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

Itai Benjamini; Hilary Finucane; Romain Tessera

doi:10.1007/s00493-015-2975-4

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

Itai Benjamini, Hilary Finucane, Romain Tessera

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	333-374
Number of pages	42
Journal	Combinatorica
Volume	37
Issue number	3
DOIs	https://doi.org/10.1007/s00493-015-2975-4
State	Published - 1 Jun 2017

All Science Journal Classification (ASJC) codes

Computational Mathematics
Discrete Mathematics and Combinatorics

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10.1007/s00493-015-2975-4

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title = "On the scaling limit of finite vertex transitive graphs with large diameter",

abstract = "Let (Xn) be an unbounded sequence of finite, connected, vertex transitive graphs such that |Xn|=O(diam(Xn)q) for some q>0. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (Xn) 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{\textquoteright}s theorem on groups with polynomial growth obtained by Breuillard, Green and Tao. If Xn is only roughly transitive and |Xn|=O diam(Xn δ) for δ >1 sufficiently small, we prove, this time by elementary means, that (Xn) converges to a circle.",

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T1 - On the scaling limit of finite vertex transitive graphs with large diameter

AU - Benjamini, Itai

AU - Finucane, Hilary

AU - Tessera, Romain

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N2 - Let (Xn) be an unbounded sequence of finite, connected, vertex transitive graphs such that |Xn|=O(diam(Xn)q) for some q>0. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (Xn) 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 Xn is only roughly transitive and |Xn|=O diam(Xn δ) for δ >1 sufficiently small, we prove, this time by elementary means, that (Xn) converges to a circle.

AB - Let (Xn) be an unbounded sequence of finite, connected, vertex transitive graphs such that |Xn|=O(diam(Xn)q) for some q>0. We show that up to taking a subsequence, and after rescaling by the diameter, the sequence (Xn) 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 Xn is only roughly transitive and |Xn|=O diam(Xn δ) for δ >1 sufficiently small, we prove, this time by elementary means, that (Xn) converges to a circle.

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

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