%0 Journal Article %1 cox1989 %A Cox, J. T. %D 1989 %I The Institute of Mathematical Statistics %J Ann. Probab. %K coalescent_theory coalescing_random_walk spatial_coalescent spatial_structure voter_model %N 4 %P 1333--1366 %R 10.1214/aop/1176991158 %T Coalescing Random Walks and Voter Model Consensus Times on the Torus in $Z^d$ %U http://dx.doi.org/10.1214/aop/1176991158 %V 17 %X Let $\eta_t$ be the basic voter model on $Z^d$ and let $\eta_t^(N)$ be the voter model on $Łambda(N)$, the torus of side $N$ in $Z^d$. Unlike $\eta_t$, $\eta_t^(N)$ (for fixed $N$) gets trapped with probability 1 as $t ınfty$ at all 0's or all 1's. We examine the asymptotic growth of these trapping or consensus times $\tau^(N)$ as $N ınfty$. To do this we obtain limit theorems for coalescing random walk systems on the torus $Łambda(N)$, including a new hitting time limit theorem for (noncoalescing) random walk on the torus.

@article{cox1989, abstract = {Let $\eta_t$ be the basic voter model on $Z^d$ and let $\eta_t^{(N)}$ be the voter model on $\Lambda(N)$, the torus of side $N$ in $Z^d$. Unlike $\eta_t$, $\eta_t^{(N)}$ (for fixed $N$) gets trapped with probability 1 as $t \to \infty$ at all 0's or all 1's. We examine the asymptotic growth of these trapping or \emph{consensus} times $\tau^{(N)}$ as $N \to \infty$. 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. }, added-at = {2015-06-15T03:34:23.000+0200}, author = {Cox, J. T.}, biburl = {https://www.bibsonomy.org/bibtex/2dd03172d7143f46bf6e77c176fb3f4f2/peter.ralph}, doi = {10.1214/aop/1176991158}, fjournal = {The Annals of Probability}, interhash = {391f74a4ba2980fadbb775d50b9ffee5}, intrahash = {dd03172d7143f46bf6e77c176fb3f4f2}, journal = {Ann. Probab.}, keywords = {coalescent_theory coalescing_random_walk spatial_coalescent spatial_structure voter_model}, month = {10}, number = 4, pages = {1333--1366}, publisher = {The Institute of Mathematical Statistics}, timestamp = {2015-06-15T03:34:23.000+0200}, title = {Coalescing Random Walks and Voter Model Consensus Times on the Torus in $\mathbb{Z}^d$}, url = {http://dx.doi.org/10.1214/aop/1176991158}, volume = 17, year = 1989 }