## Brazilian Journal of Probability and Statistics

### 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
Accepted: February 2016
First available in Project Euclid: 14 April 2017

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

Digital Object Identifier
doi:10.1214/16-BJPS312

Mathematical Reviews number (MathSciNet)
MR3635905

Zentralblatt MATH identifier
1380.60088

#### 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

#### References

• Berezin, R. and Mytnik, L. (2014). Asymptotic behaviour of the critical value for the contact process with rapid stirring. Journal of Theoretical Probability 27, 1045–1057.
• De Masi, A., Ferrari, P. A. and Lebowitz, J. L. (1986). Reaction-diffusion equations for interacting particle systems. Journal of Statistical Physics 44, 589–644.
• Durrett, R. and Neuhauser, C. (1994). Particle systems and reaction-diffusion equations. The Annals of Probability 22, 289–333.
• Griffeath, D. (1983). The binary contact path process. The Annals of Probability 11, 692–705.
• Harris, T. E. (1974). Contact interactions on a lattice. The Annals of Probability 2, 969–988.
• Katori, M. (1994). Rigorous results for the diffusive contact processes in $d$ $>$ or $=3$. Journal of Physics A: Mathematical and General 27, 7327–7341.
• Konno, N. (1995). Asymptotic behavior of basic contact process with rapid stirring. Journal of Theoretical Probability 8, 833–876.
• Lawler, G. F. and Limic, V. (2010). Random Walk: A Modern Introduction. Cambridge Studies in Advanced Mathematics, Vol. 123. Cambridge: Cambridge University Press.
• Liggett, T. M. (2013). Stochastic Interacting Systems: Contact, Voter and Exclusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 324. Berlin: Springer.
• Spitzer, F. (1970). Interaction of Markov processes. Advances in Mathematics 5, 246–290.