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**Coalescing random walks and voter model consensus times on the torus in \({\mathbb{Z}}^ d\).**
*(English)*
Zbl 0685.60100

Infinite particle systems and the corresponding finite systems often behave differently as time \(t\to \infty\). This difference can be crucial, since in statistical physics infinite systems are approximations for large, but finite systems and concerning simulation the situation is the other way round. Thus it is important to know how long it takes until this difference becomes apparent.

In the present paper this question is studied for the basic voter model on \({\mathbb{Z}}^ d\). Unlike the infinite system, the finite system eventually reaches consensus with all states being 0 or 1. The author derives the asymptotics of the consensus time for \(\Lambda\) (N), the torus of side N in \({\mathbb{Z}}^ d\), as \(N\to \infty\). For its proof he uses the duality with coalescing random walks, for which he proves limit theorems with a hitting time limit theorem for ordinary random walks. These results are also of interest for themselves.

Finally he considers the multitype voter model or stepping stone model, for which essentially the same results hold.

In the present paper this question is studied for the basic voter model on \({\mathbb{Z}}^ d\). Unlike the infinite system, the finite system eventually reaches consensus with all states being 0 or 1. The author derives the asymptotics of the consensus time for \(\Lambda\) (N), the torus of side N in \({\mathbb{Z}}^ d\), as \(N\to \infty\). For its proof he uses the duality with coalescing random walks, for which he proves limit theorems with a hitting time limit theorem for ordinary random walks. These results are also of interest for themselves.

Finally he considers the multitype voter model or stepping stone model, for which essentially the same results hold.

Reviewer: M.Mürmann

### MSC:

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

60G50 | Sums of independent random variables; random walks |