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arXiv · 1203.5624

On the scaling limit of finite vertex transitive graphs with large diameter

Itai Benjamini·Hilary Finucane·Romain Tessera
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Abstract

Let $(X_n)$ be an unbounded sequence of finite, connected, vertex transitive graphs such that $ |X_n | = o(diam(X_n)^q)$ for some $q>0$. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence $(X_n)$ converges in the Gromov Hausdorff distance to a torus of dimension $ 1$ sufficiently small, we prove, this time by elementary means, that $(X_n)$ converges to a circle.

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Itai Benjamini, Hilary Finucane, Romain Tessera. 2014-08-26. On the scaling limit of finite vertex transitive graphs with large diameter. https://arxiv.org/abs/1203.5624

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