Disconnection and level-set percolation for the Gaussian free field

We study the level-set percolation of the Gaussian free field on \mathbb Z^d, d \ge 3. We consider a level \alpha such that the excursion-set of the Gaussian free field above \alpha percolates. We derive large deviation estimates on the probability that the excursion-set of the Gaussian free field below the level \alpha disconnects a box of large side-length from the boundary of a larger homothetic box. It remains an open question whether our asymptotic upper and lower bounds are matching. With the help of a recent work of Lupu [21], we are able to infer some asymptotic upper bounds for similar disconnection problems by random interlacements, or by simple random walk.