From loop clusters of parameter 1/2 to the Gaussian free field

To a transient symmetric Markov jump process on a network is naturally associated an infinite measure on loops and the Poisson point process of loops of intensity proportional to this measure are sometimes called “loop soups”. We focus on the “loop soup” of parameter 1/2 and construct a coupling between the Poisson ensemble of loops and the Gaussian free field on the network satisfying two constraints. First of all half the square of the free field must be the occupation field of the loops. Besides that the sign of the free field must be constant on clusters of loops. This is an improvement over the relation between the Poisson ensemble of loops and the Gaussian free field obtained by Le Jan, which did not take in account the sign of of the free field. As a consequence of our coupling we deduce that loop clusters at parameter 1/2 do not percolate on periodic lattices.