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We consider the standard model of i.i.d. bond percolation on $\mathbb Z^d$ ofparameter $p$. When $p>p_c$, there exists almost surely a unique infinitecluster $\mathcal C_p$. Using the recent techniques of Cerf and Dembin, weprove that the variance of the graph distance in $\mathcal C_p$ between twopoints of $\mathcal C_p$ is sublinear. The main result extends the works ofBenjamini, Kalai and Schramm, Benaim and Rossignol and Damron, Hanson and Sosoefor the study of the variance of passage times in first passage percolationwithout moment conditions on the edge-weight distribution.

The variance of the graph distance in the infinite cluster of percolation is sublinear