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.
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.
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Tom Hutchcroft 1
@article{CRMATH_2016__354_9_944_0, author = {Tom Hutchcroft}, 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}, }
Tom Hutchcroft. Critical percolation on any quasi-transitive graph of exponential growth has no infinite clusters. Comptes Rendus. Mathématique, Volume 354 (2016) no. 9, pp. 944-947. doi : 10.1016/j.crma.2016.07.013. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.07.013/
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