Ergodicity of the voter model with dynamic anti-voter bonds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866918439540490240 |
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| author | Astoquillca, Jhon Valesin, Daniel |
| author_facet | Astoquillca, Jhon Valesin, Daniel |
| contents | The voter model with anti-voter bonds is a variant of the classical voter model in which the edges of the underlying graph are assigned signs. At each update, a voter chooses a neighbour according to a transition kernel; interactions across a positive edge follow the usual voter dynamics, so that a site adopts the current opinion of its chosen neighbour, whereas interactions across a negative edge lead to the adoption of the opposite opinion.
In this work, we introduce a new variant in which the edge signs evolve dynamically according to dynamical percolation with density parameter $p \in (0,1)$ and speed $\mathsf{v} \in (0,\infty)$, where the two states of the process represent positive and negative edges. This defines a joint spin-bond Markov process. Following Liggett's notion of ergodicity, we prove that this process is ergodic on any simple graph with countably many vertices, with an arbitrary transition kernel of adoption rates and for all choices of the parameters of the edge dynamics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10051 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ergodicity of the voter model with dynamic anti-voter bonds Astoquillca, Jhon Valesin, Daniel Probability 60J27, 60K35, 60K37 The voter model with anti-voter bonds is a variant of the classical voter model in which the edges of the underlying graph are assigned signs. At each update, a voter chooses a neighbour according to a transition kernel; interactions across a positive edge follow the usual voter dynamics, so that a site adopts the current opinion of its chosen neighbour, whereas interactions across a negative edge lead to the adoption of the opposite opinion. In this work, we introduce a new variant in which the edge signs evolve dynamically according to dynamical percolation with density parameter $p \in (0,1)$ and speed $\mathsf{v} \in (0,\infty)$, where the two states of the process represent positive and negative edges. This defines a joint spin-bond Markov process. Following Liggett's notion of ergodicity, we prove that this process is ergodic on any simple graph with countably many vertices, with an arbitrary transition kernel of adoption rates and for all choices of the parameters of the edge dynamics. |
| title | Ergodicity of the voter model with dynamic anti-voter bonds |
| topic | Probability 60J27, 60K35, 60K37 |
| url | https://arxiv.org/abs/2604.10051 |