## The Annals of Probability

### Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$

J. T. Cox

#### Abstract

Let $\eta_t$ be the basic voter model on $\mathbb{Z}^d$ and let $\eta^{(N)}_t$ be the voter model on $\Lambda(N)$, the torus of side $N$ in $\mathbb{Z}^d$. Unlike $\eta_t, \eta^{(N)}_t$ (for fixed $N$) gets trapped with probability 1 as $t \rightarrow\infty$ at all 0's or all 1's. We examine the asymptotic growth of these trapping or consensus times $\tau^{(N)}$ as $N \rightarrow\infty$. To do this we obtain limit theorems for coalescing random walk systems on the torus $\Lambda(N)$, including a new hitting time limit theorem for (noncoalescing) random walk on the torus.

#### Article information

Source
Ann. Probab., Volume 17, Number 4 (1989), 1333-1366.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176991158

Digital Object Identifier
doi:10.1214/aop/1176991158

Mathematical Reviews number (MathSciNet)
MR1048930

Zentralblatt MATH identifier
0685.60100

JSTOR
Cox, J. T. Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$. Ann. Probab. 17 (1989), no. 4, 1333--1366. doi:10.1214/aop/1176991158. https://projecteuclid.org/euclid.aop/1176991158