A lower bound for disconnection by simple random walk

We consider simple random walk on Z^d, d bigger or equal to 3. Motivated by the work of A.-S. Sznitman and the author in arXiv:1304.7477 and arXiv:1310.2177, we investigate the asymptotic behaviour of the probability that a large body gets disconnected from infinity by the set of points visited by a simple random walk. We derive asymptotic lower bounds that bring into play random interlacements. Although open at the moment, some of the lower bounds that we obtain possibly match the asymptotic upper bounds obtained in a recent article of A.-S. Sznitman. This potentially yields special significance to the tilted walks that we use in this work, and to the strategy that we employ to implement disconnection.