First-passage percolation results getdoc5332. 401KB Jun 04 2011 12:04:23 AM

2 Preliminary results

2.1 First-passage percolation results

Let us recall some classical results about first-passage percolation. We assume here that the passage times are independent identically distributed with common law ν satisfying • ν0 p c ; • for some γ 0, R [0,+∞ exp γx dνx +∞. Denote by k.k ν the norm given by the shape theorem, and by B x t the discrete ball relatively to k.k ν with center x and radius t. The first two results give large deviations and moderate deviations for fluctuations with respect to the asymptotic shape, and the third one gives the strict monotonicity result for the asymptotic shape with respect to the distribution of the passage times: Proposition 2.1 Grimmett-Kesten [9]. For any ǫ 0, there exist two strictly positive constants A, B such that ∀t 0 P € B 1 − ǫt ⊂ B t ⊂ B 1 + ǫt Š ≥ 1 − Aexp−B t. Proposition 2.2 Kesten [14], Alexander [1]. For any β 0, for any η ∈ 0, 12, there exist two strictly positive constants A, B such that ∀t 0 P € B t − β t 1 2+η ⊂ B t ⊂ B t + β t 1 2+η Š ≥ 1 − Aexp−B t η . Proposition 2.3 van den Berg-Kesten [15]. Let ν p 1 and ν p 2 be two probability measures on [0, +∞ satisfying H1, H3, H4, H5. There exists a constant C p 1 ,p 2 ∈ 0, 1 such that ∀x ∈ R d kxk p 2 ≤ C p 1 ,p 2 kxk p 1 . Note that in [15], the proof of this result is only written for the time constants. Nevertheless, it applies in any direction and computations can be followed in order to preserve a uniform control, whatever direction one considers. See for instance Garet and Marchand [6] for a detailed proof in an analogous situation. In the same way, the large deviation result of Proposition 2.1 is only stated in [9] for the time constant, but the result can be extended uniformly in any direction, as it is done in Garet and Marchand [7] for chemical distance in supercritical Bernoulli percolation. As far as Proposition 2.2 is concerned, it is a by-product of the proof of Theorem 3.1 in Alexander [1]. We include here a short proof for convenience. Proof of Proposition 2.2. The outer bound for B t follows from Kesten [14], Equation 1.19: there exist positive constants A 1 , B 1 such that for all t 0, we have PB t 6⊂ B t + β t 1 2+η ≤ A 1 exp−B 1 t η . 2125 Turning to the inner bound, we follow the lines of Alexander’s proof: for x, y ∈ Z d , let us define the travel time between x and y by T x, y = inf{t ≥ 0 : y ∈ B x t}. Then we have: PB t 6⊃ B t − β t 1 2+η ≤ PB t 6⊃ B t2 + P t 2≤kxk ν ≤t−β t 1 2+η PT 0, x ≥ t ≤ Aexp−B t + P t 2≤kxk ν ≤t−β t 1 2+η PT 0, x ≥ t, where A and B are determined by Proposition 2.1 with ǫ = 12. By Alexander [1], Theorem 3.2, there exist positive constants C ′ 4 , M such that kxk ν ≥ M =⇒ E T 0, x ≤ kxk ν + C ′ 4 kxk 1 2 ν log kxk ν . Assume t 2 ≥ M and let x with t2 ≤ kxk ν ≤ t − β t 1 2+η . Let C M = inf{2 1 2+η β − C ′ 4 y −η log y : y ≥ M }. We assume that M is so large that C M 0. We have E T 0, x ≤ t − C M kxk 1 2+η ν ≤ t − C ′ M kxk 1 2+η 2 , where C ′ M is a positive constant, and then PT 0, x ≥ t ≤ PT 0, x − E T 0, x ≥ C ′ M kxk 1 2+η 2 . By Kesten’s result [14], Equation 2.49 see also Equation 3.7 in Alexander [1], there exist positive constants C 5 , C 6 such that PT 0, x − E T 0, x ≥ C ′ M kxk 1 2+η ≤ C 5 exp−C 6 kxk η 2 , provided that M is large enough. Finally, it gives that P t 2≤kxk ν ≤t−β t 1 2+η PT 0, x ≥ t ≤ |B t|C 5 exp−C 6 t2 η ≤ C ′ 5 exp−C ′ 6 t η , where C ′ 5 , C ′ 6 are positive constants. Lemma 2.5 will ensure that the minimal time to cross the cylinder Cyl z − → x , h, r from bottom to top, using only edges in the cylinder, can not be much larger than the expected value hk − → x k ν . We need an intermediary lemma. For x, y ∈ Z d , denote by I x, y the length of the shortest path from x to y which is inside B x 1.25kx − yk ν ∩ B y 1.25kx − yk ν . Of course I x, y as the same law that I 0,x− y , and we simply write I x = I 0,x . Lemma 2.4. Let ǫ, a in 0, 1 and k.k be any norm on R d . There exists M such that for each M ≥ M , there exist ρ ∈ 0, 1 and t 0 such that kxk ∈ [aM , M a] =⇒ E exptI x − 1 + ǫkxk ν ≤ ρ. Proof. Note that by norm equivalence, we can restrict ourselves to k.k 1 . Let Y be a random variable with law ν and let γ 0 be such that E e 2 γY +∞. 2126 First, the large deviations result, Proposition 2.1, easily implies the following almost sure conver- gence: lim kxk 1 →+∞ I x kxk ν = 1. By considering a deterministic path from 0 to x with length kxk 1 , we see that I x is dominated by a sum of kxk 1 independent copies of Y denoted by Y 1 , . . . , Y kxk 1 . Thus, I x kxk 1 is dominated by 1 kxk 1 kxk 1 X k=1 Y i . By the law of large numbers, this family is equi-integrable. Hence, T x kxk 1 x∈Z d \{0} and finally I x kxk ν x∈Z d \{0} are also equi-integrable families. It follows that lim kxk 1 →+∞ E I x kxk ν = 1. 1 Note now that for every y ∈ R and t ∈ 0, γ], e t y ≤ 1 + t y + t 2 2 y 2 e t| y| ≤ 1 + t y + t 2 γ 2 e 2 γ| y| . Define ˜I x = I x − 1 + ǫkxk ν and suppose that t ∈ 0, γ]. Then, as |˜I x | ≤ I x + 2kxk ν , the previous inequality implies that e t ˜ I x ≤ 1 + t ˜I x + t 2 γ 2 e 4 γkxk ν e 2 γI x . As kxk ν ≤ kxk 1 ke 1 k ν and I x ≤ Y 1 + · · · + Y kxk 1 , if we set R = e 4 γke 1 k ν E e 2 γY , we obtain that E e t ˜ I x ≤ 1 + t E ˜ I x + t γ 2 R kxk 1 . Considering Equation 1, let M be such that kxk ≥ aM implies E I x kxk ν ≤ 1 + ǫ3. For x such that kxk 1 ≥ aM , we have E ˜I x ≤ − 2 3 ǫkxk ν , so E e t ˜ I x ≤ 1 + t − 2 3 ǫkxk ν + t γ 2 R kxk 1 . Therefore, we take t = min ¦ γ, γ 2 ke 1 k ν ǫ 3 R −M a © 0 and ρ = 1 − 1 3 ǫke 1 k ν t 1. Lemma 2.5. For z ∈ R d , − → x ∈ S , and h, r 0 large enough, we can define the point s the point s f to be the integer point in Cyl z − → x , r, h which is closest to z respectively, z + h − → x . We define the crossing time t[Cyl z − → x , h, r] of the cylinder Cyl z − → x , r, h as the minimal time needed to cross it from s to s f , using only edges in the cylinder. Then for any ǫ 0, and any function f : R + → R + with lim +∞ f = +∞, there exist two strictly positive constants A and B such that ∀z ∈ R d ∀ − → x ∈ S ∀h 0 P € t[Cyl z − → x , h, f h] ≥ k − → x k ν 1 + ǫh Š ≤ Aexp−Bh. 2127 Note that this gives the existence of a nearly optimal path from z to z + h − → x that remains at a distance less than f h of the straight line. This result can be interesting on its own as we often lack of information on the position of the real optimal paths. Proof. Let ǫ ∈ 0, 1 and consider the integer M ∈ N given by Lemma 2.4. As k.k ν is a norm, there exist two strictly positive constants c and C such that ∀ − → x ∈ S c ≤ k − → x k ν ≤ C. 2 Let M 1 ≥ M be an integer large enough to have 1 + 2ǫ ≥ 1 + ǫ 1 + 4ke 1 k ν c M 1 . 3 Consider h M 1 and set N = 1 + Inth M 1 , where Intx denotes the integer part of x, and, for each i ∈ {0, . . . , N } denote by x i the integer point in the cylinder which is the closest to z + ih − → x N . Note that ∀h ≥ M 1 M 1 2 ≤ 1 − 1 N M 1 ≤ h N ≤ M 1 . 4 and that for each i, j ∈ {0, . . . , N − 1}, ¯ ¯ ¯ ¯ ¯ kx i − x j k ν − | j − i| hk − → x k ν N ¯ ¯ ¯ ¯ ¯ ≤ 2ke 1 k ν . 5 1. Applying 5, 2 and 4, we obtain that for each i ∈ {0, . . . , N − 1}, hk − → x k ν N − 2ke 1 k ν ≤ kx i − x i+1 k ν ≤ hk − → x k ν N + 2ke 1 k ν c M 1 2 − 2ke 1 k ν ≤ kx i − x i+1 k ν ≤ C M 1 + 2ke 1 k ν . So we can find a 0 such that, by increasing M 1 if necessary, ∀i ∈ {0, . . . , N − 1} aM 1 ≤ kx i − x i+1 k ν ≤ M 1 a. 6 2. Let h 1 ≥ 0 be such that ∀h ≥ h 1 f h ≥ 2.5CM 1 + 1ke 1 k ν + 1. If we take now h larger than h 1 , and if y ∈ B x i 1.25kx i − x i+1 k ν for some i ∈ {0, . . . , N − 1}, then, with 2 and 5, d y − z, R − → x = d y, z + R − → x ≤ y − z − ih N − → x 2 ≤ k y − x i k 2 + x i − z − ih N − → x 2 ≤ 1.25Ckx i − x i+1 k ν + 1 ≤ 1.25C ‚ hk− → x k ν N + 2ke 1 k ν Œ + 1. 2128 But h N ≤ M 1 and k − → x k ν ≤ k − → x k 1 ke 1 k ν ≤ 2ke 1 k ν , thus d y − z, R − → x ≤ f h. On the other hand, 〈 y − z, − → x 〉 = 〈 y − x i , − → x 〉 + 〈x i − z + ih N − → x , − → x 〉 + 〈 ih N − → x , − → x 〉 i.e. ¯ ¯ ¯ ¯ 〈 y − z, − → x 〉 − ih N ¯ ¯ ¯ ¯ ≤ y − x i 2 + 1 ≤ 2.5CM 1 + 1ke 1 k ν + 1. We choose then i ∈ N such that: i ≥ 2 M 1 2.5CM 1 + 1ke 1 k ν + 1. Then, if h 2 is such that 1 + Inth 2 M 1 ≥ 3i , we obtain: ∀h ≥ h 2 ∀i ∈ {i , . . . , N − 1 − i } B x i 1, 25kx i − x i+1 k ν ⊂ Cyl z − → x , R, h. 7 3. There exists a deterministic path inside the cylinder from x to x i from x N −i to x N which uses less than i hk − → x k 1 N + 2 edges: we denote by L st ar t respectively, L end the random length of this path. By Equation 4, we have ∀h ≥ M 1 i hk − → x k 1 N + 2 ≤ i 2h N + 2 ≤ 2i + 1M 1 . If h h 3 = 3i +1M 1 E Y ǫk − → x k ν , Chernoff’s theorem gives the existence of two strictly positive constants A 1 , B 1 such that ∀h 0 P € L st ar t ǫhk − → x k ν Š + P € L end ǫhk − → x k ν Š ≤ A 1 e −B 1 h . 8 4. So, provided that h ≥ h 2 , we have by 8, inside the cylinder, a path from x to x N with length L st ar t + N −i −1 X i=i I x i ,x i+1 + L end . By Equation 5, if h is large enough, for each i, j ∈ {0, . . . , N − 1}, B x i 1.25kx i − x i+1 k ν ∩ B x i+1 1.25kx i − x i+1 k ν ∩ B x j 1.25kx j − x j+1 k ν ∩ B x j+1 1.25kx j − x j+1 k ν = ∅ as soon as | j − i| ≥ 2. We thus introduce, for j ∈ {0, 1}, the sums: S j = X I≤i≤N −I−1 i= j mod 2 I x i ,x i+1 . 2129 Note that, with 5, 2 and 3 for each j ∈ {0, 1}, X i ≤i≤N −i −1 i= j mod 2 kx i+1 − x i k ν ≤ N − 2i 2 ‚ hk− → x k ν N + 2ke 1 k ν Œ ≤ hk − → x k ν 2 ‚ 1 + 2N ke 1 k ν hk − → x k ν Œ ≤ hk − → x k ν 2 1 + 4ke 1 k ν c M 1 ≤ 1 + 2 ǫ 1 + ǫ hk − → x k ν 2 . Then, by independence of the terms in each S j , P ‚ S j ≥ hk − → x k ν 2 1 + 2ǫ Œ ≤ P      S j ≥ 1 + ǫ X i ≤i≤N −i −1 i= j mod 2 kx i+1 − x i k ν      ≤ E exp      t X i ≤i≤N −i −1 i= j mod 2 € I x i ,x i+1 − 1 + ǫkx i+1 − x i k ν Š      ≤ Y i ≤i≤N −i −1 i= j mod 2 E exptI x i ,x i+1 − 1 + ǫkx i+1 − x i k ν By 6, for each i, we have kx i − x i+1 k ν ∈ [aM 1 , M 1 a], so we can apply Lemma 2.4: there exists some ρ 1, such that for every h large enough, ∀ j ∈ {0, 1} P ‚ S j ≥ hk − → x k ν 2 1 + 2ǫ Œ ≤ ρ N 2 ≤ ρ h 2M 1 . Together with 8, this proves the estimate of the lemma.

2.2 Comparisons with first-passage percolation