  We consider continuous time random interlacements on $\mathbb{Z}^d$, $d \ge
3$, and characterize the distribution of the corresponding stationary random
field of occupation times. When d = 3, we relate this random field to the
two-dimensional Gaussian free field pinned at the origin by looking at scaled
differences of occupation times of long rods by random interlacements at
appropriately tuned levels. In the main asymptotic regime, a scaling factor
appears in the limit, which is independent of the free field, and distributed
as the time-marginal of a zero-dimensional Bessel process. For arbitrary $d \ge
3$, we also relate the field of occupation times at a level tending to
infinity, to the d-dimensional Gaussian free field.
