  In this paper we establish a decoupling feature of the random interlacement
process I^u in Z^d, at level u, for d \geq 3. Roughly speaking, we show that
observations of I^u restricted to two disjoint subsets A_1 and A_2 of Z^d are
approximately independent, once we add a sprinkling to the process I^u by
slightly increasing the parameter u. Our results differ from previous ones in
that we allow the mutual distance between the sets A_1 and A_2 to be much
smaller than their diameters. We then provide an important application of this
decoupling for which such flexibility is crucial. More precisely, we prove
that, above a certain critical threshold u**, the probability of having long
paths that avoid I^u is exponentially small, with logarithmic corrections for
d=3. To obtain the above decoupling, we first develop a general method for
comparing the trace left by two Markov chains on the same state space. This
method is based in what we call the soft local time of a chain. In another
crucial step towards our main result, we also prove that any discrete set can
be "smoothened" into a slightly enlarged discrete set, for which its
equilibrium measure behaves in a regular way. Both these auxiliary results are
interesting in themselves and are presented independently from the rest of the
paper.
