where the error term is uniform in ǫ ∈ 0, 1] and in ~k in a bounded subset of R
dr −1
. Moreover, for any
ǫ ∈ 0, 1], λ
ǫ
c
= 1 + OL
−d
, A
ǫ
= 1 + OL
−d
, v
ǫ
= 1 + OL
−d
, V
ǫ
= 2 − ǫ + OL
−d
. 2.5
ii Let d ≤ 4, r ≥ 2, ~k ∈ R
dr −1
, ~t ∈ 0, ∞
r −1
and let δ, L
1
, λ
T
, µ be as in Theorem 1.1ii. For
large T such that log T ≥ max
i
t
i
, ˆ
τ
r
T~t ~k
p σ
2 T
T
= 2 − ǫT
r −2
ˆ M
r−1
~t
~k + OT
−µ∧δ
, 2.6
where the error term is uniform in ǫ ∈ 0, 1] and in ~k in a bounded subset of R
dr −1
. For r = 2, the claims in Theorem 2.1 were proved in [16, Propositions 2.1–2.2]. We will only prove
the statements for r ≥ 3.
For oriented percolation for which ǫ = 1, Theorem 2.1i reproves [19, Theorem 1.2]. The unifor-
mity in ǫ in Theorem 2.1 is crucial in order for the continuum limit ǫ ↓ 0 to be performed, and to
extend the results to the contact process.
2.2 Overview of the expansion for the higher-point functions
In this section, we give an introduction to the expansion methods of Sections 3–4. For this, it will be convenient to introduce the notation
Λ = Z
d
× ǫZ
+
. 2.7
We write a typical element of Λ as x rather than x, t as was used until now. We fix
λ = λ
ǫ
c
throughout Section 2.2 for simplicity, though the discussion also applies without change when λ
λ
ǫ
c
. We begin by discussing the underlying philosophy of the expansion. This philosophy is identical to the one described in [20, Section 2.2.1].
As explained in more detail in [16], the basic picture underlying the expansion for the 2-point function is that a cluster connecting o and x can be viewed as a string of sausages. In this picture,
the strings joining sausages are the occupied pivotal bonds for the connection from o to x . Pivotal bonds are the essential bonds for the connection from o to x , in the sense that each occupied path
from o to x must use all the pivotal bonds. Naturally, these pivotal bonds are ordered in time. Each sausage corresponds to an occupied cluster from the endpoint of a pivotal bond, containing
the starting point of the next pivotal bond. Moreover, a sausage consists of two parts: the backbone, which is the set of sites that are along occupied paths from the top of the lower pivotal bond to
the bottom of the upper pivotal bond, and the hairs, which are the parts of the cluster that are not connected to the bottom of the upper pivotal bond. The backbone may consist of a single site, but
may also consist of sites on at least two bond-disjoint connections. We say that both these cases correspond to double connections. We now extend this picture to the higher-point functions.
For connections from the origin to multiple points
~x = x
1
, . . . , x
r −1
, the corresponding picture is a “tree of sausages” as depicted in Figure 2. In the tree of sausages, the strings represent the union
over i = 1, . . . , r
− 1 of the occupied pivotal bonds for the connections o −→ x
i
, and the sausages are again parts of the cluster between successive pivotal bonds. Some of them may be pivotal for
{o −→ x
j
∀ j ∈ J}, while others are pivotal only for {o −→ x
j
} for some j ∈ J. 810
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Î Î
Î Í
Í Í
Í Í
Í Í
Í Í
Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Í Í
Í Í
Í Í
Î Î
Î Î
Í Í
Í Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Í
Í Í
Í Í
Í Í
Í Í
Î Î
Î Î
Î
Í Í
Í Í
Í Í
Í Î
Î Î
Î Î
Í Í
Í Í
Í Í
Í Î
Î Î
Í Í
Í Í
Í Í
Í Í
Í Í
Í
Í Í
Í Í
Í Í
Í Í
Î Î
Î Î
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Î
Î Î
Î Î
Î Î
Í Í
Í Í
Í Í
Í Í
Í Í
Î Î
Î Î
Î Î
Í Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Î Î
Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Î
Î Î
Î Î
Î Î
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í Í
Í
⇐= ⇐=
⇐=
= ⇒
= ⇒
= ⇒
= ⇒
= ⇒
= ⇒
⇐= ⇐=
⇐=
⇐= =
⇒ ⇐=
⇐= ⇐=
o x
1
x
2
o x
1
x
2
Figure 2: a A configuration for the discretized contact process. Both Î and Í denote occupied temporal bonds; Î is connected from o, while Í is not. The arrows are occupied spatial bonds,
representing the spread of an infection to neighbours. b Schematic depiction of the configuration as a “string of sausages.”
We regard this picture as corresponding to a kind of branching random walk. In this correspondence, the steps of the walk are the pivotal bonds, while the sites of the walk are the backbones between
subsequent pivotal bonds. Of course, the pivotal bonds introduce an avoidance interaction on the branching random walk. Indeed, the sausages are not allowed to share sites with the later backbones
since otherwise the pivotal bonds in between would not be pivotal.
When d 4 or when d ≤ 4 and the range of the contact process is sufficiently large as described
in 1.7–1.8, the interaction is weak and, in particular, the different parts of the backbone in between different pivotal bonds are small and the steps of the walk are effectively independent.
Thus, we can think of the higher-point functions of the critical time-discretized contact process as “small perturbations of the higher-point functions of critical branching random walk. We will use
this picture now to give an informal overview of the expansions we will derive in Sections 3–4.
We start by introducing some notation. For r ≥ 3, let
J = {1, 2, . . . , r − 1},
J
j
= J \ { j} j ∈ J.
2.8 For I =
{i
1
, . . . , i
s
} ⊂ J, we write ~x
I
= {x
i
1
, . . . , x
i
s
} and ~x
I
− y = {x
i
1
− y, . . . , x
i
s
− y} and abuse
notation by writing p
ǫ
x = p
ǫ
x δ
t, ǫ
for x = x, t. 2.9
There may be anywhere from 0 to r − 1 pivotal bonds, incident to the sausage at the origin, for the
event
{o −→ ~x
J
} = {o −→ x
j
∀ j ∈ J}. 2.10
811
Configurations with zero or more than two pivotal bonds will turn out to constitute an error term. Indeed, when there are zero pivotal bonds, this means that o =
⇒ x
i
for each i, which constitutes an error term. When there are more than two pivotal bonds, the sausage at the origin has at least three
disjoint connections to different x
i
’s, which also turns out to constitute an error term. Therefore, we are left with configurations which have one or two branches emerging from the sausage at the
origin. When there is one branch, then this branch contains
~x
J
. When there are two branches, one branch will contain
~x
I
for some nonempty I ⊆ J
1
and the other branch will contain ~x
J \I
, where we require 1
∈ J \ I to make the identification unique. The first expansion deals with the case where there is a single branch from the sausage at the origin.
It serves to decouple the interaction between that single branch and the branches of the tree of sausages leading to
~x
J
. From now on, we write a function F on Λ
n
≡ Z
d n
× Z
n +
or on Z
d n
× R
n +
for the continuous-time model for a given n
∈ N as F
~x = F
~t
~x for
~x = ~x,~t. 2.11
The expansion writes
τ~x
J
in the form
τ~x
J
= A
~x
J
+ B
⋆
τ~x
J
= A
~x
J
+ X
v
∈Λ
Bv τ~x
J
− v, 2.12
where f
⋆
gx represents the space-time convolution of two functions f , g : Λ → R given by
f
⋆
gx =
X
y
∈Λ
f y gx − y.
2.13 For details, see Section 3, where 2.12 is derived. We have that
Bx =
π
⋆
p
ǫ
x , 2.14
where πx is the expansion coefficient for the 2-point function as derived in [16, Section 3].
Moreover, for r = 2,
Ax = πx ,
2.15 so that 2.12 becomes
τx = πx + π
⋆
p
ǫ ⋆
τx .
2.16 This is the lace expansion for the 2-point function, which serves as the key ingredient in the analysis
of the 2-point function in [16].
1
The next step is to write A ~x
J
as A
~x
J
= X
I ⊂J
1
:I 6=∅
X
y
1
By
1
, ~x
I
τ~x
J \I
− y
1
+ a ~x
J
; 1, 2.17
where, to leading order, J \ I consists of those j for which the first pivotal bond for the connection to
x
j
is the same as the one for the connection to x
1
, while for i ∈ I, this first pivotal is different. The
1
In this paper, we will use a different expansion for the 2-point function than the one used in [16]. However, the resulting
πx is the same, as πx is uniquely defined by the equation 2.16.
812
equality 2.17 is the result of the first expansion for A ~x
J
. In this expansion, we wish to treat the connections from the top of the first pivotal to
~x
J \I
as being independent from the connections from
o
to ~x
I
that do not use the first pivotal bond. In the second expansion for A ~x
J
, we wish to extract a factor
τ~x
I
− y
2
for some y
2
from the connection from o to ~x
I
that is still present in By
1
,
~x
I
. This leads to a result of the form
X
y
1
By
1
,
~x
I
τ~x
J \I
− y
1
= X
y
1
,y
2
Cy
1
, y
2
τ~x
J \I
− y
1
τ~x
I
− y
2
+ a
~x
J \I
,
~x
I
, 2.18
where a ~x
J \I
, ~x
I
is an error term, and, to first approximation, Cy
1
, y
2
represents the sausage at o together with the pivotal bonds ending at y
1
and y
2
, with the two branches removed. In particular,
Cy
1
, y
2
is independent of I. The leading contribution to Cy
1
, y
2
is p
ǫ
y
1
p
ǫ
y
2
with y
1
6= y
2
,
corresponding to the case where the sausage at o is the single point o. For details, see Section 4, where 2.18 is derived.
We will use a new expansion for the higher-point functions, which is a simplification of the expansion for oriented percolation in Z
d
× Z
+
in [20]. The difference resides mainly in the second expansion, i.e., the expansion of A
~x
J
.
2.3 The main identity and estimates
