Law of large numbers for critical first-passage percolation on the triangular lattice

We study the site version of (independent) first-passage percolation on the triangular lattice $\mathbb{T}$. Denote the passage time of the site $v$ in $\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by $a_{0,n}$ the passage time from $\textbf{0}$ to $(n,0)$, and by $b_{0,n}$ the passage time from $\textbf{0}$ to the halfplane $\{(x,y):x\geq n\}$. We prove that there exists a constant $0<\mu<\infty$ such that as $n\rightarrow\infty$, $a_{0,n}/\log n\rightarrow \mu$ in probability and $b_{0,n}/\log n\rightarrow \mu/2$ almost surely. This result confirms a prediction of Kesten and Zhang (Probab. Theory Relat. Fields \textbf{107}: 137--160, 1997). The proof relies on the existence of the full scaling limit of critical site percolation on $\mathbb{T}$, established by Camia and Newman.

The passage time between two site sets A, B is defined as T (A, B) := inf{T (γ) : γ is a path connecting some site of A with some site of B}, and a time minimizing path between A, B is called a geodesic. Denote the origin by 0.
These are called the point to point and point to line passage times respectively. It is well known (Kingman [13], Wierman and Reh [23]) that if Et(v) < ∞, there is a nonrandom constant µ = µ(F ) < ∞ such that lim n→∞ a 0,n n = lim n→∞ b 0,n n = µ a.s. and in L 1 , where µ is called the time constant. Kesten [11] showed that where p c (T, site) is the critical probability for site percolation on T. Since there is a transition of the time constant at F (0) = p c , Kesten and Zhang [12] call this "critical" FPP.
In this paper, we shall restrict ourselves to a special critical FPP, that is, we assume that P (t(v) = 0) = P (t(v) = 1) = 1 2 . (1.3) Note that we can view this model as the critical site percolation on T. Recall that it can be obtained by coloring the faces of the honeycomb lattice randomly, each cell being open (black) or closed (white) with probability 1/2 independently of the others. For this critical FPP, from (1.2) it is natural to ask whether or not the sequences in (1.1) converge to positive limits after properly normalizing. We give a historical note related to this problem here. Let θ stands for a or b. In a survey paper [10] (see the paragraph right below (3.16P) in [10]), Kesten pointed out that the results proved in [2] that Eθ 0,n lies between two positive multiples of log n would imply that {θ 0,n / log n} is a tight family, furthermore, using RSW and FKG, one may show that P (θ 0,n ≤ ε log n) is small for small ε, which implies that any limit distribution of θ 0,n / log n has no mass at zero. Later, Kesten and Zhang [12] indicated that the estimates they developed in their paper can be used to prove a strong law of large numbers (SLLN) for b 0,n : b 0,n /Eb 0,n → 1 a.s. Further, they expected that Eb 0,n / log n and Ea 0,n / log n converge to finite, strictly positive limits as n → ∞. In this paper, we continue the study from [12], the following is our main theorem: Once the existence of full scaling limit of critical bond percolation on Z 2 is established, one may derive the theorem by our strategy. Also, the limits will be the same as Theorem 1.1 because of the conjectural universality of critical percolation. Remark 1.3. One may consider more general critical FPP on T, for example, the distribution function F satisfying the conditions (1.4)-(1.6) in [12]. That is, It is expected that Theorem 1.1 still holds for the F above (with µ(F ) as a function of F ).
For each r > 0, let D r denote the Euclidean disc of radius r centered at 0 and ∂D r denote its boundary. Let D denote the unit disk for short. For v ∈ V, let B(v, r) denote the discrete ball of radius r centered at v in the triangular lattice: We denote by ∂B(v, r) its boundary, which is the set of sites in B(v, r) that have at least one neighbor outside B(v, r). For short, we let B(r) := B(0, r). Remark 1.4. We can express a 0,n and b 0,n in terms of circuits. For example, it is easy to see that a 0,n and the maximum number of disjoint closed circuits which separate 0 and (n, 0) differ by at most 2. Note that with probability 1 there is no infinite cluster for the critical site percolation on T, therefore the cluster boundaries form loops. Now we introduce two quantities for this model, which are similar as a 0,n and b 0,n respectively: Note that a 0,n is essentially introduced in [4]. Using the strategy in the present paper and the result of [18], one may get the following result, which is analogous to Theorem 1.1 but with explicit limit values: Furthermore, the convergence in (1.6) does not occur almost surely. The explicit limits above mainly relies on the work of [18]. However, it seems very hard to give the explicit value of µ in Theorem 1.1. Nevertheless, it need not much work to deduce that µ > 1/(2 √ 3π) from above. We just give a sketch of the proof here. First, let us introduce some notations from [18]. Camia and Newman defined the conformal loop ensemble CLE 6 in D (see Section 3.2 in [3]), which is almost surely a countably infinite collection of (oriented) continuum nonsimple loops and is the scaling limit of the cluster boundaries of critical site percolation on ηT ∩ D with monochromatic boundary conditions. We inductively define L k to be the outermost loop surrounding 0 in D when the loops L 1 , . . . , L k−1 are removed. Note that the loops L k exist for all k ≥ 1 with probability 1. Define A 0 = D and let A k be the component of D\L k that contains 0. If D is a simply connected planar domain and 0 ∈ D, the conformal radius of D viewed from 0 is defined to be CR(D) := |g (0)| −1 , where g is any conformal map from D to D that sends 0 to 0. For k ≥ 1, define From Proposition 1 in [18], we know that B k , k ≥ 1 are i.i.d random variables. Furthermore, it is shown (see (2) in [18]) that A well known consequence of the Schwarz Lemma and the Koebe 1/4 Theorem (see e.g., Lemma 2.1 and Theorem 3.17 in [14], see also (2.1) in [15] for a similar application) is that Let N (ε) be the number of loops surrounding 0 in D\D ε . From the definition it is clear that log CR(A n ) = − n k=1 B k . Combining above issues, one may conclude is an analog of Lemma 2.7 below. One can get analogs of Lemma 2.5 and Lemma 2.8 following our method. Combining these issues, the result would be proved.

Remark 1.5.
Consider the oriented (directed) FPP on Z 2 (see e.g., Section 12.8 in [6] for background). We assign independently to each bond e i.i.d. passage time t(e). Let p c denote the critical probability for oriented bond percolation on Z 2 . Assume P (t(e) = 0) = p c , P (t(e) = 1) = 1 − p c .
We denote by T (0, (r, θ)) the passage time from 0 to ( r sin θ , r cos θ ) by a northeast path for (r, θ) ∈ R + × [0, π/2]. Based on Conjecture 4 in [25], we conjecture that there is a constant 0 < µ < ∞, such that as r → ∞, Remark 1.6. Camia and Newman's full scaling limit plays a central role in the proof of Theorem 1.1. We want to note that the scaling limit of a critical system may help to show laws of large numbers for many different variables. For example, consider the largest winding angle (the interested reader is referred to [24] for a more general discussion and references of winding angles) θ max,n of the paths from 0 to ∂B(n) in Kesten's incipient infinite cluster (IIC) [9]. Heuristically, once the existence of an appropriate scaling limit of IIC on T is established, one may derive a SLLN for θ max,n by our strategy.
Idea of the Proof. We show a SLLN for c n := T (0, ∂B(n)), that is, c n / log n → µ/2 a.s., then Theorem 1.1 follows from this easily. First, using the estimates developed by Kesten and Zhang [12], we prove that c n /Ec n → 1 a.s. Next, we want to show Ec n / log n → µ/2, which implies the required SLLN immediately. For this, we divide the discrete ball B(n) into long annuli, which have the same shape. The summation of the passage times of these annuli approximates c n . Inspired by Beffara and Nolin [1], we express the passage time of an annulus in terms of the collection of cluster interfaces (see Fig. 2). When the annulus is very large, this quantity can be approximated well by the passage time defined analogously for the corresponding annulus with respect to Camia and Newman's full scaling limit [3] (see Fig. 1 ecp.ejpecp.org subadditive ergodic theorem we get a SLLN for the passage times of annuli, which can be used to approximate the passage times of the large and long annuli for the discrete model.
Throughout this paper, C, C 1 , C 2 , . . . denote positive finite constants that may change from line to line or page to page according to the context.

Preliminary results
We shall use some estimates developed in [12]. Let us give some notations from [12]. For 0 < m < n, define the annulus contains an open circuit surrounding 0}, Then we can write Essentially the same as the proof of (2.28), (2.29) in [12], one may get the following lemma, we omit the proof here. Note that (2.3) is stronger than (2.29) in [12], since the distribution of the passage time in [12] is more general than ours. One needs no new technique here.
We say a path γ ⊂ R(m, n) is a crossing path in R(m, n) if the endpoints of γ lie adjacent (Euclidean distance smaller than 1) to the rays of argument ± π 10 respectively. By step 3 of the proof of Theorem 5 in [1], we obtain the following lemma.
Proof. As we have discussed before Theorem 1.1, Kesten and Zhang got similar result for b 0,n in [12] (see (1.15) in [12]), but they did not give the proof. Now let us use the estimates in [12] to prove Lemma 2.5. First we claim that for each ε ∈ (0, 1 4 ), there exist constants δ 1 > 0, C 6 > 0 such that Let us prove this. First we define where C 4 > 3/C 1 and C 1 is from Lemma 2.1. By (2.1), we write Let us now estimate each term separately. For the first term, P (∆ p,q = ∆ p,q for some p ≤ q) ≤ q p=0 P (m(p) − p ≥ C 4 log q) ≤ (q + 1) exp(−C 4 C 1 log q) by (2.2). Now we estimate the second term. Similar as the second half of Lemma 1 in [12], we have: For any 0 ≤ p, r ≤ q, if |p − r| ≥ C 4 log q + 2, then ∆ p,q and ∆ r,q are independent.
As it is well discussed in [19], there are several different ways to describe the scaling limit of critical planar percolation. In the present paper, we focus on the full scaling limit constructed by Camia and Newman in [3], described in detail below.
First, we compactify R 2 as usual intoṘ 2 := R 2 ∪ {∞} S 2 . Let d S 2 be the induced metric onṘ 2 . We call a continuous map from the circle to R 2 a loop, and the loops are identified up to reparametrization by homeomorphisms of the circle with positive winding. We equip the space L of loops with the following metric: where the infimum is taken over all homeomorphisms of the circle which have positive winding. Let L be the space of countable collections of loops in L. Consider the Hausdorff topology on L induced by d L . That is, for c 1 , c 2 ∈ L, let d L := inf{ε : ∀ 1 ∈ c 1 , ∃ 2 ∈ c 2 such that d L ( 1 , 2 ) ≤ ε and vice versa}. For the critical site percolation on T, with probability 1 there is no infinite cluster, therefore the cluster boundaries form loops. We orient a loop counterclockwise if it has open sites on its inner boundary and closed sites on its outer boundary, otherwise we orient it clockwise.
The following celebrated theorem is shown in [3]: Theorem 2.6. As η → 0, the collection of all cluster boundary loops of critical site percolation on ηT converges in law, under the topology induced by d L , to a probability distribution on L, which is a continuum nonsimple loop process.
The continuum nonsimple loop process in Theorem 2.6 is just the full scaling limit introduced by Camia and Newman in [3]. Since it is also called the conformal loop ensemble CLE 6 in [20] (for the general CLE κ , 8/3 ≤ κ ≤ 8, see [20,18,21]), we just call it CLE 6 in the present paper. Although extracting geometric information is far from being straightforward from CLE 6 (according to [19]), it was used to show the uniqueness of the quad-crossing percolation limit in Subsection 2.3 in [5] and the existence of the monochromatic arm exponents in Section 4 in [1]. In fact, the key idea of the proof of (2.11) is stimulated by the latter.
Several properties of CLE 6 are established. For example, if two loops touch each other and have the same orientation, then almost surely one loop cannot lie inside the other one. Conversely, if two loops of different orientations touch each other, then one has to be inside the other one. See [3] for more details. For CLE 6 , we want to define the passage time between two circles. First, we call a sequence of loops C = 1 . . . l a chain which connects ∂D m and ∂D n , if C satisfies the following conditions: Proof. For short, let X i,j := −T (∂D 2 i , ∂D 2 j ) + 1, 0 ≤ i < j. Now we verify that X i,j , 0 ≤ i < j satisfy the conditions of the subadditive ergodic theorem (see [16]): If T (∂D 1 , ∂D 2 j ) > 0, then for any chain C = 1 . . . l connecting ∂D 1 and ∂D 2 j , clearly we have l ≥ 2 by the definition of chain. Then it is easy to see that we can find some 2 ≤ k ≤ l such that C 1 = 1 . . . k is a chain connecting ∂D 1 and ∂D 2 i , and C 2 = k−1 . . . l is a chain connecting ∂D 2 i and ∂D 2 j . Therefore, which implies the above inequality. If T (∂D 1 , ∂D 2 j ) = 0, the inequality holds obviously.
Define the scaling transformation τ k : R 2 → R 2 , x → x/2 k . Then for each configuration ω of CLE 6 , X jk,(j+1)k (ω) = X k,2k (τ j−1 k ω). Since CLE 6 is invariant under scalings, τ k is measure preserving and {X jk,(j+1)k , j ≥ 1} is stationary. Now we show that τ k is also mixing, which implies {X jk,(j+1)k , j ≥ 1} is ergodic. When A, B are events which depend only on the realization of the CLE 6 inside an annulus, then lim j→∞ P (A ∩ τ −j k B) = P (A)P (B) follows immediately. For arbitrary events A and B, one approximates A and B by events which depend only on the realization of CLE 6 inside the annulus D 1/ε \D ε , and let ε → 0. Then the result follows easily.
• The distribution of {X i,i+k , k ≥ 1} does not depend on i. CLE 6 is invariant under scalings, which implies this immediately.
Then by the subadditive ergodic theorem, there exists a constant 0 < µ 0 < ∞ such that which ends the proof. Proof. For short, define Recall the definition of T (∂D m , ∂D n ) before Lemma 2.7. For the passage time defined respectively for the discrete FPP and CLE 6 , we claim that for any fixed k ≥ 1, as i → ∞, (2.11) The proof of this claim is similar as the arguments in Section 4 in [1], but it's more complicated. First we show that for each 0 < ε < 1, there is a δ > 0, such that lim i→∞ P (for any geodesic γ connecting ∂B(2 ki + 1) and ∂B(2 k(i+1) ) in A(k, i), for any closed site x ∈ γ, dist(x, ∂B(2 ki + 1) ∪ ∂B(2 k(i+1) )) ≥ δ2 ki , for any two closed sites x, y ∈ γ, dist(x, y) ≥ δ2 ki ) ≥ 1 − ε. Therefore, if γ is a geodesic connecting ∂B(2 ki + 1) and ∂B(2 k(i+1) ) in A(k, i), then there exist T (γ) disjoint circuits which surround 0 in A(k, i) and pass through the T (γ) closed sites in γ respectively. Using the fact that the polychromatic half-plane 3-arm exponent is 2 (in fact, one needs a more general version, see Lemma 6.8 in [22]) and the polychromatic plane 6-arm exponent is larger that 2 (see e.g. [17]), one can get  Now let us introduce the definition of chain for the critical site percolation on T, which is analogous to its continuum version for CLE 6 . We call a sequence of (discrete) loops C = 1 . . . l a (discrete) chain which connects ∂B(m) and ∂B(n), if C satisfies the following conditions:  if there exists no chain connecting ∂B(2 ki + 1) and ∂B(2 k(i+1) ), let T k,i = 0. It is easy to get that lim i→∞ P (T k,i = T k,i ) = 1. (2.14) By (2.12),(2.13) and (2.14), the value of T k,i is determined by macroscopic loops with high probability as i → ∞. It has been argued in [1], two loops touch in the scaling limit is exactly the asymptotic probability that they are separated by exactly one site on discrete lattice. Therefore, using Theorem 2.6, comparing the definitions of T k,i and T (∂D 1 , ∂D 2 k ), we have T k,i → d T (∂D 1 , ∂D 2 k ).
(2.84) in [12] essentially tells us that as n → ∞, By RSW and FKG, there is a universal constant C > 0, such that P (B i ) > C. Then with probability 1 we can find an infinite sequence {i j , j ≥ 1} such that A ij happens. Conditioned on A ij , there exists a 2 ij < n(i j ) ≤ 2 ij +1 , such that a 0,n(ij ) = c 2 i j +1 . Then by (3.1) we have lim j→∞ a 0,n(ij ) log n(i j ) = µ 2 a.s.
This completes the proof.