单模随机单端平面图是sofic

Combinatorics, Probability and Computing Pub Date : 2019-10-03 DOI:10.1017/s0963548323000159

Ádám Timár

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引用次数: 2

摘要

我们证明了如果一个单模随机图几乎肯定是平面的，并且期望度有限，那么它在这个平面上有一个组合嵌入，这个组合嵌入也是单模的。这直接暗示了题目中的主张，通过Angel, Hutchcroft, Nachmias和Ray b[2]的定理。我们的单模嵌入也意味着[2]关于单模映射的所有二分结果在单端情况下扩展到单模随机平面图。

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Unimodular random one-ended planar graphs are sofic

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.

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Combinatorics, Probability and Computing

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