Concentration estimates for the isoperimetric constant of the super critical percolation cluster

We consider the Cheeger constant $ϕ(n)$ of the giant component of supercritical bond percolation on $\mathbb{Z}^d/n\mathbb{Z}^d$. We show that the variance of $ϕ(n)$ is bounded by $\fracξ{n^d}$, where $ξ$ is a positive constant that depends only on the dimension $d$ and the percolation parameter.

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