Isoperimetry is a well-studied subject that have found many applications in geometric measure theory (e.g. concentration of measure, heat-kernal estimates, mixing time, etc.) Consider the super-critical bond percolation on mathbb Zd (the d-dimensional square lattice), and φn the Cheeger constant of the super-critical percolation cluster restricted to the finite box [-n,n]d. Following several papers that proved that the leading order asymptotics of φn is of the order 1/n, Benjamini conjectured a limit to nφn exists. As a step towards this goal, Rosenthal and myself have recently shown that Var(nφn) C n2-d. This implies concentration of nφn around its mean for dimensions d2. Consider the super-critical bond percolation on mathbb Z2 (the square lattice). We prove the Cheeger constant of the super-critical percolation cluster restricted to finite boxes scale a.s to a deterministic quantity. This quantity is given by the solution to the isoperimetric problem on mathbb R2 with respect to a specific norm. The unique set which gives the solution, is the normalized Wulff shape for the same norm.
Joint work with Marek Biskup, Oren Louidor and Ron Rosenthal.