The investigation of the contact-process survival probability is deferred to the sequel [18] to this paper, in which we also discuss the implications of our results for the convergence of the critical
spread-out contact process towards super-Brownian motion, in the sense of convergence of finite- dimensional distributions [23]. See also [12] and [28] for more expository discussions of the var-
ious results for oriented percolation and the contact process for d
4, and [29] for a detailed discussion of the applications of the lace expansion. For a summary of all the notation used in this
paper, we refer the reader to the glossary in Appendix A at the end of the paper.
1.2 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
+
≡ [0, ∞, and let C = {o}. An infected site x recovers in a small time interval [t, t + ǫ]
with probability ǫ + oǫ independently of t, where oǫ is a function that satisfies lim
ǫ↓0
o ǫǫ = 0.
In other words, x ∈ C
t
recovers at rate 1. A healthy site x gets infected, depending on the status of its neighboring sites, at rate
λ P
y
∈C
t
Dx − y, where λ ≥ 0 is the infection rate. We denote the
associated probability measure by P
λ
. We assume that the function D : Z
d
→ [0, 1] is a probability distribution which is symmetric with respect to the lattice symmetries. Further assumptions on D
involve a parameter L ≥ 1 which serves to spread out the infections, and will be taken to be large. In
particular, we require that Do = 0 and kDk
∞
≡ sup
x ∈Z
d
Dx ≤ C L
−d
. Moreover, with σ defined
as σ
2
= X
x
|x|
2
Dx, 1.1
where | · | denotes the Euclidean norm on R
d
, we require that C
1
L ≤ σ ≤ C
2
L and that there exists a ∆
0 such that X
x
|x|
2+2∆
Dx ≤ C L
2+2∆
. 1.2
See [16, Section 5] for the precise assumptions on D. A simple example of D is Dx =
1
{0kxk
∞
≤L}
2L + 1
d
− 1 ,
1.3 which is the uniform distribution on the cube of radius L.
For r ≥ 2, ~t = t
1
, . . . , t
r −1
∈ R
r −1
+
and ~x = x
1
, . . . , x
r −1
∈ Z
r−1d
, we define the r-point function as
τ
λ ~t
~x = P
λ
x
i
∈ C
t
i
∀i = 1, . . . , r − 1. 1.4
For a summable function f : Z
d
→ R, we define its Fourier transform for k ∈ [−π, π]
d
by ˆ
f k = X
x ∈Z
d
e
ik ·x
f x. 1.5
By the results in [8] and the extension of [2] to the spread-out model, there exists a unique critical point
λ
c
∈ 0, ∞ such that Z
∞
dt ˆ τ
λ t
∞, if λ λ
c
, = ∞, otherwise,
lim
t ↑∞
P
λ
C
t
6= ∅ = 0,
if λ ≤ λ
c
, 0, otherwise.
1.6 We will next investigate the sufficiently spread-out contact process at the critical value
λ
c
for d 4,
as well as a local mean-field limit when d ≤ 4.
804
1.3 Previous results for the 2-point function
We first state the results for the 2-point function proved in [16]. Those results will be crucial for the current paper. In the statements,
σ is defined in 1.1 and ∆ in 1.2. Besides the high-dimensional setting for d
4, we also consider a low-dimensional setting, i.e., d
≤ 4. In this case, the contact process is not believed to be in the mean-field regime, and Gaussian asymptotics are thus not expected to hold as long as L remains finite. However, inspired by the
mean-field limit in [5] of Durrett and Perkins, we have proved Gaussian asymptotics when range and time grow simultaneously [16]. We suppose that the infection range grows as
L
T
= L
1
T
b
, 1.7
where L
1
≥ 1 is the initial infection range and T ≥ 1. We denote by σ
2
T
the variance of D = D
T
in this situation. We will assume that
α = bd + d
− 4 2
0. 1.8
Theorem 1.1 Gaussian asymptotics for the two-point function.
i Let d 4, δ ∈ 0, 1 ∧ ∆ ∧
d −4
2
and L ≫ 1. There exist positive finite constants A = Ad, L, v = vd, L and C
i
= C
i
d i = 1, 2 such that
ˆ τ
λ
c
t k
p
v σ
2
t
= A e
−
|k|2 2d
1 + O |k|
2
1 + t
−δ
+ O 1 + t
−d−42
, 1.9
1 ˆ
τ
λ
c
t
X
x
|x|
2
τ
λ
c
t
x = v σ
2
t 1 + O 1 + t
−δ
, 1.10
C
1
L
−d
1 + t
−d2
≤ kτ
λ
c
t
k
∞
≤ e
−t
+ C
2
L
−d
1 + t
−d2
, 1.11
with the error estimate in 1.9 uniform in k ∈ R
d
with |k|
2
log2 + t sufficiently small. More- over,
λ
c
= 1 + OL
−d
, A = 1 + OL
−d
, v = 1 + OL
−d
. 1.12
ii Let d ≤ 4, δ ∈ 0, 1 ∧ ∆ ∧ α and L
1
≫ 1. There exist λ
T
= 1 + OT
−µ
for some µ ∈ 0, α − δ
and C
i
= C
i
d i = 1, 2 such that, for every 0 t ≤ log T ,
ˆ τ
λ
T
T t k
p σ
2 T
T t
= e
−
|k|2 2d
1 + OT
−µ
+ O |k|
2
1 + T t
−δ
, 1.13
1 ˆ
τ
λ
T
T t
X
x
|x|
2
τ
λ
T
T t
x = σ
2
T
T t 1 + OT
−µ
+ O 1 + T t
−δ
, 1.14
C
1
L
−d
T
1 + T t
−d2
≤ kτ
λ
T
T t
k
∞
≤ e
−T t
+ C
2
L
−d
T
1 + T t
−d2
, 1.15
with the error estimate in 1.13 uniform in k ∈ R
d
with |k|
2
log2 + T t sufficiently small. In the rest of the paper, we will always work at the critical value, i.e., we take
λ = λ
c
for d 4
and λ = λ
T
as in Theorem 1.1ii for d ≤ 4. We will often omit the λ-dependence and write
τ
r
~t
~x = τ
λ ~t
~x to emphasize the number of arguments of τ
λ ~t
~x. 805
While τ
λ
c
t
x tells us what paths in a critical cluster look like, τ
λ
c
~t
~x gives us information about the branching structure of critical clusters. The goal of this paper is to prove that the suitably scaled
critical r-point functions converge to those of the canonical measure of super-Brownian motion SBM.
In [5], Durrett and Perkins proved convergence to SBM of the rescaled contact process with L
T
defined in 1.7. We now compare the ranges needed in our results and in [5]. We need that α ≡ bd +
d −4
2
0, i.e., bd
4 −d
2
. In [5], bd = 1 for all d ≥ 3, and L
2 T
∝ T log T for d = 2, which is the critical case in [5]. In comparison, we are allowed to use ranges that grow to infinity slower
than the ranges in [5] when d ≥ 3, but the range for d = 2 in our results needs to be slightly larger
than the range in [5]. It would be of interest to investigate whether a range L
2 T
∝ T log T or even smaller is possible by adapting our proofs.
1.4 The r-point function for r
