Phase transition for level-set percolation of the membrane model in dimensions $d\ge 5$

We consider level-set percolation for the Gaussian membrane model on ${\mathbb{Z}}^d$, with $d\ge 5$, and establish that as $h\in \mathbb{R}$ varies, a non-trivial percolation phase transition for the level-set above level $h$ occurs at some finite critical level $h_{\ast }$, which we show to be positive in high dimensions. Along $h_{\ast }$, two further natural critical levels $h_{\ast \ast }$ and $\bar{h}$ are introduced, and we establish that $-\mathrm{\infty }<\bar{h}\le h_{\ast }\le h_{\ast \ast }<\mathrm{\infty }$, in all dimensions. For $h>h_{\ast \ast }$, we find that the connectivity function of the level-set above $h$ admits stretched exponential decay, whereas for $h < \overline{h}$, chemical distances in the (unique) infinite cluster of the level-set are shown to be comparable to the Euclidean distance, by verifying conditions identified by Drewitz, R'ath and Sapozhnikov, see arXiv:1212.2885, for general correlated percolation models. As a pivotal tool to study its level-set, we prove novel decoupling inequalities for the membrane model.