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

We study the isoperimetric subgraphs of the giant component ${\mathbf{\text{C}}}_n$ of supercritical bond percolation on the square lattice. These are subgraphs of ${\mathbf{\text{C}}}_n$ having minimal edge boundary to volume ratio. In contrast to the work of Biskup, Louidor, Procaccia and Rosenthal, the edge boundary is taken only within ${\mathbf{\text{C}}}_n$ 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 ${\mathbf{\text{C}}}_n$ scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.