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.
Highlights
The so called “isomorphism theorems” relate the square of a Gaussian free field (GFF) on an electrical network to occupation times of symmetric Markov jump processes [19, 24]
The generalized second Ray-Knight theorem couples the squares of two GFFs with different, ordered, boundary conditions by adding the occupation times of independent Markovian excursions from boundary to boundary to the square with lower boundary conditions in order to obtain the square with higher boundary conditions
In this paper we deal with the inversion of Lupu’s isomorphism, that is to say with retrieving the conditional law of the discrete loop-soup given a discrete Gaussian free field
Summary
The so called “isomorphism theorems” relate the square of a Gaussian free field (GFF) on an electrical network to occupation times of symmetric Markov jump processes [19, 24]. In this paper we deal with the inversion of Lupu’s isomorphism, that is to say with retrieving the conditional law of the discrete loop-soup given a discrete Gaussian free field (both its absolute value and its sign). Jump Process (VRJP) provides an inversion of the generalized second Ray-Knight theorem, in the sense that it enables to retrieve the law of (lx(τux0), φ2x)x∈V conditional on lx(τux0 ).
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