Gaussian scaling for the critical spread-out contact process above the upper critical dimension

We consider the critical spread-out contact process in $\Zd$ with $d\geq 1$, whose infection range is denoted by $L\geq1$. The two-point function $\tau_t(x)$ is the probability that $x\in\Zd$ is infected at time $t$ by the infected individual located at the origin $o\in\Zd$ at time 0. We prove Gaussian behavior for the two-point function with $L\geq L_0$ for some finite $L_0=L_0(d)$ for $d>4$. When $d\leq 4$, we also perform a local mean-field limit to obtain Gaussian behaviour for $\tau_{tT}$ with $t>0$ fixed and $T\to \infty$ when the infection range depends on $T$ such that $L_T=LT^b$ for any $b>(4-d)/2d$. The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards the canonical measure of super-Brownian motion, which we defer to a sequel of this paper.


Introduction
The contact process is a model for the spread of an infection among individuals in the d-dimensional integer lattice Z d . We suppose that the origin o ∈ Z d is the only infected individual at time 0, and that every infected individual may infect a healthy individual at a distance less than L ≥ 1. We refer to this model as the spread-out contact process. The rate of infection is denoted by λ, and it is well known that there is a phase transition in λ (see e.g., [22]).
Sakai [26,27] has proved that when d > 4, the sufficiently spread-out contact process has several critical exponents which are equal to those of branching random walk. The proof by Sakai uses the lace expansion for the time-discretized contact process, and the main ingredient is the proof of the socalled infrared bound uniformly in the time discretization. Thus, we can think of his results as proving Gaussian upper bounds for the two-point function of the critical contact process. Since these Gaussian upper bounds imply the so-called triangle condition in [3], it follows that certain critical exponents take on their mean-field values, i.e., the values for branching random walk. These values also agree with the critical exponents appearing on the tree. See [22,Chapter I.4] for an extensive account of the contact process on a tree.
Recently, van der Hofstad and Slade [20] proved that for all r ≥ 2, the r-point functions for sufficiently spread-out critical oriented percolation with spatial dimension d > 4 converge to those of the canonical measure of super-Brownian motion when we scale space by n 1/2 , where n is the largest temporal component among the r points, and then take n ↑ ∞. That is, the finite-dimensional distributions of the critical oriented percolation cluster when it survives up to time n converge to those of the canonical measure of super-Brownian motion. The proof in [20] is based on the lace expansion and the inductive method of [19]. Important ingredients in [20] are detailed asymptotics and estimates of the oriented percolation two-point function. The proof for the higher-point functions then follows by deriving a lace expansion for the r-point functions together with an induction argument in r.
In this paper, we prove the two-point function results for the contact process via a time discretization. The discretized contact process is oriented percolation in Z d × εZ + with ε ∈ (0, 1], and the proof uses the same strategy as applied to oriented percolation with ε = 1, i.e., an application of the lace expansion and the inductive method. However, to obtain the results for ε 1, we use a different lace expansion from the two expansions used in [20,, and modify the induction hypotheses of [19] to incorporate the ε-dependence. In order to extend the results from infrared bounds (as in [27]) to precise asymptotics (as in [20]), it is imperative to prove that the properly scaled lace expansion coefficients converge to a certain continuum limit. We can think of this continuum limit as giving rise to a lace expansion in continuous time, even though our proof is not based on the arising partial differential equation. In the proof that the continuum limit exists, we make heavy use of convergence results in [4] which show that the discretized contact process converges to the original continuous-time contact process.
In a sequel to this paper [18], we use the results proved here as a key ingredient in the proof that the finite-dimensional distributions of the critical contact process above four dimensions converge to those of the canonical measure of super-Brownian motion, as was proved in [20] for oriented percolation.

The spread-out contact process and main results
We define the spread-out contact process as follows. Let C t ⊂ Z d be the set of infected individuals at time t ∈ R + , and let C 0 = {o}. An infected site x recovers in a small time interval [t, t + ε] with probability ε + o(ε) independently of t, where o(ε) is a function that satisfies lim ε→0 o(ε)/ε = 0. In other words, x ∈ C t recovers at rate 1. A healthy site x gets infected, depending on the status of its neighbours, at rate λ y∈Ct D(x − y), where λ ≥ 0 is the infection rate and D(x − y) represents the strength of the interaction between x and y. We denote by P λ the associated probability measure.
The function D is a probability distribution over Z d that is symmetric with respect to the lattice symmetries, and satisfies certain assumptions that involve a parameter L ≥ 1 which serves to spread out the infections and will be taken to be large. In particular, we require that there are L-independent constants C, C 1 , C 2 ∈ (0, ∞) such that D(o) = 0, sup x∈Z d D(x) ≤ CL −d and C 1 L ≤ σ ≤ C 2 L, where σ 2 is the variance of D: where | · | denotes the Euclidean norm on R d . Moreover, we require that there is a ∆ > 0 such that See Section 5.1.1 for the precise assumptions on D. A simple example of D is the uniform distribution over the cube of side length 2L, excluding its center: The two-point function is defined as In words, τ λ t (x) is the probability that at time t, the individual located at x ∈ Z d is infected due to the infection located at o ∈ Z d at time 0.
By an extension of the results in [4,10] to the spread-out contact process, there exists a unique critical value λ c ∈ (0, ∞) such that where we denote the Fourier transform of a summable function f : (1.6) We next describe our results for the sufficiently spread-out contact process at λ = λ c for d > 4.

Results above four dimensions
We now state the results for the two-point function. In the statements, σ and ∆ are defined in (1.1)-(1.2), and we write f ∞ = sup x∈Z d |f (x)| for a function f on Z d .
with the error estimate in (1.7) uniform in k ∈ R d with |k| 2 / log(2 + t) sufficiently small.
The above results correspond to [20,Theorem 1.1], where the two-point function for sufficiently spreadout critical oriented percolation with d > 4 was proved to obey similar behaviour. The proof in [20] is based on the inductive method of [19]. We apply a modified version of this induction method to prove Theorem 1.1. The proof also reveals that (1. 10) In a sequel to this paper [17], we will investigate the critical point in more detail and prove that (1.11) holds for d > 4, where D * n is the n-fold convolution of D in Z d . In particular, when D is defined by (1.3), we obtain (see [17,Theorem 1.2]) (1.12) where U is the uniform probability density over [−1, 1] d ⊂ R d , and U n is the n-fold convolution of U in R d . The above expression was already obtained in [8], but with a weaker error estimate. Let γ and β be the critical exponents for the quantities in (1.5), defined as where we use "∼" in an appropriate sense. For example, the strongest form of χ(λ) ∼ (λ c − λ) −γ is that there is a C ∈ (0, ∞) such that where o(1) tends to 0 as λ ↑ λ c . Other examples are the weaker form 15) and the even weaker form (1) . (1.16) See also [22, p.70] for various ways to define the critical exponents.
As discussed for oriented percolation in [20, Section 1. Extending the argument in [24] for oriented percolation to the continuous-time setting, we conclude that (λ c ) < ∞ implies the triangle condition of [1,2,3], under which γ and β are both equal to 1 in the form given in (1.15), independently of the value of d [3]. Since these d-independent values also arise on the tree [29,34], we call them the mean-field values. The results (1.7)-(1.8) also show that the critical exponents ν and η, defined as 1 τ λc t (0) x∈Z d |x| 2 τ λc t (x) ∼ t 2ν ,τ λc t (0) ∼ t η , (1.18) take on the mean-filed values ν = 1/2 and η = 0, in the stronger form given in (1.14). The result η = 0 proves that the statement in [22,Proposition 4.39] on the tree also holds for sufficiently spread-out contact process on Z d for d > 4. See the remark below [22,Proposition 4.39]. Furthermore, following from bounds established in the course of the proof of Theorem 1.1, we can extend the aforementioned result of [3], i.e., γ = 1 in the form given in (1.15), to the precise asymptotics as in (1.14). We will prove this in Section 2.5. So far, d > 4 is a sufficient condition for the mean-field behaviour for the spread-out contact process. It has been shown, using the hyperscaling inequalities in [28], that d ≥ 4 is also a necessary condition for the mean-field behaviour. Therefore, the upper critical dimension for the spread-out contact process is 4, and one can expect log corrections in d = 4.
In [18], we will investigate the higher-point functions of the critical spread-out contact process for d > 4. These higher-point functions are defined for t ∈ [0, ∞) r−1 and x ∈ Z d(r−1) by τ λ t ( x) = P λ (x i ∈ C t i ∀i = 1, . . . , r − 1). (1.19) The proof will be based on a lace expansion that expresses the r-point function in terms of s-point functions with s < r. On the arising equation, we will then perform induction in r, with the results for r = 2 given by Theorem 1.1. We discuss the extension to the higher point functions in somewhat more detail in Section 2.2, where we discuss the lace expansion. In order to bound the lace expansion coefficients for the higher point functions, the upper bounds in (1.7) for k = 0 and in (1.9) are crucial.

Results below and at four dimensions
We also consider the low-dimensional case, i.e., d ≤ 4. In this case, the contact process is believed not to exhibit the mean-field behaviour as long as L remains finite, and Gaussian asymptotics are not expected to hold in this case. However, we can prove local Gaussian behaviour when the range grows in time as (1.20) where L 1 ≥ 1 is the initial infection range. We denote by σ 2 T the variance of D in this situation. We assume that Our main result is the following.
such that, for sufficiently large L 1 , there are positive and finite constants C 1 = C 1 (d) and with the error estimate in (1.22) uniform in k ∈ R d with |k| 2 / log(2 + T t) sufficiently small.
The upper bound on t in the statement can be replaced by any slowly varying function. However, we use log T to make the statement more concrete. The proof of Theorem 1.2 follows the same steps as the proof of Theorem 1.1.
First, we give a heuristic explanation of how (1.21) arises. Recall that, for d > 4, (λ c ) < ∞ is a sufficient condition for the mean-field behaviour. For d ≤ 4, since (λ T ) cannot be defined in full space-time as in (1.17), we modify the triangle function as Using the upper bounds in (1.22) for k = 0 and in (1.24), we obtain which is finite for all T whenever bd > 4−d 2 . We can find a similar argument in [33,Section 14]. Next, we compare the ranges needed in our results and in the results of Durrett and Perkins [8], in which the convergence of the rescaled contact process to super-Brownian motion was proved. As in (1.21) we need bd > 4−d 2 , while in [8] bd = 1 for all d ≥ 3. For d = 2, which is a critical case in the setting of [8], the model with range L 2 T = T log T was also investigated. In comparison, we are allowed to use ranges that grow to infinity slower than the ranges in [8] when d ≥ 3, but the range for d = 2 in our results needs to be larger than that in [8]. It would be of interest to investigate whether Theorem 1.2 holds when L 2 T = T log T (or even smaller) by adapting our proofs. Finally, we give a conjecture on the asymptotics of λ T as T ↑ ∞. The role of λ T is a sort of critical value for the contact process in the finite-time interval [0, T log T ], and hence λ T approximates the real critical value λ c,T that also converges to 1 in the mean-field limit T ↑ ∞. We believe that the leading term of λ c,T − 1, say c T , is equal to that of λ T − 1. As we will discuss below in Section 5.4, λ T satisfies a type of recursion relation (5.41). We expect that, for d ≤ 4, we may employ the methods in [17] to obtain where D T equals D with range L T . (In fact, the exponent µ could be replaced by any positive number strictly smaller than α.) The integral with respect to t ∈ R + converges when d > 2, and hence we may obtain for sufficiently large T that where we use (1.20) and the fact that the sum in (1.
. Based on our belief mentioned above, this would be a stronger result than the result in [8] . However, to prove this conjecture, we may require serious further work using block constructions used in [8].

Outline of the proof
In this section, we provide an outline of the proof of our main results. This section is organized as follows. In Section 2.1, we explain what the discretized contact process is, and state the results for the discretized contact process. These results apply in particular to oriented percolation, which is a special example of the discretized contact process. In Section 2.2, we briefly explain the lace expansion for the discretized contact process, and state the bounds on the lace expansion coefficients in Section 2.3. In Section 2.4, we explain how to use induction to prove the asymptotics for the discretized contact process. In Section 2.5, we state the results concerning the continuum limit, and show that the results for the discretized contact process together with the continuum limit imply the main results in Theorems 1.1-1.2.

Discretization
By the graphical representation, the contact process can be constructed as follows. We consider Z d ×R + as space-time. Along each time line {x} × R + , we place points according to a Poisson process with intensity 1, independently of the other time lines. For each ordered pair of distinct time lines from {x} × R + to {y} × R + , we place directed bonds ((x, t), (y, t)), t ≥ 0, according to a Poisson process with intensity λ D(y − x), independently of the other Poisson processes. A site (x, s) is said to be connected to (y, t) if either (x, s) = (y, t) or there is a non-zero path in Z d × R + from (x, s) to (y, t) using the Poisson bonds and time line segments traversed in the increasing time direction without traversing the Poisson points. The law of C t defined in Section 1.2 is equivalent to that of {x ∈ Z d : (o, 0) is connected to (x, t)}. See also [22,Section I.1].
Inspired by this percolation structure in space-time and following [27], we consider an oriented percolation approximation in Z d × εZ + to the contact process, where ε ∈ (0, 1] is a discretization parameter. We call this approximation the discretized contact process, and it is defined as follows. A directed pair b = ((x, t), (y, t + ε)) of sites in Z d × εZ + is called a bond. In particular, b is a temporal bond if x = y, otherwise b is a spatial bond. Each bond is either occupied or vacant independently of the other bonds, and a bond b = ((x, t), (y, t + ε)) is occupied with probability provided that p ε ∞ ≤ 1. We denote the associated probability measure by P λ ε . It is proved in [4] that P λ ε weakly converges to P λ as ε ↓ 0. See Figure 2.1 for a graphical representation of the contact process and the discretized contact process. As explained in more detail in Section 2.2, we prove our main results by proving the results first for the discretized contact process, and then taking the continuum limit when ε ↓ 0.
We also emphasize that the discretized contact process with ε = 1 is equivalent to oriented percolation, for which λ ∈ [0, D −1 ∞ ] is the expected number of occupation bonds per site. We denote by (x, s) −→ (y, t) the event that (x, s) is connected to (y, t), i.e., either (x, s) = (y, t) or there is a non-zero path in Z d × εZ + from (x, s) to (y, t) consisting of occupied bonds. The two-point function is defined as Similarly to (1.5), the discretized contact process has a critical value λ (ε) c satisfying c . (2. 3) The main result for the discretized contact process with ε ∈ (0, 1] is the following theorem: where all error terms are uniform in ε ∈ (0, 1]. The error estimate in (2.4) is uniform in k ∈ R d with |k| 2 / log(2 + t) sufficiently small.
Proposition 2.1 is the discrete analog of Theorem 1.1. The uniformity in ε of the error terms is crucial, as this will allow us to take the limit ε ↓ 0 and to conclude the results in Theorem 1.1 from the corresponding statements in Proposition 2.1. In particular, Proposition 2.1 applied to oriented percolation (i.e., ε = 1) reproves [20,Theorem 1.1].
Note that Proposition 2.2 applies also to oriented percolation, for which ε = 1.

Expansion
The proof of Proposition 2.1 makes use of the lace expansion, which is an expansion for the two-point function. We postpone the derivation of the expansion to Section 3, and here we provide only a brief motivation. We also motivate why we discretize time for the contact process. We make use of the convolution of functions, which is defined for absolutely summable functions f, g on Z d by (2.10) We first motivate the basic idea underlying the expansion, similarly as in [20, Section 2.1.1], by considering the much simpler corresponding expansion for continuous-time random walk. For continuoustime random walk making jumps from x to y at rate λD(y − x) with killing rate 1 − λ, we have the partial differential equation where q λ t (x) is the probability that continuous-time random walk started at o ∈ Z d is at x ∈ Z d at time t. By taking the Fourier transform, we obtain In this simple case, the above equation is readily solved to yield that We see that λ = 1 is the critical value, and the central limit theorem at λ = λ c = 1 follows by a Taylor expansion of 1 −D(k) for small k, yieldinĝ where |k| 2 = d j=1 k 2 i (recall also (1.1)). The above solution is quite specific to continuous-time random walk. When we would have a more difficult function on the right-hand side of (2.12), such as −[1 − λD(k)]q λ t−1 (k), it would be much more involved to solve the above equation, even though one would expect that the central limit theorem at the critical value still holds.
A more robust proof of central limit behaviour uses induction in time t. Since time is continuous, we first discretize time. The two-point function for discretized continuous-time random walk is defined by setting q λ 0;ε (x) = δ 0,x and (recall (2.1)) To obtain a recursion relation for q λ t;ε (x), we simply observe that by independence of the underlying random walk q λ t;ε (x) = (p ε * q λ t−ε;ε )(x) (t ∈ εN). (2.16) We can think of this as a simple version of the lace expansion, applied to random walk, which has no interaction. For the discretized continuous-time random walk, we can use induction in n for all t = nε. If we can further show that the arising error terms are uniform in ε, then we can take the continuum limit ε ↓ 0 afterwards, and obtain the result for the continuous-time model. The above proof is more robust, and can for instance be used to deal with the situation where the right-hand side of (2.12) equals −[1−λD(k)]q λ t−1 (k). This robustness of the proof is quite valuable when we wish to apply it to the contact process.
The identity (2.16) can be solved using the Fourier transform to givê (2.17) We note that the limit of [q t;ε (k) −q t−ε;ε (k)]/ε exists and equals (2.12). In order to obtain the central limit theorem, we divide k by √ σ 2 t. Then, uniformly in ε > 0, we havê Therefore, the central limit theorem holds uniformly in ε > 0. We follow Mark Kac's adagium: "Be wise, discretize!" for two reasons. Firstly, discretizing time allows us to obtain an expansion as in (2.11), and secondly, it allows us to analyse the arising equation. The lace expansion, which is explained in more detail below, can be used for the contact process to produce an equation of the form whereπ λ s are certain expansion coefficients. In order to derive the equation (2.19), we use that the discretized contact process is oriented percolation, for which lace expansions have been derived in the literature [20,24,25,26,27]. Clearly, the equation (2.19) is much more complicated than the corresponding equation for simple random walk in (2.11). Therefore, a simple solution to the equation as in (2.13) is impossible. We see no way to analyse the partial differential equation in (2.19) other than to discretize time combined with induction. It would be of interest to investigate whether (2.19) can be used directly.
We next explain the expansion for the discretized contact process in more detail, following the explanation in [20, Section 2.1.1]. For the discretized contact process, we will regard the part of the oriented percolation cluster connecting (o, 0) to (x, t) as a "string of sausages." An example of such a cluster is shown in Figure 2. The difference between oriented percolation and random walk resides in the fact that for oriented percolation, there can be multiple paths of occupied bonds connecting (o, 0) to (x, t). However, for d > 4, each of those paths passes through the same pivotal bonds, which are the essential bonds for the connection from (o, 0) to (x, t). More precisely, a bond is pivotal for the connection from (o, 0) to (x, t) when (o, 0) −→ (x, t) in the possibly modified configuration in which the bond is made occupied, and (o, 0) is not connected to (x, t) in the possibly modified configuration in which the bond is made vacant (see also Definition 3.1 below). In the strings-and-sausages picture, the strings are the pivotal bonds, and the sausages are the parts of the cluster from (o, 0) in between the subsequent pivotal bonds. We expect that there are of the order t/ε pivotal bonds. For instance, the first black triangle indicates that (o, 0) is connected to (o, ε), and this bond is pivotal for the connection from (o, 0) to (x, t).
Using this picture, we can think of the oriented percolation two-point function as a kind of random walk two-point function with a distribution describing the statistics of the sausages, taking steps in both space and time. Due to the nature of the pivotal bonds, each sausage avoids the backbone from the endpoint of that sausage to (x, t), so that any connected path between the sausages is via the pivotal bonds between these sausages. Therefore, there is a kind of repulsive interaction between the sausages. The main part of our proof shows that this interaction is weak for d > 4.
Together with the initial values π λ 0;ε (x) = δ o,x and π λ ε;ε (x) = 0, the identity (2.20) gives an inductive definition of the sequence π λ t;ε (x) for t ≥ 2ε with t ∈ εZ + . However, to analyse the recursion relation (2.20), it will be crucial to have a useful representation for π λ t;ε (x), and this is provided in Section 3. Note that (2.16) is of the form (2.20) with π λ t;ε (x) = δ o,x δ 0,t , so that we can think of the coefficients π λ t;ε (x) for t ≥ 2ε as quantifying the repulsive interaction between the sausages in the "string of sausages" picture. Our proof will be based on showing that 1 ε 2 π λ t;ε (x) for t ≥ 2ε is small at λ = λ (ε) c if d > 4 and both t and L are large, uniformly in ε > 0. Based on this fact, we can rewrite the Fourier transform of (2.20) aŝ Assuming convergence of 1 ε 2π λ s;ε (k) toπ λ s (k), which will be shown in Section 2.5, we obtain (2.19). Therefore, (2.19) is regarded as a small perturbation of (2.12) when d > 4 and L 1, and this will imply the central limit theorem for the critical two-point function.
Now we briefly explain the expansion coefficients π λ t;ε (x). In Section 3, we will obtain the expression where we suppress the dependence of π (N ) t;ε (x) on λ. The idea behind the proof of (2.22) is the following. Let denote the contribution to τ λ t;ε (x) from configurations in which there are no pivotal bonds, so that where the sum over b is over bonds of the form b = ((u, s), (v, s + ε)). We write b = (u, s) for the starting point of the bond b and b = (v, s + ε) for its endpoint. Then, the probability on the right-hand side of (2.24) equals We ignore the intersection with the event that b is pivotal for (o, 0) −→ (x, t), and obtain using the Markov property that We will investigate the error term R (0) t;ε (x) further, again using inclusion-exclusion, by investigating the first pivotal bond after b to arrive at (2.22). The term π (1) t;ε (x) is the contribution to R (0) t;ε (x) where such a pivotal does not exist. Thus, in π (0) t;ε (x) for t ≥ ε and in π (1) t;ε (x) for all t ≥ 0, there is at least one loop, which, for L large, should yield a small correction only. In (2.22), the contributions from N ≥ 2 have at least two loops and are thus again smaller, even though all N ≥ 0 give essential contributions to π λ t;ε (x) in (2.22).
There are three ways to obtain the lace expansion in (2.20) for oriented percolation models. We use the expansion by Sakai [26,27], as described in (2.23)-(2.27) above, based on inclusion-exclusion together with the Markov property for oriented percolation. For unoriented percolation, Hara and Slade [11] developed an expression for π λ t;ε (x) in terms of sums of nested expectations, by repeated use of inclusion-exclusion and using the independence of percolation. This expansion, and its generalizations to the higher-point functions, was used in [20] to investigate the oriented percolation r-point functions. The original expansion in [11] was for unoriented percolation, and does not make use of the Markov property. Nguyen and Yang [24,25] derived an alternate expression for π (N ) t;ε (x) by adapting the lace expansion of Brydges and Spencer [7] for weakly self-avoiding walk. In the graphical representation of the Brydges-Spencer expansion, laces arise which give the "lace expansion" its name. Even though in many of the lace expansions for percolation type models, such as oriented and unoriented percolation, no laces appear, the name has stuck for historical reasons.
It is not so hard to see that the Nguyen-Yang expansion is equivalent to the above expansion using inclusion-exclusion, just as for self-avoiding walks [23]. Since we find the Sakai expansion simpler, especially when dealing with the continuum limit, we prefer the Sakai expansion to the Nguyen-Yang expansion. In [20], the Hara-Slade expansion was used to obtain (2.22) with a different expression for π (N ) t;ε (x). In either expansion, π (N ) t;ε (x) is nonnegative for all t, x, N , and can be represented in terms of Feynman-type diagrams. The Feynman diagrams are similar for the three expansions and obey similar estimates, even though the expansion used in this paper produces the simplest diagrams.
In [20], the Nguyen-Yang expansion was also used to deal with the derivative of the lace expansion coefficients with respect to the percolation parameter p. In this paper, we use the inclusion-exclusion expansion also for the derivative of the expansion coefficients with respect to λ, rather than on two different expansions as in [20].
We now comment on the relative merits of the Sakai and the Hara-Slade expansion. Clearly, the Hara-Slade expansion is more general, as it also applies to unoriented percolation. On the other hand, the Sakai expansion is somewhat simpler to use, and the bounding diagrams on the arising Feynman diagrams are simpler. Finally, the resulting expressions for π (N ) t;ε (x) in the Sakai expansion allow for a continuum limit, where it is not clear to us how to perform this limit using the Hara-Slade expansion coefficients.
In [18], we will adapt the expansion in Section 3 to deal with the discretized contact process and oriented percolation higher-point functions. For this, we will need ingredients from the Hara-Slade expansion to compare occupied paths living on a common time interval, with independent paths. This independence does not follow from the Markov property, and therefore the Hara-Slade expansion, which does not require the Markov property, will be crucial. The "decoupling" of disjoint paths is crucial in the derivation of the lace expansion for the higher point functions, and explains the importance of the Hara-Slade expansion for oriented percolation and the contact process.
To complete this discussion, we note that an alternative route to the contact process results is via (2.19). In [5], an approach using a Banach fixed point theorem was used to prove asymptotics of the twopoint function for weakly self-avoiding walk. The crucial observation is that a lace expansion equation such as (2.19) can be viewed as a fixed point equation of a certain operator on sequence spaces. By proving properties of this operator, Bolthausen and Ritzman were able to deduce properties of the fixed point sequence, and thus of the weakly self-avoiding walk two-point function. It would be interesting to investigate whether such an approach may be used on (2.19) as well.

Bounds on the lace expansion
In order to prove the statements in Proposition 2.1, we will use induction in n, where t = nε ∈ εZ + . The lace expansion equation in (2.20) forms the main ingredient for this induction in time. We will explain the inductive method in more detail below. To advance the induction hypotheses, we clearly need to have certain bounds on the lace expansion coefficients. The form of those bounds will be explained now. The statement of the bounds involve the small parameter (2.28) We will use the following set of bounds: The bounds on the lace expansion consist of the following estimates, which will be proved in Section 4. Proposition 2.3 (Bounds on the lace expansion for d > 4). Assume (2.29) for some λ 0 and all s ≤ t. Then, there are β 0 = β 0 (d, K) > 0 and C = C(d, K) < ∞ (both independent of ε, L) such that, for λ ≤ λ 0 , β < β 0 , s ∈ εZ + with 2ε ≤ s ≤ t + ε, q = 0, 2, 4 and ∆ ∈ [0, 1 ∧ ∆], and uniformly in ε ∈ (0, 1], The main content of Proposition 2.3 is that the bounds onτ s;ε for s ≤ t in (2.29) imply bounds on π s;ε for all s ≤ t + ε. This fact allows us to use the bounds onπ s;ε for all arising s in (2.20) in order to advance the appropriate induction hypotheses. Of course, in order to complete the inductive argument, we need that the induction statements imply the bounds in (2.29).
The proof of Proposition 2.3 is deferred to Section 4. Proposition 2.3 is probably false in dimensions d ≤ 4. However, when the range increases with T as in Theorem 1.2, we are still able to obtain the necessary bounds. In the statement of the bounds, we recall that L T is given in (1.20).
The main point in Propositions 2.3-2.4 is the fact that we need to extract two factors of ε. One can see that such factors must be present by investigating, e.g., π (0) t;ε (x), which is the probability that (o, 0) is doubly connected to (x, t). When t > 0, there must be at least two spatial bonds, one emanating from (o, 0) and one pointing into (x, t). By (2.1), these two spatial bonds give rise to two powers of ε. The proof for N ≥ 1 then follows by induction in N .

Implementation of the inductive method
Our analysis of (2.20) begins by taking its Fourier transform, which gives the recursion relation As already explained in Section 2.3, it is possible to estimateπ λ s;ε (k), for all s ≤ t, in terms of τ λ s;ε 1 ≡ x∈Z d τ λ s;ε (x) =τ λ s;ε (0) and τ λ s;ε ∞ ≤ τ λ s;ε 1 with s ≤ t − ε. Therefore, the right-hand side of (2.33) explicitly involvesτ λ s;ε (k) only for s ≤ t − ε. This opens up the possibility of an inductive analysis of (2.33). A general approach to this type of inductive analysis is given in [19]. However, here we will need the uniformity in the variable ε, and therefore we will state a version of the induction in Section 5 that is adapted to the uniformity in ε and thus the continuum limit. The advancement of the induction hypotheses is deferred to Appendix A.
Moreover, we will show that the critical point is given implicitly by the equation and that the constants A (ε) and v (ε) of Proposition 2.1 are given by where we have added an argument λ (ε) c to emphasize that λ is critical for the evaluation of π λ t;ε on the right-hand sides. Convergence of the series on the right-hand sides, for d > 4, follows from Proposition 2.3. For oriented percolation, i.e., for ε = 1, these equations agree with [20, (2.11-2.13)].
The result of induction is summarized in the following proposition: There is also a low-dimensional version of Proposition 2.5, but we refrain from stating it.

Continuum limit
In this section we state the result necessary to complete the proof of Theorems 1.1-1.2 from Propositions 2.1-2.2. In particular, from now onwards, we specialize to the contact process.
Proposition 2.6 (Continuum limit). Suppose that λ (ε) → λ and λ (ε) ≤ λ (ε) c for ε sufficiently small. Then, for every t > 0 and x ∈ Z d , there is a π λ t (x) such that Consequently, for λ ≤ λ c and q = 0, 2, 4, In Proposition 2.3, the right-hand sides of (2.30)-(2.32) are proportional to ε 2 . The main point in the proof of Proposition 2.6 is that the lace expansion coefficients, scaled by ε −2 , converge as ε ↓ 0, using the weak convergence of P λ ε to P λ [4, Proposition 2.7]. In Section 6, we will show that 1 ε 2 π λ t;ε (x) and 1 ε 2 ∂ λ π λ t;ε (x) both converge pointwise. We now show that this implies that the limit of 1 To see this, we use where we use 1 ε 2 π 0 t;ε (x) = 0 for t > 0. By the assumed pointwise convergence, the left-hand side converges to π λ t (x), while the right-hand side converges to the integral of the limit of 1 ε 2 ∂ λ π λ t;ε (x), denoted f λ t (x) for now, using the dominated convergence theorem. Therefore, for any λ ≤ λ c , . Using the uniformity in ε of the upper and lower bounds in (2.6), we obtain (1.9). (2.42) To use the pointwise convergence of τ λ (ε) c t;ε (x), we first show that the sum over x ∈ Z d in (2.42) can be approximated by a finite sum. To see this, we note that For any fixed t, we can choose δ R ≥ 0, which is ε-independent and decays to zero as R ↑ ∞, such that Therefore, the same holds for τ λ t (x), and hence we can approximate bothτ λ (ε) c t;ε (k) andτ λc t (k) by sums over t;ε (k). Using the above, we obtain which proves (1.7). Similar argument can be used for (1.8).
so that (2.47) By (2.34) and Proposition 2.6, λ c must satisfy (2.50) By the continuity in λ of ∂ λ π λ t (x) and its summability in ( Finally, we note that the above proof, where the integral is replaced with a sum over n ∈ Z + , also shows that the stronger version of γ = 1 holds for oriented percolation. The proofs of Theorems 1.1-1.2 are now reduced to the proof of Propositions 2.1-2.2 and 2.6. Proposition 2.6 will be proved in Section 6. The proof of Propositions 2.1-2.2 is reduced to Propositions 2.3-2.5, which will be proved in Sections 4-5. The advancement of the induction hypotheses is deferred to Appendix A. We start in Section 3 by deriving the lace expansion (2.20).

Expansion for the two-point function
In this section, we derive the expansion (3.28). We will also write Λ = Z d × εZ + , and use bold letters o, x, . . . to represent elements in Λ, such as o = (o, 0) and x = (x, t), and write τ ( t (x), and so on. We recall that the two-point function is defined by Before starting with the expansion, we introduce some definition: (ii) Given a configuration, we say that v is doubly connected to x, and we write v =⇒ x, if there are at least two bond-disjoint paths from v to x consisting of occupied bonds. By convention, we say that x =⇒ x for all x.
(iii) A bond is said to be pivotal for v −→ x if v −→ x in the possibly modified configuration in which that bond is made occupied, whereas v is not connected to x in the possibly modified configuration in which that bond is made vacant.
We split, depending on whether there is a pivotal bond for o −→ x, to obtain We denote so that we can rewrite (3.2) as then, by inclusion-exclusion on the event that b is pivotal for o −→ x, we arrive at The event o =⇒ b only depends on bonds with time variables less than or equal to the one of b, while the event b −→ x only depends on bonds with time variables larger than or equal to the one of b. Therefore, by the Markov property, we obtain where we abuse notation to write Therefore, we arrive at where we use " " to denote convolution in Λ, i.e., This completes the first step of the expansion, and we are left to investigate R (0) (x). For this, we need some further notation.
Definition 3.2. (i) Given a configuration and x ∈ Λ, we define C(x) to be the set of sites to which x is connected, i.e., C(x) = {y ∈ Λ : x −→ y}. Given a bond b, we also defineC b (x) to be the set of sites to which x is connected in the (possibly modified) configuration in which b is made vacant.
(ii) Given a site set C, we say that v is connected to x through C, if every occupied path connecting v to x has at least one bond with an endpoint in C. This event is written as v We then note that Therefore, The event {v C −→ x} can be decomposed into two cases depending on whether there is or is not a pivotal (3.14) See Figure 3 for a schematic representation of the event E(b, x; C). If there are pivotal bonds for v −→ x, then we take the first such pivotal bond b for which v C −→ b. Therefore, we have the partition Defining we obtain To the second term, we apply the inclusion-exclusion relation We define so that we obtain where we use that ) depends only on bonds before b 2 , while {b 2 −→ x} depends only on bonds after b 2 . By the Markov property, we end up with so that This completes the second step of the expansion.
For this subset, we will use (3.15) and (3.18) again, and follow the steps of the above proof. The expansion is completed by repeating the above steps indefinitely. To facilitate the statement and the proof of the expansion, we make a few more definitions.
where we use the convention that b 0 = o and that the empty intersection, arising when N = 1, is the whole probability space. Also, we letẼ (3.25) Using this notation, we define and denote the alternating sum by Note that the sum in (3.27) is a finite sum, as long as t x is finite, where t x denotes the time coordinate of x, since each of the bonds b 1 , . . . , b N eats up at least one time-unit ε, so that The result of the expansion is summarized as follows.
Proof. By (3.22), we are left to identify R (1) (x). For N ≥ 1, we define We prove below The equation (3.28) follows by repeated use of (3.30) until the remainder R (N +1) (x) vanishes, which must happen at least when N ε > t x . To complete the proof of Proposition 3.3, we are left to prove (3.30), which is a generalization of (3.22). (3.29). As in (3.15), this event can be decomposed into two cases, ). If there are such pivotal bonds, then we take the first bond b among these bonds and obtain (cf., (3.15 , which is the first term in the right-hand side of (3.30). For the contribution from the remaining event in (3.31), we use The last term in the above expression is the last term in the right-hand side of (3.30). Again by the Markov property and (3.26), the first term in the right-hand side of (3.32) equals the second term in the right-hand side of (3.30). This completes the proof of (3.30) and thus the proof of Proposition 3.3.

Representation for the derivative
In this section, we derive a formula for ∂ λ π(x). To state the result below, we define

35)
and where b N +1 is defined to be x.
By the same reason as in (3.27), the sum in (3.34) is a finite sum. We prove (3.34) by differentiating the expression (3.28) for τ (x) and comparing it with the expression for ∂ λ τ (x) obtained by using Russo's formula, rather than differentiating ∂ λ π t (x) directly. Possibly, one can also use direct differentiation of the expressions (3.26) for π(x), but this is cumbersome because of the complex combination of increasing and decreasing events consisting of π (N ) (x), where an increasing (respectively, decreasing) event is an event that is more (respectively, less) likely to occur as λ increases. We note that, instead of a difference of two terms due to the pivotals for the increasing and decreasing events, we only obtain a single sum over pivotals. Thus, an intricate cancellation takes place. This is further demonstrated by the fact that there is no contribution from N = 0. In particular, it is not true that Proof. In the proof it will be convenient to split Π (N,n) (x) = Π (N ;n,1) (x) + Π (N ;n,2) (x), (3.36) where Π (N ;n,1) (x) is the contribution from b = b n in (3.35), whereas Π (N ;n,2) (x) is the contribution from To obtain an expression for ∂ λ τ (x), we use Russo's formula [4,9]. Let E be an increasing event that depends only on finitely many spatial bonds. Then where we use the fact that only spatial pivotal bonds for E are responsible to the differentiation with respect to λ.
which is the set of bond configurations whose restriction on bonds (u, v) , and that, for any λ 0 ∈ [0, ∞), uniformly in λ ∈ [0, λ 0 ], which we will show at the end of this section. Therefore, we can exchange the order of lim R→∞ and ∂ λ , and obtain We follow the same strategy as in Section 3.1 to obtain a recursion relation, now for ∂ λ τ (x) rather than for τ (x). Then, (3.39) equals where b 1 <b is the sum over bonds b 1 with t b 1 < t b . The first and second terms in the brackets of the right-hand side correspond respectively to when b is or is not the first element of piv[o, x]. The contribution from the first term is the same as (3.2), apart from the factor 1 λ and the restriction that b has to be a spatial bond. Thus, the first term equals where we abuse notation to write D((y, s)) = D(y) δ s,ε . We ignore the condition that b 1 is pivotal for o −→ x and use inclusion-exclusion in the form (3.18) to make up for the arising error. Using the Markov property, the contribution from the second term in (3.40) is where we define Q (n) (x) by and we recall thatẼ (3.29), except for the factor 1 λ , the sum over spatial bonds b, and the extra condition b ∈ piv[b n , x]. Therefore, by (3.40)-(3.41) and (3.44), we have Below, we will use inclusion-exclusion to prove that, for n ≥ 1, Before proving (3.47), we complete the proof of (3.34) assuming (3.47). By repeated applications of (3.47) to (3.46) until the remainder term Q (n) (x) vanishes, we obtain Differentiating both sides of (3.28) and comparing with the above expression, we obtain Using this identity, we prove (3.34) by induction on t x /ε. Since π((x, 0)) = δ o,x and Π (N ) ((x, 0)) = 0 for all N ≥ 1, we obtain (3.34) for t x /ε = 0. Suppose that (3.34) holds for all t x /ε ≤ m. Then the contribution from the second term in the brackets of (3.49) equals the second term on the left-hand side of (3.49), and thus (3.34) for t x /ε = m + 1 holds. This completes the inductive proof of (3.34).
In order to complete the proof of Proposition 3.4, we prove (3.47). Because of the condition b nC (3.45), either the event E(b n , x;C bn (b n−1 )) occurs or there is an occupied bond b n+1 ∈ piv[b n , x] for which the event E(b n , b n+1 ;C bn (b n−1 )) occurs. The contribution from the former case to Q (n) (x) is The contribution from the latter case is, as in (3.40), where the first, second and third terms in the brackets correspond respectively to when b n+1 = b, when b n+1 is between b n and b, and when b n+1 is between b and x. The first term is similar to that in (3.40), and its contribution equals, as in (3.41), For the second term in (3.51), we apply (3.43), with b 1 and o being replaced respectively by b n+1 and b n , and use the inclusion-exclusion relation (3.18) and the Markov property. Then, the contribution from the second term equals, as in (3.44), By the inclusion-exclusion relation (3.18), the contribution from the third term equals where the first term equals, by the Markov property, For the second term in (3.55), we use the same argument as above (3.31). Because of the condition occurs. By repeated use of inclusion-exclusion and the Markov property, as above (3.31), the contribution from the second term in (3.55) equals Proof of the uniformity of (3.38). Given λ 0 ∈ [0, ∞), we prove that ∂ λ P(E R (x)) converges to the righthand side of (3.38), uniformly in λ ∈ [0, λ 0 ]. Recall The difference between ∂ λ P(E R (x)) and the right-hand side of (3.38) is bounded by where we take the sum over the spatial component of x to obtain the bound. Similarly to (2.44), this is further bounded, uniformly in λ and ε, by cδ R where c = c(λ 0 , t x ) and δ R = δ R (λ 0 ) are some finite constants satisfying lim R→∞ δ R = 0. Next, we consider the first term in (3.58). Note that, if b is pivotal for E R (x), but not pivotal for o −→ x, then there must be a detour from some y ∈ R × [0, t b ] to another z ∈ R × [t b , t x ] that passes through (Z d \ R ) × [0, t x ] without traversing b. Therefore, the event in the first term of (3.58) is a subset of where (u, v) is the first bond along the detour that crosses the boundary of R × [0, t x ], so that it is a spatial bond, and E 1 • E 2 is the event that E 1 and E 2 occur disjointly, i.e., there is a bond set B such that E 1 occurs on B and E 2 occurs on the complement of B. By the Markov property, the three events joined by ∩ are independent of each other. For the middle event, we use the van den Berg-Kesten (BK) inequality [9], which asserts that P(E 1 • E 2 ) ≤ P(E 1 ) P(E 2 ) when both E 1 and E 2 are increasing events, i.e., E 1 and E 2 are more likely to occur as λ increases, as in (3.60). Sometimes, we also make use of the van den Berg-Kesten-Reimer (BKR) inequality [6], which proves P(E 1 • E 2 ) ≤ P(E 1 ) P(E 2 ) for any events E 1 and E 2 . Then, we use (2.43) as in (3.59) and obtain that the first term in (3.58) (even when we sum over the spatial component of x) is bounded by c δ R , where c = c (λ 0 , t x , ε) and δ R = δ R (λ 0 ) are some finite constants satisfying lim R→∞ δ R = 0. This completes the proof of the uniformity in λ of (3.38).
In the above proof, we did not care about the uniformity in ε, since it has been fixed and positive in this section. In fact, the above constant c is of order O(ε −2 ) and diverges as ε → 0. This is because the contribution from (3.60) involves the sums over t b (= t b + ε), t y , t z , t v (= t u + ε) ∈ εZ + that give rise to the factor ε −4 . However, the factor ε −2 is cancelled by the bond occupation probabilities of the spatial bonds b and (u, v), and therefore c = O(ε −2 ). We could improve this to c = O(1) by using the ideas in Section 4.1, and hence obtain the uniformity in ε as well, though this is not necessary here.  (i) Let d > 4. Then, there are β 0 > 0 and C K < ∞ such that, for λ ≤ λ 0 , β < β 0 , s ∈ εZ + with 2ε ≤ s ≤ t + ε, and q = 0, 2, 4,

Bounds on the lace expansion
and t ≤ T log T . Then, there are β 0 > 0 and C K < ∞ such that, for λ ≤ λ 0 , β 1 < β 0 , s ∈ εZ + with 2ε ≤ s ≤ t + ε, and q = 0, 2, 4,  t (x) that are defined in terms of two-point functions, and prove that these diagram functions are upper bounds on π (N ) t (x) and Π (N ;n) t (x), respectively. Then, we bound these diagram functions assuming the bounds in (2.29) on the two-point function, for d > 4 in Section 4.2 and for d ≤ 4 in Section 4.3. Finally, in Section 4.4, we use these diagram functions to obtain finite-volume approximations of π (N ) t (x) and Π (N ;n) t (x), which will be used in Section 6 to prove the continuum limit ε ↓ 0.

Bounds in terms of the diagram functions
In this section, we prove that π (N ) t (x) and Π (N ;n) t (x) are bounded by certain diagram functions P (N ) t (x) andP (N ;n) t (x) that are defined below in terms of two-point functions. The strategy in this section is similar to [20, Section 4.1] for oriented percolation in Z d × Z + , where bounds on π (N ) t (x) were proved by using some diagram functions arising from the Hara-Slade lace expansion. Since the expansion used in this paper is somewhat simpler, we can use simpler diagram functions. However, to consider the case ε 1 as in [27], extra care is needed to obtain the factor ε 2 in (4.36)-(4.54).

Preliminaries
Before defining the diagram functions, we start by some preliminaries. For v = (v, s) ∈ Λ and a bond b, we write v + = (v, s+ε) and {v −→ b} = {v −→ b}∩{b occupied} (cf., Definition 3.1(i)). For convenience, we will also use abbreviations, such as  We note that I (v, y, x) for y = x equals I (v, x, x) with an extra spatial bond b being embedded (or added) along the connection from v to x. Denoting we define Our first claim is thatẼ (N ) b N (x) satisfies the following successive relations: 11) and for N > n ≥ 1, We note that the left-hand side of (4.12) is the same as that of (4.11), except that b is pivotal for b n −→ b n+1 with n < N .
Proof. The relation (4.9) follows immediately from (3.24) and (4.13) We only prove (4.10), since (4.11)-(4.12) can be proved similarly. First, we use (4.9) to obtaiñ (4.14) The event in the left-hand side of (4.14) implies existence of v ∈ C(b N −1 ) such that v −→ b N and M (b N , v; x, y) occur disjointly. Therefore, We investigate the vicinity of v ∈ Λ in (4.15), where there are two disjoint connections, v −→ b N and v −→ x. Since there is at most one temporal bond growing out of each vertex in Λ, at least one of the two connections has to use a spatial bond at v. Therefore, Substituting this relation into (4.15) and relabelling b = v in the latter event we obtain (4.14), and thus (4.10). This completes the proof.

Diagrammatic bounds
Inspired by the successive relations (4.9)-(4.12), we inductively construct the diagram functions P (N ) t (x) andP (N ;n) t (x) as follows. For b = (u, v) with u = (u, s) and v = (v, s + ε), we abuse notation to write and (4.18) We define Given an admissible line from v to x of a diagram function, say τ (x − v) for simplicity, and given y = x, Construction B spat (y) is defined to be the operation in which τ (x − v) is replaced by and Construction B temp (y) is defined to be the operation in which τ ( We note that (4.20)-(4.21) are inspired by (4.6). The sum of the results of Construction B spat (y) and Construction B temp (y) is simply said to be the result of Construction B (y). We define Construction B (s) to be the operation in which Construction B (y, s) is performed and then followed by summation over y ∈ Z d . Construction B spat (s) and Construction B temp (s) are defined similarly. We denote the result of applying Construction B (y) to a diagram function f (x) by f (x, B (y)), and define f (x, B spat (y)) and f (x, B temp (y)) similarly. We construct P (N ) (x) from P (N −1) (x) by where is the sum over the (N − 1) st admissible lines in each diagram. We define  We call the newly added lines, contained in L(v, y; x), the N th admissible lines. Finally, for N ≥ n ≥ 1, we definẽ where is the sum over the n th admissible lines. Thanks to the construction in terms of two-point functions, the diagram functions can be estimated by using (2.29), and this will be done in Sections 4.2-4.3. The following is the main statement of this section: Proof. We begin with proving π (0) (x) ≤ P (0) (x). The first term in (4.19) is the contribution from the case of o = x. If o = x, there are at least two nonzero disjoint occupied paths from o to x. As explained below (4.15), at least one of two nonzero disjoint occupied paths from o has to use a spatial bond at o. That is, We use the same observation at x: at least one of the two nonzero disjoint connections, b −→ x and o −→ x, has to use another spatial bond at x. Therefore, we can bound the right-hand side of (4.27) by λεL(o, o; x) using the BK inequality. This completes the proof of π (0) (x) ≤ P (0) (x). Next, we consider π (N ) (x) for N ≥ 1. Let By the convention in Lemma 4.2, π (0) (x, y) = P({o =⇒ x} ∩ I (o, y, x)). We prove below by induction that π (n) (x, y) ≤ (2λε) −δx,y P (n) (x, B (y)) (4.29) holds for all n ≥ 0, where is the sum over the n th admissible lines. The inequality π (N ) (x) ≤ P (N ) (x) for N ≥ 1 follows from (4.9) and (4.29) for y = x, together with the convention in (4.24), i.e., 2λεP (N ) (x) = P (N ) (x, B (x)). For n = 0, we can assume y = x, since π (0) (x) = π (0) (x, x) ≤ P (0) (x) has already been proved. By the equivalence {o =⇒ x} ∩ I (o, y, x) = {o −→ x} • I (o, y, x) and by (4.6), we obtain where we use the BK inequality to derive λεD(b) in the second sum. We only prove that the first sum in (4.30) is bounded by while the contribution due to b = (y, x) is bounded by We can estimate the case (iii) similarly, and obtain a bound which is λεL(o, y; x) with one of the two τ 's in each product τ · τ in (4.18) replaced by (4.20). Summarizing these bounds, we conclude that the first sum in (4.30) is bounded by P (0) (x, B spat (y)). This completes the proof of (4.29) for n = 0, and initializes the inductive proof for n ≥ 1.
To advance the induction hypotheses, we assume that (4.29) holds for n = N − 1, and that b:b=u holds, where we write L(u, v; x, B (x)) = 2λεL(u, v; x), similarly to the convention used in (4.24). We will prove (4.33) below. By (4.10) and the BKR inequality, together with (4.24), (4.29) and (4.33), we obtain x, B (y)). (4.34) This advances the induction hypotheses, and hence completes the proof of (4.29), assuming that (4.33) holds. It thus remains to prove (4.33). We only consider the case of u = v and x = y, since it explains why the factor 2λε is in the definition of the diagram functions. The case of u = v and x = y can be proved similarly to (4.29) for n = 0, and the proof of the remaining case is simpler because extracting the factors of ε at u = v is unnecessary. Let u = o in (4.33), by translation invariance. Then, the left-hand side of (4.33) equals the rightmost expression in (4.27), except for the condition that b is a spatial bond. By the same observation at o as in (4.27), we obtain b:b=o P (M (b, o; x, x)) = b spatial: b=o This completes the proof of (4.33), and hence the proof of the first inequality in (4.26).
To prove the second inequality in (4.26), we recall that Π (N,n) (x) = Π (N ;n,1) (x) + Π (N ;n,2) (x), where the first and second terms are the contributions to (3.35) from b = b n and from b ∈ piv[b n , b n+1 ], respectively. There are two n th admissible lines terminating at b n+1 , one from b n and the other from some vertex w. We can bound Π (N ;n,1) (x) by P (N ) (x) with the nonzero admissible line from b n , say (p τ )(b n+1 − b n ), replaced by (λεD τ )(b n+1 − b n ); if w = b n , we replace the factor p in one of the two admissible lines by λεD as above, and add both contributions. For Π (N ;n,2) (x), we use (4.11)-(4.12) to obtain the bound is the sum over the n th admissible lines. Together with the bound on Π (N ;n,1) (x), we obtain (4.25). This completes the proof the second inequality in (4.26), and hence the proof of Lemma 4.3.

Estimate of the diagram functions above four dimensions
In this section, we bound the diagram functions for d > 4 as follows: Lemma 4.4. Let d > 4 and suppose that (2.29) holds for some λ 0 and all s ≤ t. Then, there are β 0 > 0 and C K < ∞ such that, for λ ≤ λ 0 , β < β 0 , s ∈ εZ + with 2ε ≤ s ≤ t + ε, and q = 0, 2, 4, for N ≥ n ≥ 1. Lemma 4.5. Assume (2.29) for s ≤ t = nε and λ ∈ I n . Then, there is a C K = C K (d, λ) < ∞ such that the following bounds hold for s ≤ t, q = 0, 2 and for that λ: (4.40) Lemma 4.6. Assume (2.29) for s ≤ t. Let f t (x) be a diagram function that satisfies x f t (x) ≤ F (t) by assigning l 1 or l ∞ norm to each diagram line and using (2.29) to estimate those norms. Then, there is a for every s ≤ t and every admissible line .
Lemma 4.7. Let a, b ∈ R, and let κ be a positive number if a or b is 2, and zero otherwise. Then, there exists a C = C(a, b, κ) < ∞ such that (4.41) Lemmas 4.5 and 4.6 correspond respectively to Lemmas 4.3 and 4.6(a) in [20], and Lemma 4.7 corresponds to [19,Lemma 3.2]. The result of applying Lemma 4.7 is the same as [20, (4.26)] when d > 4 (see also the proof of Lemma 4.6(b) in [20]), but Lemma 4.7 can be applied to the lower dimensional case as well. We will use Lemmas 4.5-4.6 again in Sections 4.3-4.4, and Lemma 4.7 in Section 4.3 and Appendix A.
First, we prove Lemma 4.4 assuming Lemmas 4.5-4.7, and then prove these lemmas. We will use c to denote a finite positive constant whose exact value is unimportant and may change from line to line. For s ≤ s < t, we use Lemma 4.5 to obtain for t ≥ 2ε, this implies (4.36) with q = N = 0. By Lemma 4.6, we also obtain where δ 0,t is the contribution from the first term in (4.19).
To prove (4.37), we first use Lemma 4.6 to obtain x P (N ) where is an n th admissible line. Then, we sum the bound over s ∈ [0, t] ∩ εZ + to obtain the desired bound in (4.37). Note that the factor ε is used in an approximation by the Riemann sum. This completes the proof.
Proof of Lemma 4.5. The inequality (4.38) immediately follows from (2.29) and the properties of D. To prove (4.39)-(4.40), we use and where (1 − ε) n is the probability that (o, 0) is connected to (o, nε) along the temporal axis. Since the first term in (4.44) and the second term in (4.45), multiplied by |x| q , are both bounded by cσ q β (1 + s) −(d−q)/2 for any x ∈ Z d , we only need to consider the last terms in (4.44)-(4.45). We fix r ∈ (0, 1) and use (2.29) to bound the second term in (4.44) by (4.46) By the same argument, the third term in (4.45) can be bounded by cβ (1 + s) −d/2 . This completes the proof of (4.39)-(4.40) for q = 0. For |x| 2 times the second term in (4.44) or the third term in (4.45), we have Proof of Lemma 4.6. By the convention used in (4.24), for s = t we have x,y where is the sum over the admissible lines arriving at (x, t). For s < t, Construction B (s) replaces the diagram line , say τ t (x), by λε(τ s * D * τ t−s−ε )(x) + λε(1 − ε)(τ s * τ t−s−ε )(x). By Lemma 4.5 and (2.29), we obtain λε τ s * D * τ t−s−ε 1 = λε τ s * D 1 τ t−s−ε 1 ≤ λεC 4.5 K, (4.49) where C 4.5 is the constant in Lemma 4.5, and we use s ∨ (t − s − ε) ≥ t/2 to obtain the last inequality. The l 1 and l ∞ norms of λε(1 − ε)(τ s * τ t−s−ε )(x) can be estimated in the same way. Therefore, the effect of Construction B (s) is to obtain, at worst, an additional constant C K ε = 2 1+d/2 C 4.5 λε in a bound. This completes the proof.
Proof of Lemma 4.7. We prove (4.41) for a ∧ b ≥ 0 and for a ∧ b < 0 separately. Let a ∧ b ≥ 0. Separating the sum over s 1 into • 0≤s 1 ≤t/2 and • t/2<s 1 ≤t , and using s 2 ≥ t − s 1 ≥ t/2 in the former sum, we can bound the left-hand side of (4.41) by where the first term is bounded by c (1 + t) −b−(a−2)∧0+κa , where κ a is an arbitrarily small but positive number if a = 2, otherwise κ a = 0. Also, the double sum in (4.51) is where κ b is an arbitrarily small but positive number if b = 2, otherwise κ b = 0. This completes the proof of (4.41) for a ∧ b ≥ 0. Next, we consider the case a ∧ b < 0. Due to the symmetry of the left-hand side of (4.41) in terms of s 1 , s 2 , we suppose b < a ∧ 0. Then, we use the trivial inequality (1 + s 2 ) −b ≤ (1 + t) −b . The remaining term equals ε • 0≤s 1 ≤t s 1 (1 + s 1 ) −a and is bounded by c (1 + t) −(a−2)∧0+κa . This completes the proof of (4.41) for a ∧ b < 0, and hence the proof of Lemma 4.7.

Estimate of the diagram functions at and below four dimensions
In this section, we bound the diagram functions for d ≤ 4 by using their inductive construction in (4.19) and (4.24)-(4.25) as well as Lemmas 4.5-4.7, as in the proof of Lemma 4.4 in Section 4.2, but we replace σ and β in Lemma 4.5 by σ T and β T , respectively. Lemma 4.1(ii) is an immediate consequence of Lemma 4.3 and the following lemma: ) and t ≤ T log T , and suppose that (2.29) holds for some λ 0 and all s ≤ t. Then, there are β 0 > 0 and C K < ∞ such that, for λ ≤ λ 0 , β 1 < β 0 , s ∈ εZ + with 2ε ≤ s ≤ t + ε, and q = 0, 2, 4, Proof. The proof is almost the same as that of Lemma 4.4. The only difference arises when we apply Lemma 4.7. Let N ≥ 1 and suppose that the quantity in the first brackets in (4.43) is bounded by ). Then, by Lemma 4.7 and (4.42) with β replaced by β T , the right-hand side of (4.43) for d ≤ 4 is bounded by where κ is an arbitrarily small but positive number if d = 4, otherwise κ = 0. Since t ≤ T log T and −bd + 4−d 2 = −α < −µ, the factor in front of the fraction in the right-hand side is bounded by C KβT if C K is sufficiently large, and thus we obtain (4.53) with q = 0.

Finite containment
In this section, we prove that π (N ) t (x) can be approximated by We will also use the abbreviation C s = C s (o). More precisely, we prove below that where o(1) is independent of ε and decays to zero as R → ∞, by using the estimates for the diagram functions in Sections 4.1-4.3. This is a refined version of the finite containment argument used in proving the uniformity of (3.38), and will be useful in dealing with the continuum limit in Section 6.
Proof. First, we note that Note that ((u, s − ε), (v, s)) is a spatial bond, since u ∈ R and v / ∈ R . The event in the rightmost expression is included in the union of two events: The contribution from the case (i) and from the case (ii) with (w, r) = (u, s − ε) can be bounded where is the sum over all admissible lines (i.e., the sum over n = 1, . . . , N of the sum over the n th admissible lines), and where we modified Construction B spat (v, s) by fixing the second endpoint (v, s), instead of fixing the first endpoint (u, s − ε) as defined in (4.20). This bound, divided by ε 2 , decays as R → ∞ uniformly in ε, since the sum over x ∈ Z d of the unrestricted sum • s P (N ) (x, B (s)) is bounded, by using Lemma 4.6, by cε 2 t (1 + t) −d/2 . The contribution from the case (ii) with (w, r) = (u, s − ε) can be bounded by where we relabelled the second endpoint of the spatial bond in (4.21) as (w, r). For w ∈ R/2 , we use and for w ∈ R \ R/2 , we use x, B (w, r)). By Lemma 4.5, we have τ s−ε−r * D 1 ≤ C K , so that z / ∈ R/2 (τ s−ε−r * D)(z) decays to zero as R → ∞, independently of ε. In addition, by Lemma 4.6, we have P (N ) (x, B (r)) ≤ cε 3 (1 + t) −d/2 , so that w / ∈ R/2 P (N ) (x, B (w, r)) ≤ o(1) ε 3 (1 + t) −d/2 . Since there is another factor of ε in the summand of (4.61), while there are two summations over εZ + in (4.61), we conclude that (4.61) is o(1) ε 2 t 2 (1 + t) −d/2 . This completes the proof of (4.59).

Inductive argument
In this section, we prove Proposition 2.1 by applying the inductive method of [19] for self-avoiding walk in Z d and for oriented percolation in Z d × Z + , to the recursion equation (5.1) for oriented percolation in Z d × εZ + . To consider the case of ε 1, we will modify the induction hypotheses to incorporate the dependence on ε. We expect that a similar method could be used for continuous-time weakly self-avoiding walk above its upper critical dimension.
First, we consider d > 4 in Sections 5.1-5.3. In Section 5.1, we give the modified version of the induction hypotheses. In Section 5.2, we show several consequences of the induction hypotheses, mainly the bounds in (2.29). In Section 5.3, we prove Proposition 2.1. We complete this section by proving the results for d ≤ 4 in Section 5.4. Since a similar strategy applies to the lower-dimensional case, we only discuss the necessary changes.

General assumptions
In Section 3.1, we derived the recursion equation (3.28) for the two-point function. Taking the Fourier transform with respect to the spatial component, we obtain (2.33), i.e., The probability distribution D : Z d → [0, 1] satisfies the assumptions in Section 1.2. In addition, we assume that there exists η > 0 such that where a b means that the ratio a/b is bounded away from zero and infinity. These assumptions correspond to Assumption D in [19].
If we replace t by nε, and write f n (k) =τ nε (k), e n (k) =π nε (k), g n+1 (k) =π nε (k)p ε (k), (5.3) where the dependence on λ is left implicit, then (5.1) equals with f 0 (k) = 1. This is equivalent to the recursion relation (1.1) in [19]. The only difference is whereas in [19], f 1 (k) = g 1 (k) = λD(k). This change leads to a modification of the induction hypotheses in [19], the main reason being that we need to prove uniformity in ε. Further technical changes are explained in Section 5.1.2.

Statement of the induction hypotheses
Fix γ, δ and ρ according to We define λ n recursively by λ 0 = λ 1 = 1 and, for n ≥ 2, where we explicitly write the dependence on λ n−1 of g l (0). Let and define v n = v n (λ) by v 0 = v 1 = λ and, for n ≥ 2, .
Let K 1 , . . . , K 5 be some positive and finite constants that are independent of β and ε, and are related by The induction hypotheses are that the following (H1)-(H4) hold for all λ ∈ I n and m = 1, . . . , n. (H1)-(H2) where r l (k) obeys the bounds Instead of (5.12), we can alternatively write f m (k) as where The induction hypothesis (H3) implies In some cases, we will use (5.15)-(5.17), instead of (5.12)-(5.13). Moreover, by (5.15) and spatial symmetry, we obtain The advancement of the induction hypotheses is a small modification of that in [19], which we add to keep this paper self-contained. The advancement is deferred to Appendix A.

Consequences of the induction hypotheses
We assume β 1 and use c to denote a positive and finite constant that may depend on d, γ, δ, ρ, but not on K i , k, n, β, ε. The value of c may change from line to line.

Discussion on changes below and at four dimensions
In dimensions d ≤ 4, the induction analysis in Sections 5.1-5.3 no longer works as long as the infection range is fixed, and we need to incorporate the factor L T = L 1 T b into the induction hypotheses.
. We again define λ n = λ n (T ) and v n = v n (λ) by (5.7) and (5.9), respectively, where we emphasize the dependence on T of λ n . However, we replace (5.6), (5.8), (5.11) and (5.13), respectively, by −(2 + ρ) < 0 < d 2 − (2 + ρ) < γ < γ + δ < ω ∧ ∆, (5.36) The induction hypotheses are that (H1)-(H4) hold for all λ ∈ I n and m = 1, . . . , n, where we assume that nε ≤ T log T . It suffices to prove the main statement for sufficiently small β 1 > 0, i.e., for sufficiently large initial infection ranges L 1 . We now discuss the induction hypotheses. One of the key ingredients in the induction is the fact that the intervals I n are decreasing in n ≤ T ε log T . This implies that we can use the bounds following from (H1)-(H4) in the advancement of the induction hypotheses. One would expect that one could choose I n = λ n + K 1 β T (1 + nε) −(d−2)/2 [−1, 1], i.e., by simply replacing β in (5.8) by β T . However, to obtain a decreasing sequence of I n , it is required for the power exponent (d − 2)/2 in the width of I n to be greater than 1, and it is not the case when d ≤ 4 (cf., the proof of Lemma 5.1 in Appendix A). To satisfy this requirement, we transfer some power exponent of β T as for T ≥ 1, where we use nε ≤ T log T and −bd + 4−d 2 + ω = −(α − ω) < −µ. This is the motivation of the changes in (5.37)-(5.39).
By the above changes, (5.22)-(5.23) are modified by replacing β withβ T . We have that λ and v m (λ) are both 1+O(β T ) for λ ∈ I m and m = 1, . . . , n with nε ≤ T log T . Similarly, we replace β in Lemmas 5.2-5.3 and Lemma 5.4 byβ T and β T , respectively, although the proofs of these lemmas remain unchanged. However, the proof of the main result does change, due to the fact that the constants A and v for d > 4 are replaced by 1, and the fact that there is no unique limit of n m=1 I m .
Proof of Proposition 2.2. Let n ≤ T ε log T and λ ∈ I n . In particular, the following results hold at λ = λ T , which is defined as which proves that the asymptotic expected number of infected individuals is 1. Also, using (5.23) and (5.38), we have v n = 1 + O(β T ), which means that the asymptotic diffusion constant is 1. of the integral when ε ↓ 0 is bounded by a multiple of λ − λ 0 . By taking the limit λ 0 ↑ λ and using the fact that ∂ λ π λ t (x) is continuous in λ, the first expression in (2.36) follows. Therefore, we are left to prove convergence of 1 ε 2 π λ t;ε (x) and 1 ε 2 ∂ λ π λ t;ε (x) for every λ ≤ λ c and the continuity in λ of ∂ λ π λ t (x). We prove below that, for every N ≥ 0, λ ≤ λ c and (x, t), there is a π (N ) t (x) such that where we suppress the dependence on λ. That is, we will deal with pointwise convergence, rather than the uniform bounds in Section 4 which are valid for all (x, t) and ε ≤ 1, and hence all terms which are o(1) as ε ↓ 0 will be estimated away. By this pointwise convergence, together with the uniform bounds in Section 4 and the dominated convergence theorem, we have This completes the proof of the pointwise convergence of 1 ε 2 π λ t;ε (x). The proof of the convergence of 1 ε 2 ∂ λ π λ t;ε (x) is similar, and we will only discuss the necessary changes. The proof of (6.3) is divided into several steps.
Statement of the induction hypothesis. Given a site set C ⊂ Z d (which may be an empty set), we define where we recallẼ (0) b 0 (x, t) = {(o, 0) =⇒ (x, t)}, b 0 = (o, 0) and the notation (4.58) for C t (b N ). We will use induction in N to prove that, for every t > 0, there is a π (N ) t (x; C) such that lim ε↓0 1 ε 2 π (N ) t;ε (x; C) = π (N ) t (x; C). (6.6) The claim for π (N ) t;ε (x) in (6.3) then follows by summing over C ⊂ Z d , together with the fact that the main contribution comes from C ⊂ R by the finite containment property in Section 4.4.
Initialization of the induction. First, we investigate N = 0. For S 1 , S 2 , A ⊂ Z d × εZ + , we denote and define Using the Markov property at time ε, we arrive at Since every p ε (a) for a = o gives rise to a factor of ε, we immediately see that the main contribution comes from A = {o, y} for some y = o. Therefore, we obtain where the second equality is due to the fact that ((o, 0), (o, ε)) or ((y, 0), (y, ε)) is vacant (with probability (2 − ε)ε) in the symmetric difference between the events on both sides of the equality, and the third equality is due to the fact that the double connection from (o, ε) or (y, ε) gives rise to an extra factor of ε.
We repeat the same observation around (x, t) and obtain that, for every y ∈ Z d and C ⊂ Z d , Substituting (6.11) into (6.10) and using the weak convergence of P λ ε towards P λ as formulated in [4, Proposition 2.7], we obtain Here and in the rest of this section, we use "−→" and "=⇒" to denote connections in Z d × R + , via the graphical representation in Section 2.1. This completes the proof of (6.6) for N = 0, with π (0) t (x; C) for t > 0 defined to be the right-hand side of (6.12).
Preliminaries for the advancement. To advance the induction hypothesis, we first note that, by using the finite containment property of Section 4.4 and the Markov property at the time component of b N , we have where, similarly to (6.8), we writeC b N (C × {s − ε}) = v∈C C b N (v, s − ε), and o(1) is independent of ε and decays to zero as R → ∞. We now investigate the second probability in (6.13) when C = {y} and when C {y}, separately.
When C = {y}, we recall the definitions (3.13)-(3.14) and use the Markov property at time s, similarly to the discussion around (6.9)-(6.10), to obtain that where o(1) decays to zero as ε ↓ 0, and the first probability in the brackets is the contribution from the case in which b N is the temporal bond ((y, s − ε), (y, s)), while the second probability is the contribution from the case in which b N is the spatial bond ((y, s − ε), (y , s)). In (6.14), we also use the fact that, with To deal with the event E ((y, s), (x, t); A × {s}) for A ⊂ Z d \ {y} in (6.14)-(6.15), we introduce some notation. We define the set of sites that are connected from (y, s) via a path which does not go through v byC v (y, s) = b=( · ,v)C b (y, s). (6.16) We also define (6.18) By this notation, it is not hard to see that E ((y, s), (x, t); A × {s}) is rewritten as The contribution from R s,t (y, x; A) has an extra factor of ε, due to the fact that there are at least two spatial bonds at v (one before and one after v), which leads to an error term as ε ↓ 0. Therefore, we only need to focus on the contribution from E s,t (y, x; A), i.e., Generalizing the definition (6.17) from a single end (x, t) to a pair {(x, t), (z, t)} with x = z as E s,t (y, {x, z}; A) (6.21) and following the argument in (6.11) (see also the discussion around (6.9)-(6.10)), we obtain Bounds on π λ t in (2.37) and convergence of A (ε) and v (ε) . The bound on x∈Z d |x| q π λ t (x) follow immediately from the pointwise convergence of 1 ε 2 π λ t;ε (x), together with the uniform bounds in Proposition 2.3 and dominated convergence for the sum over x.
To prove convergence of A (ε) and v (ε) , we first note that by [27, Section 3.1], λ (ε) c → λ c . Convergence of A (ε) and v (ε) follows by dominated convergence, together with the identification of A (ε) and v (ε) in (2.35). Thus, we obtain that Proof of Lemma 5.2. By (5.12)-(5.13) and the trivial inequality 1 + x ≤ e x , This completes the proof.
Proof of Lemma 5.4.
where R 2 is empty if m 1. Then, On R 1 , we consider the cases of mε < 1 and mε ≥ 1 separately. If mε < 1, we use Lemma 5.2 and obtain If mε ≥ 1, we use the inequalityD 2 (k) ≤ 1, Lemma 5.2, and then the assumption a(k) L 2 |k| 2 for k ∞ ≤ L −1 , and obtain Summarizing both cases, we obtain the desired bound on the contribution from R 1 .
On R 2 , we use Lemma 5.2 and the assumption a(k) > η for k ∞ > L −1 to conclude that there exists an r > 1 independently of β such that Since r −mε ≤ c (1 + mε) −d/2 , we obtain the desired bound on the contribution from R 2 . On R 3 and R 4 , we use (H4). Then, the contribution from these two regions is bounded by It thus suffices to bound the integral by cβ. On R 3 , we use the inequalityD(k) 2 ≤ 1 and the assumption a(k) L 2 |k| 2 for k ∞ ≤ L −1 . Since d > 2(2 + ρ) (cf., (5.6)), we obtain On R 4 , we use the assumption a(k) > η for k ∞ > L −1 and the fact that (2π) dD (k) 2 ≤ β, to obtain the desired bound cβ on the integral over R 4 . This completes the proof. By (A.11), we obtain that, for m ≤ n + 1, Since λ ∈ I n+1 , (H2) holds for n + 1, if β 1 and K 2 > 2C K .