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We consider the Cheeger constant $\phi(n)$ of the giant component ofsupercritical bond percolation on $\mathbb{Z}^d/n\mathbb{Z}^d$. We show thatthe variance of $\phi(n)$ is bounded by $\frac{\xi}{n^d}$, where $\xi$ is apositive constant that depends only on the dimension $d$ and the percolationparameter.

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