From loop clusters and random interlacement to the free field

Understand

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.