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We study the isoperimetric subgraphs of the giant component $\textbf{C}_n$ ofsupercritical bond percolation on the square lattice. These are subgraphs of$\textbf{C}_n$ having minimal edge boundary to volume ratio. In contrast to thework of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is takenonly within $\textbf{C}_n$ instead of the full infinite cluster. Theisoperimetric subgraphs are shown to converge almost surely, after rescaling,to the collection of optimizers of a continuum isoperimetric problem emergingnaturally from the model. We also show that the Cheeger constant of$\textbf{C}_n$ scales to a deterministic constant, which is itself anisoperimetric ratio, settling a conjecture of Benjamini in dimension two.

Intrinsic isoperimetry of the giant component of supercritical bond percolation in dimension two