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

Weizmann Institute of Science Israel
Massachusetts Institute of Technology
Université Paris-Saclay

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

10 Citations (Scopus)

Résumé

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.

langue originale	Anglais
Pages (de - à)	333-374
Nombre de pages	42
journal	Combinatorica
Volume	37
Numéro de publication	3
Les DOIs	https://doi.org/10.1007/s00493-015-2975-4
état	Publié - 1 juin 2017
Modification externe	Oui

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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.",

author = "Itai Benjamini and Hilary Finucane and Romain Tessera",

note = "Publisher Copyright: {\textcopyright} 2017, J{\'a}nos Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.",

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doi = "10.1007/s00493-015-2975-4",

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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

PY - 2017/6/1

Y1 - 2017/6/1

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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DO - 10.1007/s00493-015-2975-4

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SP - 333

EP - 374

JO - Combinatorica

JF - Combinatorica

IS - 3

ER -

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

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