Inverting the coupling of the signed Gaussian free field with a loop-soup

Abstract

Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction relies on the previous work by Sabot and Tarrès, which inverts the coupling from the square of the GFF rather than the signed GFF itself. As a consequence, we also deduce an inversion of the coupling between the random current and the FK-Ising random cluster models introduced by Lupu and Werner.

Inverting the Ray-Knight identity on the line

“…In Section 2 we will give some properties of our self-repelling diffusion. In Section 3 we will recall how the self-repelling jump processes of Theorem 1.4 appear in the inversion of the Ray-Knight identity on discrete subsets of R. This is a result obtained in [20]. Using this we will prove in Section 4 the Theorem 1.5 and the particular case of Theorem 1.4 whenλ 0 pxq "λ0 pxq " φ paq pxq 2 .…”

“…As described in Section 2 in [20] and in particular in Theorem 8, the conditional probability that O J ‚ q ptx 1 , x 2 uq " 0, given pX J ‚ r , λ J ‚ r pxqq xPJ ‚ ,0ďrďq , and that O J ‚ 0 ptx 1 , x 2 uq " 0, and that X J ‚ r has not crossed the edge tx 1 , x 2 u before time q, equals…”

“…This is known as second generalized Ray-Knight identity. The inversion in the discrete setting was done in [27,20]. This inversion involves a nearest neighbor self-repelling jump process on the network.…”

“…Next we state that the joint process pX J ‚ q , λ J ‚ q , O J ‚ q q qě0 is Markovian and give the transitions rates. For details we refer to Theorem 8 in [20].…”

On coupling and “vacant set level set” percolation

“…The percolative properties of V u and of {ϕ > a} have been the object of much interest, see for instance [3], [4], [5], and the references therein. There are strong links between the two models that result from Dynkin-type isomorphism theorems, see [12], [7], and in a broader context [9], [11], [8]. The consideration of cable graphs and the resulting extended couplings constructed in [7], later refined in [13], provide in good cases efficient tools to compare the two percolation models.…”