Isoperimetry in supercritical bond percolation in dimensions three and higher

Julian Gold*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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.

Original languageEnglish (US)
Pages (from-to)2092-2158
Number of pages67
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume54
Issue number4
DOIs
StatePublished - Nov 2018

Keywords

  • Cheeger constant
  • Isoperimetry
  • Limit shapes
  • Percolation

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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