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

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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 some finite dimensional torus equipped with some invariant Finsler metric. The proof relies on a recent quantitative version of Gromov’s theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If X n is only roughly transitive and |X n |=O diam(X n δ) for δ >1 sufficiently small, we prove, this time by elementary means, that (X n ) converges to a circle.

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Author information

Authors and Affiliations

  1. Weizmann Institute of Science, Rehovot, Israel

    Itai Benjamini

  2. MIT Department of Mathematics Cambridge, Cambridge MA, France

    Hilary Finucane

  3. Laboratoire de Mathématiques d’Orsay Univ. Paris-Sud, CNRS Université Paris-Saclay, 91405, Orsay, France

    Romain Tessera

Authors
  1. Itai Benjamini
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  2. Hilary Finucane
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  3. Romain Tessera
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Corresponding author

Correspondence to Itai Benjamini.

Additional information

This work was done while she was a student at the Weizmann Institute of Science, Rehovot, Israel.

Supported by ANR-09-BLAN-0059 and ANR-10-BLAN 0116.

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Benjamini, I., Finucane, H. & Tessera, R. On the scaling limit of finite vertex transitive graphs with large diameter. Combinatorica 37, 333–374 (2017). https://doi.org/10.1007/s00493-015-2975-4

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  • DOI: https://doi.org/10.1007/s00493-015-2975-4

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