[La percolation critique sur les graphes quasi-transitifs à croissance exponentielle ne possède pas de composante connexe infinie]
Nous démontrons que la percolation critique sur les graphes quasi transitifs à croissance exponentielle ne possède pas de composante connexe infinie unique. En utilisant certains résultats antérieurs, ceci nous permet de déduire la non-existence d'une composante connexe infinie pour la percolation critique sur de tels graphes. Ce résultat était auparavant inconnu pour les cas moyennable et non unimodulaire.
We prove that critical percolation on any quasi-transitive graph of exponential volume growth does not have a unique infinite cluster. This allows us to deduce from earlier results that critical percolation on any graph in this class does not have any infinite clusters. The result is new when the graph in question is either amenable or nonunimodular.
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@article{CRMATH_2016__354_9_944_0,
author = {Hutchcroft, Tom},
title = {Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters},
journal = {Comptes Rendus. Math\'ematique},
pages = {944--947},
publisher = {Elsevier},
volume = {354},
number = {9},
year = {2016},
doi = {10.1016/j.crma.2016.07.013},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.07.013/}
}
TY - JOUR AU - Hutchcroft, Tom TI - Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters JO - Comptes Rendus. Mathématique PY - 2016 SP - 944 EP - 947 VL - 354 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.07.013/ DO - 10.1016/j.crma.2016.07.013 LA - en ID - CRMATH_2016__354_9_944_0 ER -
%0 Journal Article %A Hutchcroft, Tom %T Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters %J Comptes Rendus. Mathématique %D 2016 %P 944-947 %V 354 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.07.013/ %R 10.1016/j.crma.2016.07.013 %G en %F CRMATH_2016__354_9_944_0
Hutchcroft, Tom. Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. Comptes Rendus. Mathématique, Tome 354 (2016) no. 9, pp. 944-947. doi : 10.1016/j.crma.2016.07.013. http://www.numdam.org/articles/10.1016/j.crma.2016.07.013/
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