  We study percolative properties of excursion processes and the discrete
Gaussian free field (dGFF) in the planar unit disk. We consider discrete
excursion clouds, defined using random walks as a two-dimensional version of
random interlacements, as well as its scaling limit, defined using Brownian
motion. We prove that the critical parameters associated to vacant set
percolation for the two models are the same and equal to $\pi/3.$ The value is
obtained from a Schramm-Loewner evolution (SLE) computation. Via an isomorphism
theorem, we use a generalization of the discrete result that also involves a
loop soup (and an SLE computation) to show that the critical parameter
associated to level set percolation for the dGFF is strictly positive and
smaller than $\sqrt{\pi/2}.$ In particular this entails a strict inequality of
the type $h_*<\sqrt{2u_*}$ between the critical percolation parameters of the
dGFF and the two-dimensional excursion cloud. Similar strict inequalities are
conjectured to hold in a general transient setup.
