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The random interlacements (at level u) is a one parameter family of randomsubsets of Z^d introduced by Sznitman in arXiv:0704.2560. The vacant set atlevel u is the complement of the random interlacement at level u. In thispaper, we study the effect of small quenched noise on connectivity propertiesof the random interlacement and the vacant set. While the random interlacementinduces a connected subgraph of Z^d for all levels u, the vacant set has anon-trivial phase transition in u, as shown in arXiv:0704.2560 andarXiv:0808.3344.  For a positive epsilon, we allow each vertex of the random interlacement(referred to as occupied) to become vacant, and each vertex of the vacant setto become occupied with probability epsilon, independently of the randomness ofthe interlacement, and independently for different vertices. We prove that forany d>=3 and u>0, almost surely, the perturbed random interlacement percolatesfor small enough noise parameter epsilon. In fact, we prove the strongerstatement that Bernoulli percolation on the random interlacement graph has anon-trivial phase transition in wide enough slabs. As a byproduct, we show thatany electric network with i.i.d. positive resistances on the interlacementgraph is transient, which strengthens our result in arXiv:1102.4758. As for thevacant set, we show that for any d>=3, there is still a non-trivial phasetransition in u when the noise parameter epsilon is small enough, and we giveexplicit upper and lower bounds on the value of the critical threshold, whenepsilon tends to 0.

The effect of small quenched noise on connectivity properties of random interlacements