Percolation on the stationary distributions of the voter model with stirring
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| Format: | Preprint |
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2024
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| _version_ | 1866916032174620672 |
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| author | Astoquillca, Jhon Severo, Franco Szabó, Réka Valesin, Daniel |
| author_facet | Astoquillca, Jhon Severo, Franco Szabó, Réka Valesin, Daniel |
| contents | The voter model with stirring is a variant of the classical voter model on $\mathbb{Z}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf v \ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d \ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{ μ_{α,\mathsf{v}}: α\in [0,1] \}$, where~$α$ is the density of voters with opinion~1. Sampling a configuration~$ξ$ from~$μ_{α, \mathsf v}$, we study~$ξ$ as a site percolation model on~$\mathbb{Z}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$ξ$. Letting~$α_c(\mathsf v)$ be the supremum of all the values of~$α$ for which percolation does not occur~$μ_{α, \mathsf v}$-a.s., we prove that $α_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf v$ large enough, the model exhibits a non-trivial phase transition in~$α$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_09532 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Percolation on the stationary distributions of the voter model with stirring Astoquillca, Jhon Severo, Franco Szabó, Réka Valesin, Daniel Probability 60K35 and 82C22 and 82B43 The voter model with stirring is a variant of the classical voter model on $\mathbb{Z}^d$ with two possible opinions (0 and 1) that, in addition to copying neighbouring opinions at rate 1, allows voters to interchange their opinions at rate~$\mathsf{v}$ where~$\mathsf v \ge 0$ is the stirring parameter. This model was considered in \cite{Astoquillca24}, where it was proved that for~$d \ge 3$ and for any~$\mathsf{v}$ the set of extremal stationary measures is given by a family~$\{ μ_{α,\mathsf{v}}: α\in [0,1] \}$, where~$α$ is the density of voters with opinion~1. Sampling a configuration~$ξ$ from~$μ_{α, \mathsf v}$, we study~$ξ$ as a site percolation model on~$\mathbb{Z}^d$, where the set of occupied sites is the set of voters with opinion 1 in~$ξ$. Letting~$α_c(\mathsf v)$ be the supremum of all the values of~$α$ for which percolation does not occur~$μ_{α, \mathsf v}$-a.s., we prove that $α_c(\mathsf{v})$ converges to~$p_c$, the critical density for classical Bernoulli site percolation, as~$\mathsf{v}$ tends to infinity. As a consequence, for $\mathsf v$ large enough, the model exhibits a non-trivial phase transition in~$α$. |
| title | Percolation on the stationary distributions of the voter model with stirring |
| topic | Probability 60K35 and 82C22 and 82B43 |
| url | https://arxiv.org/abs/2412.09532 |