Asymptotic behavior of the Cheeger constant of super-critical percolation in the square lattice

Опубликовано 28 июля 2016, 23:16

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 $[-n,n]^d$. Following several papers that proved that the leading order asymptotics of $\phi_n$ is of the order $1/n$, Benjamini conjectured a limit to $n\phi_n$ exists. As a step towards this goal, Rosenthal and myself have recently shown that $Var(n\phi_n) C n^{2-d}$. This implies concentration of $n\phi_n$ around its mean for dimensions $d2$. Consider the 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 scale 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. 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.