We consider the Gaussian free field on the cable system corresponding to a transient weighted graph and study the percolation problem associated to its excursion sets above a given real-valued height h. Owing to certain `integrability properties’ of this setup, involving the strong Markov property of the field and the central role played by a certain capacity observable, we rigorously determine the behavior of several quantities related to the (near-)critical picture for this problem. In particular, our results apply in case the base graph is the three-dimensional cubic lattice, and unveil the values of the associated critical exponents.
Joint work with A. Drewitz and A. Prévost.