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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.07145 |
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| _version_ | 1866913832171995136 |
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| author | Pete, Gábor Rokob, Sándor |
| author_facet | Pete, Gábor Rokob, Sándor |
| contents | In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $ν$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($λ$) copies of them at each vertex. If the expected volume of the animals w.r.t. $ν$ is infinite, then the whole $G$ is covered for any $λ>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $λ$ the union of the animals has only finite clusters, while for $λ$ large enough there are also infinite clusters. Here we show that:
1. If $G$ is a nonamenable free product, then for ANY $ν$ with infinite second but finite first moment and any $λ>0$, there will be infinite clusters, despite having arbitrarily low density.
2. The same result holds for ANY nonamenable $G$, when the lattice animals are worms: random walk pieces of random finite length.
It remains open if the result holds for ANY nonamenable Cayley graph with ANY lattice animal measure $ν$ with infinite second moment.
3. We also give a Poisson zoo example $ν$ on $\mathbb{T}_d \times \mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $λ>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07145 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Nonamenable Poisson zoo Pete, Gábor Rokob, Sándor Probability Mathematical Physics Group Theory In the Poisson zoo on an infinite Cayley graph $G$, we take a probability measure $ν$ on rooted finite connected subsets, called lattice animals, and place i.i.d. Poisson($λ$) copies of them at each vertex. If the expected volume of the animals w.r.t. $ν$ is infinite, then the whole $G$ is covered for any $λ>0$. If the second moment of the volume is finite, then it is easy to see that for small enough $λ$ the union of the animals has only finite clusters, while for $λ$ large enough there are also infinite clusters. Here we show that: 1. If $G$ is a nonamenable free product, then for ANY $ν$ with infinite second but finite first moment and any $λ>0$, there will be infinite clusters, despite having arbitrarily low density. 2. The same result holds for ANY nonamenable $G$, when the lattice animals are worms: random walk pieces of random finite length. It remains open if the result holds for ANY nonamenable Cayley graph with ANY lattice animal measure $ν$ with infinite second moment. 3. We also give a Poisson zoo example $ν$ on $\mathbb{T}_d \times \mathbb{Z}^5$ with finite first moment and a UNIQUE infinite cluster for any $λ>0$. |
| title | Nonamenable Poisson zoo |
| topic | Probability Mathematical Physics Group Theory |
| url | https://arxiv.org/abs/2505.07145 |