## Brazilian Journal of Probability and Statistics

- Braz. J. Probab. Stat.
- Volume 31, Number 2 (2017), 254-274.

### Improved asymptotic estimates for the contact process with stirring

#### Abstract

We study the contact process with stirring on $\mathbb{Z}^{d}$. In this process, particles occupy vertices of $\mathbb{Z}^{d}$; each particle dies with rate 1 and generates a new particle at a randomly chosen neighboring vertex with rate $\lambda$, provided the chosen vertex is empty. Additionally, particles move according to a symmetric exclusion process with rate $N$. For any $d$ and $N$, there exists $\lambda_{c}$ such that, when the system starts from a single particle, particles go extinct when $\lambda<\lambda_{c}$ and have a chance of being present for all times when $\lambda>\lambda_{c}$. Durrett and Neuhauser proved that $\lambda_{c}$ converges to 1 as $N$ goes to infinity, and Konno, Katori and Berezin and Mytnik obtained dimension-dependent asymptotics for this convergence, which are sharp in dimensions 3 and higher. We obtain a lower bound which is new in dimension 2 and also gives the sharp asymptotics in dimensions 3 and higher. Our proof involves an estimate for two-type renewal processes which is of independent interest.

#### Article information

**Source**

Braz. J. Probab. Stat., Volume 31, Number 2 (2017), 254-274.

**Dates**

Received: September 2015

Accepted: February 2016

First available in Project Euclid: 14 April 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.bjps/1492156962

**Digital Object Identifier**

doi:10.1214/16-BJPS312

**Mathematical Reviews number (MathSciNet)**

MR3635905

**Zentralblatt MATH identifier**

1380.60088

**Keywords**

Interacting particle systems contact process contact process with rapid stirring

#### Citation

Levit, Anna; Valesin, Daniel. Improved asymptotic estimates for the contact process with stirring. Braz. J. Probab. Stat. 31 (2017), no. 2, 254--274. doi:10.1214/16-BJPS312. https://projecteuclid.org/euclid.bjps/1492156962