  We investigate certain large deviation asymptotics concerning random
interlacements in Z^d, d bigger or equal to 3. We find the principal
exponential rate of decay for the probability that the average value of some
suitable non-decreasing local function of the field of occupation times,
sampled at each point of a large box, exceeds its expected value. We express
the exponential rate of decay in terms of a constrained minimum for the
Dirichlet energy of functions on R^d that decay at infinity. An application
concerns the excess presence of random interlacements in a large box. Our
findings exhibit similarities to some of the results of van den
Berg-Bolthausen-den Hollander in their work on moderate deviations of the
volume of the Wiener sausage. An other application relates to recent work of
the author on macroscopic holes in connected components of the vacant set in
arXiv:1802.05255v2.
