1 Introduction and results
1.1 Introduction
The contact process is a model for the spread of an infection among individuals in the d-dimensional integer lattice Z
d
. Suppose that the origin o ∈ Z
d
is the only infected individual at time 0, and assume for now that every infected individual may infect a healthy individual at a distance less than
L ≥ 1. We refer to this type of 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 at a critical value λ
c
∈ 0, ∞ see, e.g., [24].
In the previous paper [16], and following the idea of [25], we proved the 2-point function results for the contact process for d
4 via a time discretization, as well as a partial extension to d ≤ 4. The discretized contact process is a version of oriented percolation in Z
d
× ǫZ
+
, where ǫ ∈ 0, 1] is
the time unit and Z
+
is the set of nonnegative integers: Z
+
= {0} ˙ ∪ N. The proof is based on the
strategy for ordinary oriented percolation ǫ = 1, i.e., on the application of the lace expansion and
an adaptation of the inductive method so as to deal with the time discretization. In this paper, we use the 2-point function results in [16] as a key ingredient to show that, for any
r ≥ 3, the r-point functions of the critical contact process for d 4 converge to those of the canon-
ical measure of super-Brownian motion, as was proved in [20] for ordinary oriented percolation. We follow the strategy in [20] to analyze the lace expansion, but derive an expansion which is dif-
ferent from the expansion used in [20]. The lace expansion used in this paper is closely related to the expansion in [15] for the oriented-percolation survival probability. The latter was used in
[14] to show that the probability that the oriented-percolation cluster survives up to time n decays proportionally to 1
n. Due to this close relation, we can reprove an identity relating the constants arising in the scaling limit of the 3-point function and the survival probability, as was stated in [13,
Theorem 1.5] for oriented percolation. The main selling points of this paper in comparison to other papers on the topic are the following:
1. Our proof yields a simplification of the expansion argument, which is still inherently difficult, but has been simplified as much as possible, making use of and extending the combined
insights of [9; 15; 16; 20]. 2. The expansion for the higher-point functions yields similar expansion coefficients to those for
the survival probability in [15], thus making the investigation of the contact-process survival probability more efficient and allowing for a direct comparison of the various constants arising
in the 2- and 3-point functions and the survival probability. This was proved for oriented percolation in [13, Theorem 1.5], which, on the basis of the expansion in [19], was not
directly possible.
3. The extension of the results to certain local mean-field limit type results in low dimensions, as was initiated in [5] and taken up again in [16].
4. A simplified argument for the continuum limit of the discretized model, which was performed in [16] through an intricate weak convergence argument, and which in the current paper is
replaced by a soft argument on the basis of subsequential limits and uniformity of our bounds.
803
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
Kategori