Critical scaling for an anisotropic percolation system on $\mathbb{Z}^2$

In this article, we consider an anisotropic finite-range bond percolation model on $\mathbb{Z}^2$. On each horizontal layer $H_i = \{(x,i): x \in \mathbb{Z}\}$ we have edges $\langle(x, i),(y, i)\rangle$ for $1 \leq |x - y| \leq N$. There are also vertical edges connecting two nearest neighbor vertices on distinct layers $\langle(x, i), (x, i+1)\rangle$ for $x, i \in\mathbb{Z}$. On this graph we consider the following anisotropic independent percolation model: horizontal edges are open with probability $1/(2N)$, while vertical edges are open with probability $\epsilon$ to be suitably tuned as $N$ grows to infinity. The main result tells that if $\epsilon=\kappa N^{-2/5}$, we see a phase transition in $\kappa$: positive and finite constants $C_1, C_2$ exist so that there is no percolation if $\kappaC_2$. The question is motivated by a result on the analogous layered ferromagnetic Ising model at mean field critical temperature [J. Stat. Phys. (2015), 161, 91-123] where the authors showed the existence of multiple Gibbs measures for a fixed value of the vertical interaction and conjectured a change of behavior in $\kappa$ when the vertical interaction suitably vanishes as $\kappa\gamma^b$, where $1/\gamma$ is the range of the horizontal interaction. For the product percolation model we have a value of $b$ that differs from what was conjectured in that paper. The proof relies on the analysis of the scaling limit of the critical branching random walk that dominates the growth process restricted to each horizontal layer and a careful analysis of the true horizontal growth process. This is inspired by works on the long range contact process [Probab. Th. Rel. Fields (1995), 102, 519-545]. A renormalization scheme is used for the percolative regime.


Introduction
In this article, we consider an anisotropic finite-range bond percolation model on the plane. For this we let Z 2 = (V, E) be the graph with vertex set V = {v = (x, i) : x ∈ Z, i ∈ Z} and edge set E = E N = {e = v 1 , v 2 : v k = (x k , i k ), k = 1, 2; |i 1 − i 2 | = 1 for x 1 = x 2 and 1 ≤ |x 1 − x 2 | ≤ N for i 1 = i 2 }. The edge set can be partitioned into two disjoint subsets E = E v ∪ E h . E v is the set of vertical edges, s.t. E v = {e = v 1 , v 2 : x 1 = x 2 } and E h denotes the set of horizontal edges, s.t. E h = {e = v 1 , v 2 : i 1 = i 2 } (here (x k , i k ) corresponds to v k for k = 1, 2). Each vertical edge is open with probability ǫ and each horizontal edge is open with probability 1/2N , and they are all independent of each other. Our main purpose is to study the existence of percolation in this system, with ǫ = ǫ(N ) that tends to zero as N grows to infinity.
The basic motivation for this paper comes from a question raised in [12], where the authors investigated the existence of phase transition for an anisotropic Ising spin system on the square lattice Z 2 . On each horizontal layer {(x, i) : x ∈ Z} the {−1, +1}-valued spins σ(x, i) interact through a ferromagnetic Kac potential at the mean field critical temperature, i.e. the interaction between the spins σ(x, i) and σ(y, i) is given by where J γ (x, y) = c γ γJ(γ(x − y)), and one assumes J(r), r ∈ R, to be smooth and symmetric with support in [−1, 1], J(0) > 0, J(r)dr = 1, and moreover c γ is the normalization constant (c γ → 1 as γ → 0). To this one adds a small nearest neighbor vertical interaction −ǫσ(x, i)σ(x, i + 1), and the authors proved in [12] that given any ǫ > 0, for all γ > 0 small µ + γ = µ − γ , where µ + γ and µ − γ denote the Dobrushin-Lanford-Ruelle (DLR) measures obtained as thermodynamic limits of the Gibbs measures with +1, respectively −1 boundary conditions.
One of the questions left open in [12] has to do with the following: how small can we take ǫ = ǫ(γ) and still observe a phase transition of the Ising model (for all γ small). Following various considerations, the authors conjectured that if ǫ = ǫ(γ) = κγ 2/3 we might see a different behavior while varying κ. This is the problem that motivates this paper. Our technique does not give an answer to the Ising system, but considering the related product percolation model and usual Fortuin-Kasteleyn-Ginibre (FKG) comparison, it yields a partial answer to the question, and shows that the conjecture has to be modified. (See Remark (b) after the statement of Theorem 1.1.) Indeed, the original problem just described could be formulated in terms of percolation for a Fortuin-Kasteleyn (FK) measure with shape parameter q = 2. Here we treat a simpler case by considering a corresponding anisotropic percolation model on Z 2 . Since a Kac potential can be taken as an interaction of strength γ with range γ −1 (which we fix as N ) we consider the edge percolation problem where horizontal edges of length at most N are open with probability 1/2N and the vertical edges between sites at distance 1 are open with probability ǫ = κN −b .
With respect to layer 0, we denote C 0 0 as the cluster containing (0, 0), C 0 0 = {x : (0, 0) = 0 → (x, 0) with all the edges along the path in Z × {0}}, where v 1 → v 2 means there is an open path from v 1 to v 2 . We can speak of generations on each horizontal layer (we will consider the horizontal behaviour on layer 0 for simplicity). x ∈ C 0 0 is of k-th generation if the shortest path from (0, 0) to (x, 0) is of length k. Denote G 0 k as the collection of vertices that can be reached from 0 at k-th generation. The sites of {G 0 k } k≥0 form a process very close to a branching random walk starting from 0. The difference between {G k } k≥0 and a critical branching process is the domain of the state function. Denote {ξ k (x)} k≥0 as the critical branching random walk. At each time n, particles of occupied sites branch following Binomial(2N, 1/2N ) and move to its 2N neighbours uniformly (note that it is different from the model in [16] discussed in the next paragraph). The state function ξ k (x) ∈ Z + . However, the process {G k } k≥0 only includes if the site is occupied or not, which takes value in {0, 1}. In section 2, we show that these two processes are not too different. This intrigues us to consider the asymptotic density on each horizontal layer and use it to derive the cumulated occupied sites over generations. But the introduction of generations will cause a problem in the percolation problem if we only consider the branching random walk. Because, the vertical interaction is between two nearest vertical sites, the vertical interaction should be considered only once over the generations. Therefore, the true process we are considering is a branching random walk with attrition. The attrition means that if any site has been visited during the propagation, it cannot be visited again since the vertical interaction has already been considered for this site.
The way of dealing with horizontal propagation is motivated by the work of Lalley [16] on the scaling limit spatial epidemics to Dawson-Watanabe process with killings. The process considered in [16] is as follows. At each site i, there is a fixed population (or village) of N individuals and each of them can be either susceptible, infected or recovered. The model runs in discrete time; an infected individual recovers after a unit of time and cannot be infected again. An infected individual may transmit the infection to a randomly selected (susceptible) individual in the same or in the neighboring villages. The transmission probabilities are chosen to be critical (i.e. the expected number of individuals to be infected by a unique infected individual when everyone else is susceptible is one). The evolution of this SIR dynamics can be studied with the help of a branching random walk envelope: with any particle at site i and time t lives for one unit and reproduces, placing a random number of particles at a nearest site j with |j − i| ≤ 1, where the random number is of law Binomial(N, C d /N ). The particles are categorized into Susceptible, Infected and Recovered (SIR) and any recovered particle is immune and will not be infected again. They studied the scaling limit (space factor N β/2 and time factor N β ) of this system by considering the cluster of particles at each site as a village and calculating the log-likelihood functions. The recovered particles do matter only when β = 2/5, which corresponds to the attrition part of our process. To study the scaling limit of our problem on horizontal level, we first need to do two scaling on the approximate density. First, we have to scale the space with N , then the movement of the edges from x will have a uniform displacement on x + [−1, 1]/{0}. Then, to get the weak convergence, we will renormalize the space and time with N α and N 2α respectively. The state of the process at time n ∈ Z + is given byξ n (·) : Z/N 1+α → {0, 1}.ξ n (x) = 0 indicates that the site x is vacant and ξ n (x) = 1 indicates that the site x is occupied. Two sites are neighbors in the scaled space, denoted by We are going to derive the asymptotic approximate density The method in [16] is to calculate the log-likelihood function with respect to a branching envelope with known asymptotic density. However, we do not have the log-likelihood function in our case. A more standard argument is to show the weak convergence of the rescaled continuous-time particle system by verifying the tightness criteria [11] like in [18], [5] and [9]. We will mainly refer to the way of Mueller and Tribe [18] dealing with long-range contact process and long-range voter model and adapt it to our discrete model to get the asymptotic stochastic PDEs. Our strategy on the horizontal level is to derive the asymptotic density of the branching random walk without attrition dominating the true system, where the states are denoted by ξ(x). In the branching random walk, the case of multiple particles at one site is allowed. But we can show that the probability of multiple particles is small with order O(N α−1 ). Then the state can be reduced from N-valued to {0, 1}-valued. We will then derive the asymptotic density of the true process. Since we are considering the existence of percolation, to consider the infinite cluster containing (0, 0) is equivalent to consider 2N 2α initial particles on {−N 1+α , . . . , 0, . . . , N 1+α } with distance N 1−α . Notice that if we denote [−r, r] N = [−r, r] ∩ Z/N 1+α as the rescaled discrete interval, then the initial condition is A(ξ 0 ) = ½ [−1,1]N whose linear interpolation tends to f = I [−1,1] .
When showing percolation and adding the vertical connections, by a renormalization argument (ref. [10]) we can reduce our layered system to an oriented percolation. We can define a site as open if its corresponding block has a certain amount of cumulated density, since we have already taken into account the attrition in the true system. After building the renormalization argument, we are able to use the criteria in [7] to determine the existence of percolation. The main result of this article is as follows. Remark. (a) The critical value α = 1/5 can be guessed by standard coupling as in [16]. First, we build a critical branching random walk with the same initial conditions, namely 2N 2α particles with at most one on each site distributed uniformly on 2N 1+α sites in [−1, 1] N . At each time, particles of the branching random walk produce offsprings at their neighbourhoods following a Binomial distribution Bin (2N, 1/2N ). The branching random walk will finally become extinct as we know [1]. The existence of percolation is a meaningful problem if we introduce a vertical interaction. In the beginning, there are O(N α−1 ) particles at each site on average. Since the branching random walk is critical, i.e. the expectation of offspring is exactly 1, this average behaviour will not change too much during the propagation. Next, we colour the particles as red or blue according to they are alive or dead respectively. The attrition means that if any site that has been visited, then it cannot be visited again. Initially, all the particles are red. The offspring of blue particles are blue and the choice of colour of red particles is as follows. if a site x has been occupied in the past, then the offsprings of red particles that are produced at x become blue. The branching random walk will last for O(N 2α ) generations (ref. [1]). Up to extinction, the chance of dying for any particle is O(N 3α−1 ). The total attrition at each generation is O(N 5α−1 ). Hence if α = 1/5, then the total attrition per generation is O(1).
(b) As already mentioned above, the original problem that motivated this paper can be formulated in terms of the existence of percolation for a corresponding Fortuin-Kasteleyn measure with shape parameter q = 2 and edge probabilities of By the FKG inequality, the probability of percolation for q = 2 is bounded from above by that when q = 1 (product measure). As a consequence, we conclude that there is no phase transition if ǫ(γ) = κγ 2/5 , for all γ small provided κ > 0 is sufficiently small.
(c) The organization of this paper is as follows. We will provide necessary lemmas and use them to show the weak convergence of the dominating envelope in Section 2. Behaviours related to the true horizontal process like the asymptotic density, Girsanov transformation and cumulated density are shown in Section 3. The killing property of the attrition part can help us show the case when κ < C 1 in Subsection 4.1. With the properties of the true process, the oriented percolation construction is built up in Subsection 4.2 and we can show the existence of percolation when κ > C 2 .

The envelope process
Before studying the asymptotic behaviour of the process, we first study that of an envelope process. In this section, we consider the state function ξ n (·) : Z/N 1+α → Z + . The mechanism of this envelope process is as follows. The number of particles at site x will increase by 1 if one of its neighbours branches following Binomial(2N, 1/2N ) and then chooses x uniformly among the 2N neighbours. It can be written as ∼ Bernoulli(1/2N ). The horizontal processξ n (·) : Z/N 1+α → {0, 1} analysed in Section 3 is dominated by this envelope process in two senses: ξ n (·) does not allow multiple particles at one site and any visited site cannot be visited again. At the end of this section, we will show that the probability of multiple particles at one site is quite small, of order O(N α−1 ) which is negligible when α < 1.
The main result of this section is that the asymptotic density of the dominating envelope with initial condition A(ξ 0 ) = ½ [−1,1]N follows the following stochastic partial differential equation (SPDE). 1] andẆ is the space-time white noise.
The idea of the proof is to write the mechanism as a martingale problem, then introduce a Green function to simplify the approximate density. The tightness criteria in [11] can be applied to get the weak convergence. We will follow the blueprint of [18] to show the tightness.
Before starting the proof, we first explain the notation used in the following sections. In the discrete case, the inner product between two functions is defined as (f, g) = 1 N 1+α Define the discrete measure generated by ξ n as x ξ n (x)δ x and the inner product between function f and measure ν is defined as In Lemma 2.2, we will see that for any test function f which is bounded with compact support We define the amplitude of a function around a neighbourhood as In examining the Green function, we use the norm where e λ (x) = e λ|x| .

Martingale Problem
Rephrasing the mechanism of ξ n (x), we have (2. 2) The first term will contribute to the space-time white noise part and the second term will contribute to the Laplacian in the SPDE. Choose test function φ : Summation by parts and (2.2) give .
2N y∼x (f (y) − f (x)). By summation by parts to the second term again, we can obtain . Summing up n from 1 to m, we get a semimartingale decomposition For any x ∈ Z/N 1+α , let ψ z i (x) ≥ 0 be the solution to ∆ D can be seen as the generator of this symmetric random walk X n with steps of variance c3 We apply (2.4) with test function φ k = ψ n−k for k ≤ n − 1, so that the first drift term vanishes and (ν N n , φ n ) = (ν N n , ψ x 0 ) = A(ξ n )(x). Thus we obtain an approximation Proving the tightness of A(ξ ⌊tN 2α ⌋ ) is equivalent to prove that of M ⌊tN 2α ⌋ . Some estimations on ψ n and the moments of A(ξ n ) used to show the tightness are stated in the appendix A. We skip the proofs which are very similar to those in [18].
Tightness of A(ξ ⌊tN 2α ⌋ ), N ≥ 1 follows from Lemma 2.1. Proof. Therefore, This implies the tightness of (ν N ⌊tN 2α ⌋ , φ), N ≥ 1 and then implies the tightness of ν N ⌊tN 2α ⌋ , N ≥ 1 as a measure-valued process under vague topology. Hence, with probability one, for all T > 0, λ > 0, and test function φ with compact support, we can have a subsequence of is a martingale and every term on the right-hand side converges almost surely by Lemma 2.
which is continuous since every term on the right-hand side is continuous. Moreover, from (2.5), is also a continuous local martingale. (2.11) and (2.12) prove that any subsequential weak limit ν t (dx) = u t (x)dx solves (2.1). The uniqueness follows from [6] which finish the proof of Theorem 2.1.

Multiple particles at one site
In this subsection, we show the probability of multiple particles at one site is negligible in the branching envelope. Then, the state function can be reduced from its number of particles to that it is occupied or vacant. We will first show a property on the weak limit of the envelope process, that is Lemma 2.3, then use it to prove this reduction of the discrete state function. Let X t denote the total mass of this system, that is We then have that and therefore its quadratic variation is For k ≥ 1, let T k denote the stopping time given by

Lemma 2.3. For a fixed initial condition
Proof. P(T k < ∞) can be decomposed as (2.13) If we denote S 0 as the first hitting time of zero then P(T ′ k < ∞) = P(T ′ k < S 0 ), hence the first term on the RHS of (2.13) is simply By using the Markov property of X t , (2.14) The total mass satisfies Hence we can get From this, the second term in (2.14) can be bounded by using Markov inequality Therefore, After plugging in P( are finite with probability one.
Then for the discrete state function, we have: For any k ∈ N and any x ∈ Z/N 1+α , there exists constant C, Proof. In the discrete system, we have Hence, The notation P k−1 (·) means conditional probability given F k−1 .
Since the branching envelope dominates the true horizontal process, this property will also hold for the true horizontal process.

The true horizontal process
The true process we consider is dominated by the branching random walk in the proceeding section, which means that at each time step, the particles will move and produce following the mechanism of ξ k . But if the site x has been occupied by some particle before, then it cannot be occupied again. We denote {ξ k (x) ∈ {0, 1}, k ∈ Z + , x ∈ Z/N 1+α } as the mechanism of the true process. It can be expressed aŝ The main goal of this section is to describe the limiting behaviour of the true horizontal process, summarized in the following result: Remark. The initial condition The proof is given in the next two subsections: we first prove the tightness and that any weak limit satisfies (3.2), and in (sub)section 3.2 we prove the uniqueness.

Limit behaviour of the rescaled horizontal process
xξ k (x)δ x as the measure generated byξ k . Choose test function φ satisfying (2.3) and sum by parts, where the error term and martingale terms Summing k from 1 to n, we can get a semimartingale decomposition where we use the fact that j≤kξ j (x) ∈ {0, 1} to get the first inequality. We first show that the error term in (3.3) is negligible.
Proof. By Hölder inequality, where in the second inequality, we used the facts that 1 − j≤kξ j (x) ≤ 1 and given F k , are independent. Following similar reason as Corollary 2.4, The result follows by using the properties of test functions (2.3).
Choosing φ k = ψ n−k as the Green function in Section 2.1, we can obtain , the estimations in Lemma A.3 also hold forξ k (x). As in Section 2.2, we will use the estimations in Lemma A.2 and Lemma A.3 to get the tightness of A(ξ ⌊tN 2α ⌋ )(x). We assume that the linear Proof. We first deal with the error term and the left parts will be shown as in the proof of Lemma 2.1, where we decompose this difference into space and time differences.
The error term is Moreover, is a martingale. Hence, By using BDG inequality, we have To get the estimation of space difference, first we need to deal with where the last inequality is because of α = 1/5. Next, we will use BDG inequality to estimate As the argument in (3.4), (3.7) Therefore, by using the fact that α = 1/5, Similarly, for the time difference, we first deal with the drift term By (b), (e) of Lemma A.2, (c) of Lemma A.3 and the fact that α = 1/5, the p-th moment of the first term above can be bounded by By (b), (c) of Lemma A.2, (c) of Lemma A.3 and the fact that α = 1/5, the p-th moment of the second term above can be bounded by To deal with the part of M ⌊sN 2α ⌋ (ψ y ⌊sN 2α ⌋−· ), we can separate it into two parts and use BDG inequality.
The first part is M Using (3.7) gives us The second part is M Using (3.7) again gives us Combining with Lemma 2.1, we get (3.6).
The tightness of A(ξ ⌊tN 2α ⌋ ) follows from Lemma 3.2, which means that we can find a subsequence with a limitû t . Since the true process is dominated by the branching envelope, we easily see that Lemma 2.2 also holds for the true horizontal process. This implies the tightness ofν N ⌊tN 2α ⌋ under vague topology. Letν t be a weak limit. By substituting φ k = φ in the semimartingale decomposition (3.3), if α = 1/5, we can see that is a martingale and every term on the right-hand side converges almost surely by Lemma 3.2. HenceM N ⌊tN 2α ⌋ (φ) converges to a local martingalê which is continuous since every term on the right-hand side is continuous. Moreover, from (3.4), is also a continuous local martingale. (3.8) and (3.9) prove that any subsequential weak limitν t (dx) =û t (x)dx solves (3.2).

Girsanov transformation. Proof of the uniqueness in Theorem 3.1
As is discussed in Section 2.2, the envelope measure ν t solves the martingale problem: ∀φ ∈ C 2 0 (R) test function twice differentiable with compact support, the process is a continuous local martingale with quadratic variation process From this, we know that is a continuous local martingale. Using the duality method in sec. 4.4 of [11], we can choose triplet (f, 0, 0) on the space M F × C 2 0 , where M F is the collection of finite Borel measures and f (·, ·) is defined as s is the solution to the deterministic equation (3.10) {u * t } t≥0 is the dual process of the solution to the martingale problem. The existence of solution to (3.10) gives the uniqueness of {ν t } t≥0 .
Let m(ds, dx) be the orthogonal martingale measure of m t (·), which means that it is of intensity measure for any Borel measurable set A ⊂ R. Then the Radon-Nykodym derivative of the true process with respect to the envelope is where the drift term The uniqueness of {ν t } t≥0 follows directly from the uniqueness of {ν t } t≥0 . This concludes the proof of Theorem 3.1.

Existence of Percolation
In the past two sections, we have shown that α = 1/5 (in the sense of Theorem 3.1) is a critical exponent for the horizontal process. The envelope process on each horizontal layer follows the law with asymptotic approximate density (2.1).
In the anisotropic percolation model, the horizontal movement has an attrition compared to the envelope process. The attrition comes from two parts: • In the envelope process, it is allowed to have multiple particles at each site. However, in the true mechanism, we only consider if a site is occupied or not hence the configuration at each site can only 0 or 1. Fortunately, the probability of multiple particles is negligible when α = 1/5 (Corollary 2.4).
• As was explained in the Introduction, the vertical interaction should be only considered once for any site in the anisotropic percolation. When we consider the horizontal movement, any site that has been visited before cannot be visited again. Under the critical exponent α = 1/5, this attrition becomes significant and leads to the part −û t t 0û s ds in the asymptotic approximate density.
In this section, we investigate the occurrence (or not) of percolation for large (small) values of κ for the true model, i.e., with attrition.

The case κ < C 1
As we have discussed in Section 1, the occupied sites at each layer follows a horizontal process with attrition whose asymptotic approximate density follows the SPDE (3.2). Here we abuse the notation C i x as the cluster starting from x at layer i. in the rescaled space Z/N 6/5 × Z. The main theorem to show in this subsection is as follows.

Theorem 4.1. For the horizontal process with attrition, there exists a constant L such that the cumulated number of occupied sites (or the cluster size) starting from zero satisfies
E|C 0 0 | ≤ LN 2/5 . Before proving the main theorem, let us show how it implies that there is no percolation when κ < C 1 for C 1 small enough. In the first vertical movements, these gray sites at layer 0 can connect with sites (green) at layer ±1. Before the second vertical movement, these open sites at layer ±1 will produce following the law C 0 0 distributed (gray) at layer ±1 which will connect with sites (green) at layer ±2, 0. S m can be constructed inductively by considering the total number of horizontal connected sites and then their vertical movements.
Due to attrition, we only consider whether a site is occupied or not rather than the number of particles at each site, the cardinality {|S m |} m≥0 is stochastically dominated (in the sense of Definition II.2.3 of [17]) by a branching process {Z m } m≥0 following the law Theorem 4.1 gives the upper bound of E|C 0 0 |, therefore {Z m } m≥0 is stochastically dominated by a branching process {Z m } m≥0 following the law When κ is small enough to make 2κL < 1, {Z m } m≥0 is a sub-critical branching process which will die out (ref. [1]). Therefore, there exists positive constant C 1 = (2L) −1 such that for κ < C 1 , there is no percolations in this layered system.
We now move to the proof of Theorem 4.1. For this we will need two inequalities (Lemma 4.1 and Proposition 4.1 below) which concern the following hitting times for the branching envelope and for the true horizontal process: which is just the discrete version of the hitting time T k in Section 2.3, and Lemma 4.1. For a initial condition ξ 0 (x) = δ 0 (x), we have Proposition 4.1. Let integer k 0 be defined by 2 k0 ≤ N 2/5 < 2 k0+1 . There exists M 1 large such that for any k = k 0 + log 2 M 1 + r, r ≥ 0, we have Postponing the proofs of these estimates, we first see how they all to conclude the proof of Theorem 4.1.
Proof of Theorem 4.1. The proof is given in the following steps. As we can see in the proof of Theorem 3.2, the attrition part is negligible when α < 1/5 becomes significant when α = 1/5. Because of attractiveness, we only need to consider the attrition once the total mass is of order O(N 2/5 ). So we consider a process that dominates the horizontal process, which follows the pure branching random walk before the total mass reaches M 1 N 2/5 for some M 1 large and includes the attrition part after that. We are first interested in the crossing time of x ξ n (x) over level M 1 N 2/5 .
The dominating process that we consider in this subsection follows {ξ k (x)} beforeT k0+log 2 M1 and follows {ξ k (x)} afterT k0+log 2 M1 . The reason of separating the time is as follows. The size of cluster containing the origin satisfies The third inequality is by Lemma 4.1 and the fact that 2 k0 ≤ N 2/5 . The last work is to bound the second term in the last inequality. By Proposition 4.1, the size of cluster containing zero can be bounded by This finishes the proof of Theorem 4.1.
In the following part of this subsection, we will show Lemma 4.1 and Proposition 4.1.
Proof of Lemma 4.1. The proof is similar as in Lemma 2.3. It is followed by replacing the corresponding part in the proof of Lemma 2.3 that Then we can use the similar martingale technique and the fact that is a martingale. Denote as the total mass (in discrete sense) at time n. The desired probability can be decomposed as Denote S 0 as the first hitting time of zero, then P(T ′ k < ∞) = P(T ′ k < S 0 ), and this is simply By Markov property of {X n } n≥0 , For m < n, we have Letting n =T ′ j+1 and m =T ′ j gives that Therefore, To show Proposition 4.1, we need two properties of the branching processes: on the large deviations and the next one is on the population size of the critical branching process.
The r.h.s. reaches the minimum if t satisfies But for γ ≥ 1, e t ≈ n γ−1 and where Y i ∼ Binomial (2N, 1/2N ). For any ǫ > 0, we have that for n large enough, Proof. The generating function of Z n is The result follows from Theorem I.10.1 of Harris [13].
With the help of these two properties, we can prove Proposition 4.1 used in the proof of Theorem 4.1.

Proof of Proposition 4.1.
Here we consider the mechanism after time k 0 +log 2 M 1 when the attrition becomes significant. We will construct a Lyapunov function and use optional sampling theorem to get the result.
Given thatT k0+log 2 M1 < ∞, we first construct a function onT k used in the proof. For k = k 0 + log 2 M 1 + r, we define The second case of (4.2) corresponds to {T k =T ′′ k < ∞} and the last two cases of (4.2) Our aim is to show that which will imply, by optional sampling theorem: In the following proof, we will separate the l.h.s. of (4.3) into three terms and estimate them. The case when U r = 0 is trivial, so we will assume that U r = 0. Under this assumption, (4.3) can be expressed as Estimation of (4.6): this term is easier to handle because of Lemma 4.2.
where c(k) → 0 as k → ∞. Hence we can take M 1 large enough to make c(k) ≤ 1/3 for k ≥ k 0 + log 2 M 1 . Estimation of (4.5): the probability P U r+1 = 2 k+2 | U r = 0 is estimated based on the attrition property. U r = 0 means that . In case (i), the envelope process, with probability larger than 1−2 −r /(2M 1 ), these 2 k particles will not die out untilT k + N 2/5 /2. Given that the system does not die out, suppose that the particle at x at timeT k survives untilT k + N 2/5 /2. Lemma 4.3 gives us that (4.7) Theorem 1.1 of Kesten [15] showed the distribution of maximal displacement of a critical branching random walk. In our case, if we denote {M n } n≥0 as the maximal displacement at time n, then in the renormalized space i.e. Z/N 6/5 , we have since the envelope branching process has finite moment for any order α. This theorem shows that given it survives until N 2/5 /2, these ǫN 2/5 particles will concentrate in x + [−ǫ −1 , ǫ −1 ] until N 2/5 with probability 1 − ǫ 2 . We can find a interval with length 2ǫ, without loss of generality denoted as (x − ǫ, x + ǫ) such that there are ǫ 3 N 2/5 particles. Therefore, For another particle that moves inside (x − ǫ, x + ǫ), the chance that it will be killed because of attrition is ǫ 2 /N 2/5 . Without loss of generality, we assume that x = 0 and call the (−ǫ, ǫ) a killing zone. By the compactness of solution to the Dawson-Watanabe process [14], we can choose an M 2 large enough such that (4.10) For any x ∈ [−M 2 , M 2 ], consider a random walk {X i } i≥0 starting from X 0 = x and each step it moves to one of its neighbourhood y (|y − x| ≤ N −1/5 ) with probability 1/2N . If this random walk moves into the zone (−ǫ, ǫ), the chance of survival caused by attrition is bounded by For the random walk over M 3 N 2/5 steps, this is bounded by It is less than As N → ∞, (4.11) tends to (4.12) By Lemma 4.4 and Corollary 4.2 (below), we can find an M 3 large enough such that Therefore at timeT k + N 2/5 + M 3 N 2/5 , the expectation of number of remaining particles (here the superscript means the total mass in discrete case) Hence by Markov property and gambler's ruin argument that afterT k + (M 3 + 1)N 2/5 ,X will hit U r+1 > 2 k+1 before hitting zero is with probability The term (4.5) is δ and ǫ can be chosen small enough to ensure that 2(2δ + e −ǫ −1 ) ≤ 1/3. Case (ii) can be dealt with by compact support property and the definition ofT ′′ k because in this caseT k =T ′′ k . The detail of analysis is put into case (ii) in the estimation of (4.4). The only difference is that after we get which is similar as (4.9), we can follow the same procedure as case (i).
Estimation of (4.4): We need to consider two cases that {U r = 0} corresponds to: The cumulated number of particles from 0 toT ′ k is less than 2 2k . To reach 2 2k+2 , the increment of the cumulated occupied sites should be no less than 3 · 2 2k . After time (M 3 + 1)N 2/5 ≥ (M 3 + 1)2 k0 , the increment of each time step is The gambler's ruin arguments gives that In case (ii), U r = 2 k means thatT k =T ′′ k < ∞. Notice that atT ′′ k , the cumulated occupied sites have already reached 2 2k . In this case, And it also means thatT k =T ′′ k ≤T ′ k , hence sup 1≤i≤T ′′ kX i ≤ 2 k . AtT k , the cumulated occupied sites Since {ξ n (·)} n≥0 is dominated by {ξ n (·)} n≥0 , the compact support property of Dawson-Watanabe process (4.10) also holds for {ξ n (·)} n≥0 . With probability larger than 1 − δ, these occupied sites concentrate on The same killing box analysis procedure as we deal with the estimation of (4.5), the expectation of number of remaining occupied sites To reachT ′′ k+1 , the number of cumulated occupied sites fromT k =T ′′ k toT ′′ k+1 is larger than (2 2(k+1) − 2 2k − 2 k ). After time M 3 N 2/5 ≥ M 3 2 k0 , the increment of each time step is The gambler's ruin argument gives that Up to now, we can choose suitable M 1 and M 3 to guarantee that This finish the proof of (4.3) and concludes this proposition.

Remark.
Notice that without considering the attrition, we can have the probability P(T k < ∞) ≤ C2 −k . This is not enough in the proof of Theorem 4.1. However, the existence of attrition that those sites who have been visited before cannot be visited again can help us. Even in a very small killing box (x− ǫ, x+ ǫ) in the proof above, many particles will be killed in a finite but large time period.
The following two results, Lemma 4.4 for continuous case (Brownian motion) and Corollary 4.2 (used in the proof of Proposition 4.1) analyse the killing property of process with attrition and estimate the remaining density after a large but finite time.
Proof. Since x is within distance K from the origin, with probability 1, the Brownian motion will hits 0 with finite time. Without loss of generality, we can assume that this Brownian motion (B t ) t≥0 is zero at time 0. We can define a sequence of stopping times such that τ 0 is the first time after 0 that hits ǫ or −ǫ, and σ 0 be the first time after τ 0 that hits zero, Inductively, By the Itô isometry of Brownian motion, we have We can find a V large enough, such that by gambler's ruin, after τ k , the probability of hitting zero before hitting V or −V is 1 − ǫ/V . Then the number of times of hitting zero before hitting V of −V follows a geometric distribution. If we let V >> ǫ −4 and M >> V 2 , then with probability larger than 1 − ǫ 3 , it will hit zero V /ǫ times before M . Therefore, the expectation of time spent between (−ǫ, ǫ) will be with order ǫ −3 in the time length of M .
In the discrete sense, we have the following corollary.

The case κ > C 2
In this case, we will prove some properties of the true process, and then lead to an oriented percolation construction. The first step is to show that the difference between the solution to (3.2) and the solution to deterministic heat equation is quite small. Suppose under Q, u(t, x) is the solution to (3.2) and under P, u(t, x) is the solution to (2.1). The Radon-Nykodym derivative of Q with respect to P is (3.11). Let the initial condition be f (x) = I [−1,1] (x) and define the difference where G t is the transition function of Brownian motion with diffusion coefficient 1/3. By Lemma 4.2 of [19] (also ref. Lemma 4 of [16]), This property also holds for Q:
By Hölder inequality, Similarly, Therefore, and we have the expected result.
The previous result helps to get a lower bound for the total density in a small time period which is our first desired property.
The second property is based on the work of Kesten [15]. Denote M n as the maximal displacement of our process with attrition, i.e. M n = sup{x ∈ Z/N 1+α :ξ n (x) = 0}.
Since our unscaled branching random walk satisfies infinite moment assumption and it dominates the process with attrition, by Theorem 1.1. of [15], we have In our original percolation model, the edges are not directed. However, it suffices to show percolation in the related model where the vertical edges are directed upward. For this we shall build a block argument, reducing the analysis to that of an oriented percolation model. Here, we keep the notation as in [7].
L 0 is made into a graph by drawing oriented edge from (m, n) to (m − 1, n + 1) or (m + 1, n + 1). Random variables ω(m, n) ∈ {0, 1} are to indicate whether (m, n) is open (ω(m, n) = 1) or close (ω(m, n) = 0). We say that there is a path from (m, n) to (m ′ , n ′ ) if there is a sequence of points m = x n , . . . , n = x n ′ so that |x l − x l−1 | = 1 for n < l ≤ n ′ and ω(x l , l) = 1 for n ≤ l ≤ n ′ . Let Proof. With out loss of generality, we only need to prove the case when r = 1 and K = 1. In each sub-interval I 1 i , we can denote the center of this interval as x i . Since [−1, 1] N is uniformly occupied with density 1, for any 1 For any x ∈ [−1, 1] N , we can find an x i such that x ∼ x i . It means that |x − x i | ≤ N −1/5 , so by the Hölder continuity in space of the approximate density (Lemma 3.2),

The proof is finished by Chebychev inequality.
This proposition shows that with probability greater than 1−N −1/40 , [−r, r] N is uniformly occupied with density K can be regarded as Aξ = K½ [−r,r]N . The idea of proving the infinite cluster is to show that the certain amount of occupied sites will expand larger and larger as the layer increases. Our next step is to prove that with κ large enough, a good interval [−r, r] at layer i will result in a good interval [−r − δ 5/2 , r + δ 5/2 ] at layer i + 1 with large probability. We have already considered the attrition in the true mechanism, so our idea is to show that over a small period of time, the cumulated amount of occupied sites can reach a reasonable amount and then parts of them will move to a higher level with large probability. Before proving this development, we give a preliminary result on the large deviation property of a Binomial distribution which represents the selection of occupied sites to higher level. Lemma 4.6. Suppose that S ∼ Binomial(⌊c 1 δ 5 N 3/5 ⌋, κN −2/5 ). S has the following large deviation property Proof. We only show the large deviation of the right tail here.
The following proposition guarantees that the goodness of an interval will result, with large probability, in the goodness of a larger interval at higher level. Proof. Without loss of generality, we only need to consider the case when r = 1 and i = 0. Suppose at layer 0, the interval [−1, 1] N is uniformly occupied with density 1 (actually the initial condition in our settings, it is exactly a uniform I [−1,1] initial condition). By Corollary 4.3, we know that there is a constant c 1 such that with probability greater than 1 − δ 7/2 , Recall that the approximate density at site x is where the center of each interval is denoted as x i , 1 ≤ i ≤ (1 + δ 5/2 )N 1/5 . With probability greater than 1 − δ 7/2 , for any 1 ≤ i ≤ (1 + δ 5/2 )N 1/5 , It means that the cumulated number of occupied sites in each I 1 i over this period is greater than c 1 δ 5 N 3/5 . Lemma 4.6 gives that if κ >> c −1 1 δ −5 , then outside an event of negligible probability N 1/5 e −N 1/10 , the interval [−1−δ 5/2 , 1+δ 5/2 ] N is uniform occupied with density κc 1 δ 5 >> 1 at layer ±1.
The initial condition in our settings is I [−1,1] , which means that [−1, 1] contains a pile. The proposition above shows that with probability greater than 1 − δ 7/2 , there is a pile in [−1 − δ 5/2 , 1 + δ 5/2 ]. After 2δ −5/2 steps, the piles will expand to [−3, 3] at layer 2δ −5/2 by the following Corollary. Proof. Without loss of generality, we assume that r = 1. By FKG inequality and Proposition 4.3, the probability of this expansion is greater than This property holds for larger compact interval by the proposition above. Therefore, the renormalized space is L 0 = {(m, n) ∈ Z 2 : m + n is even , n ≥ 0} and make L 0 into a graph G = (V, E) by drawing oriented edges from (m, n) to (m ± 1, n + 1). The percolation process (ψ(e)) e∈E is called d-dependent percolation with density p if for a sequence of vertices v i = (m i , n i ), 1 ≤ i ≤ I with v i − v j ∞ > d, i = j connected by a sequence of edges e i , 1 ≤ I − 1, P(ψ(e i ) = 0, 1 ≤ i ≤ I − 1) ≤ (1 − p) I−1 .
The initial condition is ω(0, 0) = 1. By using the comparison argument Theorem 4.3 in [7] and Corollary 4.5, we have the following result. The theorem of existence of percolation for d-dependent oriented percolation (Theorem 4.1 in [7]) shows that if 2δ < 6 −4·9 , there is a percolation. Figure 2 shows this renormalization construction.

A.1 Estimations of characteristic function
First, we need some bounds on the distribution function of X t . Recall that This directly gives that for u ≤ N α /2, |ρ(u)| ≤ exp − c 3 u 2 12N 2α , and for u ≥ N α /2, |ρ(u)| ≤ 23/24. Moreover, with the help of Theorem 8.5 of [3], for u ≤ N α /2, Follow the inversion formula, The difference satisfies |ρ t (u)| + e − c 3 tu 2 6N 2α du Therefore, we get the bound Because of (a) in the next lemma and p(t, x) ≤ t −1/2 , we have

A.2 Estimations of ψ t
With the help of Lemma A.1, we can get the estimations on ψ n .
(d) For |x − y| ≤ 1, The second statement is because P (X t = y) ≤ c/N for any y.