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 $\mathbb{Z}^d$ (the d-dimensional square lattice), and $\phi_n$ the Cheeger constant of the super-critical percolation cluster restricted to the finite box $[-n,n]^d$....