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We introduce a model of random interlacements made of a countable collection of doubly infinite paths on Z d , d ≥ 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 (Z/N Z) d−1 × Z by simple random walk, or the set of points visited by simple random walk on the discrete torus (Z/N Z) d at times of order uN d . In particular 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 u * such that the vacant set percolates for u < u * and does not percolate for u > u * . Our main results show that u * is finite when d ≥ 3 and strictly positive when d ≥ 7.

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Brownian motion, obstacles and random media 1 Alain-Sol Sznitman. p. em. --(Springer monographs in mathematics) Includes bibliographical referenees and index.

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