Isoperimetry in two-dimensional percolation

Consider super-critical bond percolation on $\mathbb{Z}^2$ (the square lattice). We prove the Cheeger constant of the super-critical percolation cluster restricted to finite boxes scales a.s to a deterministic quantity. This quantity is given by the solution to the isoperimetric problem on $\mathbb{R}^2$ with respect to a specific norm. We also prove every set achieving the Cheeger constant converges to the deterministic Wulff crystal, with respect to the same norm.

Joint work with Marek Biskup, Oren Louidor and Ron Rosenthal.