# Complete convergence theorem for a two-level contact process

## TL;DR

This paper proves a complete convergence theorem for a two-level contact process modeling fleas on animals, showing that fleas survive only if the underlying animal process is supercritical, with implications for their extinction at criticality.

## Contribution

It establishes the complete convergence theorem for a two-level contact process with a novel block construction method.

## Key findings

- Fleas survive only if the animal process is supercritical.
- Fleas die out at their critical value.
- The proof uses a block construction approach.

## Abstract

We study a two-level contact process. We think of fleas living on a species of animals. The animals are a supercritical contact process in $\mathbb{Z}^d$. The contact process acts as the random environment for the fleas. The fleas do not affect the animals, give birth at rate $\mu$ when they are living on a host animal, and die at rate $\delta$ when they do not have a host animal. The main result is that if the contact process is supercritical and the fleas survive then the complete convergence theorem holds. This is done using a block construction so as a corollary we conclude that the fleas die out at their critical value.