##
**Central limit theorem for the contact process.**
*(English)*
Zbl 0615.60095

This tightly written paper proves a central limit theorem for a cylindrical function f: P(\({\mathbb{Z}})\to {\mathbb{R}}\) and a supercritical contact process \(\xi^ A(t)\) starting on an initial configuration \(| A| =\infty\) in \({\mathbb{Z}}\). Briefly, as \(T\to \infty\), if \(\mu\) is the nontrivial invariant measure, then
\[
T^{1/2}[T^{- 1}\int^{T}_{0}f(\xi^ A(t))dt-\int f d\mu]
\]
approaches a (possibly trivial) normal distribution with 0 mean.

The main steps of the proof, motivated by methods developed by J. T. Cox and D. Griffeath [Ann. Probab. 11, 876-893 (1983; Zbl 0527.60095)] for occupation time limit theorems in the voter process, is as follows. (\(\zeta\) (t),t\(\geq 0)\), the contact process with random initial distribution with measure \(\mu\), is introduced. A similar limit theorem is proved with \(\zeta\) (t) replacing \(\xi^ A(t)\) by a skeleton argument involving integrals related to \(\zeta\) (t). An associated FKG set of variables bounds the correlations over time of the integrals. The bound depends on a decay time for the correlations of f(\(\zeta\) (t)), derived by considering the dual percolation structure for the contact process. The proof is completed by a standard theorem showing that the difference between the limit theorems involving \(\zeta\) (t) and \(\xi^ A(t)\) approaches 0 in probability.

The main steps of the proof, motivated by methods developed by J. T. Cox and D. Griffeath [Ann. Probab. 11, 876-893 (1983; Zbl 0527.60095)] for occupation time limit theorems in the voter process, is as follows. (\(\zeta\) (t),t\(\geq 0)\), the contact process with random initial distribution with measure \(\mu\), is introduced. A similar limit theorem is proved with \(\zeta\) (t) replacing \(\xi^ A(t)\) by a skeleton argument involving integrals related to \(\zeta\) (t). An associated FKG set of variables bounds the correlations over time of the integrals. The bound depends on a decay time for the correlations of f(\(\zeta\) (t)), derived by considering the dual percolation structure for the contact process. The proof is completed by a standard theorem showing that the difference between the limit theorems involving \(\zeta\) (t) and \(\xi^ A(t)\) approaches 0 in probability.

Reviewer: J.Spouge

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60F05 | Central limit and other weak theorems |