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It was shown by Le Jan that the occupation field of a Poisson ensemble ofMarkov loops ("loop soup") of parameter one-half associated to a transientsymmetric Markov jump process on a network is half the square of the Gaussianfree field on this network. We construct a coupling between these loops and thefree field such that an additional constraint holds: the sign of the free fieldis constant on each cluster of loops. As a consequence of our coupling wededuce that the loop clusters of parameter one-half do not percolate onperiodic lattices. We also construct a coupling between the randominterlacement on $\mathbb{Z}^{d}$, $d\geq 3$, introduced by Sznitman, and theGaussian free field on the lattice such that the set of vertices visited by theinterlacement is contained in a level set of the free field. We deduce aninequality between the critical level for the percolation by level sets of thefree field and the critical parameter for the percolation of the vacant set ofthe random interlacement. Both in the case of loops and of the randominterlacement, the couplings are constructed by replacing discrete graphs bymetric graphs. Le Jan's and Sznitman's isomorphism theorems between theGaussian free field and the occupation field of trajectories can be extended tothe metric graph setting on which the intermediate value principle forcontinuous fields holds.

From loop clusters and random interlacement to the free field