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