We study the isoperimetric subgraphs of the giant component Cn of supercritical bond percolation on the square lattice. These are subgraphs of Cn with minimal edge boundary to volume ratio. In contrast to the work of , the edge boundary is taken only within Cn instead of the full infinite cluster. The isoperimetric subgraphs are shown to converge almost surely, after rescaling, to the collection of optimizers of a continuum isoperimetric problem emerging naturally from the model. We also show that the Cheeger constant of Cn scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.
- Cheeger constant
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty