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 , we are able to infer some asymptotic upper bounds for similar disconnection problems by random interlacements, or by simple random walk.
"Disconnection and level-set percolation for the Gaussian free field." J. Math. Soc. Japan 67 (4) 1801 - 1843, October, 2015. https://doi.org/10.2969/jmsj/06741801