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

We study the isoperimetric subgraphs of the giant component C_n of supercritical bond percolation on the square lattice. These are subgraphs of C_n with minimal edge boundary to volume ratio. In contrast to the work of [8], the edge boundary is taken only within 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 C_n scales to a deterministic constant, which is itself an isoperimetric ratio, settling a conjecture of Benjamini in dimension two.