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

Abstract

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 $[0,\hat{u})$, where $\hat{u}$ is positive and plausibly coincides with the critical level $u_{\ast }$ 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 $\nu$ of sites that do not belong to an infinite cluster of the vacant set. When $u$ is smaller than $\hat{u}$, we describe a regime of "small excess" for $\nu$ where all minimizers of the constrained minimization problem remain strictly below the natural threshold value ${\sqrt{u}}_{\ast }-\sqrt{u}$ for the variational problem.

arXiv comment20 pages, 1 figure, appears in the special volume in memory of Vladas Sidoravicius