From loop clusters and random interlacement to the free field

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