Spatial moments for high-dimensional critical contact process, oriented percolation and lattice trees

Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.

1 Introduction and results

Introduction
It is by now well established that super-Brownian motion arises as the scaling limit in a number of critical lattice models above the upper critical dimension, including the voter model, the contact process, oriented percolation, percolation, and lattice trees (see, e.g., [6,8,13,27,28]). There are various ways, of differing strengths, of stating such convergence results. A particularly strong statement is that the scaling limit of the range of the critical lattice model is the range of super-Brownian motion, with the convergence with respect to the Hausdorff metric on the set of compact subsets of R d . Recently, Holmes and Perkins [22,23] have identified conditions which imply this strong form of convergence. These conditions also imply an asymptotic formula for the probability of exiting a large ball, i.e., for the extrinsic one-arm probability.
One of the substantial conditions of [22] is an estimate on a spatial moment of degree higher than four. Holmes and Perkins have proved the required bound on the sixth moment for spreadout lattice trees in dimensions d > 8, and Chen and Sakai have proved asymptotic formulas for all spatial moments for spread-out critical oriented percolation in dimensions d + 1 > 4 + 1 (their case α = ∞, see [5, p.510]). These results are sufficient to establish the spatial moment condition of Holmes and Perkins in these contexts.
In this paper, we prove, in a unified and relatively simple fashion, estimates on all spatial moments for critical spread-out models of the contact process in dimensions d > 4, oriented percolation in dimensions d + 1 > 4 + 1, and lattice trees in dimensions d > 8. Our results for the contact process are new, and, together with the other conditions verified in [22,23], yield the conclusions of Holmes and Perkins. As one of their conclusions, Holmes and Perkins have shown that the upper bounds on the spatial moments can be promoted to asymptotic formulas. We discuss this last point in detail in Section 1.5.
Our method of proof simplifies the bound of [22] on the sixth moment for lattice trees and extends it to all moments. It also provides a simpler approach to bounding the moments for oriented percolation than the method of [5] (who however did obtain asymptotic formulas for the moments). Our proof is based on the lace expansion, which has been applied to study critical models of the contact process (e.g., [15,17,26]), oriented percolation (e.g., [3-5, 19, 24, 25]) and lattice trees (e.g., [7,12,20,21]). We emphasise the contact process throughout the paper, because it is the greatest novelty in our work, and because it is the most delicate of the three models to analyse.
Our spread-out models are formulated in terms of a probability measure D on Z d , which is defined as follows. Let h : R d → [0, ∞) be bounded, continuous almost everywhere, invariant under the symmetries of Z d , and such that h(x) d d x = 1, |x| n h(x) d d x < ∞ (n ∈ N). (1.1) It follows that, for any n ∈ N, We use big-O notation in the standard way: We focus in this paper on spread-out models because we rely for diagrammatic estimates on the results of the inductive approach to the lace expansion [14,18], which has only been developed for spread-out models. The spread-out models also have the advantage that they can be analysed using our methods for all dimensions above the upper critical dimension (4 for oriented percolation and the contact process, 8 for lattice trees). We believe that our results could be extended, with more effort, to nearest-neighbour models in sufficiently high dimensions, but we do not pursue that here.

Oriented percolation
We begin with oriented percolation, which also serves as a discretisation of the contact process. We will use this discretisation to analyse the contact process.
Spread-out oriented percolation is defined on the graph with vertex set Z d × {0, 1, 2, . . .} and directed bonds ((x, n), (y, n+1)), for x, y ∈ Z d and n ≥ 0. To the directed bonds ((x, n), (y, n+1)), we associate independent random variables taking the value 1 with probability pD(y − x) and 0 with probability 1 − pD(y − x). We say a bond is occupied when its random variable takes the value 1, and vacant when its random variable is 0. The parameter p ∈ [0, D −1 ∞ ] is the expected number of occupied bonds per vertex (it is not a probability). The joint probability distribution of the bond variables is P p , with expectation denoted E p .
We say that (x, n) is connected to (y, m), and write (x, n) −→ (y, m), if there is an oriented path from (x, n) to (y, m) consisting of occupied bonds, or if (y, m) = (x, n). This requires that n ≤ m. Let C(x, n) denote the set of sites (y, m) such that (x, n) −→ (y, m); its cardinality is denoted by |C(x, n)|. The critical point p c is defined to be the supremum of the set of p ∈ [0, D −1 ∞ ] for which E p |C(0, 0)| < ∞. It is known that p c = 1 + O(L −d ) (and more) [16]. Let τ n (x) = P pc ((0, 0) −→ (x, n)). (1.4) Our main result for oriented percolation is the following theorem.
Stronger results than Theorem 1.1 have been proved previously. For s = 0, 2, the existence of the limit was proved in [25,Theorem 3] and also in [19,Theorem 1.1], and this was extended to general s ≥ 0 in [5] in the case where |x| s is replaced by |x 1 | s .

Contact process
The contact process is a continuous-time Markov process with state space The state of the contact process is determined by a variable ξ x ∈ {0, 1}, for each x ∈ Z d . When ξ x = 0, then x is healthy, and when ξ x = 1, then x is infected. An infected particle spontaneously becomes healthy at rate 1, and, given p > 0, a healthy particle at x becomes infected at rate p y∈Z d ξ y D(x − y).
We assume that at time zero there is a single infected individual at the origin, with all others healthy. Let C t denote the set of infected particles at time t ≥ 0. The susceptibility is defined by (1.7) The critical point is defined by p c = sup{p : χ(p) < ∞}. It is known that p c = 1 + O(L −d ) (and more) [16]. The following theorem is our main result for the contact process.
There is an L 0 = L 0 (d) such that for L ≥ L 0 , and for any s ≥ 0, there is a constant c s = c s (L) such that The dichotomy in (1.8) does not arise for integer-time models such as oriented percolation and lattice trees. For s = 0, 2, the existence of the limit was proved in [15, Theorem 1.1].
It is well known that the contact process can be approximated by an oriented percolation model (see, e.g., [1,2,15,17,26]). For this, we replace the time interval [0, ∞) by εZ + ≡ {0, ε, 2ε, 3ε, . . . }, and define an oriented percolation model on Z d ×εZ + , as follows. Bonds have the form ((x, t), (y, t+ ε)) with t ∈ εZ + and x, y ∈ Z d . A bond is occupied with probability 1 − ε if x = y, and with probability εpD(y − x) if x = y. Bonds with x = y are called temporal and bonds with x = y are called spatial. Let P ε p denote the probability measure for the oriented percolation model on Z d × εZ + . Then P ε p converges weakly as ε → 0 + to the contact process measure P p , and the critical value p ε c of the discretised model converges to the critical value p c of the contact process [2,26] (see also [29]). In particular, (1.10) where ⌊·⌋ denotes the floor function. In order to deal with all models simultaneously, we adopt the notational convenience (1.11)

Lattice trees
For lattice trees, we assume that  A standard subadditivity argument implies that there exists p c > 0 such that the 1-point function N p N has radius of convergence p c . Critical exponents and the scaling limit of lattice trees in dimensions d > 8 are discussed in, e.g., [7,20,28]. Here, we rely on results from [20]. Our assumption that D has finite range is to conform with [20]; we expect that this restriction is actually unnecessary. It is known that which is finite since t The following theorem is our main result for lattice trees.
For s = 0, 2, the existence of the limit was proved in [20,Theorem 3.7], under the assumptions of Theorem 1.3. The cases s = 0, 2, 4, 6 of (1.15) are proved in detail in [22]. Our method of proof is simpler than that of [22], and it applies to all s ≥ 0.

Asymptotic formulas
According to [23,Remark 2.8], the conclusions of Holmes and Perkins include the promotion of the upper bounds on the spatial moments, once they have been established, to precise asymptotic formulas. For oriented percolation and lattice trees obeying the hypotheses of Theorems 1.1 and 1.3, this implies existence of the limits Similarly, for the contact process obeying the hypotheses of Theorem 1.2, we have existence of the limits The limits A s of course depend on the model. However, in each case, they can be expressed in terms of A 0 , A 2 and the dimension d as where Z is a d-dimensional vector whose components are each standard normal random variables. For the contact process and oriented percolation, the amplitudes A 0 , A 2 obey and where σ 2 is the variance of the probability distribution D of (1.2) [15,19]. For lattice trees, A 0 is within a factor [1 + O(L −d/2 )] of the critical 1-point function, and If we instead consider moments where |x| s is replaced by |x 1 | s , then, as pointed out in [23], the formula (1.19) holds instead for Z a 1-dimensional standard normal, which obeys E[|Z| s ] = Γ(s + 1)/[Γ( s 2 + 1)2 s/2 ]. Using the notation for oriented percolation and lattice trees, this gives the limiting value , (1.20) and similarly for the contact process. To compare this with the results of [5] for oriented percolation, we recall from [5, (1.6)] and [5, (1.9)] that there exist positive C I , C II such that . This is consistent with (1.20), with A 0 = C I and A 2 = C I C II σ 2 .

Sufficient condition on generating function
For any one of the three models under consideration, let The Fourier transform of an absolutely summable function f : We use the Fourier-Laplace transformt . Then, for |z| < 1 and r ∈ N, A key element of our proof is the following minor extension of [7, Lemma 3.2(i)], which is a kind of Tauberian theorem (see also [9]). Lemma 2.1. Suppose that, for N ≥ 1 and for u j ≥ 1, v j ≥ 0, the power series f (z) = ∞ n=0 a n z n has radius of convergence at least 1 and obeys Then there is a constant c depending only on u 1 , . . . , u N such that Proof. Let Γ n denote the circle of radius r n = 1− 1 n centred at the origin and with counterclockwise orientation. By the Cauchy Integral Formula, Therefore, by hypothesis, and since (1 − 1/n) −n is bounded above by a universal constant K, The integral on the right-hand side is analysed in the proof of [7, Lemma 3.2(i)], where it is shown to be bounded by n u j −1 if u j > 1, or by log n if u j = 1, times a constant depending only on u j . This completes the proof.
We will prove the following proposition for the discretised contact process. The proposition also applies for oriented percolation for d > 4 and lattice trees for d > 8, with the parameter ε given simply by ε = 1.
By Lemma 2.1, Proposition 2.2 implies that there is an ε-independent constant c ′ 2r such that, for all r, n ∈ N, (2.10) When ε = 1 this gives a bound c 2r n r and proves Theorems 1.1 and 1.3 for even integer powers.
For the contact process, the bound becomes By (1.10), this implies the conclusion of Theorem 1.2 for even integer powers. In fact, it suffices to prove Theorems 1.1 and 1.3 for s a non-negative even integer, since the general case then follows by Hölder's inequality (using the known results for the zeroth moment). In detail, if s is not an even integer then let r be the smallest integer such that s < 2r. Let a = 2r/s > 1 and define b by 1 a + 1 b = 1. Then sa = 2r is an even integer and, for oriented percolation and lattice trees (ε = 1), (2.12) For the contact process, the above argument gives instead, by using (2.11) for the (2r) th moment, the upper bound (with t = nε) For t ≤ 1, the right-hand side is O(1), consistent with (1.8). For t ≥ 1, since s/2r ≤ s/2 the right-hand side is O(t s/2 ), which is again consistent with (1.8). Thus, for Theorem 1.2 it also suffices to consider even integer powers s.
3 Oriented percolation: proof of Theorems 1.1-1.2 We fix d > 4, L sufficiently large, and p = p ε c throughout this section. We are generally not concerned with the L-dependence of constants, and usually allow them to depend on L. As discussed above, we can restrict to even integer powers s, and since the cases s = 0, 2 are already well established in previous papers, we can restrict attention to even integers s ≥ 4.

Lace expansion for oriented percolation
The proof uses the lace expansion, which provides a formula for the coefficients π n (x) in the convolution equation with the convention that an empty sum is zero. Here (f * g)(x) = y∈Z d f (y)g(x − y), π 0 (x) = π 1 (x) = 0 for all x, and There are three different versions of the lace expansion for oriented percolation, which provide different representations for π n (x) [10,24,26]. See [28] for a discussion of these three representations. We require few details here.
The following proposition is proved in [15] (see also [19,25] for ε = 1, and note that the amplitudes A (ε) , v (ε) in [ There is an L 0 > 0 and a finite C, both independent of ε, such that for L ≥ L 0 and for n ≥ 0, We use the Fourier-Laplace transforms, for complex z and for k ∈ [−π, π] d , We will also have use for the definitionŝ By (3.3) with r = 0, the power seriest z (k) converges in the open unit disk |z| < 1 in the complex plane. It is known that the limit A = lim n→∞τn (0) = 1+O(L −d ) exists for d > 4 and L sufficiently large [15,19]. Since, by the Markov property,τ m+n (0) ≤τ m (0)τ n (0), by Fekete's Lemma the limit µ = lim n→∞τn (0) 1/n also exists andτ n (0) ≥ µ n for all n. It follows that µ must be 1 and that t z (0) ≥ ∞ n=0 z n = (1 − z) −1 for z ∈ [0, 1). In particular,t 1 (0) = ∞. On the other hand, the following corollary to Proposition 3.1 shows thatΠ z and some of its derivatives remain bounded on the closed disk |z| ≤ 1.
The Fourier-Laplace transform converts the spatio-temporal convolutions in (3.1) into products, which leads to the following lemma.
Proof. We take the Fourier transform of (3.1), multiply by z n+1 , and sum over n ≥ 0, to obtain The left-hand side ist z − 1, and the first and last terms on the right-hand side are respectively zqt z andΠ z (recall that π n = 0 for n = 0, 1). After interchange of summation, the middle term on the right-hand side becomes zqt zΠz . Thus we havê as required.
Finally, there can be no division by zero on the right-hand side of (3.16), sincet z (k) converges for |z| < 1, and the numerator is close to 1 by (3.8).

Induction
The proof of Proposition 2.2 is by induction on r ∈ N. The case r = 1 is provided by (3.22). To advance the induction, we first observe that, by Hölder's inequality, Then, for a Z d -symmetric nonnegative function f on Z d , we have Therefore, to prove Proposition 2.2, it suffices to show (2.9) for a single component, i.e., where ∂ 1 is an abbreviation for ∂ ∂k 1 . By applying Leibniz's rule to the first equality in (3.16) and using the spatial symmetry, we obtain (This reproduces the first equality of (3.22) when r = 1.) The right-hand side of (3.26) only involves lower-order derivatives oft than the left-hand side. This opens up the possibility of an inductive proof, though we must deal with the fact that the right-hand side does involve ∂ 2r 1Π z . The idea is similar to that of [5], where for oriented percolation an asymptotic formula for x∈Z d |x 1 | s t n (x) is derived for any s > 0. We only prove upper bounds in this paper, and can be less careful in dealing with the recursion relation (3.26).
The next lemma shows that ℓ 1 estimates on |x| q τ n (x) imply corresponding ℓ ∞ estimates.
Lemma 3.4. Assume the same setting as Proposition 2.2, and let q ∈ N. Suppose there is an ε-independent constant C such that x∈Z d |x| q τ n (x) ≤ C(1 + nε) q/2 holds for all n ≥ 0. Then there is an ε-independent constant C ′ such that sup x∈Z d |x| q τ n (x) ≤ C ′ (1 + nε) (q−d)/2 also holds for all n ≥ 0.  Figure 1: Allocation of |x| 2 and |x| 2r to a diagram bounding π (1) n (x) for oriented percolation (left) and the contact process (right).
Proof. We write m = ⌊n/2⌋. Since (a + b) q ≤ 2 q (a q + b q ) for any a, b > 0, and since τ n (x) ≤ y τ m (y)τ n−m (x − y), we have By hypothesis, each of the two sums is bounded by C(1 + nε) q/2 . By (3.4) for r = 0, each ℓ ∞ norm is bounded above by a multiple of (1 . This completes the proof. The next lemma is a key step in advancing the induction. It shows that a bound on the (2r) th moment of τ n implies a bound on the (2r + 2) th moment of π n . The proof uses standard diagrammatic estimates on π n , which are explained in detail in [15, (4.26)] (and [19, (4.10)] for ε = 1). Figure 1 illustrates the diagrams for π (1) n (x), which is one contribution to π n (x). The oriented percolation case ε = 1 (left figure) needs less care, as compared to the contact process case ε < 1 (right figure). For the latter, correct factors of ε must be extracted. Those factors can be obtained by noting that, at each intersection of two bond-disjoint paths, at least one must use a spatial bond (red arrows) to leave or enter that point. Summation over the temporal location of the two middle red bonds consumes their ε factors to form a Riemann sum, leaving the two ε factors from the top and bottom red bonds; these are responsible for the factor ε 2 in the bound (3.5) on π n (x). In the diagrammatic estimates of [15,19], it was also necessary to extract an inverse power of L for each diagram loop in order to guarantee convergence of the sum over diagrams with increasing complexity. Once that work has been done, it is not necessary for us to consider L-dependence here, since our operations modify the estimates essentially only in a single loop.
Proof. We make the split |x| 2r+2 = |x| 2 |x| 2r and multiply |x| 2 on the left side and |x| 2r to the right side of the diagrams bounding π n (see Figure 1 for an example of a diagram). Then we use the triangle inequality to decompose |x| 2 along the left side of the diagram and |x| 2r along the right side of the diagram. The proof of the r = 1 case of (3.5) uses the ℓ 1 and ℓ ∞ estimates (3.3)-(3.4) for r = 0, 1 to obtain an upper bound of order ε 2 (1 + nε) 1−d/2 , where ε 2 arises from the occupied spatial bonds at (0, 0) and (x, nε). By Lemma 3.4 and the hypothesis, we also have sup x |x| 2r τ n (x) ≤ C ′ (1 + nε) r−d/2 . If the same bounds as in the proof of the r = 1 case of (3.5) are applied with one line having weight |x| 2r , then our assumption tells us that this line contributes an additional factor (1 + mε) r where mε is the temporal displacement of the line. Since m ≤ n, we obtain an upper bound of order (1 + nε) r+1−d/2 , as required.
Example 3.6. We illustrate the diagrammatic estimate underlying the proof of Lemma 3.5 with an example for ε = 1 and r = 2. Suppose that x |x| 4 τ n (x) ≤ O(n 2 ), and consider the left diagram in Figure 1 weighted with |x| 6 and summed over x, which is a prototype for a contribution to x |x| 6 |π n (x)|. We write |x| 6 = |x| 2 |x| 4 and decompose each factor along the two sides of the diagram using the triangle inequality. One contribution that results is (3.28) By hypothesis and Lemma 3.4, sup z |z| 4 τ n (z) ≤ O(n 2−d/2 ). Therefore, by (3.3), The elementary verification that the sum on the right-hand side decays like n −d/2 is carried out in detail in [19,Example 4.4]. This gives an overall bound n 3−d/2 and illustrates the origin of the bound in the conclusion of Lemma 3.5. Note that the factor n 3 in the overall bound results from the inequalities (n − l) 2 ≤ n 2 and n − m ≤ n used above to estimate the factors arising from the spatial moments.
The next lemma promotes the bounds of Lemma 3.5 to bounds on generating functions.
Lemma 3.8. Suppose that the bound (3.30) holds for all r + 1 ≤ R + 1. Then, for 2 ≤ j ≤ R + 1, Proof. We again drop the argument "0" from the transforms. We use the fact that ∂ 2j For j > d/2, (3.36) needs to be modified as This completes the proof.
Proof of Proposition 2.2. We again drop the argument "0." We use induction on r ∈ N to prove (3.25), which asserts that By (3.22), the initial case r = 1 is already confirmed. Suppose that (3.38) holds for all positive integers r ≤ R. By Lemma 2.1, there are εindependent constants C r such that, for all nonnegative integers n, x |x| 2r τ n (x) ≤ C r (1 + nε) r .
(3.39) By Lemma 3.7, this provides the estimate (3.30) on ∂ 2r+2 1Π z for all r ≤ R. Suppose first that R + 1 ≤ d/2. By (3.26) with r = R + 1, By Corollary 3.2 and Lemmas 3.7-3.8, followed by application of the induction hypothesis, It is then an exercise in bookkeeping to verify that this implies that, as required, (3.43) The above is of the correct form to advance the induction if d > 4 is even. If instead d is odd (in which case R + 1 ≥ (d + 1)/2), then we use and we obtain a result of the correct form to advance the induction also in this case. We write the absolute value of the additional term in (3.35) as X j . It contributes to the last two terms of the first line of (3.41) an amount Proof. The bound (4.6) is given in [20,Proposition 5.1]. The bounds on the sum in (4.5) follow from [20,Theorem 3.7], and then the bounds on the supremum follow from Lemma 3.4 (which applies equally well to lattice trees).
By (4.6), |Π z (k)| and |∆Π z (k)| are uniformly bounded in |z| ≤ 1. The bound on the 1-point function mentioned above then gives a bound onĥ z (k) in the same closed disk. By the bound on 1 −Φ z (0) provided by [7,Lemma 5.3(ii)] (their ζ is our z), as in (3.21) we obtain |t z (0)| ≤ C |1 − z| (|z| < 1); (4.7) in fact much more is known (see [7, (2.5)]). We have already noted that Lemma 3.4 applies to lattice trees. Lemma 3.5 is replaced by the following lemma, whose proof we discuss below. The proof of Proposition 2.2 is essentially the same but is simpler because now we can set ε = 1, and it applies also to prove Theorem 1.3. It remains only to discuss the proof of Lemma 4.2. This is carried out in detail for r = 1 in the proof of [20, Proposition 5.1], and for r = 2 in the proof of [22, (9.32)]. The same method applies more generally to handle higher values of r. Briefly, the proof goes as follows.
Proof of Lemma 4.2. The diagrammatic estimate in the proof of Lemma 3.5 must be replaced by an estimate for the diagrams that arise for lattice trees (the diagrams are discussed in [22, Section 9.2] -see, in particular, [22, Figure 2]). We divide |x| 2r+2 as |x| 2 |x| 2r−2 |x| 2 , distribute one |x| 2 factor along the top of a diagram via the triangle inequality, and distribute the other |x| 2 factor along the bottom of the diagram (see Figure 2 for an example of a 3-loop diagram). This leads to terms with one line on the top of the diagram weighted with the displacement squared, and one line on the bottom similarly weighted. The factor |x| 2r−2 is distributed along the backbone, which includes lines on top and bottom of the diagram, which may or may not be already weighted with the displacement squared. Thus, altogether, we have one line weighted with |y| 2 and a different line (which must lie on the backbone) weighted with |y| 2 |y| 2r−2 = |y| 2r , or we have two lines weighted with |y| 2 and a third line (which must lie on the backbone) weighted with |y| 2r−2 . The case r = 1 is handled in [20, Proposition 5.1] by using the bounds (4.5) on the backbone lines. For r > 1, we first apply Hölder's inequality (as in (2.12)) to see that the hypothesis on the (2r) th moment of τ n implies x∈Z d |x| 2r−2 τ n (x) = O(n r−1 ). With Lemma 3.4, these bounds on the (2r) th and (2r − 2) th moments of τ n imply corresponding ℓ ∞ bounds. Together, these imply that the estimate for the (2r + 2) th moment of π n will be at most n r−1 times larger than the fourth moment estimate of (4.6), i.e., as required.