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We consider random interlacements on $\mathbb{Z}^d$, $d \ge 3$. We show thatthe percolation function that to each $u \ge 0$ attaches the probability thatthe origin does not belong to an infinite cluster of the vacant set at level$u$, is $C^1$ on an interval $[0,\^u)$, where $\^u$ is positive and plausiblycoincides with the critical level $u_*$ for the percolation of the vacant set.We apply this finding to a constrained minimization problem that conjecturallyexpresses the exponential rate of decay of the probability that a large boxcontains an excessive proportion $\nu$ of sites that do not belong to aninfinite cluster of the vacant set. When $u$ is smaller than $\^u$, we describea regime of "small excess" for $\nu$ where all minimizers of the constrainedminimization problem remain strictly below the natural threshold value$\sqrt{u}_* - \sqrt{u}$ for the variational problem.

On the $C^1$-property of the percolation function of random interlacements and a related variational problem