  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.
