  We introduce a model of random interlacements made of a countable collection
of doubly infinite paths on Z^d, d bigger or equal to 3. A non-negative
parameter u measures how many trajectories enter the picture. This model
describes in the large N limit the microscopic structure in the bulk, which
arises when considering the disconnection time of a discrete cylinder with base
a d-1-dimensional discrete torus of side-length N, or the set of points visited
by simple random walk on the d-dimensional discrete torus of side-length N by
times of order uN^d. We study the percolative properties of the vacant set left
by the interlacement at level u, which is an infinite, connected, translation
invariant random subset of Z^d. We introduce a critical value such that the
vacant set percolates for u below the critical value, and does not percolate
for u above the critical value. Our main results show that the critical value
is finite when d is bigger or equal to 3, and strictly positive when d is
bigger or equal to 7.
