (Z n (L0 )) (defined in (2.15)) and Ins := Ins (Z n (L0 )). By Lemma 2.3, for sufficiently small c > 0 one can find b1 > 0 so that n ∞ P(McL ≥ zr) ≤ P(McL ≥ (z − 1)r) ≤ e−b1 zr . 0 zr 0 zr
(2.26)
By (2.24), P(|γr (Pn )|
P(|γr (Pn )| ≥ zr) ≤ n ≥ zr , |Ir/L0 | ≤ cL0 z(r/L0 )) + P(|Inr/L0 | > cL0 z(r/L0 )) n P(McL ≥ zr) + P(Inr/L0 > cL0 z(r/L0 )) . 0 zr
≤
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
13
Together with (2.26) and (2.25), this yields (2.22). Remark: the connective function on disordered graphs. Problems related to selfavoiding paths are connected with various branches of applied mathematics such as long chain polymers, percolation and ferromagnetism (Kesten [11], Hammersley [5]). One fundamental problem is the asymptotic behavior of the connective function κr defined by the logarithm of the number of self-avoiding paths (on some graph G) starting at v and with r steps. For planar and periodic graphs subadditivity arguments yields that r −1 κr (v) converges, when r → ∞, to some value κ ∈ (0, ∞) (the connectivity constant) independent of the initial vertex v. In disordered planar graphs subadditivity is lost but, if the underline graph possess some statistical symmetries (ergodicity), we may believed that the rescaled connective function still converges to some constant. From Proposition 2.1 we obtain a linear upper bound for the connective function of the Delaunay triangulation, where we set v to be the closest vertex to the origin: there exist a constant c > 0 so that κr lim sup ≤ c. (2.27) r r→∞ To prove (2.27) recall we have associate to each self-avoiding path γ and ordered sequence A(γ) = (z1 , ..., zn ) of nearest neighbors integer sites, which maybe seen as animal in Z2 (Figure 3). The number of self-avoiding paths associated to an ordered sequence (z1 , ..., zn ) should be at at most |B1/2,1 ∩ P|! × · · · × |B1/2,1 ∩ P|! z1 zn (n! is factorial of n). Now, every self-avoiding path with r steps intersects at most Gr boxes, and thus Nr ≤ 4Gr max {|B1/2,1 ∩ P|! × · · · × |B1/2,1 z1 z|A| ∩ P|!} , |A|≤Gr
which implies that κr = log Nr ≤ (log 4)Gr + max { |A|≤Gr
|A| X i=1
log |B1/2,1 ∩ P|!} . zi
(2.28)
Denote Xz := log |B1/2,1 ∩ P|! , z 2 and let X := {Xz : z ∈ Z } and Ms := Ms (X) (see (2.15)). By (2.28), if a > 1 and c < 1/2 log 4 then
P(κr > ar) ≤ P(Gr > car) + P(Mcar > ar/2) .
(2.29)
14
LEANDRO P. R. PIMENTEL
Since (2.16) holds for X, by Lemma 2.3 there exist b1 , b2 > 0 such that if a > 1 and c < min{1/2 log 4, 1/2b1 } then 2 P(Mcar > ar/2) ≤ e−b2 car .
(2.30)
By Proposition 2.1 we can find c3 , c4 > 0 so that, if ca > c3 then P(Gr > car) ≤ e−c4 r . Combining this with (2.29) and (2.30) we obtain that for a > c3 / min{1/2 log 4, 1/2b1 } P(κr > ar) ≤ e−b2 car + e−c4 r .
Together with Borel-Cantelli lemma, this proves (2.27). 3. Density of open edges in percolation and the time constant In this section we prove a sufficient condition on the passage time distribution F to ensure that µ(F) > 0. We also show that the same condition ensures the existence of geodesics. These results will follow from the subsequent propositions. Proposition 3.1. There exist p0 ∈ (0, 1) so that if p ∈ (p0 , 1) then η ∗ (p) = lim inf P∗p (AR ) = 1 . R→∞
In particular,
p∗c
< 1.
Let tr := inf{
X e∈γ
τe : γ ∈ Cr (Pn )} .
(3.31)
Proposition 3.2. If F(0) < 1 − p∗c then there exist constants cj > 0 such that for all n sufficiently large, for all r ≥ 1 P(tr ≤ c1 r) ≤ e−c2 r . 1/2,L
Proof of Proposition 3.1. Now we say that Bz if the following holds:
is a good box, or equivalently Vz (L, p) = 1,
1/2,L
1/2,L
• B¯z is a full box, with respect to P, and |B¯z |¯ z − z|∞ ≤ 1 ; 1/2,L • Xe∗ = 1 for all e∗ ∈ Ve which intersects B¯z .
∩ P| ≤ 4L2 for all ¯z with
We denote B1 (z, L) and B2 (z, L) the event specified in the first and in the second item above, respectively. The law of large numbers implies that lim P(B1 (z, L)) = 1 .
L→∞
(3.32)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
15
1/2,L
Since B¯z is a full box for all z¯ with |¯ z − z|∞ ≤ 1, by Lemma 2.1, the set of edges 3/2,L 1/2,L ∗ is contained in B¯z . Let V , E and F be the number of e ∈ Ve intersecting B¯z 3/2,L vertexes, edges and faces of the Delaunay triangulation on B¯z , respectively. Then 3E = 2F and, by the Euler formula, V −E +F = 2. 3/2,L has the same order as of vertexes in the This yields that the number of edges in B¯z same box. Thus, for some constant b > 0, P(B1 (z, L) ∩ B2 (z, L)c ) ≤ bL2 (1 − p) . Together with (3.32), this yields that for all δ > 0 we can chose L sufficiently large, and then p = p(L, δ) sufficiently close to 1, to have that P(B1/2,L is a good box ) > 1 − δ . z
(3.33)
From Lemma 2.2 we have that V := {Vz (L, p) : z ∈ Z2 } is a collection of 3-dependent random variables. Notice that by a similar argument to construct γr (see subsection 2.3) and by using Lemma 2.1 one can get that [∃ a crossing composed by good boxes of [0, 3R/L] × [0, R/L]] ⊆ [∃ an open crossing, with respect to X, of [0, 3R] × [0, R]] .
Combining (3.33) with 3-dependence, one can find L0 large enough and then p0 sufficiently close to 1 so that the probability of the first event in the above inclusion goes to 1 as R goes to infinity, which finishes the proof of Proposition 3.1. Proof of Proposition 3.2. First we claim it suffices to prove Proposition 3.2 if P(τe = 1) = p = 1 − P(τe = 0). To see this, assume that F(0) < 1 − p∗c . Since F is right-continuous, we can chose ǫ > 0 small so that F(ǫ) < 1 − p∗c . Let τeǫ := I(τ≥ ǫ) and denote tǫr the infimun defined in (3.31) but considering the travel times τeǫ . Therefore
which yields the last claim.
P(τeǫ = 0) < 1 − p∗c and ǫtǫr ≤ tr ,
Now assume that P(τe = 1) = p = 1 − P(τe = 0). Let L > 0 and z ∈ Z2 and define that is a good box if the following holds:
1/2,L Bz
1/2,L
• For all z′ ∈ Z2 with |z − z′ | = 2, Bz′
is a full box;
16
LEANDRO P. R. PIMENTEL 1/2,L
• For all γ ∈ C(Bz
3/2,L
, ∂Bz
) we have t(γ) ≥ 1 .
Let Wzn (L) := I(B1/2,L is a good box ) , z and p(n, L) := inf2 P(Wzn (L) = 1) . z∈Z
Lemma 3.1. W n (L) := {Wzn (L) : z ∈ Z2 } is 5-dependent for all L > 0. Moreover, if P(τe = 0) < 1 − p∗c then for all ǫ > 0 there exists Lǫ > 0 (independent of n) so that for all n ≥ Lǫ 1 − ǫ < p(n, Lǫ ) ≤ 1 . Proof of Lemma 3.1. 5-dependence follows from Lemma 2.2. Now consider the event 1/2,L
HzL,n := [Bz′
is a full box (with respect to Pn ) ∀ |z − z′ | = 2]
and the event Then
GL,n := [t(γ) ≥ 1 ∀ γ ∈ C(Bz1/2,L , ∂Bz3/2,L )] . z c P(Wzn (L) = 0) ≤ P((HzL,n )c ) + P((GL,n z ) ).
(3.34)
As we have seen before, P((HzL,n )c ) → 0 when L → ∞ uniformly in n .
This yields Lemma 3.1 if we prove that if P(τe = 0) < 1 − p∗c then for all ǫ > 0 there exists Lǫ > 0 so that for all n ≥ Lǫ 1 − ǫ < P(GLz ǫ ,n ) ≤ 1 . (3.35) We first prove (3.35) for n = ∞. Let Xe∗ := τe , where e∗ is the edge in Ve (with point configuration P) dual to e. Then {Xe∗ : e∗ ∈ Ve } defines a bond percolation model on V with law P∗p . Let RL1 := [L/2, 3L/2] × [−3L/2, 3L/2] , RL2 := [−3L/2, 3L/2] × [L/2, 3L/2] , RL3 := [−3L/2, −L/2] × [−3L/2, 3L/2] , RL4 := [−3L/2, 3L/2] × [−3L/2, −L/2] .
Let AiL be the event that there exists an open crossing in V, as in the definition of the event AL , but now translate to the rectangle RLi . 1/2,L
Denote by FL the event that an open circuit σ ∗ in V which surrounds B0 3/2,L inside B0 does not exist. Thus one can see that ∩4i=1 AiL ⊆ (FL )c .
and lies
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
17 1/2,L
Notice also that, if there exists an open circuit σ ∗ in V which surrounds B0 and lies 3/2,L 1/2,L 3/2,L inside B0 then every path γ connecting B0 to ∂B0 has an edge crossing with σ ∗ , and thus t(γ) ≥ 1. Together with translation invariance, this yields P((GL,∞ )c ) ≤ P∗p (FL ) ≤ 4(1 − P∗p (AL )) . 0
(3.36)
Since p > p∗c , by using (3.36) and the definition of p∗c one can obtain (3.35) for n = ∞. Now,
2δ
P(P∞ 6= Pn in B5/2,L ) ≤ c1 L2 e−c2 n . z for some constant c1 > 0. Thus, for all n ≥ L 2δ
c L,∞ c P((GL,n ) ) + c1 L2 e−c2 L . z ) ) ≤ P((Gz
(3.37)
By (3.36) and (3.37), given ǫ > 0 we can find Lǫ so that for all n ≥ Lǫ P((GLz ǫ ,n )c ) ≤ ǫ ,
which finishes the proof of this lemma.
Now consider the random variable m5r (W n (L)) defined by (2.18). Combining Lemma 2.4 with Lemma 3.1, one obtains that there exists L0 > 0 and b1 , b2 > 0, whose value depends only on L0 , such that for all n ∈ [1, ∞], for all r > 0 P(m5r (W n (L0 )) ≤ b1 r) ≤ e−b2 r .
(3.38)
Let γ ∈ Cr and consider the animal A(γ) previously defined. Then A(γ) contains a 1/2,L 1/2,L subset B ∈ Ξ5A with m5 (A(γ), W n (L)) good boxes. If Bz1 and Bz2 are 5-distant good boxes in A(γ) then (by definition) for each j = 1, 2 there exist a piece of γ, say γj , 1/2,L 3/2,L connecting ∂Bzj to ∂Bzj and so that t(γj ) ≥ 1. Since these good boxes are 5-distant, by Lemma 2.1, γ1 and γ2 must be disjoints. This yields that t(γ) ≥ 2. By repeating this argument inductively, one gets that m5 (A(γ), W n (L0 )) ≤ t(γ) ≤ r . Therefore, P(tr ≤ cr) ≤ P(gr < br) + P(m5br (W n (L)) ≤ cr) . Together with (3.38) and Proposition 2.1, this yields Proposition 3.2. Corollary 3.1. If F(0) < 1 − p∗c then there exist constants cj > 0 such that for all n sufficiently large, for all r > 0 P(T (0, r) < c1 r) ≤ e−c2 r .
In particular, if F(0) < 1 − p∗c then µ(F) > 0.
(3.39)
18
LEANDRO P. R. PIMENTEL
Proof of Corollary 3.1. If we have a path γ connecting 0 to r then it intersects at least r 1/2,1 unit boxes Bz . Thus, if b > 0 then P(T (0, r) < c1 r) ≤ P(Gbr > r) + P(tbr ≤ c1 r) . Combining Proposition 3.2 with Proposition 2.1, one can get (3.1). The second part follows from (3.1), together with Borel-Cantelli Lemma. Corollary 3.2. If F(0) < 1 − p∗c then for all x, y ∈ R2 P-a.s. there exist at least one geodesic connecting x to y. Further, if F is continuous, then we have P-a.s. uniqueness. Proof of Corollary 3.2. Let ¯ ) : |¯ dr (v) := inf{T (v, v v| ≥ r} . By Proposition 2.1 and Proposition 3.2, we can find sufficiently small c > 0 so that P-a.s. 0 < c < lim inf r
dr (v) . r
In particular, dr (v) → ∞ when r → ∞.
¯ ∈ Pn then we can find r large enough so that dr (v) > T (v, v ¯ )+1. Therefore, if we fix v, v ¯ will be attained by a path in the finite This implies that the passage time between v and v ¯ and lying inside the ball centered at v and with radius collection of paths connecting v to v r. To complete the proof,P notice thatPif F is a continuous function then P-a.s. there are no finite path γ and γ¯ with e∈γ τe = e∈¯γ τe .
4. Martingales with bounded increments and the rate of convergence We formulate the abstract martingale estimate used by Howard and Newman [8]. Lemma 4.1. Let (Ω, F , P) be a probability space and let {Fk }k≥0 be an increasing family of σ-algebras of measurable sets. Let {Mk }k≥0 , M0 = 0, be a martingale with respect to the filtration {Fk }k≥0 and let {Uk }k≥0 be a collection of positive random variables that are F -measurable. Assume that the increments ∆k = Mk − Mk−1 satisfy |∆k | ≤ c, for some constant c ≥ 0,
(4.40)
E(∆2k | Fk−1) ≤ E(Uk | Fk−1).
(4.41)
and
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
19
Assume further that for some finite constants 0 < c1 , υ, x0 with x0 ≥ c2 , we have that for all x ≥ x0 , X P( Uk > x) ≤ c1 exp(xυ ). (4.42) k≥1
Then almost surely M = limk→∞ Mk exists. Moreover, there exists constants 0 < c2 , c3 , whose values do not depend on c and x0 , such that for all x ≤ xυ0 , √ P (|M| > x x0 ) ≤ c2 exp(−c3 x). (4.43) Proof of Lemma 4.1. See the proof of Lemma 5.2 in Howard and Newman [8] We construct the underline probability space as follows. For each k ≥ 1 let and denote
Ik := {(a, b, c) : a ≥ 1, b ≥ k and c ≥ 1 if b 6= k or c > a if b = k} , b,c ωk := (Nk , (Uk,a)a≥1 , (τk,a )(a,b,c)∈Ik )
where Nk is a Poisson random variable with intensity n2δ , (Uk,a )a≥1 is a sequence of inde1/2,nδ b,c )(a,b,c)∈Ik is a sequence of pendent points uniformly distributed in Bnk = Buk , and (τk,a independent random variables with distribution F. Let (Ωk , Pk , Ak ) be the probability space induced by ωk and set ω = (ωk )k≥1 and ∞ Y Pk . P= k=1
To determine the point process P, we put Nk points in the box Bnk given by Uk,1 , . . . , Uk,Nk . For each e ∈ De we know there exists k ≥ 1 and (a, b, c) ∈ Ik so that e = (Uk,a , Ub,c ) and b,c so we set τe := τk,a . We shall use the following σ-fields: F0 := {∅, Ω}
Fk := σ-field generated by ω1 , . . . , ωk . Let Tn := T (0, n). The martingale representation of Tn − ETn is ∞ X Tn − ETn = {E(Tn | Fk ) − E(Tn | Fk−1)} . k=1
This representation is valid because M0 := 0 and Ml :=
l X k=1
{E(Tn | Fk ) − E(Tn | Fk−1 )}
= E(Tn | Fl ) − ETn
20
LEANDRO P. R. PIMENTEL
defines a Fk -martingale that, under (1.5), converges a.s. (and also in L2 ), to Tn − ETn . The increments of Ml are
∆k = E(Tn | Fk ) − E(Tn | Fk−1) ,
and the main step is to estimate E(∆2k | Fk−1 ).
4.1. Successive approximations. In order to satisfies the prescription of Lemma 4.1 we shall modified the configuration ω. We define ω ¯ k := ω ¯ k (n, ωk ) as follows: we change the point configuration P ∩ Bnk to Pn ∩ Bnk ; the travel time configuration is changed by putting travel times 3 b,c b,c τ¯k,a := min{τk,a , 8b−1 log n} .
¯ (¯ We note that Proposition 3.2 still holds for tr (P, τe )D¯e ) (with constants not depending on n), since for large enough n, τ¯e ≥ min{τe , 1}.
We denote ω ¯ := (¯ ωk )k≥1 . When a variable, such as travel times, geodesics and increments, is a function of ω ¯ we will decorate it with a bar. For instance we write T¯r (ω) := Tr (¯ ω ) and ρ¯r := ρr (¯ ω ). Lemma 4.2. If F(0) < 1 − p∗c and (1.5) holds then, and
2δ
P(|Tn − T¯n | > x) ≤ e−c1 n + e−c2 x
(4.44)
E({Tn − T¯n }2 ) ≤ c3 .
(4.45)
Proof of Lemma 4.2. To prove (4.44) we first claim that (for sufficiently large c > 0).
P(¯ ρn 6⊆ [−cn, cn]2 ) ≤ e−c2 n
(4.46)
Lemma 4.3. Under (1.5), there exists x0 , c0 > 0 so that for all x > x0 P(T¯r > xr) ≤ e−c0 xr . Proof of Lemma 4.3. By Proposition 2.2, there exist c1 , z0 > 0 so that if cx > z0 then P(T¯r > xr) ≤ P(T¯r > xr , |¯ γr | ≤ cxr) + e−c1 cxr . (4.47) By independence, P(T¯r > xr , |¯ γr | ≤ cxr) ≤ 3b
> 0 is given by (1.5)
cxr X j=0
P(
X
e∈¯ γr
τ¯e > xr | |¯ γr | = j)P(|¯ γ | = j) ≤
ASYMPTOTICS FOR FIRST-PASSAGE TIMES cxr X j=0
P(
j X i=0
21
τ¯i > xr)P(|¯ γ | = j) ≤ e−xar (Eeaτ )crx ≤ e−axr/2
aτ
if c < a log Ee . Together with 4.47, this yields Lemma 4.3. To prove (4.46) notice that, P(¯ ρn 6⊆ [−cn, cn]2 ) ≤ P(|¯ ρn | > bn) + P(∃ γ ∈ C bn : γ 6⊆ [−cn, cn]2 ) .
(4.48)
Now, P(|¯ ρn | > bn) ≤ P(T¯n > an) + P(T¯n ≤ an and |¯ ρn | > bn) a = P(T¯n > an) + P(tbn ≤ bn) . b Together with Proposition 3.2 and Lemma 4.3, this yield that if we take a > x0 , b > 0 so that a/b < c1 (given by Proposition 3.2) then for some constant c2 > 0.
P(|¯ ρn | > bn) ≤ ec2 n ,
(4.49)
By Proposition 2.1, if we take c sufficiently large, P(∃ γ ∈ C bn : γ 6⊆ [−cn, cn]2 ) ≤ P(Gbn ≥ cn)
will also decays exponentially fast with n. Together with (4.48) and (4.49), this proves (4.46). Turning back to the proof of (4.44), assume that ρ¯n ⊆ [−cn, cn]2 and P = Pn . Thus |T¯n − Tn | ≤ |T¯n | + |Tn | ≤ 2t(¯ γ )n ) .
This yields,
P(|T¯n − Tn | > x) ≤ P(P = 6 Pn in [−c1 n, c1 n]2 )+ P(¯ ρn 6⊆ [−cn, cn]2 ) + P(
X
[−c1 n,c1
τe I(τe > 8a−1 log n) > x) .
(4.50)
n]2
Notice that, since the mean number of Poisson points inside Bkn is n2δ , by using simple large deviations results for Poisson process we have that for each b > 0 there exists a constant b1 = b1 (b) > 0 such that for all n ∈ [1, ∞) 2δ
P(P ∩ [−bn, bn]2 6= Pn ∩ [−bn, bn]2 ) ≤ e−b1 n .
(4.51)
Let B ⊆ R2 and denote EB (P) the number of edges (v, v′) in D so that v, v′ ∈ B. Since the number of edges is comparable with the number of vertexes (see the proof of
22
LEANDRO P. R. PIMENTEL
Proposition 3.1) we have that for each b > 0 there exists b1 = b1 (b) > 0 such that for all r>0 2 (4.52) P(E[−br,br]2 (P) > 2(br)2 ) ≤ e−b1 r . To estimate the last term in the right hand side of (4.50) notice that, by (4.52), X τe I(τe > 8a−1 log n) > x) ≤ P( P(
X
[−c1 n,c1 n]2
[−c1 n,c1 n]2
2
τe I(τe > 8a−1 log n) > x | E[−c1 n,c1n]2 ≤ b1 n2 ) + eb2 n .
By using a similar procedure presented in Kesten [13] (see (2.37) there) one can prove that X τe I(τe > 8a−1 log n) > x | E[−c1 n,c1n]2 ≤ b1 n2 ) ≤ e−c1 x . P( [−c1 n,c1 n]2
Together with (4.46), (4.50) and (4.51), this yields (4.44). By Lemma 4.3, we also have that P(|T¯n − Tn | > ny) ≤ P(T¯n > ny/2) + P(Tn > ny/2) ≤ e−byn ,
if y ≥ y0 . Thus
E({T¯n − Tn }2 ) = 2n2
Z
(4.53)
∞
P(|T¯n − Tn | > ny)ydy 0 Z y0 Z ∞ 2 ¯ = 2n [ P(|Tn − Tn | > ny)ydy + P(|T¯n − Tn | > ny)ydy] . 0
y0
By (4.44) and (4.50), the last term is bounded by a constant and this completes the proof of Lemma 4.2 4.2. Main estimate. We introduce the the following notation: if ω = (ωk ), σ = (σk ) ∈ Ω then we denote [ω, σ]k := (ω1 , . . . , ωk , σk+1 , σk+2 , . . . ) . We also set ∞ Y Pk . νk := j=k
Since
Z
we have ¯k = ∆
T¯n [ω, σ]k νk+1 (dσ) = Z
Z
T¯n [ω, σ]k νk+1 (dσ) −
T¯n [ω, σ]k νk (dσ) , Z
T¯n [ω, σ]k−1νk (dσ)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
=
Z
23
{T¯n [ω, σ]k − T¯n [ω, σ]k−1}νk (dσ) .
Let Fk be the event that a geodesic from 0 to n has a vertex in Bnk . Let Ik be the indicator function of Fk and denote I¯k (ω) := Ik (¯ ω ). Lemma 4.4. |T¯n [ω, σ]k − T¯n [ω, σ]k−1| ≤ c1 n2δ log n × max{I¯k [ω, σ]k , I¯k [ω, σ]k−1} . Proof of Lemma 4.4. By Lemma 2.1, if I¯k [ω, σ]k = 0 then ρ¯n [ω, σ]k is also a path in D[ω, σ]k−1. Thus Tn [ω, σ]k−1 ≤ t(ρn [ω, σ]k ) = Tn [ω, σ]k . Analogously, if I¯k [ω, σ]k−1 = 0 then
Tn [ω, σ]k ≤ t(ρn [ω, σ]k−1) = Tn [ω, σ]k−1 . Therefore, max{I¯k [ω, σ]k , I¯k [ω, σ]k−1} = 0 ⇒ |T¯n [ω, σ]k − T¯n [ω, σ]k−1| = 0 .
(4.54)
Now assume I¯k [ω, σ]k = 1. Let vf (resp. vl ) be the first (resp. the last) vertex v ∈ δ ρn [ω, σ]k so that Cv ∩ Bz13,n . Define k ψn := ρ(0, vf )γ(vf , vu )ρ(vf , n)
which is the concatenation of these three paths. δ
By Lemma 2.1 and the definition of Pn , which ensures that Bu13,n has enough points k 13,nδ c n to isolate Bk from GPn ((Buk ) ) (see sub-section 2.1), D[ω, σ]k and D[ω, σ]k−1 coincide 2δ . Thus, ψn is also a path in D[ω, σ]k−1 and so outside Bu13,n k Tn [ω, σ]k−1 ≤ t(ψn ) = T (0, vf )[ω, σ]k + t(γ(vf , vl ))[ω, σ]k−1 + T (vl , n)[ω, σ]k ≤ Tn [ω, σ]k + t(γ(vf , vl ))[ω, σ]k−1 .
Analogously, if I¯k [ω, σ]k−1 = 1, then
Tn [ω, σ]k ≤ t(ψn ) = T (0, vf )[ω, σ]k−1 + t(γ(vf , vl ))[ω, σ]k + T (vl , n)[ω, σ]k−1 ≤ Tn [ω, σ]k−1 + t(γ(vf , vl ))[ω, σ]k .
Therefore,
|T¯n [ω, σ]k − T¯n [ω, σ]k−1| ≤
max{t(γ(vf , vl ))[ω, σ]k , t(γ(vf , vl ))[ω, σ]k−1} max{I¯k [ω, σ]k−1, I¯k [ω, σ]k } . Lemma 2.1 and the definition of Pn also ensures that,
max{|γ(vf , vl ))[ω, σ]k |, |γ(vf , vl ))[ω, σ]k−1|} ≤ c1 n2δ ,
(4.55)
24
LEANDRO P. R. PIMENTEL
for some constant c1 > 0. Together with and (4.55), this implies Lemma 4.4.
τ¯e ≤ 8a−1 log n
Let Uk := 2(c1 )2 n4δ (log n)2 I¯k , where c1 is the constant given by Lemma 4.4. Lemma 4.5. If F(0) < 1 − p∗c and (1.5) holds then for all sufficiently large n ∞ X E( Uk ) ≤ n1+5δ .
(4.56)
k=1
Further, for all δ ∈ (0, 1/8) for all y ≥ n1+7δ , P(
∞ X k=1
Uk > y) ≤ e−c2 y
1/2
.
Proof of Lemma 4.5. By definition ∞ ∞ X X 4δ 2 Ik ≤ c1 n4δ (log n)2 sup |¯ ρn | . Uk = c1 n (log n) k=1
k=1
Thus P(
∞ X k=1
Ik > ny) ≤ P(sup |¯ ρn | > ny)
≤ P(T¯n ≤ cny) + P(tny < cny) . By Proposition 3.2, if we chose c > 0 sufficiently small, P(tny < cny) ≤ ec¯yn . Therefore, E(
∞ X k=1
Ik ) ≤ E(sup |¯ ρn |) = n
≤n
Z
0
∞
Z
P(T¯n ≥ cny)dy +
∞ 0
Z
P(sup |¯ ρn | > ny)dy ∞
P(tny < cny)dy
0
= E(T¯n ) + c˜n , which proves (4.56). From Proposition 3.2 and Lemma 4.3, if x > c0 n then P(sup |¯ ρn | > x) ≤ P(T¯n ≥ cx) + P(tx < cx) ≤ ebx . Together with (4.56), this yields (4.57).
(4.57)
(4.58)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
25
Proof of Theorem 1. (1.4) follows from Corollary 3.1. Now we prove (1.6): by Lemma 4.4, the Schwarz inequality and the Fubini theorem, Z Z 2 ¯ E(∆k | Fk−1) = { {T¯n [ω, σ]k − T¯n [ω, σ]k−1}νk (dσ)}2 Pk (dω) ≤
c21 n4δ (log n)2 ≤
Z Z { max{I¯k [ω, σ]k , I¯k [ω, σ]k−1}νk (dσ)}Pk (dω)
2c21 n4δ (log n)2
Z
I¯k [ω, σ]k−1νk (dσ) = E(Uk | Fk−1 ) .
(4.59)
From Lemma 4.5 and (4.59), for all n sufficiently large VT¯n =
∞ X k=1
¯ 2 ) ≤ E( E(∆ k
∞ X k=1
Uk ) ≤ n1+5δ .
(4.60)
Now, Lemma 4.2 and the Schwarz inequality implies that VTn − VT¯n = E({Tn − T¯n }{Tn + T¯n }) + E(T¯n − Tn )E(T¯n + Tn ) ≤ 2{E({Tn − T¯n }2 )E({Tn + T¯n }2 )}1/2 ≤ 2{c3 E({Tn + T¯n }2 )}1/2
(c3 is given by Lemma 4.2).
By Lemma 4.3, E({Tn + T¯n }2 ) is at most of order n2 . Together with (4.60), this proves (1.6). To prove (1.7), let c = c(n) := c21 n2δ log n and x0 := n1+7δ , where c1 is the constant appearing in Lemma 4.4. Then x0 ≥ c2 (for large n). By Lemma 4.4 and 4.59, ¯k ≤ c ∆
and
¯ 2 | Fk−1 ) ≤ E(Uk | Fk−1) . E(∆ k
Together with Lemma 4.5, this yields (4.40), (4.41) and (4.42) (with υ = 1/2). By applying Lemma 4.1, one obtain that for all large n and x ≤ n(1+7δ)/2 P(|T¯n − ET¯n | > xn(1+7δ)/2 ) ≤ c2 e−c3 x .
On the other side, by Lemma 4.2, if x > 1 and n is large, P(|Tn − ETn | > xn(1+7δ)/2 ) ≤ P(|Tn − T¯n | > xn(1+7δ)/2 /3) + P(|T¯n − ET¯n | > xn(1+7δ)/2 /3) .
(4.61)
26
LEANDRO P. R. PIMENTEL
Combining this with Lemma 4.2 and (4.61) one get that for some constants b2 , b3 > 0, for n large and x ≤ n(1+7δ)/2 , P(|Tn − ETn | > xn(1+7δ)/2 ) ≤ b2 e−b3 x
4δ/(1+7δ)
.
(4.62)
Therefore, given κ ∈ (1/2, 1) we set δ := (2κ − 1)/7 and obtain (1.7) from (4.62).
(1.8) will follow from the same argument presented by Howard and Newman [8] to deal with the same inequality, but in the euclidean FPP context (see the proof of (4.3) there). For this reason we just sketch this proof and leave further details for the reader. By Lemma 4.2 of Howard and Newman [8], (1.8) will follows if we prove that, for some constant c1 , ET (0, 2n) ≥ 2ET (0, n) − c1 nκ (log n)1/ν (4.63) To prove (4.63) take δ < κ/2 and pick x1 := n, x2 , . . . , xl(n) in the boundary of Dn so that every x ∈ Dn is within distance nδ of one of the xj , and l(n) ≤ cn1−δ for some ¯ n := 2n − Dn . ¯ j := 2n − xj (the radial reflection of xj about n) D constant c > 0. Let x
Let ρ2n be a geodesic connecting 0 to 2n. Then ρ2n exits Dn for the last time at some ¯ n at some some point y ¯ n . Let xj(n) ∈ {x1 , . . . , xn } point yn and after that it enters in D ¯ ∈ {¯ ¯ l } be the closest points to yn and y ¯ n respectively. and xj(n) x1 , . . . , x Thus,
T (0, 2n) ≥ T (0, yn ) + T (¯ yn , 2n) ≥ T (0, xj(n) + T (¯ xn , 2n) − [(T (0, xj(n) − T (0, yn )) + (T (¯ xj(n) , 2n) − T (¯ yn , 2n))] . By sub-additivity, T (0, xj(n) ) − T (0, yn ) ≤ T (xj(n) , yn ) ≤ max sup T (xj , y) j
y−xj ≤nδ
and ¯ n ) ≤ max sup T (¯ T (¯ xj(n) , 2n) − T (¯ yn , 2n) ≤ T (¯ xj(n) , y xj , y) . j
y−xj ≤nδ
Therefore, T (0, 2n) ≥ ¯ j ) − [max sup T (xj , y) + max sup T (¯ min T (0, xj ) + min T (0, x xj , y)] . j
j
j
j
y−xj ≤nδ
y−xj ≤nδ
By symmetry, this implies that ET (0, 2n) ≥ 2ET (0, n) − 2[E(max{ET (0, xj ) − T (0, xj )}) − E(max sup T (xj , y))] . j
j
(4.64)
y−xj ≤nδ
Together with Lemma 4.3 of Howard and Newman [8], (1.7) yields E(max{ET (0, xj ) − T (0, xj )}) ≤ c2 nκ (log n)1/ν . j
(4.65)
ASYMPTOTICS FOR FIRST-PASSAGE TIMES
27
By noting that the expected number of edges inside {y : y − xj ≤ nδ } is of order n2δ and by using Lemma 4.3, one can prove that E(max sup T (xj , y)) ≤ c3 n2δ ≤ c3 nκ . j
y−xj ≤nδ
Combining this with (4.64) and (4.65), one gets (4.63). Acknowledgment. Part of this work was develop during my doctoral studies at Impa and I would like to thank my adviser, Prof. Vladas Sidoravicius, for his dedication and encouragement during this period. I also thank the whole administrative staff of IMPA for their assistance and CNPQ for financing my doctoral studies, without which this work would have not been possible. This work also amends some mistaken passages of my thesis (Pimentel [17]) and, for this reason, I should also express gratitude for the assistance given by Prof. Thomas Mountford. References [1] Alexander, K. S. (1997) Approximations of subadditive functions and convergence rates in limitingshape results, Ann.Probab. 25 30-55. [2] Baik, J.; Deift, P.; Johansson, K. (1999) On the distribution of the longest increasing subsequence in a random permutation, J. Amer. Math. Soc. 12 1119-1178. [3] Cox., J. T.; Gandolfi, A.; Griffin, P. S.; Kesten, H. (1993) Greedy lattice animals I: upper bounds, Ann. Appl. Probab. 3, 1151-1169. [4] Grimmett, G. (1999) Percolation (second edition), Springer. [5] Hammersley, J.M. (1961) On the rate of convergence to the connective function of the hypercubical lattice,Quart. J. Math. Oxford, Ser. 2 12, 250-256. [6] Hammersley, J.M.; Welsh, D.J.A. (1965) First-passage percolation, sub-additive process, stochastic network and generalized renewal theory, Springer-Verlag, 61-110. [7] Howard, C. D.; Newman, C. M. (1997) Euclidean models of first-passage percolation, Probab. Theory Related Fields 108, 153-170. [8] Howard, C. D.; Newman, C. M. (2001) Geodesics and spanning trees for Euclidean first-passage percolation, Ann. Probab. 29, 577-623. [9] Johansson, K. (2000) Shape fluctuations and random matrices, Comm. Math. Phys. 209, 437-476. [10] Kardar, K.; Parisi, G.; Zhang Y. Z. (1986) Dynamic scaling of growing interfaces Phys. Rev. Lett. 56, 889-892. [11] Kesten, H. (1963) On the number of self-avoiding walks J. Math. Phys. 4, 960-969. [12] Kesten, H. (1986) Aspects of first-passage percolation, Lectures Notes in Math. 1180, Springer-Verlag, 125-264 [13] Kesten, H. (1993) On the speed of convergence in first-passage percolation Ann. Appl. Probab. 3, 296-338. [14] Krug, J.; Spohn, H. (1991) Solids far from Equilibrium, (C. Godreche, ed.) Cambridge Univ. Press, Cambridge. [15] Ligget, T.M.; Schonmann, R.H.; Stacey, A.M. (1997) Domination by product measures, Ann. Probab. 25 , 71-95. [16] Moller, J. (1991) Lectures on random Voronoi tessellations, Lectures Notes in Stat. 87, SpringerVerlag.
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LEANDRO P. R. PIMENTEL
[17] Pimentel, L. P. R. (2004) Competing growth, interfaces and geodesics in first-passage percolation on Voronoi tilings. Phd Thesis, IMPA, Rio de Janeiro. [18] Talagrand, M. (1995), Concentration of measure and isoperimetric inequalities in product spaces Inst. Hautes Etudes Sci. Publ. Math. 81, 73-205. [19] Vahidi-Asl, M.Q.; Wierman, J.C. (1990) First-passage percolation on the Voronoi tessellation and Delaunay triangulation, Random Graphs 87 (M. Karonske, J. Jaworski and A. Rucinski, eds.), 1990, 341-359. [20] Vahidi-Asl, M.C.; Wierman, J.C. (1992) A shape result for first-passage percolation on the Voronoi tessellation and Delaunay triangulation, Random Graphs 89 (A. Frieze and T. Luczak, eds.), Wiley, 247-262. ´ ´matiques, Ecole Institut de Mathe Polytechinique F´ ed´ erale de Lausanne, CH-1015 Lausanne, Switzerland. E-mail address: [email protected] URL: http://ima.epfl.ch/∼lpimente/
