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

Percolation for two-dimensional excursion clouds and the discrete Gaussian free field