A unimodular random graph with large upper growth and no growth
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866910718799904768 |
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| author | Mester, Péter Timár, Ádám |
| author_facet | Mester, Péter Timár, Ádám |
| contents | We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least $\sqrt{d-1}$, then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non-hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular Riemannian surfaces of bounded negative curvature always have growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18465 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A unimodular random graph with large upper growth and no growth Mester, Péter Timár, Ádám Probability We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least $\sqrt{d-1}$, then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non-hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular Riemannian surfaces of bounded negative curvature always have growth. |
| title | A unimodular random graph with large upper growth and no growth |
| topic | Probability |
| url | https://arxiv.org/abs/2411.18465 |