Unimodular random one-ended planar graphs are sofic
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arXiv
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| Format: | Preprint |
| Published: |
2019
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| _version_ | 1866915149753876480 |
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| author | Timar, Adam |
| author_facet | Timar, Adam |
| contents | We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_01307 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Unimodular random one-ended planar graphs are sofic Timar, Adam Probability Combinatorics We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs. |
| title | Unimodular random one-ended planar graphs are sofic |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/1910.01307 |