## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Infinite particle systems finite particle systems random walks voter model

#### Citation

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