The talk will focus on the percolation problem associated to excursion sets of the Gaussian free field. The cable-system version of this percolation model enjoys a certain integrability, which permits unprecedented conclusions at and near the critical point, especially in low transient dimensions. We will give a gentle introduction to this circle of ideas, with the aim to discuss the tail behaviour of the cluster size of a point, as well as the largest cluster size. Based on work with A. Drewitz and A. Prévost.