@article{779b675e023043dfa42f81080f62ac0d,
title = "Isoperimetry in supercritical bond percolation in dimensions three and higher",
abstract = "We study the isoperimetric subgraphs of the infinite cluster C∞ for supercritical bond percolation on Zd with d ≥ 3. Specifically, we consider subgraphs of C∞ ∩ [−n, n]d having minimal open edge boundary to volume ratio. We prove a shape theorem for these subgraphs: when suitably rescaled, they converge almost surely to a translate of a deterministic shape. This deterministic shape is itself an isoperimetric set for a norm we construct. As a corollary, we obtain sharp asymptotics on a natural modification of the Cheeger constant for C∞ ∩ [−n, n]d, settling a conjecture of Benjamini for the version of the Cheeger constant defined here.",
keywords = "Cheeger constant, Isoperimetry, Limit shapes, Percolation",
author = "Julian Gold",
note = "Funding Information: I thank my advisor Marek Biskup for suggesting this problem, for his guidance and his support. I am deeply indebted to Rapha{\"e}l Cerf for patiently sharing his expertise and insight during my time in Paris. I likewise thank Eviatar Procaccia for his guidance. I am grateful to Vincent Vargas, Claire Berenger and Shannon Starr for making it possible for me to attend the IHP Disordered Systems trimester. I thank Yoshihiro Abe, Ian Charlesworth, Arko Chatterjee, Hugo Duminil-Copin, Aukosh Jagannath, Ben Krause, Sangchul Lee, Tom Liggett, Jeff Lin, Peter Petersen, Jacob Rooney and Ian Zemke for helpful conversations. Finally, I wish to express my gratitude to anonymous referees for their helpful comments. This research has been partially supported by the NSF grant DMS-1407558. Publisher Copyright: {\textcopyright} Association des Publications de l'Institut Henri Poincar{\'e}, 2018",
year = "2018",
month = nov,
doi = "10.1214/17-AIHP866",
language = "English (US)",
volume = "54",
pages = "2092--2158",
journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",
issn = "0246-0203",
publisher = "Institute of Mathematical Statistics",
number = "4",
}