Ergodicity and Percolation for Variants of One-dimensional Voter Models

Details Ergodicity and Percolation for Variants of One-dimensional Voter Models Ergodicity and Percolation for Variants of One-dimensional Voter Models

2011-12-08

arxiv.org/abs/1112.1893v2

Mathematics

Probability

20 pages, 8 figures

Scientific paper

We study variants of one-dimensional q-color voter models in discrete time. In addition to the usual voter model transitions in which a color is chosen from the left or right neighbor of a site there are two types of noisy transitions. One is bulk nucleation where a new random color is chosen. The other is boundary nucleation where a random color is chosen only if the two neighbors have distinct colors. We prove under a variety of conditions on q and the magnitudes of the two noise parameters that the system is ergodic, i.e., there is convergence to a unique invariant distribution. The methods are percolation-based using the graphical structure of the model which consists of coalescing random walks combined with branching (boundary nucleation) and dying (bulk nucleation).

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